11.4 Factorization of univariate polynomials Joachim von zur Gathen . . . . . 30011.5 Factorization of multivariate polynomials Erich Kaltofen and Gregoire Lecerf 301
17.1 Finite Fields in Biology Franziska Hinkelmann and Reinhard Laubenbacher 68317.1.1 Polynomial dynamical systems as framework for discrete models in sys-
17.2 Finite fields in quantum information theory Arne Winterhof . . . . . . . . 69317.3 Finite fields in engineering Jonathan Jedwab and Kai-Uwe Schmidt . . . . 694
.Michael D. Fried, University of California Irvine
9.7.1 Rational function definitions
9.7.1 Remark (Extend values) The historical functions of this section are polynomials and rationalfunctions: f(x) = Nf (x)/Df (x) with Nf and Df relatively prime (nonzero) polynomials,denoted f ∈ F (x), F a field (almost always Fq or a number field). The subject takes offby including functions f – covers – where the domain and range are varieties of the samedimension. Still, we emphasize functions between projective algebraic curves (nonsingular),often where the target and domain are projective 1-space.
9.7.2 Definition The degree of f ∈ F (x), deg(f), is the maximum of deg(Nf ) and deg(Df ).Add a point at∞ to F , F ∪∞ = P1
x(F ), to get the F points of projective 1-space.
9.7.3 Remark (Plug in ∞) Using Definition 9.7.2 requires plugging in and getting out ∞. Wesometimes use the notion of value sets Vf and their cardinality #Vf (Section 8.3).
1. The value of f(x′) for x′ ∈ F is ∞ if x′ is a zero of Df (x).
2. The value of f(∞) is respectively ∞, 0, or the ratio of the Nf and Df leadingcoefficients, if the degree of Nf is greater, less than, or equal to the degree of Df .
If z is a variable indicating the range, this gives f as a function from P1x(F ) to P1
z(F ). Weabbreviate this as f : P1
x → P1z.
9.7.4 Definition (Mobius equivalence) Denote the group – under composition – of Mobiustransformations x 7→ ax+b
cx+d with ad − bc 6= 0, a, b, c, d ∈ F by PGL(F ). Refer tof1, f2 ∈ F (x) as Mobius equivalent if f2 = α f1 β for α, β ∈ PGL(F ).
9.7.5 Example If f(x) = xn, with gcd(n, q − 1) = 1, then #Vf = qk + 1 on P1x(Fqk) exactly for
those infinitely many k with gcd(n, qk − 1) = 1.
9.7.6 Remark Initial motivation came from Schur’s Conjecture Thm. 9.7.32, which starts over anumber field K – a finite extension of Q, the rational numbers – with its ring of integersOK . That asks about Vf over residue class fields, OK/ppp of prime ideals ppp, denoting thisVf (O/ppp) (Vf (Fp) if O = Z). Assume Nf and Df have coefficients in OK . Avoid ppp – it is abad prime – if it contains the leading coefficient of either Nf or Df .
9.7.7 Definition For f ∈ F (x), if f = f1 f2 with f1, f2 ∈ F (x), deg(fi) > 1, i = 1, 2, we say fdecomposes over F . Then, the fi s are composition factors of f .
9.7.8 Definition (Cofinite) For B a subset of A, we say B is cofinite in A if A \B is finite.
9.7.9 Proposition [1638, p. 390] Consider X ′h = (x, y) | h(x, y) = 0, an algebraic curve, definedby h ∈ K[x, y]. Then, there is a unique nonsingular curve Xh – the normalization of X ′h– and a morphism µh : X ′h → Xh that is an isomorphism on the complement of a finitesubset of points in Xh. Indeed, every variety X ′h has such a unique normalization, but inhigher dimensions it may be singular, and µh is an isomorphism off a codimension 1 set.
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Actual appearance, pages 290–302
244 Handbook of Finite Fields
9.7.10 Definition (Components) A definition field for an algebraic set W is a field containing allcoefficients of all polynomials defining W . Components of W over F are algebraicsubsets which are not the union of two closed non-empty proper algebraic subsetsover F [1054, p. 3]. We say W is a variety if it has just one component. It isabsolutely irreducible if it has just one component over F , an algebraic closure ofF .
9.7.11 Remark (Points on varieties) [1054, Chapters 1 and 2] and [1638, §2] introduce affine andprojective algebraic sets, and their components (Definition 9.7.10), except they are over analgebraically closed field. For perfect fields F (including finite fields and number fields) thisextends for normal varieties. Since their components do not meet, taking any disjoint unionof distinct varieties under the action of the absolute Galois group of F defines componentsin general. Points on an algebraic set X over F refers here to geometric points: points withcoordinates in F . It is an F point if its coordinates are in F .
9.7.12 Definition A general f : X → Z is a cover means it is a finite, flat morphism (see Definition9.7.25) of quasi-projective varieties [1638, p. 432, Proposition 2].
9.7.13 Lemma Definition 9.7.12 simplifies for curves, because all our varieties will be normal, andso for curves, nonsingular. Then, any nonconstant morphism is a cover: That includes anynonconstant rational function f : P1
x → P1z.
9.7.14 Example If f : X → Z is finite and X and Z are nonsingular, generalizing what happensfor curves, and no matter their dimension, then f is automatically flat [1054, p. 266, 9.3a)].This doesn’t extend to weakening nonsingular to normal varieties. [1638, p. 434] has a finitemorphism, where X is nonsingular (it is affine 2-space), and Z is normal. But, the fiberdegree is 2 over each z ∈ Z, excluding one point where it is 3.
9.7.15 Remark (Assuming normality) Starting with §9.7.2 all results assume that the algebraicsets are normal. Some constructions (especially Def. 9.7.45) momentarily produce nonnormalsets, that we immediately replace with their normalizations.
9.7.2 MacCluer’s Theorem and Schur’s Conjecture
9.7.16 Definition An f ∈ Fq(x) is exceptional if it maps one-one on P1(Fqk) for infinitely manyk. Similarly, with K a number field, f ∈ K(x) is exceptional if it is exceptionalmod ppp for infinitely many primes ppp.
9.7.17 Remark We use K, allowing decoration, for a number field. §8.3 refers to the splitting field,Ωf (resp. Ωf F ), of f(x) − z over F (z) (resp. over F (z)). The automorphism group of theextension Ωf/F (z) (resp. Ωf,F /F (z)) is the arithmetic (resp. geometric) monodromy groupA (resp. G) of a separable function (Definition 9.7.25) f ∈ F (x). When there are severalfunctions, we denote these Af and Gf . They act on the zeros, x1, . . . , xn (often denoted1, . . . , n), of f(x)− z, giving a natural permutation representation on n symbols.
9.7.18 Definition Every cover f : X → Z over a field F with X irreducible has an associatedextension of function fields that determines the cover up to birational morphisms(see Lemma 9.7.43).
9.7.19 Remark Essentially all the Galois theory of fields translates to useful statements about acover f : X → Z (over F ) of an irreducible variety Z. It does this by corresponding to f the
Special functions over finite fields 245
composite of the function field extensions F (X ′)/F (Z) where X ′ runs over the componentsof X [1638, p. 396]. Several papers in our references (say, [842, §0.C]) give oft-used examples,with Lemma 9.7.20 a simple archetype.
9.7.20 Lemma (See Remark 9.7.21) Any separable cover f : X → Y over F has a Galois closure
cover f : X → Z over F . Then, Af is the group of f with its natural permutation repre-sentation TAf (of degree the degree of f). Do this over F to get the geometric monodromyGf . Then, X is irreducible (resp. absolutely irreducible) if and only if TAf (resp. TGf ) istransitive. For f a rational function it is automatic that TGf (and so TAf ) is transitive.
9.7.21 Remark [846, §2.1] explains how to form the Galois closure cover of a cover using fiberproducts (see Remark 9.7.54). This shows how to form the Galois closure cover of anycollection of covers as in Lemma 9.7.50.
9.7.22 Remark Normalization gives a nearly invertible process to Remark 9.7.19: going from fieldextensions of F (Z) to covers of Z. While this doesn’t translate all arithmetic cover problemsto Galois theory, we apply the phrase “monodromy precision” (Remark 9.7.26) to when itdoes. Example: It does in the topic of exceptional covers, as in Proposition 9.7.28.
9.7.23 Definition Denote the elements of a group G, under a representation TG, that fix 1 by G(1).When TG is transitive, refer to TG as primitive (resp. doubly transitive) if there isno group properly between G(1) and G (resp. G(1) is transitive on 2, . . . , n).
9.7.24 Theorem [840, Theorem 1]: An f ∈ Fq(x) is exceptional if and only if the following holdsfor each orbit O of Af (1) on 2, . . . , n:
O breaks into strictly smaller orbits under Gf (1). (9.23)
Denote the projective normalization of (x, y) | f(x)−f(y)x−y = 0 by Xf,f \∆. Also equivalent
to (9.23): Each Fq component of Xf,f \∆ has at least 2 components over Fq.Similarly, an f ∈ K(x), K a number field, is exceptional if and only if (9.23) holds for
f mod ppp for infinitely many primes ppp.
9.7.25 Definition (Covers) Let f ∈ Fq(x) be nonconstant and separable: not g(xp) for someg ∈ Fq(x). Then, f : P1
x(Fq)→ P1z(Fq) by x 7→ f(x) has these cover properties.
1. Excluding a finite set z1, . . . , zr ⊂ P1z(Fq), branch points of f , there are
exactly n = deg(f) points over z′.
2. For z′ a branch point, counting zeros, x′ of f(x)− z′ with multiplicity, the sumat all x′ s over z′ is still n. An x′ ∈ P1
x with multiplicity > 1 is a ramified point.
For K a number field, the same properties hold, without any separable condition.
9.7.26 Remark (MacCluer’s Theorem) Theorem 9.7.24 has a surprise: (9.23) implies exceptionalityover Fq. An error term in applying Chebotarev’s density theorem with branch points (as in§8.3, in §8.3.3) vanishes. A ramified point with p not dividing its multiplicity is tame.
Macluer’s thesis [1472] responded to a Davenport-Lewis conjecture [577] by showingTheorem 9.7.24 for a polynomial tame at every point. We say: MacCluer’s Theorem showstame polynomial exceptional covers exhibit monodromy precision [847, §3.2.1]. Proposition9.7.28 shows monodromy precision holds for general exceptional covers.
9.7.27 Example A polynomial f over Fq for which p|deg(f) is not tame at ∞.
9.7.28 Proposition [840] combined with [846, Principle 3.1]: Let f : X → Z be any cover (Defi-nition 9.7.12) over Fq with X absolutely irreducible. Then [846, Corollary 2.5]:
246 Handbook of Finite Fields
1. the extended meaning of (9.23) is that the 2-fold fiber product (§9.7.3) of f minusthe diagonal has no absolutely irreducible Fq components; and
2. (9.23) is equivalent to f being exceptional: X(Fqk) → Y (Fqk) is one-one (andonto) for infinitely many k.
9.7.29 Remark As noted in [846, Comments on Principle 3.1], the proof of [840] applies withoutchange to give Proposition 9.7.28 Part 2 when X and Z are non-singular; indeed, it appliesto pr-exceptionality (Definition 9.7.93). Without, however, this nonsingularity assumption,there are complications considered in [847, §A.4.1] (see Example 9.7.14).
9.7.30 Definition Let f in Proposition 9.7.28 over Fq be an exceptional cover. Denote values kwhere (9.23) holds with Fqk replacing Fq, by Ef,q: the exceptionality set of f .
Similarly, for f satisfying the hypotheses of Proposition 9.7.28 over a numberfield K, denote those primes ppp where f mod ppp has Ef,O/ppp infinite, by Ef,K .
9.7.31 Definition The equation Tu(cos(θ)) = cos(uθ) defines the u-th Chebychev polynomial, Tu.From it define a Chebychev conjugate: α Tu α−1 with α(x) = αz′(x) = z′x andeither z′ = 1, or z′ and −z′ are conjugate in a quadratic extension of K.
9.7.32 Theorem (Schur’s Conjecture) [837, Theorem 2]: With K a number field, the f ∈ O[x]for which Ef,K is infinite are compositions with maps a 7→ ax + b (affine) over K withpolynomials of the following form for some odd prime u:
xu (cyclic) or, α Tu α−1, u > 3, a Chebychev conjugate. (9.24)
9.7.33 Remark Many still refer to Theorem 9.7.32 as Schur’s Conjecture, though Schur conjecturedit only over Q. [837] refers to all Chebychev conjugates as Chebychev polynomials, ratherthan Dickson as in Remark 9.7.34. [1441] assiduously distinguishes Dickson polynomials.
Here is a simple branch point Chebychev Conjugate characterization: f has two finite(6= ∞) branch points, ±z′ ∈ P1
z(Q), which identify with the unique unramified points (inP1x(Q)) over the branch points, as in [837, Proof of Lemma 9].
1. A corollary of [853, Theorem 3.5] is that any cover with a unique totally andtamely ramified point decomposes over F if and only if it decomposes over F .This applies if f ∈ F [x] has deg(f) prime to the characteristic of F .
2. If f from Part 1 is indecomposable, then Gf is primitive (see Definition 9.7.23)and it contains an n-cycle.
3. If f ∈ K[x] is exceptional, since (9.23) says Gf cannot be doubly transitive, upto composing with K affine maps, f from Part 2 is in (9.24).
9.7.34 Remark (Dickson doppelgangers, see §9.6) Each Chebychev conjugate is a constant timesa Dickson polynomial [846, Proposition 5.3]. The Remark 9.7.33 characterization – bylocating their branch points – avoids using equations. That is the distinction at the laststep between the proof of Theorem 9.7.32 and [1441, Chapter 6].
9.7.35 Remark Use the notation in Theorem 9.7.32. Suppose f ∈ OK [x] is an exceptional poly-nomial. Define nf,c (resp. nf,C) to be the product of distinct primes s for which f has adegree s cyclic (resp. Chebychev conjugate) composition factor. The referee of [1507] notedCorollary 9.7.36 follows from 9.7.28 combined with 9.7.32.
9.7.36 Corollary For f ∈ OK [x] an exceptional polynomial, one can determine Ef,K (excludingbad primes, Remark 9.7.6) from nc,f and nf,C by congruences. WhenOK = Z, then p ∈ Ef,Qif and only if gcd(p− 1, s) = 1 for each s|nf,c and gcd(p2 − 1, s) = 1 for each s|nf,C .
Special functions over finite fields 247
9.7.37 Example (Infinite Ef,Q) It is necessary that gcd(2, nc) = 1 and gcd(6, nC) = 1 for thereto be infinitely many p that satisfy the conclusion of Corollary 9.7.36. But it is sufficient,too. Without loss, assume gcd(nc, nC) = 1. If 3 6 |nc, then Dirichlet’s Theorem on primesin arithmetic progressions gives an infinite set of p ≡ 3 mod ncnC . They are in Ef,Q. If3|nc, the Chinese remainder theorem gives an arithmetic progression of p satisfying p ≡ 3mod nC and p ≡ −1 mod nc. So, Ef,Q is infinite whenever it has a chance to be.
9.7.38 Remark Combine [841, Lemma 1] with monodromy precision in Proposition 9.7.39. Thisshows, the Proposition 9.7.28 fiber product statement is equivalent to f ∈ K(x) being ex-ceptional, and therefore permutation, mod ppp. If OK/ppp is sufficiently large, the fiber productstatement is also necessary for f to be permutation (well-known, for example [837, proof ofTheorem 2, last paragraph]).
9.7.39 Proposition (Permutation functions) From Remark 9.7.38, for f ∈ Fq(x), those k where fpermutes P1(Fqk) contains Ef,Fq as a cofinite subset. Similarly, for K a number field, thoseppp where f functionally permutes P1(O/ppp) contains Ef,K as a cofinite subset.
9.7.40 Remark §8.1 shows permutation polynomials are abundant. Exceptional polynomials sat-isfy a much stronger property, but Corollary 9.7.36 shows they are abundant, too. Onedifference: §9.7.3 combines them in ways with no analog for permutation polynomials.
9.7.41 Corollary An analog of Theorem 9.7.32 holds over Fq to characterize exceptional polyno-mials of degree prime to p ([848, Introduction to §5] or [846, Proposition 5.1]). There, z′ inαz′ is either 1 or in the unique quadratic extension of Fq. Consider a Chebychev conjugateαz′ Tn α−1
z′ as a permutation polynomial on Fqk with gcd(q2k − 1, n) = 1. Then, when
n ·m ≡ 1 ( mod q2k − 1), αz′ Tm α−1z′ is its functional inverse.
9.7.3 Fiber product of covers
9.7.42 Definition For any field extension F1/F2 containing Fp, there is the notion of being sep-arable [849, p. 111]. For f ∈ Fq(x), the extension Fq(x)/Fq(f(x)) being separableis equivalent to f is separable (Definition 9.7.25). Many of our examples inheritseparableness from this special case.
9.7.43 Lemma (Curve covering maps [1054, Chapter I, §6]) Any nonsingular projective algebraiccurve X over a perfect field F has a field of functions F (X) that uniquely determines X upto isomorphism over F .
Each non-constant element f ∈ F (X) determines a finite map X → P1z over F [1054,
Chapter I, Exercise 6.4]. If F (X)/F (f) is separable, then f has the covering propertiesof (9.7.25): finite number of branch points, and uniform count of points in a fiber over F(including multiplicity in the fiber) [1054, Chapter IV, Proposition 2.2].
9.7.44 Definition Refer to any f in the conclusion of Lemma 9.7.43 as a nonsingular cover of P1z.
9.7.45 Definition (Fiber product) Let fi : Xi → P1z, i = 1, 2, be two nonsingular covers of P1
z. Theset theoretic fiber product consists of the algebraic curve
(x1, x2) ∈ X1 ×X2|f1(x1) = f2(x2).
Denote this X1 ×setP1zX2. Its normalization (Proposition 9.7.9), X1 ×P1
zX2, is the
fiber product of f1 and f2.
248 Handbook of Finite Fields
9.7.46 Remark Definition 9.7.45 works equally for any covers Xi → Z, i = 1, 2, with Z a normalprojective variety. Then, X1 ×Z X2 is normal and projective (possibly with several compo-nents) with natural maps pri : X1 ×Z X2 → Xi, i = 1, 2, given by its projection on eachfactor. The functions fi pri, i = 1, 2 are identical, giving a well-defined map:
(f1, f2) : X1 ×Z X2 → Z. (9.25)
9.7.47 Remark (Fiber equations) Consider x′ ∈ X1 ×Z X2 that is simultaneously over x′i ∈ Xi,i = 1, 2, where both x′i s ramify over pr1(x′1) = pr2(x′2). Then, f1(x1) = f2(x2), with xi ina neighborhood of x′i, is not a correct local description around x′.
There is another complication when Z is not a curve (dimension 1). The fiber productmight be singular even when the Xi s are not. So (f1, f2) in (9.25) may not be a coverbecause it is not flat (Remark 9.7.67).
9.7.48 Example Consider two polynomials, f1, f2 ∈ K[x], of the same degree n. They definefj : P1
xj → P1z, j = 1, 2. Then, there are n points over z =∞ on P1
y1×P1
zP1y2
, but only onepoint on the set theoretic fiber product over∞. [839, Proposition 1] gives the generalizationof this, showing – when the covers are tame – how to compute the genus of the fiber productcomponents from the covers fj , j = 1, 2.
9.7.49 Definition The fiber product Xf,f = X ×Z X for a cover f : X → Z of degree exceeding 1has at least two components. One is the diagonal : the set ∆(X) = (x, x) | x ∈ X.The normal variety X ×Z X \∆(X) generalizes the set in Theorem 9.7.24.
9.7.50 Lemma (Fiber product monodromy [846, §2.1.3]) Consider the covers in Definition (9.7.45).To each fj there is an arithmetic (resp. geometric) monodromy group Afj (resp. Gfj ),j = 1, 2. Similarly, for (f1, f2) in (9.25). Then, A(f1,f2) maps naturally, surjectively, to Afjby homomorphisms pr∗j , j = 1, 2. There is a largest simultaneous quotient, H, of both Afj sgiven by homomorphisms mi : Afj → H, j = 1, 2, so that
A(f1,f2) = (σ1, σ2) ∈ Af1 ×Af2 | m1(σ1) = m2(σ2).
Similarly with geometric replacing arithmetic monodromy.
9.7.51 Corollary (Components) With the hypotheses of Lem. 9.7.50, let 1j , . . . , nj, be integerson which Aj acts, j = 1, 2. Then, A(f1,f2) acts on the pairs (i1, i2) and on each of the sets1j , . . . , nj separately. If X1 is absolutely irreducible, then the components of X1 ×Z X2
over F (resp. F ) correspond to the orbits of A(f1,f2)(11) (resp. G(f1,f2)(11); see Definition9.7.23) on 12, . . . , n2. Note: The degrees n1 and n2 may be different.
9.7.52 Definition (Absolute components) Given X1×ZX2 in Corollary 9.7.51, denote the union ofits absolutely irreducible F components by X1×abs
Z X2. Denote the complementaryset, X1 ×Z X2 \X1 ×abs
Z X2, of components by X1 ×cpZ X2.
9.7.53 Theorem (Explicit Ef,q – see Remark 9.7.54) Let f : X → Z (as in Proposition 9.7.28) bean exceptional cover over Fq. For X ′i, an Fq component of X ×cp
Z X, denote the number ofcomponents in its breakup over Fq by si, i = 1, . . . , u.
With sexc = lcm(s1, . . . , su), Ef,q = k mod sexc | gcd(k, si) < si, i = 1, . . . , u.
The group G(Fqsexc/Fq) is naturally a quotient of Af/Gf . We can interpret all quantitiesusing Af and Gf .
Special functions over finite fields 249
9.7.54 Remark All but the last sentence of Theorem 9.7.53 is [846, Corollary 2.8]. The last sentenceis from [846, Lemma 2.6], using that the Galois closure cover of f is a(ny) component (overFq) of the deg(f) = n-fold fiber product of f with itself. Project that fiber product ontothe 2-fold fiber product of f over Fq to finish. Corollary 9.7.51 shows the orbit lengths ofAf (1) on 2, . . . , n divided by the corresponding orbit lengths of Gf (1), give the si s.
9.7.55 Theorem (Explicit Ef,K – see Remark 9.7.56): Now change Fq to K (number field) in thefirst sentence of Theorem 9.7.53. For each cyclic subgroup C ≤ Af/Gf denote those σ ∈ Afthat map to C by AC . As previously, denote the stabilizers of 1 in the representation byAC(1) and GC(1). Consider this set, Cf,K , of cyclic C (as in (9.23)):
C | each orbit of AC(1) on 2, . . . , n breaks into strictly smaller orbits under GC(1).
Then, f is exceptional over K if and only if Cf,K is nonempty. Further, Ef,K consists ofthose primes ppp for which the Frobenius attached to ppp is a generator of some C ∈ Cf,K .
9.7.56 Remark Theorem 9.7.55 comes from applying [841, §2] exactly as in Remark 9.7.28. If Ef,Kis infinite, then X ×Z X \ ∆(X) has no absolutely irreducible component. The converse,however, does not hold.
9.7.4 Combining exceptional covers; the (Fq, Z) exceptional tower
9.7.57 Definition (Category of exceptional covers) For Z absolutely irreducible over Fq, denotethe collection of exceptional covers of Z over Fq by TZ,Fq .
9.7.58 Theorem [846, §4.1]: Given (fi, Xi) ∈ TZ,Fq , i = 1, 2, X1 ×absZ X2 (Definition 9.7.52) has
one component. We conclude that:
(f1 pr1, X1 ×absZ X2) ∈ TZ,Fq .
Also, there is at most one morphism between any two objects in TZ,Fq .
9.7.59 Remark (When f1 = f2 in Theorem 9.7.58) We definitely include the fiber product ofa cover in TZ,Fq with itself. Then, the only absolutely irreducible component of the fiberproduct is the diagonal (Definition 9.7.49), which is equivalent to the original cover.
9.7.60 Definition We call X1 ×absZ X2 the fiber product of f1 and f2 in TZ,Fq , and continue to
denote its morphism to Z by (f1, f2). This defines TZ,Fq as a category with fiberproducts. Theorem 9.7.53 shows Ef1,q ∩ Ef2,q = E(f1,f2),Fq is infinite.
9.7.61 Remark Consider (fi, Xi) ∈ TZ,Fq , i = 1, 2, for which there exists ψ : X1 → X2 over Fqthat factors through f2: f2 ψ = f1. Then, Theorem 9.7.58 says ψ is unique.
9.7.62 Corollary For (f,X) ∈ TZ,Fq , denote the group of the Galois closure cover of f over X byAf (1). Then, Af has the representation Tf by acting on cosets of Af (1). If (fi, Xi) ∈ TZ,Fq ,i = 1, 2, we write (f1, X1) > (f2, X2) if f1 factors through X2. [846, Prop. 4.3] producesfrom these pairs a canonical group AZ,Fq with a profinite permutation representation TZ,Fq .
9.7.63 Remark (A projective limit) Given (fi, Xi) ∈ TZ,Fq , i = 1, 2, there is a 3rd (f,X) ∈ TZ,Fq ,given by the fiber product, that factors through both. This is the condition defining aprojective sequence. So, AZ,Fq in Corollary 9.7.62 is a projective limit.
250 Handbook of Finite Fields
9.7.64 Definition (AZ,Fq , TZ,Fq ) is the (arithmetic) monodromy group, in its natural permutationrepresentation, of the Exceptional Tower TZ,Fq .
9.7.65 Theorem Let fi : Xi → Z, i = 1, 2, be exceptional covers over K: Efi,K is infinite, i = 1, 2.Then, X1 ×Z X2 is exceptional in the sense that
X1 mod ppp×absZ mod ppp X2 mod ppp is exceptional for infinitely ppp
if and only if Ef1,K ∩ Ef2,K is infinite.
9.7.66 Remark Theorem 9.7.65 forces considering if there is an infinite intersection of two excep-tionality sets over K. As Theorem 9.7.53 shows, this is automatic over Fq. Example 9.7.37shows it is not automatic over a number field. §9.7.5 and §9.7.6 have examples along theselines: If both fi s, i = 1, 2, are exceptional rational functions, then their composition is againexceptional over K if and only if Ef1,K ∩ Ef2,K is infinite. Beyond cyclic and Chebychevsituations, it is very difficult to decide when this intersection is infinite.
9.7.67 Remark The same definition for exceptional works for any finite, surjective, map of normalvarieties over Fq. Such maps may not be flat (say, when Remark 9.7.14 doesn’t apply),so they may not be covers. Normalization of any projective variety is projective: Segre’sEmbedding [1638, Thm. 4, p. 400].
For irreducible X, flatness says the multiplicity sum of points in the fiber over z isconstant in z: the function field extension degree, [K(X) : K(Z)] [1638, Proposition 2,p. 432]. That is, Definition 9.7.25, Part 2, holds. For finite morphisms that characterizesflatness [1638, Corollary p. 432]. With normality, but not flatness, this may hold only outsidea codimension 2 set in the target. [847, Appendix A.4] has a liesurely discussion. See Example9.7.14.
9.7.5 Exceptional rational functions; Serre’s Open Image Theorem
mapping ±z′ to 0,∞, with a = (z′)2 ∈ K, z′ 6∈ K. Then, for n odd, characterizeRn,a = (lz′)
−1 (lz′(x))n, a cyclic conjugate, by these conditions:
±z′ are its sole ramified points, Rn,a(±z′) = ±z′ and it maps ∞ 7→ ∞. (9.26)
9.7.69 Remark According to [1441, Chapter 2, §5]), Rn,a in Definition 9.7.68 is a Redei function.From [1441, Theorem 3.11]: Under the hypotheses on z′, the exceptionality set ERn,a,K is
ppp | (|OK/ppp| − 1, n) = 1 if z′ is a quadratic residue mod ppp, andppp | (|OK/ppp|+ 1, n) = 1 if not.
9.7.70 Remark (Addendum Remark 9.7.69) Quadratic reciprocity determines nonempty arithmeticprogressions for which z′ is a quadratic residue and those for which it is not. If z′ in Definition9.7.68 were in K, then – of course – the exceptional set is the same as for xn. Whether ornot z′ ∈ K, we refer to Rn,a as a cyclic conjugate.
9.7.71 Definition [846, §4.2] Suppose a collection C of covers from an exceptional tower TY,Fq isclosed under the categorical fiber product. We say C is a subtower. We also speakof the (minimal) subtower any collection generates under fiber product.
Special functions over finite fields 251
9.7.72 Remark [846, §4.3] uses that the fiber product of two unramified covers is unramified to cre-ate cryptographic exceptional subtowers. [846, §5.2.3] computes the arithmetic monodromyattached to the Dickson subtower generated by all the exceptional Chebychev conjugatesover Fq. The analog of Remark 9.7.69 over Fq gives a similar – Redei – subtower of TP1
z,Fqgenerated by exceptional cyclic conjugates.
9.7.73 Remark Theorem 9.7.65 requires common exceptional intersection (Remark 9.7.66) to formfiber products in TZ,K , Z absolutely irreducible over a number field K. For fiber products(or composites) of Chebychev and cyclic conjugates, we easily decide if exceptional sets haveinfinite intersection. Exceptional rational functions from Serre’s O(pen) I(mage) T(heorem)give much harder versions of such problems.
9.7.74 Definition (j-line P1j ) A special copy of projective 1-space, the j-line, occurs in the study
of modular curves (see Theorem 9.7.76). Each j ∈ P1j \ ∞(Q) = A1
j (Q) has anattached isomorphism class of elliptic curves Ej . For each integer n > 0, consider aspecial case of a modular curve, µ0(n) : X0(n)→ P1
j , with its cover of P1j . Denote
the points of X0(n) not lying over j =∞ by Y0(n).
9.7.75 Definition For E an elliptic curve, denote by E → E/C an isogeny from quotienting Eby a (finite) torsion subgroup C of E. When C is a cyclic, generated by e′ ∈ E(resp. all torsion points killed by multiplication by n), write C = 〈e′〉 (resp. Cn).
9.7.76 Theorem There are two approaches to giving “meaning” to each algebraic point y ∈ Y0(n),whose image in P1
j is jy.
1. [1718, p. 108] or [843, p. 158]: y 7→ [Ejy → Ejy/〈e′y〉] with e′y ∈ Ejy of order nwhere brackets, [ ] , indicate an isomorphism class of isogenies.
2. [843, Lemma 2.1]: y 7→ fy ∈ Q(x) (up to Mobius equivalence) of degree n.
9.7.77 Theorem [843, Theorem 2.1]: Suppose f ∈ K(x) is exceptional and of prime degree u.Then, f is Mobius equivalent over K to either:
1. a cyclic (Remark 9.7.70) or a Chebychev (Remark 9.7.34) conjugate;
2. or to some fy (u = n) in Theorem 9.7.76, Part 2.
9.7.78 Definition For a dense set of j′ ∈ A1j , we say the corresponding Ej′ is of CM -type if its ring
of isogenies, tensored by Q, has dimension 2 over Q. Such isogenies form a complexquadratic extension of Q (containing j′, which is an algebraic integer; [1904, II-28]or [1928, Chapter 2, §5.2]). Otherwise, j′ is of GL2-type.
9.7.79 Theorem [843, (2.10)]: Continue the notation of Theorem 9.7.77. Except for the two caseswhere jy is one of the two finite branch points of µ0(u), the geometric monodromy Gfy isthe order 2u dihedral group Du, and fy has four branch points (Definition 9.7.25). For u inTheorem 9.7.77, Part 2, for which Ej′ has good reduction, the coordinates of e′y generate aconstant extension of K with group Afy/Gfy (explained in Theorem 9.7.85).
9.7.80 Theorem [843, §2.B]: For j′ of CM-type, complex multiplication theory gives (an infinite)Efy,K . Computing this would use [1010, §6.3.1-§6.3.2].
9.7.81 Remark (Addendum to Theorem 9.7.80) Using adelic (modular) arithmetic gives analogsof Corollary 9.7.36; and Corollary 9.7.41 for explicitly finding the functional inverse of aCM-type reduced modulo a prime in the exceptional set Efy,K . If K = Q(j′), then Efy,Kdepends on the congruence defining the Frobenius in the (cyclic of degree u−1 over K)
252 Handbook of Finite Fields
constant field. Only finitely many j′ in Q have CM-type, corresponding to class number 1for complex quadratic extensions.
9.7.82 Problem Take one of the CM-type j s in Q. Then, consider two allowed values of u, ui,i = 1, 2, denoting the corresponding fy s by fi, i = 1, 2. Test for explicitness in Remark9.7.81 as to whether Ef1,Q ∩ Ef2,Q is infinite.
9.7.83 Definition (Composition factor definition field) For f ∈ F (x) consider a minimal fieldFf (ind) over which f decomposes into composition factors indecomposable over F .Similarly, denote the minimal field over which Xf,f \∆ in Theorem 9.7.24 breaks
into absolutely irreducible components by Ff (2).
9.7.84 Proposition [846, Proposition 6.5]: If f :X → Z is a cover over F , then Ff (ind) ⊂ Ff (2).
9.7.85 Theorem See Remark 9.7.87: Assume j′ ∈ A1j is of GL2-type. For K = Q(j′), consider
C = Cu in Definition 9.7.75 with u a prime. The corresponding fu ∈ K(x) has degree u2.Use the monodromy groups of Definition 9.7.17.
There is a constant M1,j′ so that if u > M1,j′ , then the arithmetic/geometric monodromyquotient Afu/Gfu is GL2(Z/u)/±1. Further, fu decomposes into two degree u rationalfunctions over Kf (ind), but it is indecomposable over K.
9.7.86 Theorem [846, Proposition 6.6]: Continue Theorem 9.7.85 hypotheses. For a second constantM2,j′ , and for any prime ppp of OK with |OK/ppp| > M2,j′ assume Appp ∈ GL2(Z/u)/〈±1〉represents the conjugacy class of the Frobenius for ppp. Then, fu mod ppp is an exceptionalindecomposable rational function, and it decomposes over the algebraic closure of OK/ppp,precisely when 〈Appp〉 acts irreducibly on (Z/p)2 = Vp. This holds for infinitely many primesppp. In particular, fu is exceptional over K (Definition 9.7.30).
9.7.87 Remark (Using Serre’s OIT ) [1904] lays the groundwork for [1905]. The latter has theexistence of the constant M1,j′ . [1904, App. A.1, §3.2] proves it exists when j′ ∈ A1
j (Q)is not an algebraic integer. Then, the computation of Mi,j′ , i = 1, 2, in Theorems 9.7.85and 9.7.86 is effective. Even after all these years, there is no effective computation of theseconstants when j is not CM-type, but is an algebraic integer. [843, §2] gets Theorem 9.7.85from the OIT using the relation between Parts 1 and 2 in Theorem 9.7.76.
9.7.88 Remark (More elementary, but less precise, Theorem 9.7.86) [843, Theorem 2.2] shows,for every K and any prime u > 3, the j′ ∈ K, with fu satisfying the exceptionality anddecomposability conclusions of Theorem 9.7.86, are dense. Applying the [841, Theorem 3](or [849, Theorem 12.7]) version of Hilbert’s Irreducibility Theorem to X0(u) gives thecorresponding M2,j′ explicitly.
9.7.89 Example (M1,j′ effectiveness?) [1904, App. A.1, §3.3] gives Ogg’s example [1717] withj′ ∈ Q. [846, §6.2.2] reviews this case, where M2,j′ = 6, to show how to pick an Appp actingirreducibly as in Theorem 9.7.86 (for infinitely many ppp), assuring that Efu,Q is infinite foru > M2,j′ .
[846, §6.3.2] – still Ogg’s case – aims at finding an automorphic function, a la Langland’sProgram, that would characterize the primes in Efu,Q. This is akin to the unrelated examplesof [1913], but uses results on automorphic functions in [1908, Theorem 22]. Primes of Efu,Qdo not lie in arithmetic progressions. So, Problem 9.7.90 is much harder than Problem9.7.82.
9.7.90 Problem (Analog of Problem 9.7.82) For the Ogg curve in Example 9.7.89, consider twoallowed values of u, ui, i = 1, 2, denoting the corresponding fy s by fi, i = 1, 2. Test forexplicitness in Remark 9.7.81 as to whether Ef1,Q ∩ Ef2,Q is infinite.
Special functions over finite fields 253
9.7.91 Remark [852] connects “variables separated factors” of Xf,f and composition factors of f .[61] et. al. used this to effectively test for composition factors (and primitivity) of covers.
9.7.92 Theorem [1010, Chapter 3]: Excluding finitely many degrees, all indecomposable excep-tional f ∈ K(x) (K a number field) are Mobius equivalent to a cyclic or Chebychev conju-gate, or to a CM function from Theorem 9.7.77 of prime degree; or they are from Theorem9.7.86 and of prime degree squared.
9.7.6 Davenport pairs and Poincare series
9.7.93 Definition [846, Definition 2.2] Consider f : X → Z, a cover of normal varieties over Fq,with Z absolutely irreducible, but X possibly reducible. Then f is pr-exceptionalif it is surjective on Fqk points for infinitely many k. There is a similar defini-tion extending Definition 9.7.30 over a number field, and for both a notation forexceptional sets.
9.7.94 Definition Use the value set notation of Remark 9.7.3. We say fi ∈ Fq(x), i = 1, 2, is aDavenport pair over Fq if Vf1
(P1(Fqk)) = Vf2(P1(Fqk)) for infinitely many k. So,
take f2(x) = x to see Davenport pairs generalize exceptional functions. The notionapplies to any pair of covers fi : Xi → Z, i = 1, 2. For K a number field, thissimilarly generalizes Definition 9.7.30: f1, f2 ∈ K(x) are a Davenport pair if theyare a Davenport pair for infinitely many residue class fields.
9.7.95 Theorem [846, Corollary 3.6]: Monodromy precision (Definition 9.7.26) applies to pr-exceptional covers and so to Davenport pairs. That is, generalizing Theorems 9.7.53and 9.7.55, a precise monodromy statement generalizes MacCluer’s Theorem (Proposition9.7.28) to pr-exceptional covers and to Davenport pairs.
9.7.96 Theorem [846, §3.1.2]: With the notation of Definition 9.7.93, a pr-exceptional cover overFq is exceptional if and only if X is absolutely irreducible.
9.7.97 Remark The proof of Schur’s Conjecture began the solution of Davenport’s problem forpolynomial pairs (f1, f2) over a number field, the main result of [838]. [846, §3.2] shows theexceptional set characterization for Davenport pairs in general is given by the intersection ofexceptionality sets for pr-exceptionality correspondences. A full description of many authors’results that came from the solution of Davenport’s problem – especially the study of generalzeta functions attached to diophantine problems – is in [847, §7.3].
9.7.98 Remark (The Genus 0 Problem) Geometric monodromy groups of rational functions areseverely limited. The mildest statement for f ∈ Q(x) is that excluding cyclic and alternatinggroups the composition factors of Gf fall among a finite set of simple groups. That is theoriginal genus 0 problem.
There is a large literature distinguishing between geometric monodromy of f ∈ Q(x) andthose in Fq(x), because of wild (not tame; Remark 9.7.26) ramification. The contrast startsfrom the [1631, §8.1.2, Guralnick’s Optimistic Conjecture] list of all primitive monodromygroups of indecomposable f ∈ Q[x].
9.7.99 Example (Davenport pairs) A significant part of the exceptional primitive monodromygroups (Remark 9.7.98), without cyclic or alternating group composition factors, camefrom the finitely many possible degrees of Davenport pairs f1, f2 ∈ K[x] (polynomials) overnumber fields, with f1 indecomposable and Vf1
(OK/ppp) = Vf2(OK/ppp).
Important hints about what to expect for primitive monodromy groups of f ∈ Fq(x)came also from Davenport pairs. [846, §3.3.3] (explicitly in [247]): Over every Fq, there are
254 Handbook of Finite Fields
infinitely many degrees of Davenport pairs, where (deg(f1), p) = 1, f1 is indecomposable,and Vf1
(Fqk) = Vf2(Fqk) for all k.
9.7.100 Example [512, Theorem 14.1] described the geometric monodromy (PSL2(pa), p = 2, 3, aodd) of the only possible exceptional polynomials over Fp whose degrees were neither primeto p or a power of p. Then, [848] produced these: the first exceptional polynomials overfinite fields with nonsolvable monodromy.
9.7.101 Remark (Zeta functions attached to problems) [849, Chapter 25 and 26] details how Dav-enport pairs led to attaching Poincare series – based on the Galois stratification procedureof [845] – to counting the values of parameters for any diophantine problem interpretableover all extensions of Fq, or for infinitely many primes ppp of K.
9.7.102 Example Denote w1, . . . , wu by www. Suppose f(www, x), g(www, y) ∈ Fq[www, x, y]. Denote the car-dinality of www′ ∈ Au(Fqk) with
V (f(www′, x))(P1(Fqk)) = V (g(www′, x))(P1(Fqk)) (9.27)
by Nf,g,k. Define Pf,g,Fq (t) to be the Poincare series∑∞i=1Nf,g,kt
k.
9.7.103 Example With notation over Z, as in Example 9.7.102, suppose f(www, x), g(www, y) ∈ Z[www, x, y].Denote the cardinality of www′ ∈ Au(Fpk) with (9.27) holding over Fpk by Nf,g,Z/p,k. Define
Pf,g,Z/p(t) to be∑∞i=1Nf,g,Z/p,kt
k.
9.7.104 Theorem [849, Chapter 25], [847, §7.3.3]: For any diophantine problem over Fq expressedin a first order language, the attached Poincare series is a rational function. Further, thereis an effective computation of the coefficients of its numerator and denominator based onexpressing those coefficients in p-adic Dwork cohomology.
9.7.105 Theorem [Theorem 9.7.104 continued] Given a diophantine problem D over Z (or OK)expressed in a first order language, there is an effective split of the primes of Q (or over K)into two sets: LD,1 and LD,2, with LD,2 finite. Further, there is a set of varieties V1, . . . , Vsover Z, from which we produce linear equations in variables Y1, . . . , Ys′ that serve as thecoefficients of the numerator and denominator of a rational function PD(t). To each (p, Yi),p ∈ LD,1 there is a universal attachment of a p-adic Dwork cohomology group, H(p, Yi),computed in the category of such Dwork cohomology attached to V1, . . . , Vs.
The corresponding Poincare series PD,p at p ∈ LD,1 comes by substituting H(p, Yi) foreach Y1, . . . , Ys′ in PD(t). Then apply the Frobenius operator at p to these coefficients.
9.7.106 Remark [608]: In Theorem 9.7.105 it is possible to take V1, . . . , Vs to be nonsingular projec-tive varieties with Yi representing a Chow motive (over Q). Applying the Frobenius operatorat p is meaningful as Chow motives are formed from etale cohomology groups of V1, . . . , Vs.
9.7.107 Remark The effectiveness of Theorem 9.7.104 is based on Dwork cohomology [713], and theexplicit calculations of [256]. Theorem 9.7.105 and Remark 9.7.106 both rest on the Galoisstratification procedure of [845] or [849, Chapter 24].
On the plus side, the uniform use of etale cohomology from characteristic 0 produceswonderful invariants – like, Euler characteristics – attached to diophantine problems. Onthe negative, all the effectiveness disappears. In particular, the relation between the setsdenoted LD,1 in the two results is a mystery.
9.7.108 Remark Relating exceptional covers (and Davenport pairs) and other problems about al-gebraic equations is a running theme in [846] and [847]. Detecting these relations comesfrom pr-exceptional correspondences [846, §3.2]. We catch the possible appearance of suchcorrespondences when two Poincare series have infinitely many identical coefficients.
Special functions over finite fields 255
9.7.109 Example Example: An exceptional cover, X → P1z, over Q, will be a curve whose Poincare
series is the same as that of P1z at infinitely many primes. The systematic use of such
characterizations combines monodromy precision (where it applies) and Theorem 9.7.110.
9.7.110 Theorem [847, Proposition 7.17], based on [775]: The zero support of the difference of twoPoincare series consists of the union of arithmetic progressions.
See Also
§8.1 Discusses the large literature on permutation polynomials (as in Proposition9.7.39). This contrasts with the use of a cover given by an exceptional polyno-mial, where one fixed polynomial works for infinitely many finite fields.
§8.3 Section §8.3.3 mentions several explicit Chebotarev density theorem errorterms. Such error terms have improved over time, but, like Proposition 9.7.28,this sections’ results exhibit monodromy precision: the error term vanishes.
§9.6 Discusses Dickson polynomials in detail, including their various combinatorialformulas. This contrasts with the Remark 9.7.34 formula free characterization.
[1] Groupes de monodromie en geometrie algebrique. II. Lecture Notes in Mathematics,Vol. 340. Springer-Verlag, Berlin, 1973. Seminaire de Geometrie Algebrique duBois-Marie 1967–1969 (SGA 7 II), Dirige par P. Deligne et N. Katz. [122, 124,127, 401, 402]
[2] Theorie des topos et cohomologie etale des schemas. Tome 3. Lecture Notes inMathematics, Vol. 305. Springer-Verlag, Berlin, 1973. Seminaire de GeometrieAlgebrique du Bois-Marie 1963–1964 (SGA 4), Dirige par M. Artin, A.Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. [20, 385, 386, 392, 393]
[3] Cohomologie l-adique et fonctions L. Lecture Notes in Mathematics, Vol. 589.Springer-Verlag, Berlin, 1977. Seminaire de Geometrie Algebrique du Bois-Marie 1965–1966 (SGA 5), Edite par Luc Illusie. [20, 385, 388, 393]
[4] 1998. [635, 643, 644, 647]
[5] M. Abdon and F. Torres. On maximal curves in characteristic two. ManuscriptaMath., 99(1):39–53, 1999. [366, 367]
[6] R. J. R. Abel. Some new BIBDs with block size 7. J. Combin. Des., 8(2):146–150,2000. [509]
[7] R. J. R. Abel and M. Buratti. Some progress on (v, 4, 1) difference families andoptical orthogonal codes. J. Combin. Theory Ser. A, 106(1):59–75, 2004. [506]
[8] R. J. R. Abel, N. J. Finizio, G. Ge, and M. Greig. New Z-cyclic triplewhist framesand triplewhist tournament designs. Discrete Appl. Math., 154:1649–1673,2006. [558]
[9] R. J. R. Abel and G. Ge. Some difference matrix constructions and an almostcompletion for the existence of triplewhist tournaments TWh(v). European J.Combin., 26(7):1094–1104, 2005. [557, 558]
[10] F. Abu Salem, S. Gao, and A. G. B. Lauder. Factoring polynomials via polytopes.In ISSAC ’04: Proceedings of the 2004 International Symposium on Symbolicand Algebraic Computation, pages 4–11, New York, 2004. ACM Press. [308,311]
[11] F. K. Abu Salem. An efficient sparse adaptation of the polytope method over Fp anda record-high binary bivariate factorisation. J. Symbolic Comput., 43(5):311–341, 2008. [307, 311]
[12] W. W. Adams and P. Loustaunau. An introduction to Grobner bases. AmericanMathematical Society, Providence, RI, first edition, 1994. [55]
[13] L. Adleman and H. W. Lenstra, Jr. Finding irreducible polynomials over finitefields. STOC ’86: Proceedings of the eighteenth annual ACM symposium onTheory of computing, Nov. 1986. [297, 298, 299]
[14] L. M. Adleman, J. DeMarrais, and M.-D. Huang. A subexponential algorithmfor discrete logarithms over the rational subgroup of the Jacobians of largegenus hyperelliptic curves over finite fields. In Algorithmic number theory(Ithaca, NY, 1994), volume 877 of Lecture Notes in Comput. Sci., pages 28–40.Springer, Berlin, 1994. [360]
[15] L. M. Adleman and M.-D. Huang. Counting points on curves and abelian varietiesover finite fields. J. Symbolic Comput., 32(3):171–189, 2001. [404, 406]
Miscellaneous applications 703
[16] A. Adolphson and S. Sperber. On unit root formulas for toric exponential sums.Alg. Num. Th., to appear. [397]
[17] A. Adolphson and S. Sperber. p-adic estimates for exponential sums and the theoremof Chevalley-Warning. Ann. Sci. Ecole Norm. Sup. (4), 20(4):545–556, 1987.[157, 158, 395, 402]
[18] A. Adolphson and S. Sperber. On the degree of the L-function associated with anexponential sum. Compositio Math., 68(2):125–159, 1988. [126, 388, 391, 393]
[19] A. Adolphson and S. Sperber. Exponential sums and Newton polyhedra: cohomol-ogy and estimates. Ann. of Math. (2), 130(2):367–406, 1989. [122, 127, 154,158, 391, 393, 397, 402]
[20] A. Adolphson and S. Sperber. On twisted exponential sums. Math. Ann.,290(4):713–726, 1991. [397]
[21] A. Adolphson and S. Sperber. Twisted exponential sums and Newton polyhedra.J. Reine Angew. Math., 443:151–177, 1993. [397]
[22] A. Adolphson and S. Sperber. On the zeta function of a complete intersection. Ann.Sci. Ecole Norm. Sup. (4), 29(3):287–328, 1996. [154, 158, 396]
[23] A. Adolphson and S. Sperber. Exponential sums on An. III. Manuscripta Math.,102(4):429–446, 2000. [121, 127]
[24] A. Adolphson and S. Sperber. On the zeta function of a projective complete inter-section. Illinois J. Math., 52(2):389–417, 2008. [396]
[25] S. Agou. Factorisation sur un corps fini Fpn des polynomes composes f(Xpr − aX)lorsque f(X) est un polynome irreductible de Fpn(X). J. Number Theory,9(2):229–239, 1977. [35]
[26] S. Agou. Irreductibilite des polynomes f(Xp2r − aXpr − bX) sur un corps fini Fps .J. Number Theory, 10(1):64–69, 1978. [35]
[27] S. Agou. Irreductibilite des polynomes f(∑mi=0 aiX
pri) sur un corps fini Fps . Canad.Math. Bull., 23(2):207–212, 1980. [35]
[28] S. Ahmad. Cycle structure of automorphisms of finite cyclic groups. J. Combina-torial Theory, 6:370–374, 1969. [184, 185]
[29] O. Ahmadi. Self-reciprocal irreducible pentanomials over F2. Des. Codes Cryptogr.,38(3):395–397, 2006. [37, 38, 41]
[30] O. Ahmadi. On the distribution of irreducible trinomials over F3. Finite FieldsAppl., 13(3):659–664, 2007. [38]
[31] O. Ahmadi. The trace spectra of polynomial bases for F2n . Appl. Algebra Engrg.Comm. Comput., 18(4):391–396, 2007. [77, 79]
[32] O. Ahmadi. Generalization of a theorem of carlitz. Finite Fields Appl., 2011. [28,30]
[33] O. Ahmadi and R. Granger. An efficient deterministic test for Kloosterman sumzeros. 2011. submitted. [111, 118]
[34] O. Ahmadi, F. Luca, O. A., and S. I. E. On stable quadratic polynomials. Preprint,2010. [288, 289]
[35] O. Ahmadi and A. Menezes. On the number of trace-one elements in polynomialbases for F2n . Des. Codes Cryptogr., 37(3):493–507, 2005. [77, 79]
[36] O. Ahmadi and A. Menezes. Irreducible polynomials of maximum weight. Util.Math., 72:111–123, 2007. [41]
[37] O. Ahmadi and I. Shparlinski. Bilinear character sums and sum-product problemson elliptic curves. Proc. Edinb. Math. Soc. (2), 53(1):1–12, 2010. [130]
704 Handbook of Finite Fields
[38] O. Ahmadi, I. E. Shparlinski, and J. F. Voloch. Multiplicative order of gauss periods.Int. J. Number Theory, 6(4):877–882, 2010. [70]
[39] O. Ahmadi and G. Vega. On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields. Finite Fields Appl., 14(1):124–131,2008. [40]
[40] W. Aitken. On value sets of polynomials over a finite field. Finite Fields Appl.,4(4):441–449, 1998. [191, 192]
[41] M. Ajtai, H. Iwaniec, J. Komlos, J. Pintz, and E. Szemeredi. Construction of a thinset with small Fourier coefficients. Bull. London Math. Soc., 22(6):583–590,1990. [148]
[42] A. Akbary, S. Alaric, and Q. Wang. On some classes of permutation polynomials.Int. J. Number Theory, 4(1):121–133, 2008. [178, 179, 185]
[43] A. Akbary, D. Ghioca, and Q. Wang. On permutation polynomials of prescribedshape. Finite Fields Appl., 15(2):195–206, 2009. [174, 175, 185]
[44] A. Akbary, D. Ghioca, and Q. Wang. On constructing permutations of finite fields.Finite Fields Appl., pages 1–17, 2010. [176, 177, 180, 181, 185]
[45] A. Akbary and Q. Wang. On some permutation polynomials over finite fields. Int.J. Math. Math. Sci., (16):2631–2640, 2005. [178, 185]
[46] A. Akbary and Q. Wang. A generalized Lucas sequence and permutation binomials.Proc. Amer. Math. Soc., 134(1):15–22 (electronic), 2006. [174, 178, 185]
[47] A. Akbary and Q. Wang. On polynomials of the form xrf(x(q−1)/l). Int. J. Math.Math. Sci., pages Art. ID 23408, 7, 2007. [177, 178, 179, 185]
[48] S. Akiyama. On the pure Jacobi sums. Acta Arith., 75(2):97–104, 1996. [103, 118]
[49] M.-L. Akkar, N. T. Courtois, R. Duteuil, and L. Goubin. A fast and secure imple-mentation of Sflash. In Public key cryptography—PKC 2003, volume 2567 ofLecture Notes in Comput. Sci., pages 267–278. Springer, Berlin, 2002. [656]
[50] E. Aksoy, A. Cesmelioglu, W. Meidl, and A. Topuzoglu. On the Carlitz rank ofpermutation polynomials. Finite Fields Appl., 15(4):428–440, 2009. [184, 185]
[51] A. A. Albert. Symmetric and alternate matrices in an arbitrary field. I. Trans.Amer. Math. Soc., 43(3):386–436, 1938. [421, 424]
[52] A. A. Albert. Finite division algebras and finite planes. In Proc. Sympos. Appl.Math., Vol. 10, pages 53–70. American Mathematical Society, Providence, R.I.,1960. [226, 229]
[53] A. A. Albert. Generalized twisted fields. Pacific J. Math., 11:1–8, 1961. [227, 229]
[54] A. A. Albert. Isotopy for generalized twisted fields. An. Acad. Brasil. Ci., 33:265–275, 1961. [227, 229]
[55] R. Albert and H. G. Othmer. The topology of the regulatory interactions pre-dicts the expression pattern of the segment polarity genes in drosophilamelanogaster. J. Theoret. Biol., 223(1):1–18, 2003. [683, 692]
[56] N. Ali. Stabilite des polynomes. Acta Arith., 119(1):53–63, 2005. [287, 289]
[57] J.-P. Allouche and J. Shallit. Automatic sequences. Cambridge University Press,Cambridge, 2003. Theory, applications, generalizations. [458]
[58] J.-P. Allouche and D. S. Thakur. Automata and transcendence of the Tate periodin finite characteristic. Proc. Amer. Math. Soc., 127(5):1309–1312, 1999. [458]
[59] N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83–96, 1986. Theory ofcomputing (Singer Island, Fla., 1984). [539, 545]
[60] N. Alon and F. R. K. Chung. Explicit construction of linear sized tolerant net-
Miscellaneous applications 705
works. In Proceedings of the First Japan Conference on Graph Theory andApplications (Hakone, 1986), volume 72, pages 15–19, 1988. [534, 545]
[61] C. Alonso, J. Gutierrez, and T. Recio. A rational function decomposition algorithmby near-separated polynomials. J. Symbolic Comput., 19(6):527–544, 1995.[253, 255]
[62] H. Aly, R. Marzouk, and W. Meidl. On the calculation of the linear complexityof periodic sequences. In Finite fields: theory and applications, volume 518 ofContemp. Math., pages 11–22. Amer. Math. Soc., Providence, RI, 2010. [274,281]
[63] H. Aly and W. Meidl. On the linear complexity and k-error linear complexity over Fpof the d-ary Sidel′nikov sequence. IEEE Trans. Inform. Theory, 53(12):4755–4761, 2007. [279, 281]
[64] H. Aly and A. Winterhof. On the linear complexity profile of nonlinear congruen-tial pseudorandom number generators with Dickson polynomials. Des. CodesCryptogr., 39(2):155–162, 2006. [278, 281]
[65] P. R. Amestoy, T. A. Davis, and I. S. Duff. Algorithm 837: AMD, an approximateminimum degree ordering algorithm. ACM Trans. Math. Software, 30(3):381–388, 2004. [434, 436]
[66] G. An. In silico experiments of existing and hypothetical cytokine-directed clinicaltrials using agent-based modeling. Crit Care Med, 32(10):2050–2060, Oct. 2004.[689, 692]
[67] V. Anashin and A. Khrennikov. Applied algebraic dynamics, volume 49 of de GruyterExpositions in Mathematics. Walter de Gruyter & Co., Berlin, 2009. [282, 283,289]
[68] H. E. Andersen and O. Geil. Evaluation codes from order domain theory. FiniteFields Appl., 14:92–123, 2008. [605, 612]
[69] B. A. Anderson and K. B. Gross. A partial starter construction. Congress. Numer.,21:57–64, 1978. [554]
[70] G. W. Anderson. t-motives. Duke Math. J., 53(2):457–502, 1986. [457]
[71] G. W. Anderson. Log-algebraicity of twisted A-harmonic series and special valuesof L-series in characteristic p. J. Number Theory, 60(1):165–209, 1996. [453]
[72] G. W. Anderson, W. D. Brownawell, and M. A. Papanikolas. Determination of thealgebraic relations among special Γ-values in positive characteristic. Ann. ofMath. (2), 160(1):237–313, 2004. [458]
[73] G. W. Anderson and D. S. Thakur. Multizeta values for Fq[t], their period interpre-tation, and relations between them. Int. Math. Res. Not. IMRN, (11):2038–2055, 2009. [456]
[74] I. Anderson. Combinatorial Designs: Construction Methods. Ellis Horwood Ltd.,Chichester, 1990. [20, 558]
[75] I. Anderson. A hundred years of whist tournaments. J. Combin. Math. Combin.Comput., 19:129–150, 1995. [557]
[76] I. Anderson. Some cyclic and 1-rotational designs. In J. W. P. Hirschfeld, editor,Surveys in Combinatorics, 2001, pages 47–73. Cambridge Univ. Press, London,2001. [557, 558]
[77] I. Anderson and N. J. Finizio. Some new Z-cyclic whist tournament designs. DiscreteMath., 293(1-3):19–28, 2005. [557, 558]
[78] I. Anderson, N. J. Finizio, and P. A. Leonard. New product theorems for Z-cyclicwhist tournaments. J. Combin. Theory A, 88:162–166, 1999. [557, 558]
706 Handbook of Finite Fields
[79] J. Andre. Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe.Math. Z., 60:156–186, 1954. [479, 486]
[80] B. Angles and C. Maire. A note on tamely ramified towers of global function fields.Finite Fields Appl., 8(2):207–215, 2002. [367]
[81] J.-C. Angles d’Auriac, J.-M. Maillard, and C. M. Viallet. On the complexity ofsome birational transformations. J. Phys. A, 39(14):3641–3654, 2006. [282,289]
[82] ANSI. The elliptic curve digital signature algorithm (ECDSA). WorkingDraft American National Standard: Public Key Cryptography for the Fi-nancial Services Industry X9.62-1998, American National Standards Institute,Sept. 1998. Available at http://grouper.ieee.org/groups/1363/private/x9-62-09-20-98.zip. [667]
[83] ANSI. Key agreement and key transport using elliptic curve cryptography. Work-ing Draft American National Standard: Public Key Cryptography for the Fi-nancial Services Industry X9.63-199x, American National Standards Institute,Jan. 1999. Available at http://grouper.ieee.org/groups/1363/private/
x9-63-01-08-99.zip. [667]
[84] N. Anuradha and S. A. Katre. Number of points on the projective curves aY l =bX l + cZl and aY 2l = bX2l + cZ2l defined over finite fields, l an odd prime. J.Number Theory, 77(2):288–313, 1999. [166, 170]
[85] N. Aoki. Abelian fields generated by a Jacobi sum. Comment. Math. Univ. St.Paul., 45(1):1–21, 1996. [103, 118]
[86] N. Aoki. On the purity problem of Gauss sums and Jacobi sums over finite fields.Comment. Math. Univ. St. Paul., 46(2):223–233, 1997. [102, 103, 118]
[87] N. Aoki. A finiteness theorem on pure Gauss sums. Comment. Math. Univ. St.Pauli, 53(2):145–168, 2004. [102, 118]
[88] N. Aoki. On the zeta function of some cyclic quotients of Fermat curves. Comment.Math. Univ. St. Pauli, 57(2):163–185, 2008. [103, 118]
[89] N. Aoki. On multi-quadratic Gauss sums. Comment. Math. Univ. St. Pauli,59(2):97–117, 2010. [106, 118]
[90] K. T. Arasu and K. J. Player. A new family of cyclic difference sets with Singerparameters in characteristic three. Des. Codes Cryptogr., 28(1):75–91, 2003.[516, 519]
[91] V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. Faradzev. The economicalconstruction of the transitive closure of an oriented graph. Dokl. Akad. NaukSSSR, 194:487–488, 1970. [425, 436]
[92] C. Armana. Torsion des modules de Drinfeld de rang 2 et formes modulaires deDrinfeld. C. R. Math. Acad. Sci. Paris, 347(13-14):705–708, 2009. [457]
[93] C. Armana. Coefficients of drinfeld modular forms and hecke operators. J. NumberTheory, 131:1435–1460, 2011. [457]
[94] M. A. Armand. Multisequence shift register synthesis over commutative rings withidentity with applications to decoding cyclic codes over integer residue rings.IEEE Trans. Inform. Theory, 50(1):220–229, 2004. [275, 281]
[95] F. Armknecht and M. Krause. Algebraic attacks on combiners with memory. In Ad-vances in cryptology—CRYPTO 2003, volume 2729 of Lecture Notes in Com-put. Sci., pages 162–175. Springer, Berlin, 2003. [665]
[96] V. I. Arnold. Dynamics, statistics and projective geometry of Galois fields. Cam-bridge University Press, Cambridge, 2011. Translated from the Russian, With
Miscellaneous applications 707
words about Arnold by Maxim Kazarian and Ricardo Uribe-Vargas. [20]
[97] E. Artin. Quadratische Korper im Gebiete der hoheren Kongruenzen. I. Math. Z.,19(1):153–206, 1924. [356, 360]
[98] E. Artin. Quadratische korper im gebiete der hoheren kongruenzen, ii. Math. Z.,19:207–246, 1924. [43, 49]
[99] E. Artin. Quadratische Korper im Gebiete der hoheren Kongruenzen. II. Math. Z.,19(1):207–246, 1924. [409, 414]
[100] E. Artin. Galoissche Theorie. Verlag Harri Deutsch, Zurich, 1973. Ubersetzung nachder zweiten englischen Auflage besorgt von Viktor Ziegler, Mit einem Anhangvon N. A. Milgram, Zweite, unveranderte Auflage, Deutsch-Taschenbucher, No.21. [19, 20, 83]
[101] E. F. Assmus, Jr. and J. D. Key. Designs and their codes, volume 103 of CambridgeTracts in Mathematics. Cambridge University Press, Cambridge, 1992. [20,261, 262]
[102] E. F. Assmus, Jr. and H. F. Mattson, Jr. New 5-designs. J. Combinatorial Theory,6:122–151, 1969. [590, 602]
[103] Y. Aubry and P. Langevin. On the weights of binary irreducible cyclic codes. InCoding and cryptography, volume 3969 of Lecture Notes in Comput. Sci., pages46–54. Springer, Berlin, 2006. [109, 118]
[104] J.-P. Aumasson, M. Finiasz, W. Meier, and S. Vaudenay. A hardware-oriented trap-door cipher. In J. Pieprzyk, H. Ghodosi, and E. Dawson, editors, InformationSecurity and Privacy, volume 4586 of Lecture Notes in Computer Science, pages184–199. Springer Berlin / Heidelberg, 2007. [521, 531]
[105] J. Ax. Zeroes of polynomials over finite fields. Amer. J. Math., 86:255–261, 1964.[157, 158, 395, 402]
[106] M. Ayad and D. L. McQuillan. Irreducibility of the iterates of a quadratic polyno-mial over a field. Acta Arith., 93(1):87–97, 2000. [287, 289]
[107] M. Ayad and D. L. McQuillan. Corrections to: “Irreducibility of the iterates ofa quadratic polynomial over a field” [Acta Arith. 93 (2000), no. 1, 87–97;MR1760091 (2001c:11031)]. Acta Arith., 99(1):97, 2001. [287, 289]
[108] M. Baake, J. A. G. Roberts, and A. Weiss. Periodic orbits of linear endomorphismson the 2-torus and its lattices. Nonlinearity, 21(10):2427–2446, 2008. [282, 289]
[109] L. Babai. The fourier transform and equations over finite abelian groups. PrivateCommunication. [261]
[110] L. Babai. Spectra of Cayley graphs. J. Combin. Theory Ser. B, 27(2):180–189,1979. [537, 545]
[111] C. Bajaj, J. Canny, T. Garrity, and J. Warren. Factoring rational polynomials overthe complex numbers. SIAM J. Comput., 22(2):318–331, 1993. [305, 311]
[112] R. D. Baker, C. Culbert, G. L. Ebert, and K. E. Mellinger. Odd order flag-transitiveaffine planes of dimension three over their kernel. Adv. Geom., (suppl.):S215–S223, 2003. Special issue dedicated to Adriano Barlotti. [481, 486]
[113] R. D. Baker, J. M. Dover, G. L. Ebert, and K. L. Wantz. Hyperbolic fibrations ofPG(3, q). European J. Combin., 20(1):1–16, 1999. [485, 486]
[114] R. D. Baker, J. M. Dover, G. L. Ebert, and K. L. Wantz. Baer subgeometrypartitions. J. Geom., 67(1-2):23–34, 2000. Second Pythagorean Conference(Pythagoreion, 1999). [483, 486]
[115] R. D. Baker and G. L. Ebert. Nests of size q − 1 and another family of translationplanes. J. London Math. Soc. (2), 38(2):341–355, 1988. [480, 486]
708 Handbook of Finite Fields
[116] R. D. Baker and G. L. Ebert. A new class of translation planes. In Combinatorics ’86(Trento, 1986), volume 37 of Ann. Discrete Math., pages 7–20. North-Holland,Amsterdam, 1988. [480, 486]
[117] R. D. Baker and G. L. Ebert. Filling the nest gaps. Finite Fields Appl., 2(1):42–61,1996. [480, 486]
[118] R. D. Baker and G. L. Ebert. Two-dimensional flag-transitive planes revisited.Geom. Dedicata, 63(1):1–15, 1996. [481, 486]
[119] R. D. Baker, G. L. Ebert, K. H. Leung, and Q. Xiang. A trace conjecture andflag-transitive affine planes. J. Combin. Theory Ser. A, 95(1):158–168, 2001.[481, 486]
[120] R. D. Baker, G. L. Ebert, and T. Penttila. Hyperbolic fibrations and q-clans. Des.Codes Cryptogr., 34(2-3):295–305, 2005. [485, 486]
[121] R. D. Baker, G. L. Ebert, and K. L. Wantz. Regular hyperbolic fibrations. Adv.Geom., 1(2):119–144, 2001. [485, 486]
[122] R. D. Baker, G. L. Ebert, and K. L. Wantz. Enumeration of nonsingular Buekenhoutunitals. Note Mat., 29(1):69–90, 2009. [484, 486]
[123] R. D. Baker, G. L. Ebert, and K. L. Wantz. Enumeration of orthogonal Buekenhoutunitals. Des. Codes Cryptogr., 55(2-3):261–283, 2010. [484, 486]
[124] J. Balakrishnan, J. Belding, S. Chisholm, K. Eisentrager, K. E. Stange, and E. Teske.Pairings on hyperelliptic curves. Fields Inst. Commun., 58:1–34, 2010. [359,360]
[125] R. Balasubramanian and N. Koblitz. The improbability that an elliptic curve hassubexponential discrete log problem under the Menezes-Okamoto-Vanstonealgorithm. J. Cryptology, 11(2):141–145, 1998. [671]
[126] S. Ball. On the size of a triple blocking set in PG(2, q). European J. Combin.,17(5):427–435, 1996. [474, 475]
[127] S. Ball. The number of directions determined by a function over a finite field. J.Combin. Theory Ser. A, 104(2):341–350, 2003. [470, 475]
[128] S. Ball. On the graph of a function in many variables over a finite field. Des. CodesCryptogr., 47(1-3):159–164, 2008. [471, 475]
[129] S. Ball. The polynomial method in galois geometries. In Current Research Topics inGalois Geometry, Mathematics Research Developments. Nova, 2011, to appear.[475]
[130] S. Ball and A. Blokhuis. On the size of a double blocking set in PG(2, q). FiniteFields Appl., 2(2):125–137, 1996. [474, 475]
[131] S. Ball, A. Blokhuis, and F. Mazzocca. Maximal arcs in Desarguesian planes of oddorder do not exist. Combinatorica, 17(1):31–41, 1997. [484, 486]
[132] S. Ball and A. Gacs. On the graph of a function over a prime field whose smallpowers have bounded degree. European J. Combin., 30(7):1575–1584, 2009.[471, 475]
[133] S. Ball, A. Gacs, and P. Sziklai. On the number of directions determined by a pairof functions over a prime field. J. Combin. Theory Ser. A, 115(3):505–516,2008. [471, 475]
[134] S. Ball and M. Zieve. Symplectic spreads and permutation polynomials. In Finitefields and applications, volume 2948 of Lecture Notes in Comput. Sci., pages79–88. Springer, Berlin, 2004. [185]
[135] A. Balog. Many additive quadruples. In Additive combinatorics, volume 43 of CRMProc. Lecture Notes, pages 39–49. Amer. Math. Soc., Providence, RI, 2007.
Miscellaneous applications 709
[130]
[136] A. Balog and E. Szemeredi. A statistical theorem of set addition. Combinatorica,14(3):263–268, 1994. [130]
[137] J. Bamberg, A. Betten, C. Praeger, and A. Wassermann. Unitals in the Desargue-sian projective plane of order sixteen. International Conference on Design ofExperiments (ICODOE, 2011). [484, 486]
[138] W. D. Banks, A. Conflitti, J. B. Friedlander, and I. E. Shparlinski. Exponentialsums over Mersenne numbers. Compos. Math., 140(1):15–30, 2004. [132]
[139] W. D. Banks, J. B. Friedlander, S. V. Konyagin, and I. E. Shparlinski. Incompleteexponential sums and Diffie-Hellman triples. Math. Proc. Cambridge Philos.Soc., 140(2):193–206, 2006. [148]
[140] H. W. Bao. On two exponential sums and their applications. Finite Fields Appl.,3(2):115–130, 1997. [62]
[141] I. Baoulina. On the number of solutions to certain diagonal equations over finitefields. Int. J. Number Theory, 6(1):1–14, 2010. [165, 170]
[142] B. Barak, G. Kindler, R. Shaltiel, B. Sudakov, and A. Wigderson. Simulatingindependence: new constructions of condensers, Ramsey graphs, dispersers,and extractors. J. ACM, 57(4):Art. 20, 52, 2010. [133]
[143] M. Bardet, J.-C. Faugere, and B. Salvy. On the complexity of Grobner basis com-putation of semi-regular overdetermined algebraic equations. In Proceedingsof the International Conference on Polynomial System Solving, pages 71–74,2004. Previously INRIA report RR-5049. [664]
[144] A. Barlotti. Un’estensione del teorema di Segre-Kustaanheimo. Boll. Un. Mat. Ital.(3), 10:498–506, 1955. [500]
[145] P. S. L. M. Barreto, S. D. Galbraith, C. O’hEigeartaigh, and M. Scott. Efficientpairing computation on supersingular abelian varieties. Designs, Codes andCryptography, 42:239–271, 2007. [672]
[146] S. Barwick and G. Ebert. Unitals in projective planes. Springer Monographs inMathematics. Springer, New York, 2008. [483, 486]
[147] S. G. Barwick and W.-A. Jackson. Geometric constructions of optimal linear perfecthash families. Finite Fields Appl., 14(1):1–13, 2008. [552]
[148] S. G. Barwick, W.-A. Jackson, and C. T. Quinn. Optimal linear perfect hash familieswith small parameters. J. Combin. Des., 12(5):311–324, 2004. [552]
[149] L. Batina, S. B. rs, B. Preneel, and J. Vandewalle. Hardware architectures for publickey cryptography. Integration, the VLSI Journal, 34(1-2):1 – 64, 2003. [79]
[150] L. D. Baumert. Cyclic difference sets. Lecture Notes in Mathematics, Vol. 182.Springer-Verlag, Berlin, 1971. [20, 512, 515, 519]
[151] B. Beckermann and G. Labahn. Fraction-free computation of matrix rational in-terpolants and matrix GCDs. SIAM J. Matrix Anal. Appl., 22(1):114–144(electronic), 2000. [436]
[152] E. Bedford and K. Kim. Continuous families of rational surface automorphismswith positive entropy. Math. Ann., 348(3):667–688, 2010. [283, 289]
[153] E. Bedford and T. T. Truong. Degree complexity of birational maps related tomatrix inversion. Comm. Math. Phys., 298(2):357–368, 2010. [282, 283, 289]
[154] P. Beelen and I. I. Bouw. Asymptotically good towers and differential equations.Compos. Math., 141(6):1405–1424, 2005. [368, 372]
[155] D. Behr. Searchable magic book contents. main site:http://archive.denisbehr.
710 Handbook of Finite Fields
de. http://archive.denisbehr.de/archive/route/entries.php?url=10,
50,1036. [531]
[156] K. Belabas, M. van Hoeij, J. Kluners, and A. Steel. Factoring polynomials overglobal fields. J. Theor. Nombres Bordeaux, 21(1):15–39, 2009. [304, 311]
[157] J. Belding, R. Brker, A. Enge, and K. Lauter. Computing Hilbert class polynomi-als. In A. van der Poorten and A. Stein, editors, Algorithmic Number TheoryANTS-VIII, volume 5011 of Lecture Notes in Computer Science, pages 282–295, Berlin, 2008. Springer-Verlag. [669]
[158] M. Bellare and P. Rogaway. Minimizing the use of random oracles in authenticatedencryption schemes. In Y. Han, T. Okamoto, and S. Qing, editors, Informa-tion and Communications Security, volume 1334 of Lecture Notes in ComputerScience, pages 1–16, Berlin, 1997. Springer-Verlag. [667]
[159] M. P. Bellon and C.-M. Viallet. Algebraic entropy. Comm. Math. Phys., 204(2):425–437, 1999. [282, 283, 289]
[160] M. Ben-Or. Probabilistic algorithms in finite fields. In Proc. 22nd IEEE Symp.Foundations Computer Science, pages 394–398, 1981. [296, 299]
[161] T. D. Bending and D. Fon-Der-Flaass. Crooked functions, bent functions, anddistance regular graphs. Electron. J. Combin., 5:Research Paper 34, 14 pp.(electronic), 1998. [211, 213]
[162] A. T. Benjamin and C. D. Bennett. The probability of relatively prime polynomials.Math. Mag., 80:196–202, 2007. [51, 55, 423, 424]
[163] C. H. Bennett and G. Brassard. Quantum cryptography: Public key distributionand coin tossing. In International Conference on Computers, Systems & SignalProcessing (Bangalore, India, 2004). 1984. [633, 634]
[164] T. P. Berger, A. Canteaut, P. Charpin, and Y. Laigle-Chapuy. On almost perfectnonlinear functions over Fn2 . IEEE Trans. Inform. Theory, 52(9):4160–4170,2006. [208, 211, 213]
[165] E. R. Berlekamp. Factoring polynomials over finite fields. Bell System Tech. J.,46:1853–1859, 1967. [294, 299, 653]
[166] E. R. Berlekamp. Algebraic coding theory. McGraw-Hill Book Co., New York, 1968.[19, 20, 37, 38, 41, 163, 561, 591, 593, 602]
[167] E. R. Berlekamp, editor. Key papers in the development of coding theory. IEEEPress [Institute of Electrical and Electronics Engineers, Inc.], New York, 1974.IEEE Press Selected Reprint Series. [601, 602]
[168] E. R. Berlekamp. Bit-serial Reed-Solomon encoders. IEEE Trans. Inf. Theory,28:869–874, 1982. [79]
[169] E. R. Berlekamp, R. J. McEliece, and H. C. A. van Tilborg. On the inherentintractability of certain coding problems. IEEE Trans. Information Theory,IT-24(3):384–386, 1978. [633, 634]
[170] E. R. Berlekamp, H. Rumsey, and G. Solomon. On the solution of algebraic equa-tions over finite fields. Information and Control, 10:553–564, 1967. [38]
[171] P. Berman and G. Schnitger. On the performance of the minimum degree orderingfor Gaussian elimination. SIAM J. Matrix Anal. Appl., 11(1):83–88, 1990. [434,436]
[172] L. Bernardin. On square-free factorization of multivariate polynomials over a finitefield. Theoret. Comput. Sci., 187(1-2):105–116, 1997. [303, 311]
[173] L. Bernardin. On bivariate Hensel lifting and its parallelization. In ISSAC ’98:Proceedings of the 1998 International Symposium on Symbolic and Algebraic
Miscellaneous applications 711
Computation, pages 96–100, New York, 1998. ACM Press. [304, 311]
[174] L. Bernardin and M. B. Monagan. Efficient multivariate factorization over fi-nite fields. In Applied algebra, algebraic algorithms and error-correcting codes(Toulouse, 1997), volume 1255 of Lecture Notes in Comput. Sci., pages 15–28.Springer-Verlag, 1997. [306, 311]
[175] B. C. Berndt, R. J. Evans, and K. S. Williams. Gauss and Jacobi sums. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. CanadianMathematical Society Series of Monographs and Advanced Texts. [19, 20, 137]
[176] B. C. Berndt, R. J. Evans, and K. S. Williams. Gauss and Jacobi sums. CanadianMathematical Society Series of Monographs and Advanced Texts. John Wiley& Sons Inc., New York, 1998. A Wiley-Interscience Publication. [96, 98, 99,100, 102, 104, 105, 106, 107, 108, 113, 117, 118]
[177] D. J. Bernstein, J. Buchmann, and E. Dahmen, editors. Post-quantum cryptography.Springer-Verlag, Berlin, 2009. [19, 20, 633, 634]
[178] D. J. Bernstein, T. Lange, and C. Peters. Attacking and defending the McEliececryptosystem. In Post-quantum cryptography (Cincinnati, Ohio, 2008), volume5299 of Lecture Notes in Comput. Sci., pages 31–46. Springer, Berlin, 2008.[634]
[179] P. Berthelot. Cohomologie rigide et theorie de Dwork: le cas des sommes exponen-tielles. Asterisque, (119-120):3, 17–49, 1984. p-adic cohomology. [126]
[180] P. Berthelot, S. Bloch, and H. Esnault. On Witt vector cohomology for singularvarieties. Compos. Math., 143(2):363–392, 2007. [158]
[181] P. Berthelot and A. Ogus. Notes on crystalline cohomology. Princeton UniversityPress, Princeton, N.J., 1978. [396, 402]
[182] J. Berthomieu and G. Lecerf. Convex-dense bivariate polynomial factor-ization. Manuscript available from http://hal.archives-ouvertes.fr/
hal-00526659, to appear in Math. Comp., 2010. [308, 311]
[183] T. Beth and Z. D. Dai. On the complexity of pseudo-random sequences—or: Ifyou can describe a sequence it can’t be random. In Advances in cryptology—EUROCRYPT ’89 (Houthalen, 1989), volume 434 of Lecture Notes in Comput.Sci., pages 533–543. Springer, Berlin, 1990. [280, 281]
[184] T. Beth and W. Geiselmann. Selbstduale Normalbasen uber GF(q). Arch. Math.(Basel), 55(1):44–48, 1990. [420, 424]
[185] T. Beth, D. Jungnickel, and H. Lenz. Design theory. Cambridge University Press,Cambridge, 1986. [20, 135]
[186] T. Beth, D. Jungnickel, and H. Lenz. Design theory. Vol. I, volume 69 of Ency-clopedia of Mathematics and its Applications. Cambridge University Press,Cambridge, second edition, 1999. [20, 511, 512, 513, 517, 519, 558]
[187] T. Beth, D. Jungnickel, and H. Lenz. Design theory. Vol. II, volume 78 of En-cyclopedia of Mathematics and its Applications. Cambridge University Press,Cambridge, second edition, 1999. [20, 511, 512, 513, 519, 558]
[188] D. Betten and D. G. Glynn. Uber endliche planare Funktionen, ihre zugehorendenSchiebebenen, und ihre abgeleiteten Translationsebenen. Results Math., 42(1-2):32–36, 2002. [231, 234]
[189] C. Bey and G. M. Kyureghyan. On Boolean functions with the sum of every two ofthem being bent. Des. Codes Cryptogr., 49(1-3):341–346, 2008. [206, 213]
[190] J. Bezerra, A. Garcia, and H. Stichtenoth. An explicit tower of function fields overcubic finite fields and Zink’s lower bound. J. Reine Angew. Math., 589:159–199,
712 Handbook of Finite Fields
2005. [367, 371, 372]
[191] M. Bhargava and M. E. Zieve. Factoring Dickson polynomials over finite fields.Finite Fields Appl., 5(2):103–111, 1999. [236, 242]
[192] A. Bhattacharyya, S. Kopparty, G. Schoenebeck, M. Sudan, and D. Zuckerman. Op-timal testing of reed-muller codes (report no. 86). In Proceedings of ElectronicColloquium on Computational Complexity (2009). [200, 204]
[193] K. Bibak. Additive combinatorics with a view towards computer science and cryp-tography: An exposition. arXiv:1108.3790. [132, 133]
[194] F. Bien. Constructions of telephone networks by group representations. NoticesAmer. Math. Soc., 36(1):5–22, 1989. [532, 539, 545]
[195] J. Bierbrauer. Introduction to coding theory. Discrete Mathematics and its Ap-plications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005. [19,20]
[196] J. Bierbrauer. A direct approach to linear programming bounds for codes andtms-nets. Des. Codes Cryptogr., 42:127–143, 2007. [374, 383]
[197] J. Bierbrauer. A family of crooked functions. Des. Codes Cryptogr., 50(2):235–241,2009. [211, 213]
[198] J. Bierbrauer. New commutative semifields and their nuclei. In Applied algebra,algebraic algorithms, and error-correcting codes, volume 5527 of Lecture Notesin Comput. Sci., pages 179–185. Springer, Berlin, 2009. [233, 234]
[199] J. Bierbrauer. New semifields, PN and APN functions. Des. Codes Cryptogr.,54(3):189–200, 2010. [233, 234]
[200] J. Bierbrauer, Y. Edel, and W. C. Schmid. Coding-theoretic constructions for(t,m, s)-nets and ordered orthogonal arrays. J. Combin. Des., 10:403–418,2002. [375, 378, 383]
[201] J. Bierbrauer and G. M. Kyureghyan. Crooked binomials. Des. Codes Cryptogr.,46(3):269–301, 2008. [211, 213]
[202] E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. J.Cryptology, 4(1):3–72, 1991. [205, 213]
[203] M. Biliotti, V. Jha, and N. L. Johnson. Foundations of translation planes, volume243 of Monographs and Textbooks in Pure and Applied Mathematics. MarcelDekker Inc., New York, 2001. [478, 486]
[204] O. Billet and H. Gilbert. Cryptanalysis of rainbow. In Security and Cryptographyfor Networks, volume 4116 of LNCS, pages 336–347. Springer, September 2006.[655, 656]
[205] O. Billet, M. J. B. Robshaw, and T. Peyrin. On building hash functions frommultivariate quadratic equations. In J. Pieprzyk, H. Ghodosi, and E. Dawson,editors, ACISP, volume 4586 of Lecture Notes in Computer Science, pages82–95. Springer, 2007. [665]
[206] Y. Bilu and N. Linial. Lifts, discrepancy and nearly optimal spectral gap. Combi-natorica, 26(5):495–519, 2006. [543, 544, 545]
[207] G. Bini and F. Flamini. Finite commutative rings and their applications. The KluwerInternational Series in Engineering and Computer Science, 680. Kluwer Aca-demic Publishers, Boston, MA, 2002. With a foreword by Dieter Jungnickel.[17, 18, 19]
[208] B. J. Birch. How the number of points of an elliptic curve over a fixed prime fieldvaries. J. London Math. Soc., 43:57–60, 1968. [341, 351]
[209] B. J. Birch and H. P. F. Swinnerton-Dyer. Note on a problem of Chowla. Acta
Miscellaneous applications 713
Arith., 5:417–423 (1959), 1959. [190, 192]
[210] A. Biro. On polynomials over prime fields taking only two values on the multiplica-tive group. Finite Fields Appl., 6(4):302–308, 2000. [190, 192]
[211] R. R. Bitmead and B. D. O. Anderson. Asymptotically fast solution of Toeplitz andrelated systems of linear equations. Linear Algebra Appl., 34:103–116, 1980.[434, 436]
[212] R. Blache. First vertices for generic newton polygons, and p-cyclic coverings of theprojective line. [399, 402]
[213] R. Blache. Newton polygons for character sums and p??incare series. Int. J. NumberTh., to appear. [400, 402]
[214] R. Blache. p-density, exponential sums and artin-schreier curves. [395, 399, 402]
[215] R. Blache and E. Ferard. Newton stratification for polynomials: the open stratum.J. Number Theory, 123(2):456–472, 2007. [399, 402]
[216] R. Blache, E. Ferard, and H. J. Zhu. Hodge-Stickelberger polygons for L-functionsof exponential sums of P (xs). Math. Res. Lett., 15(5):1053–1071, 2008. [399,402]
[217] S. R. Blackburn. A generalisation of the discrete Fourier transform: determiningthe minimal polynomial of a periodic sequence. IEEE Trans. Inform. Theory,40(5):1702–1704, 1994. [274, 281]
[218] S. R. Blackburn, T. Etzion, and K. G. Paterson. Permutation polynomials, de Bruijnsequences, and linear complexity. J. Combin. Theory Ser. A, 76(1):55–82, 1996.[274, 281]
[219] S. R. Blackburn, D. Gomez-Perez, J. Gutierrez, and I. E. Shparlinski. Predictingthe inversive generator. In Cryptography and coding, volume 2898 of LectureNotes in Comput. Sci., pages 264–275. Springer, Berlin, 2003. [283, 289]
[220] S. R. Blackburn, D. Gomez-Perez, J. Gutierrez, and I. E. Shparlinski. Predictingnonlinear pseudorandom number generators. Math. Comp., 74(251):1471–1494(electronic), 2005. [283, 289]
[221] S. R. Blackburn, D. Gomez-Perez, J. Gutierrez, and I. E. Shparlinski. Reconstruct-ing noisy polynomial evaluation in residue rings. J. Algorithms, 61(2):47–59,2006. [283, 289]
[222] S. R. Blackburn and P. R. Wild. Optimal linear perfect hash families. J. Combin.Theory Ser. A, 83(2):233–250, 1998. [552]
[223] R. E. Blahut. Transform techniques for error control codes. IBM J. Res. Develop.,23(3):299–315, 1979. [273, 281]
[224] R. E. Blahut. Theory and practice of error control codes. Addison-Wesley PublishingCompany Advanced Book Program, Reading, MA, 1983. [19, 20, 561, 563, 580,588, 589, 591, 593, 602]
[225] I. F. Blake, editor. Algebraic coding theory: history and development. DowdenHutchinson & Ross Inc., Stroudsburg, Pa., 1973. Benchmark Papers in Elec-trical Engineering and Computer Science. [601, 602]
[226] I. F. Blake, S. Gao, and R. J. Lambert. Construction and distribution problems forirreducible trinomials over finite fields. In Applications of finite fields (Egham,1994), volume 59 of Inst. Math. Appl. Conf. Ser. New Ser., pages 19–32. OxfordUniv. Press, New York, 1996. [58]
[227] I. F. Blake, S. Gao, and R. C. Mullin. Specific irreducible polynomials with linearlyindependent roots over finite fields. Linear Algebra Appl., 253:227–249, 1997.[83, 94]
714 Handbook of Finite Fields
[228] I. F. Blake and T. Garefalakis. A transform property of Kloosterman sums. DiscreteAppl. Math., 158(10):1064–1072, 2010. [44, 49]
[229] I. F. Blake and R. C. Mullin. The mathematical theory of coding. Academic Press[A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London,1975. [19, 20, 561, 582, 587, 602]
[230] I. F. Blake, G. Seroussi, and N. P. Smart. Elliptic curves in cryptography, volume265 of London Mathematical Society Lecture Note Series. Cambridge UniversityPress, Cambridge, 2000. Reprint of the 1999 original. [19, 20, 666, 678]
[231] I. F. Blake, G. Seroussi, and N. P. Smart. Advances in Elliptic Curve Cryptography,volume 317 of London Mathematical Society Lecture Note Series. CambridgeUniversity Press, Cambridge, 2005. [19, 20, 666, 667, 670]
[232] D. Blessenohl and K. Johnsen. Eine Verscharfung des Satzes von der Normalbasis.J. Algebra, 103(1):141–159, 1986. [83]
[233] D. Blessenohl and K. Johnsen. Stabile Teilkorper galoisscher Erweiterungen undein Problem von C. Faith. Arch. Math. (Basel), 56(3):245–253, 1991. [84]
[234] A. Blokhuis. On the size of a blocking set in PG(2, p). Combinatorica, 14(1):111–114,1994. [471, 472, 475]
[235] A. Blokhuis. Blocking sets in Desarguesian planes. In Combinatorics, Paul ErdHosis eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages133–155. Janos Bolyai Math. Soc., Budapest, 1996. [471, 472, 475]
[236] A. Blokhuis, S. Ball, A. E. Brouwer, L. Storme, and T. SzHonyi. On the number ofslopes of the graph of a function defined on a finite field. J. Combin. TheorySer. A, 86(1):187–196, 1999. [470, 475]
[237] A. Blokhuis, A. E. Brouwer, and T. SzHonyi. The number of directions determinedby a function f on a finite field. J. Combin. Theory Ser. A, 70(2):349–353,1995. [470, 475]
[238] A. Blokhuis, A. E. Brouwer, and H. A. Wilbrink. Blocking sets in PG(2, p) forsmall p, and partial spreads in PG(3, 7). Adv. Geom., (suppl.):S245–S253,2003. Special issue dedicated to Adriano Barlotti. [473, 475]
[239] A. Blokhuis, A. A. Bruen, and J. A. Thas. Arcs in PG(n, q), MDS-codes and threefundamental problems of B. Segre—some extensions. Geom. Dedicata, 35(1-3):1–11, 1990. [498]
[240] A. Blokhuis, R. S. Coulter, M. Henderson, and C. M. O’Keefe. Permutationsamongst the Dembowski-Ostrom polynomials. In Finite fields and applications(Augsburg, 1999), pages 37–42. Springer, Berlin, 2001. [180, 185]
[241] A. Blokhuis, D. Jungnickel, and B. Schmidt. Proof of the prime power conjecturefor projective planes of order n with abelian collineation groups of order n2.Proc. Amer. Math. Soc., 130(5):1473–1476 (electronic), 2002. [230, 234, 485,486]
[242] A. Blokhuis, M. Lavrauw, and S. Ball. On the classification of semifield flocks. Adv.Math., 180(1):104–111, 2003. [229]
[243] A. Blokhuis, L. Lovasz, L. Storme, and T. SzHonyi. On multiple blocking sets inGalois planes. Adv. Geom., 7(1):39–53, 2007. [474, 475]
[244] A. Blokhuis, R. Pellikaan, and T. SzHonyi. Blocking sets of almost Redei type. J.Combin. Theory Ser. A, 78(1):141–150, 1997. [471, 475]
[245] A. Blokhuis, L. Storme, and T. SzHonyi. Lacunary polynomials, multiple blockingsets and Baer subplanes. J. London Math. Soc. (2), 60(2):321–332, 1999. [474,475]
Miscellaneous applications 715
[246] C. Blondeau, A. Canteaut, and P. Charpin. Differential properties of power func-tions. Int. J. Inf. Coding Theory, 1(2):149–170, 2010. [213]
[247] A. W. Bluher. Explicit formulas for strong Davenport pairs. Acta Arith., 112(4):397–403, 2004. [253, 255]
[248] A. W. Bluher. A Swan-like theorem. Finite Fields Appl., 12(1):128–138, 2006. [37,38]
[249] G. Bockle. An eichler-shimura isomorphism over function fields between drinfeldmodular forms and cohomology classes of crystals. preprint, 2002. [457]
[250] G. Bockle. Global L-functions over function fields. Math. Ann., 323(4):737–795,2002. [454]
[251] A. Bodin. Number of irreducible polynomials in several variables over finite fields.Amer. Math. Monthly, 115:653–660, 2008. [50, 55]
[252] A. Bodin. Generating series for irreducible polynomials over finite fields. FiniteFields Appl., 16:116–125, 2010. [50, 52, 55]
[253] A. Bodin, P. Debes, and S. Najib. Indecomposable polynomials and their spectrum.Acta Arith., 139:79–100, 2009. [53, 54, 55]
[254] E. Bombieri. On exponential sums in finite fields. Amer. J. Math., 88:71–105, 1966.[120, 127, 388, 391, 393]
[255] E. Bombieri. Counting points on curves over finite fields (d’apres S. A. Stepanov). InSeminaire Bourbaki, 25eme annee (1972/1973), Exp. No. 430, pages 234–241.Lecture Notes in Math., Vol. 383. Springer, Berlin, 1974. [391, 393]
[256] E. Bombieri. On exponential sums in finite fields. II. Invent. Math., 47(1):29–39,1978. [121, 127, 254, 255, 388, 391, 393]
[257] E. Bombieri and S. Sperber. On the estimation of certain exponential sums. ActaArith., 69(4):329–358, 1995. [121, 127]
[258] D. Bonchev, S. Thomas, A. Apte, and L. B. Kier. Cellular automata modelling ofbiomolecular networks dynamics. SAR and QSAR in Environmental Research,21(1):77–102, 2010. [685]
[259] D. Boneh and M. Franklin. Identity-based encryption from the Weil pairing. SIAMJ. Comput., 32(3):586–615, 2003. [631, 634]
[260] D. Boneh, E.-J. Goh, and K. Nissim. Evaluating 2-DNF formulas on ciphertexts. InJ. Kilian, editor, Theory of Cryptography — TCC 2005, volume 3378 of LectureNotes in Computer Science, pages 325–341, Berlin, 2005. Springer-Verlag. [674]
[261] D. Boneh, B. Lynn, and H. Shacham. Short signatures from the Weil pairing. J.Cryptology, 17(4):297–319, 2004. [632, 634]
[262] D. Boneh and R. Venkatesan. Rounding in lattices and its cryptographic applica-tions. In Proceedings of the Eighth Annual ACM-SIAM Symposium on DiscreteAlgorithms (New Orleans, LA, 1997), pages 675–681, New York, 1997. ACM.[141]
[263] D. Boneh and R. Venkatesan. Breaking rsa may not be equivalent to factoring. InK. Nyberg, editor, Advances in Cryptology EUROCRYPT’98, volume 1403 ofLecture Notes in Computer Science, pages 59–71. Springer Berlin / Heidelberg,1998. [141]
[264] T. J. Boothby and R. W. Bradshaw. Bitslicing and the method of four russians overlarger finite fields, Jan. 2009. arXiv:0901.1413v1 [cs.MS]. [425, 436]
[265] P. Borwein, K.-K. S. Choi, and J. Jedwab. Binary sequences with merit factorgreater than 6.34. IEEE Trans. Inform. Theory, 50(12):3234–3249, 2004. [269]
716 Handbook of Finite Fields
[266] S. Bosch, U. Guntzer, and R. Remmert. Non-Archimedean analysis, volume 261 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences]. Springer-Verlag, Berlin, 1984. A systematic approachto rigid analytic geometry. [449]
[267] R. C. Bose. On the application of the properties of galois fields to the constructionof hyper-graeco-latin squares. Sankhya, 3:323–338, 1938. [466]
[268] R. C. Bose. On the construction of balanced incomplete block designs. Ann. Eu-genics, 9:353–399, 1939. [504]
[269] R. C. Bose. On some connections between the design of experiments and informationtheory. Bull. Inst. Internat. Statist., 38:257–271, 1961. [520, 531]
[270] R. C. Bose and R. C. Burton. A characterization of flat spaces in a finite geometryand the uniqueness of the Hamming and the MacDonald codes. J. Combina-torial Theory, 1:96–104, 1966. [471, 475]
[271] R. C. Bose and D. K. Ray-Chaudhuri. On a class of error correcting binary groupcodes. Information and Control, 3:68–79, 1960. [578, 601, 602]
[272] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The userlanguage. J. Symbolic Comput., 24(3-4):235–265, 1997. [306, 311]
[273] A. Bostan, C.-P. Jeannerod, and E. Schost. Solving structured linear systems withlarge displacement rank. Theoret. Comput. Sci., 407(1-3):155–181, 2008. [434,436]
[274] A. Bostan, G. Lecerf, B. Salvy, E. Schost, and B. Wiebelt. Complexity issues inbivariate polynomial factorization. In ISSAC ’04: Proceedings of the 2004International Symposium on Symbolic and Algebraic Computation, pages 42–49, New York, 2004. ACM Press. [304, 311]
[275] A. Bostan, F. Morain, B. Salvy, and E. Schost. Fast algorithms for computingisogenies between elliptic curves. Mathematics of Computation, 77(263):1755–1778, 2008. [670]
[276] A. Bostin, P. Flajolet, B. Salvy, and E. Schost. Fast computation of special re-sultants. Journal of Symbolic Computation, 41(1):1–29, Jan. 2006. [297, 298,299]
[277] A. Bottcher and B. Silbermann. Introduction to large truncated Toeplitz matrices.Universitext. Springer-Verlag, New York, 1999. [422, 424]
[278] J. Bourgain. Estimates on exponential sums related to the Diffie-Hellman distribu-tions. Geom. Funct. Anal., 15(1):1–34, 2005. [132, 148]
[279] J. Bourgain. Mordell’s exponential sum estimate revisited. J. Amer. Math. Soc.,18(2):477–499 (electronic), 2005. [132]
[280] J. Bourgain. More on the sum-product phenomenon in prime fields and its appli-cations. Int. J. Number Theory, 1(1):1–32, 2005. [133]
[281] J. Bourgain. Multilinear exponential sums in prime fields under optimal entropycondition on the sources. Geom. Funct. Anal., 18(5):1477–1502, 2009. [129,130, 131, 137, 141]
[282] J. Bourgain. On exponential sums in finite fields. In An irregular mind: Szemerediis 70, pages 219–242. Springer, 2010. [132]
[283] J. Bourgain and M.-C. Chang. A Gauss sum estimate in arbitrary finite fields. C.R. Math. Acad. Sci. Paris, 342(9):643–646, 2006. [98, 118]
[284] J. Bourgain and A. Gamburd. Uniform expansion bounds for Cayley graphs ofSL2(Fp). Ann. of Math. (2), 167(2):625–642, 2008. [134]
[285] J. Bourgain, A. Gamburd, and P. Sarnak. Affine linear sieve, expanders, and sum-
[286] J. Bourgain and M. Z. Garaev. On a variant of sum-product estimates and explicitexponential sum bounds in prime fields. Math. Proc. Cambridge Philos. Soc.,146(1):1–21, 2009. [129, 130, 149]
[287] J. Bourgain and A. Glibichuk. Exponential sum estimate over subgroup in anarbitrary field. J. Analyse Math., 115(1):51–70, 2011. [130, 133]
[288] J. Bourgain, A. A. Glibichuk, and S. V. Konyagin. Estimates for the number ofsums and products and for exponential sums in fields of prime order. J. LondonMath. Soc. (2), 73(2):380–398, 2006. [128, 130, 131, 133, 137, 141]
[289] J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, andapplications. Geom. Funct. Anal., 14(1):27–57, 2004. [128]
[290] H. Boylan and N.-P. Skoruppa. Explicit formulas for Hecke Gauss sums in quadraticnumber fields. Abh. Math. Semin. Univ. Hambg., 80(2):213–226, 2010. [117,118]
[291] C. Bracken, E. Byrne, N. Markin, and G. McGuire. Determining the nonlinearityof a new family of APN functions. In Applied algebra, algebraic algorithms anderror-correcting codes, volume 4851 of Lecture Notes in Comput. Sci., pages72–79. Springer, Berlin, 2007. [211, 213, 261, 262]
[292] C. Bracken, E. Byrne, N. Markin, and G. McGuire. On the walsh spectrum of anew APN function. In Cryptography and Coding, volume 4887 of Lecture Notesin Comput. Sci., pages 92–98. Springer, Berlin, 2007. [211, 213]
[293] C. Bracken, E. Byrne, N. Markin, and G. McGuire. New families of quadratic almostperfect nonlinear trinomials and multinomials. Finite Fields Appl., 14(3):703–714, 2008. [209, 211, 213]
[294] C. Bracken, E. Byrne, N. Markin, and G. McGuire. Fourier spectra of binomialAPN functions. SIAM J. Discrete Math., 23(2):596–608, 2009. [211, 213, 261,262]
[295] C. Bracken, E. Byrne, N. Markin, and G. McGuire. A few more quadratic APNfunctions. Cryptogr. Commun., 3(1):43–53, 2011. [208, 211, 213]
[296] C. Bracken, E. Byrne, G. McGuire, and G. Nebe. On the equivalence of quadraticAPN functions. Des. Codes Cryptogr., 61(3):261–272, 2011. [211, 213]
[297] A. Braeken, C. Wolf, and B. Preneel. A study of the security of unbalanced oil andvinegar signature schemes. In Topics in cryptology—CT-RSA 2005, volume3376 of Lecture Notes in Comput. Sci., pages 29–43. Springer, Berlin, 2005.[663]
[298] N. Brandstatter and A. Winterhof. Some notes on the two-prime generator of order2. IEEE Trans. Inform. Theory, 51(10):3654–3657, 2005. [279, 281]
[299] N. Brandstatter and A. Winterhof. Linear complexity profile of binary sequenceswith small correlation measure. Period. Math. Hungar., 52(2):1–8, 2006. [281]
[300] J. V. Brawley and L. Carlitz. Irreducibles and the composed product for polynomialsover a finite field. Discrete Math., 65(2):115–139, 1987. [35, 38]
[301] J. V. Brawley and L. Carlitz. A test for additive decomposability of irreduciblesover a finite field. Discrete Math., 76(1):61–65, 1989. [35, 38]
[302] J. V. Brawley, L. Carlitz, and J. Levine. Scalar polynomial functions on the n× nmatrices over a finite field. Linear Algebra and Appl., 10:199–217, 1975. [183,185]
[303] J. V. Brawley and G. L. Mullen. Infinite Latin squares containing nested sets ofmutually orthogonal finite Latin squares. Publ. Math. Debrecen, 39(1-2):135–
718 Handbook of Finite Fields
141, 1991. [463, 467]
[304] J. V. Brawley and G. E. Schnibben. Infinite algebraic extensions of finite fields,volume 95 of Contemporary Mathematics. American Mathematical Society,Providence, RI, 1989. [19, 20, 93, 463, 467]
[305] R. P. Brent and P. Zimmermann. Ten new primitive binary trinomials. Math.Comp., 78(266):1197–1199, 2009. [66, 68]
[306] R. P. Brent and P. Zimmermann. The great trinomial hunt. Notices Amer. Math.Soc., 78(2):233–239, 2011. [66, 68]
[307] J. Brewster Lewis, R. Ini Liu, A. H. Morales, G. Panova, S. V. Sam, and Y. Zhang.Matrices with restricted entries and q-analogues of permutations. ArXiv e-prints, Nov. 2010. [416, 424]
[308] F. Brezing and A. Weng. Elliptic curves suitable for pairing based cryptography.Des. Codes Cryptogr., 37(1):133–141, 2005. [676]
[309] M. Brinkmann and G. Leander. On the classification of APN functions up todimension five. Des. Codes Cryptogr., 49(1-3):273–288, 2008. [209, 213]
[310] D. R. L. Brown. Generic groups, collision resistance, and ECDSA. Designs, Codesand Cryptography, 35:119–152, 2005. [667]
[311] M. R. Brown. Ovoids of pg(3,q),q even, with a conic section. J. London Math. Soc.,62(2):569–582, 2000. [500]
[312] M. R. Brown, G. L. Ebert, and D. Luyckx. On the geometry of regular hyperbolicfibrations. European J. Combin., 28(6):1626–1636, 2007. [485, 486]
[313] K. A. Browning, J. F. Dillon, R. E. Kibler, and M. T. McQuistan. APN polynomialsand related codes. Journal of Combinatorics Information and System Sciences,34(1-4):135–159, 2009. [209, 213]
[314] K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J. Wolfe. An APN per-mutation in dimension six. In Finite Fields: theory and applications, volume518 of Contemp. Math., pages 33–42. Amer. Math. Soc., Providence, RI, 2010.[185, 208, 213]
[315] R. H. Bruck. Difference sets in a finite group. Trans. Amer. Math. Soc., 78:464–481,1955. [482, 486]
[316] R. H. Bruck. Quadratic extensions of cyclic planes. In Proc. Sympos. Appl. Math.,Vol. 10, pages 15–44. American Mathematical Society, Providence, R.I., 1960.[482, 486]
[317] R. H. Bruck. Construction problems of finite projective planes. In CombinatorialMathematics and its Applications (Proc. Conf., Univ. North Carolina, ChapelHill, N.C., 1967), pages 426–514. Univ. North Carolina Press, Chapel Hill,N.C., 1969. [479, 480, 486]
[318] R. H. Bruck and R. C. Bose. The construction of translation planes from projectivespaces. J. Algebra, 1:85–102, 1964. [478, 479, 486]
[319] R. H. Bruck and H. J. Ryser. The nonexistence of certain finite projective planes.Canadian J. Math., 1:88–93, 1949. [513, 519]
[320] A. Bruen. Blocking sets in finite projective planes. SIAM J. Appl. Math., 21:380–392, 1971. [471, 472, 475]
[321] A. A. Bruen and R. Silverman. On the nonexistence of certain M.D.S. codes andprojective planes. Math. Z., 183(2):171–175, 1983. [499]
[322] A. A. Bruen and R. Silverman. Arcs and blocking sets. II. European J. Combin.,8(4):351–356, 1987. [471, 475]
Miscellaneous applications 719
[323] A. A. Bruen and J. A. Thas. Blocking sets. Geometriae Dedicata, 6(2):193–203,1977. [471, 475]
[324] A. A. Bruen, J. A. Thas, and A. Blokhuis. On M.D.S. codes, arcs in PG(n, q) with qeven, and a solution of three fundamental problems of B. Segre. Invent. Math.,92(3):441–459, 1988. [498, 499]
[325] L. Brunjes. Forms of Fermat equations and their zeta functions. World ScientificPublishing Co. Pte. Ltd., Hackensack, NJ, 2004. [386, 393]
[326] B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassen-ringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck,1965. [664]
[327] J. Buchmann, D. Cabarcas, J. Ding, and M. S. E. Mohamed. Flexible partialenlargement to accelerate grobner basis computation over 2. In D. J. Bern-stein and T. Lange, editors, AFRICACRYPT, volume 6055 of Lecture Notesin Computer Science, pages 69–81. Springer, 2010. [664]
[328] J. Buchmann and H. C. Williams. A key-exchange system based on imaginaryquadratic fields. J. Cryptology, 1(2):107–118, 1988. [630, 634]
[329] L. Budaghyan and C. Carlet. Classes of quadratic APN trinomials and hexanomialsand related structures. IEEE Trans. Inform. Theory, 54(5):2354–2357, 2008.[209, 211, 213]
[330] L. Budaghyan, C. Carlet, and G. Leander. Two classes of quadratic APN binomialsinequivalent to power functions. IEEE Trans. Inform. Theory, 54(9):4218–4229, 2008. [209, 211, 213]
[331] L. Budaghyan, C. Carlet, and G. Leander. Constructing new APN functions fromknown ones. Finite Fields Appl., 15(2):150–159, 2009. [211, 213]
[332] L. Budaghyan, C. Carlet, and A. Pott. New classes of almost bent and almostperfect nonlinear polynomials. IEEE Trans. Inform. Theory, 52(3):1141–1152,2006. [209, 211, 213]
[333] L. Budaghyan and T. Helleseth. New perfect nonlinear multinomials over Fp2k forany odd prime p. In Sequences and their applications—SETA 2008, volume5203 of Lecture Notes in Comput. Sci., pages 403–414. Springer, Berlin, 2008.[233, 234]
[334] L. Budaghyan and T. Helleseth. New commutative semifields defined by new PNmultinomials. Cryptogr. Commun., 3(1):1–16, 2011. [233, 234]
[335] F. Buekenhout. Existence of unitals in finite translation planes of order q2 with akernel of order q. Geometriae Dedicata, 5(2):189–194, 1976. [483, 486]
[336] F. Buekenhout. An introduction to incidence geometry. In Handbook of incidencegeometry, pages 1–25. North-Holland, Amsterdam, 1995. [20]
[337] F. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, andJ. Saxl. Linear spaces with flag-transitive automorphism groups. Geom. Ded-icata, 36(1):89–94, 1990. [482, 486]
[338] J. Buhler and N. Koblitz. Lattice basis reduction, Jacobi sums and hyperellipticcryptosystems. Bull. Austral. Math. Soc., 58(1):147–154, 1998. [405, 406]
[339] B. Bukh and J. Tsimerman. Sum-product estimates for rational functions. Proc.London Math. Soc. [130]
[340] J. R. Bunch and J. E. Hopcroft. Triangular factorization and inversion by fastmatrix multiplication. Math. Comp., 28:231–236, 1974. [429, 436]
[341] Bundesnetzagentur fur Elektrizitat, Gas, Telekommunikation, Post und Eisenbah-nen. Bekanntmachung zur elektronischen Signatur nach dem Signaturgesetz
720 Handbook of Finite Fields
und der Signaturverordnung (Ubersicht uber geeignete Algorithmen). Bundes-anzeiger, 85, June 7:2034, 2011. [667]
[342] M. Buratti. Improving two theorems of Bose on difference families. J. Combin.Des., 3(1):15–24, 1995. [506]
[343] M. Buratti. On simple radical difference families. J. Combin. Des., 3(2):161–168,1995. [507]
[344] M. Buratti. Old and new designs via difference multisets and strong differencefamilies. J. Combin. Des., 7(6):406–425, 1999. [510]
[345] M. Buratti. Existence of Z-cyclic triplewhist tournaments for a prime number ofplayers. J. Combin. Theory A, 90:315–325, 2000. [558]
[346] K. Burde. Zur Herleitung von Reziprozitatsgesetzen unter Benutzung von endlichenKorpern. J. Reine Angew. Math., 293/294:418–427, 1977. [138]
[347] D. A. Burgess. On character sums and primitive roots. Proc. London Math. Soc.(3), 12:179–192, 1962. [147]
[348] J. F. Burkhart, N. J. Calkin, S. Gao, J. C. Hyde-Volpe, K. James, H. Maharaj,S. Manber, J. Ruiz, and E. Smith. Finite field elements of high order arisingfrom modular curves. Des. Codes Cryptogr., 51(3):301–314, 2009. [70]
[349] M. V. D. Burmester. On the commutative non-associative division algebras of evenorder of L. E. Dickson. Rend. Mat. e Appl. (5), 21:143–166, 1962. [227, 229]
[350] J. F. Buss, G. S. Frandsen, and J. O. Shallit. The computational complexity of someproblems of linear algebra (extended abstract). In STACS 97 (Lubeck), volume1200 of Lecture Notes in Comput. Sci., pages 451–462. Springer, Berlin, 1997.[662]
[351] M. Butler. On the reducibility of polynomials over a finite field. Quart. J. Math.Oxford, 5:102–107, 1954. [294, 299]
[352] M. C. R. Butler. The irreducible factors of f(xm) over a finite field. J. LondonMath. Soc., 30:480–482, 1955. [31, 34]
[353] K. A. Byrd and T. P. Vaughan. Counting and constructing orthogonal circulants.J. Combinatorial Theory Ser. A, 24(1):34–49, 1978. [420, 424]
[354] A. Cafure and G. Matera. Improved explicit estimates on the number of solutionsof equations over a finite field. Finite Fields Appl., 12(2):155–185, 2006. [152,158]
[355] E. Cakcak and F. Ozbudak. Subfields of the function field of the Deligne-Lusztigcurve of Ree type. Acta Arith., 115(2):133–180, 2004. [365, 366, 367]
[356] C. Caliskan and G. E. Moorhouse. Subplanes of order 3 in Hughes planes. Electron.J. Combin., 18(1):Paper 2, 8, 2011. [483, 486]
[357] C. Caliskan and B. Petrak. Subplanes of order 3 in Figueroa planes. [483, 486]
[358] J. Calmet and R. Loos. An improvement of Rabin’s probabilistic algorithm for gen-erating irreducible polynomials over GF (p). Information Processing Letters,11(2):94–95, Oct. 1980. [295, 299]
[359] P. J. Cameron and J. J. Seidel. Quadratic forms over GF (2). Nederl. Akad. Weten-sch. Proc. Ser. A 76=Indag. Math., 35:1–8, 1973. [163]
[360] P. J. Cameron and J. H. van Lint. Designs, graphs, codes and their links, volume 22of London Mathematical Society Student Texts. Cambridge University Press,Cambridge, 1991. [20]
[361] P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas. Calabi-Yau manifolds overfinite fields. II. In Calabi-Yau varieties and mirror symmetry (Toronto, ON,
[362] R. Canetti, J. Friedlander, S. Konyagin, M. Larsen, D. Lieman, and I. Shparlinski.On the statistical properties of Diffie-Hellman distributions. Israel J. Math.,120(part A):23–46, 2000. [147, 148]
[363] R. Canetti, J. Friedlander, and I. Shparlinski. On certain exponential sums and thedistribution of Diffie-Hellman triples. J. London Math. Soc. (2), 59(3):799–812,1999. [132, 147, 148]
[364] A. Canteaut. Analyse et conception de chiffrements a clef secrete. Memoired’habilitation a diriger des recherches, Universite Paris 6, Septembre 2006.[206, 213]
[365] A. Canteaut. Open problems related to algebraic attacks on stream ciphers. InCoding and cryptography, volume 3969 of Lecture Notes in Comput. Sci., pages120–134. Springer, Berlin, 2006. [202, 204]
[366] A. Canteaut, C. Carlet, P. Charpin, and C. Fontaine. On cryptographic propertiesof the cosets of R(1,m). IEEE Trans. Inform. Theory, 47(4):1494–1513, 2001.[199, 204]
[367] A. Canteaut, P. Charpin, and H. Dobbertin. A new characterization of almostbent functions. In Fast Software Encryption 99, volume 1636 of LNCS, pages186–200. Springer-Verlag, 1999. [213]
[368] A. Canteaut, P. Charpin, and H. Dobbertin. Binary m-sequences with three-valuedcrosscorrelation: a proof of Welch’s conjecture. IEEE Trans. Inform. Theory,46(1):4–8, 2000. [207, 213]
[369] A. Canteaut, P. Charpin, and H. Dobbertin. Weight divisibility of cyclic codes,highly nonlinear functions on F2m , and crosscorrelation of maximum-lengthsequences. SIAM J. Discrete Math., 13(1):105–138 (electronic), 2000. [210,212, 213]
[370] A. Canteaut, P. Charpin, and G. M. Kyureghyan. A new class of monomial bentfunctions. Finite Fields Appl., 14(1):221–241, 2008. [206, 213, 219, 224]
[371] A. Canteaut (ed.), D. Augot, C. Cid, H. Englund, H. Gilbert, M. Hell, T. Johansson,M. Parker, T. Pornin, B. Preneel, C. Rechberger, and M. Robshaw. D.STVL.9- ongoing research areas in symmetric cryptography. ECRYPT – EuropeanNoE in Cryptology, July 2008. 108 pages. [206, 213]
[372] D. G. Cantor. Computing in the Jacobian of a hyperelliptic curve. Math. Comp.,48(177):95–101, 1987. [356, 360]
[373] D. G. Cantor and H. Zassenhaus. A new algorithm for factoring polynomials overfinite fields. Math. Comp., 36(154):587–592, 1981. [653]
[374] W. Cao, L. Hu, J. Ding, and Z. Yin. Kipnis-shamir attack on unbalanced oil-vinegarscheme. In F. Bao and J. Weng, editors, ISPEC, volume 6672 of Lecture Notesin Computer Science, pages 168–180. Springer, 2011. [663]
[375] X. Cao. A note on the moments of Kloosterman sums. Appl. Algebra Engrg. Comm.Comput., 20(5-6):447–457, 2009. [111, 118]
[376] X. Cao and L. Hu. New methods for generating permutation polynomials over finitefields. Finite Fields Appl., in press. [172, 185]
[377] A. Capelli. Sulla redutibilita delle equasioni algebrique. Rend. Acad. Sci. Fis. Mat.Napoli, 3:243–252, 1897. [31, 34]
[378] M. Car. Le probleme de Waring pour l’anneau des polynomes sur un corps fini. C.R. Acad. Sci. Paris Sr. A-B, 273:A141–A144, 1971. [413, 414]
722 Handbook of Finite Fields
[379] M. Car. Distribution des polynomes irreductibles dans Fq[T ]. Acta Arith.,88(2):141–153, 1999. [44, 46, 49]
[380] M. Car. New bounds on some parameters in the Waring problem for polynomialsover a finite field. In Finite fields and applications, volume 461 of Contemp.Math., pages 59–77. Amer. Math. Soc., Providence, 2008. [413, 414]
[381] M. Car and L. Gallardo. Sums of cubes of polynomials. Acta Arith., 112(1):41–50,2004. [413, 414]
[382] M. Car and L. Gallardo. Waring’s problem for polynomial biquadrates over a finitefield of odd characteristic. Funct. Approx. Comment. Math., 37(1):39–50, 2007.[413, 414]
[383] J.-P. Cardinal. On a property of Cauchy-like matrices. C. R. Acad. Sci. Paris Ser.I Math., 328(11):1089–1093, 1999. [434, 436]
[384] I. Cardinali, O. Polverino, and R. Trombetti. Semifield planes of order q4 withkernel Fq2 and center Fq. European J. Combin., 27(6):940–961, 2006. [228,229]
[385] C. Carlet. Recursive lower bounds on the nonlinearity profile of Boolean functionsand their applications. IEEE Trans. Inform. Theory, 54(3):1262–1272, 2008.[200, 204]
[386] C. Carlet. Boolean functions for cryptography and error correcting codes. InY. Crama and P. L. Hammer, editors, Boolean Models and Methods in Math-ematics, Computer Science, and Engineering, pages 257–397. Cambridge Uni-versity Press, 2010. [145]
[387] C. Carlet. Boolean Functions for Cryptography and Error Correcting Codes(Chapter 8). In Y. Crama and P. L. Hammer, editors, Boolean Mod-els and Methods in Mathematics, Computer Science, and Engineering,pages 257–397. Cambridge University Press, Prel. version: http://www-roc.inria.fr/secret/Claude.Carlet/pubs.html, 2010. [197, 198, 199, 202, 203,204]
[388] C. Carlet. Boolean Models and Methods in Mathematics, Computer Science, andEngineering, chapter Vectorial boolean functions for cryptography, pages 398–469. Cambridge University Press,Yves Crama and Peter L. Hammer (eds.),2010. [205, 206, 213]
[389] C. Carlet, P. Charpin, and V. Zinoviev. Codes, bent functions and permutationssuitable for DES-like cryptosystems. Des. Codes Cryptogr., 15(2):125–156,1998. [207, 209, 210, 211, 212, 213]
[390] C. Carlet and S. Dubuc. On generalized bent and q-ary perfect nonlinear func-tions. In Finite fields and applications (Augsburg, 1999), pages 81–94. Springer,Berlin, 2001. [221, 224]
[391] C. Carlet and K. Feng. An infinite class of balanced functions with optimal algebraicimmunity, good immunity to fast algebraic attacks and good nonlinearity. InAdvances in cryptology—ASIACRYPT 2008, volume 5350 of Lecture Notes inComput. Sci., pages 425–440. Springer, Berlin, 2008. [203, 204]
[392] C. Carlet and P. Gaborit. Hyper-bent functions and cyclic codes. J. Combin. TheorySer. A, 113(3):466–482, 2006. [221, 224]
[393] C. Carlet and S. Mesnager. On Dillon’s class h of bent functions, Niho bent functionsand o-polynomials. Journal of Combinatorial Theory. Series A, 2011. Toappear. [220, 221, 224]
[394] C. Carlet and A. Pott, editors. Sequences and Their Applications—SETA 2010,
Miscellaneous applications 723
volume 6338 of Lecture Notes in Computer Science, Berlin, 2010. Springer.[20]
[395] C. Carlet and B. Sunar, editors. Arithmetic of finite fields, volume 4547 of LectureNotes in Computer Science, Berlin, 2007. Springer. [20]
[396] C. Carlet and J. L. Yucas. Piecewise constructions of bent and almost optimalBoolean functions. Des. Codes Cryptogr., 37(3):449–464, 2005. [163]
[397] L. Carlitz. The arithmetic of polynomials in a Galois field. Amer. J. Math., 54:39–50, 1932. [50, 55]
[398] L. Carlitz. Primitive roots in a finite field. Trans. Amer. Math. Soc., 73:373–382,1952. [92]
[399] L. Carlitz. A theorem of Dickson on irreducible polynomials. Proc. Amer. Math.Soc., 3:693–700, 1952. [25, 26, 30, 43, 48, 49]
[400] L. Carlitz. Invariantive theory of equations in a finite field. Trans. Amer. Math.Soc., 75:405–427, 1953. [186, 188]
[401] L. Carlitz. Permutations in a finite field. Proc. Amer. Math. Soc., 4:538, 1953. [194]
[402] L. Carlitz. Representations by quadratic forms in a finite field. Duke Math. J.,21:123–137, 1954. [421, 424]
[403] L. Carlitz. Representations by skew forms in a finite field. Arch. Math. (Basel),5:19–31, 1954. [422, 424]
[404] L. Carlitz. Solvability of certain equations in a finite field. Quart. J. Math. OxfordSer. (2), 7:3–4, 1956. [167, 170]
[405] L. Carlitz. Some theorems on irreducible reciprocal polynomials over a finite field.J. Reine Angew. Math., 227:212–220, 1967. [28, 30, 238, 242]
[406] L. Carlitz. Kloosterman sums and finite field extensions. Acta Arith., 16:179–193,1969/1970. [112, 118]
[407] L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Straus. Polynomials over finite fieldswith minimal value sets. Mathematika, 8:121–130, 1961. [189, 192]
[408] L. Carlitz and S. Uchiyama. Bounds for exponential sums. Duke Math. J., 24:37–41,1957. [267]
[409] L. Carlitz and C. Wells. The number of solutions of a special system of equationsin a finite field. Acta Arith, 12:77–84, 1966/1967. [174, 185]
[410] R. Carls and D. Lubicz. A p-adic quasi-quadratic time point counting algorithm.Int. Math. Res. Not. IMRN, (4):698–735, 2009. [406]
[411] P. Cartier. Une nouvelle operation sur les formes differentielles. C. R. Acad. Sci.Paris, 244:426–428, 1957. [401, 402]
[412] R. Casse. Projective geometry: an introduction. Oxford University Press, Oxford,2006. [476, 486]
[413] J. W. S. Cassels. Diophantine equations with special reference to elliptic curves. J.London Math. Soc., 41:193–291, 1966. [334, 351]
[414] J. W. S. Cassels. Lectures on elliptic curves, volume 24 of London MathematicalSociety Student Texts. Cambridge University Press, Cambridge, 1991. [19, 20,334, 351]
[415] G. Castagnoli, S. Brauer, and M. Herrmann. Optimization of cyclic redundancy-check codes with 24 and 32 parity bits. Communications, IEEE Transactionson, 41(6):883 –892, jun 1993. [522, 524, 527, 528, 531]
[416] G. Castagnoli, J. Ganz, and P. Graber. Optimum cycle redundancy-check codeswith 16-bit redundancy. Communications, IEEE Transactions on, 38(1):111
724 Handbook of Finite Fields
–114, jan 1990. [524, 531]
[417] F. N. Castro and C. J. Moreno. Mixed exponential sums over finite fields. Proc.Amer. Math. Soc., 128(9):2529–2537, 2000. [125, 127]
[418] F. N. Castro, I. Rubio, P. Guan, and R. Figueroa. On systems of linear and diagonalequation of degree pi+1 over finite fields of characteristic p. Finite Fields Appl.,14(3):648–657, 2008. [169, 170]
[419] F. N. Castro, I. Rubio, and J. M. Vega. Divisibility of exponential sums and solv-ability of certain equations over finite fields. Q. J. Math., 60(2):169–181, 2009.[168, 170]
[420] W. Castryck, J. Denef, and F. Vercauteren. Computing zeta functions of nonde-generate curves. IMRP Int. Math. Res. Pap., pages Art. ID 72017, 57, 2006.[406]
[421] K. Cattell, C. R. Miers, F. Ruskey, J. Sawada, and M. Serra. The number ofirreducible polynomials over GF(2) with given trace and subtrace. J. Combin.Math. Combin. Comput., 47:31–64, 2003. [27, 48, 49]
[422] A. Cauchy. Recherches sur les nombres. Ecole Polytechnique, 9:99–116, 1813. [168,170]
[423] S. R. Cavior. A note on octic permutation polynomials. Math. Comp., 17:450–452,1963. [179, 185]
[424] C. Cazacu and D. Simovici. A new approach of some problems concerning poly-nomials over finite fields. Information and Control, 22:503–511, 1973. [37,38]
[425] A. Cesmelioglu, W. Meidl, and A. Topuzoglu. On the cycle structure of permutationpolynomials. Finite Fields Appl., 14(3):593–614, 2008. [184, 185]
[426] F. Chabaud and S. Vaudenay. Links between differential and linear cryptanalysis. InAdvances in cryptology—EUROCRYPT ’94 (Perugia), volume 950 of LectureNotes in Comput. Sci., pages 356–365. Springer, Berlin, 1995. [205, 207, 213]
[427] W. Chambers. Solution of Welch-Berlekamp key equation by Euclidean algorithm.Electronics Letters, 29:1031, 1993. [595, 602]
[428] A. Chambert-Loir. Compter (rapidement) le nombre de solutions d’equations dansles corps finis. Asterisque, (317):Exp. No. 968, vii, 39–90, 2008. SeminaireBourbaki. Vol. 2006/2007. [406]
[429] D. B. Chandler and Q. Xiang. The invariant factors of some cyclic difference sets.J. Combin. Theory Ser. A, 101(1):131–146, 2003. [109, 118]
[430] C.-Y. Chang, M. A. Papanikolas, D. S. Thakur, and J. Yu. Algebraic independenceof arithmetic gamma values and Carlitz zeta values. Adv. Math., 223(4):1137–1154, 2010. [458]
[431] C.-Y. Chang and J. Yu. Determination of algebraic relations among special zetavalues in positive characteristic. Adv. Math., 216(1):321–345, 2007. [454, 458]
[432] M.-C. Chang. On a question of Davenport and Lewis and new character sum boundsin finite fields. Duke Math. J., 145(3):409–442, 2008. [134]
[433] M.-C. Chang and C. Z. Yao. An explicit bound on double exponential sums relatedto Diffie-Hellman distributions. SIAM J. Discrete Math., 22(1):348–359, 2008.[132, 148]
[434] Y. Chang, W.-S. Chou, and P. J.-S. Shiue. On the number of primitive polynomialsover finite fields. Finite Fields Appl., 11(1):156–163, 2005. [59]
[435] R. Chapman. Completely normal elements in iterated quadratic extensions of finitefields. Finite Fields Appl., 3(1):1–10, 1997. [31, 34, 94, 238, 242]
Miscellaneous applications 725
[436] P. Charpin. Handbook of Coding Theory, chapter Open problems on cyclic codes,pages 963–1063. elsevier, V.S. Pless and C.W. Huffman (eds.), R.A. Brualdi(ass. ed., 1998. [210, 212, 213]
[437] P. Charpin and G. Gong. Hyperbent functions, Kloosterman sums, and Dicksonpolynomials. IEEE Trans. Inform. Theory, 54(9):4230–4238, 2008. [221, 223,224]
[438] P. Charpin, T. Helleseth, and V. Zinoviev. Divisibility properties of classical binaryKloosterman sums. Discrete Math., 309(12):3975–3984, 2009. [111, 118]
[439] P. Charpin and G. Kyureghyan. When does G(x)+γTr(H(x)) permute Fpn? FiniteFields Appl., 15(5):615–632, 2009. [181, 185]
[440] P. Charpin and G. M. Kyureghyan. On a class of permutation polynomials overF2n . In Sequences and their applications—SETA 2008, volume 5203 of LectureNotes in Comput. Sci., pages 368–376. Springer, Berlin, 2008. [181, 185]
[441] S. Chatterjee and A. Menezes. On cryptographic protocols employing asymmetricpairings – the role of ψ revisited. To appear in Discrete Applied Mathematics,http://eprint.iacr.org/2009/480/, 2011. [674]
[442] H. Chen. Fast algorithms for determining the linear complexity of sequences overGF(pm) with period 2tn. IEEE Trans. Inform. Theory, 51(5):1854–1856, 2005.[274, 281]
[443] H. Chen. Reducing the computation of linear complexities of periodic sequencesover GF(pm). IEEE Trans. Inform. Theory, 52(12):5537–5539, 2006. [274,281]
[444] J. Chen and T. Wang. On the Goldbach problem. Acta Math. Sinica, 32(5):702–718,1989. [411]
[445] J.-M. Chen and T.-T. Moh. On the Goubin-Courtois attack on TTM. CryptologyePrint Archive, 2001. http://eprint.iacr.org/2001/072. [657]
[446] J.-M. Chen and B.-Y. Yang. A more secure and efficacious TTS signature scheme.In Information security and cryptology—ICISC 2003, volume 2971 of LectureNotes in Comput. Sci., pages 320–338. Springer, Berlin, 2004. [655, 658]
[447] J.-M. Chen, B.-Y. Yang, and B.-Y. Peng. Tame transformation signatures withtopsy-yurvy hashes. In IWAP’02, pages 1–8, 2002.http://dsns.csie.nctu.edu.tw/iwap/proceedings/proceedings/sessionD/7.pdf.[658]
[448] K. Chen and L. Zhu. Existence of APAV(q, k) with q a prime power ≡ 3 (mod 4)and k odd > 1. J. Combin. Des., 7(1):57–68, 1999. [551]
[449] Y. Chen. The Steiner system S(3, 6, 26). J Geometry, 2:7–28, 1972. [501]
[450] Y. Q. Chen. A construction of difference sets. Des. Codes Cryptogr., 13(3):247–250,1998. [518, 519]
[451] Q. Cheng. Constructing finite field extensions with large order elements. SIAM J.Discrete Math., 21(3):726–730 (electronic), 2007. [69, 70]
[452] Q. Cheng, S. Gao, and D. Wan. Constructing high order elements through subspacepolynomials. To appear in ACM-SIAM Symposium on Discrete AlgorithmsSODA’12. [70]
[453] G. Cheze. Des methodes symboliques-numeriques et exactes pour la factorisationabsolue des polynomes en deux variables. PhD thesis, Universite de Nice-SophiaAntipolis (France), 2004. [305, 311]
[454] G. Cheze and G. Lecerf. Lifting and recombination techniques for absolute factor-ization. J. Complexity, 23(3):380–420, 2007. [303, 311]
726 Handbook of Finite Fields
[455] K. Chinen and T. Hiramatsu. Hyper-Kloosterman sums and their applications tothe coding theory. Appl. Algebra Engrg. Comm. Comput., 12(5):381–390, 2001.[111, 118]
[456] A. Chistov. Polynomial time construction of a finite field. In In Abstracts of Lecturesat 7th All-Union Conference in Mathematical Logic, page 196, Novosibirsk,USSR, 1984. In Russian. [297, 299]
[457] H. T. Choi and R. Evans. Congruences for sums of powers of Kloosterman sums.Int. J. Number Theory, 3(1):105–117, 2007. [114, 118]
[458] B. C. Chong and K. M. Chan. On the existence of normalized room squares. NantaMath., 7(1):8–17, 1974. [554]
[459] W. S. Chou. Permutation polynomials on finite fields and their combinatorial ap-plications, Ph.D. Thesis, Penn. State Univ., University Park, PA. PhD thesis,1990. [184, 185]
[460] W. S. Chou. The period lengths of inversive pseudorandom vector generations.Finite Fields Appl., 1(1):126–132, 1995. [184, 185]
[461] W.-S. Chou. The factorization of Dickson polynomials over finite fields. FiniteFields Appl., 3(1):84–96, 1997. [236, 242]
[462] W.-S. Chou and S. D. Cohen. Primitive elements with zero traces. Finite FieldsAppl., 7(1):125–141, 2001. Dedicated to Professor Chao Ko on the occasion ofhis 90th birthday. [62, 65]
[463] W. S. Chou, J. Gomez-Calderon, and G. L. Mullen. Value sets of Dickson polyno-mials over finite fields. J. Number Theory, 30(3):334–344, 1988. [191, 192]
[464] S. Chowla and H. J. Ryser. Combinatorial problems. Canadian J. Math., 2:93–99,1950. [513, 519]
[465] S. Chowla and H. Zassenhaus. Some conjectures concerning finite fields. NorskeVid. Selsk. Forh. (Trondheim), 41:34–35, 1968. [184]
[466] W. Chu and C. J. Colbourn. Optimal frequency-hopping sequences via cyclotomy.IEEE Trans. Inform. Theory, 51(3):1139–1141, 2005. [698, 701]
[467] F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2):187–196,1989. [534, 545]
[468] F. R. K. Chung, J. A. Salehi, and V. K. Wei. Optical orthogonal codes: design,analysis, and applications. IEEE Trans. Inform. Theory, 35(3):595–604, 1989.[696, 697, 701]
[469] J.-H. Chung and K. Yang. Bounds on the linear complexity and the 1-error linearcomplexity over Fp of M -ary Sidel′nikov sequences. In Sequences and theirapplications—SETA 2006, volume 4086 of Lecture Notes in Comput. Sci., pages74–87. Springer, Berlin, 2006. [279, 281]
[470] J. Cilleruelo. Combinatorial problems in finite fields and sidon sets. Combinatorica.[133]
[471] S. M. Cioaba. Eigenvalues, expanders and gaps between primes. ProQuest LLC,Ann Arbor, MI, 2006. Thesis (Ph.D.)–Queen’s University (Canada). [538,544, 545]
[472] S. M. Cioaba. Eigenvalues of graphs and a simple proof of a theorem of Greenberg.Linear Algebra Appl., 416(2-3):776–782, 2006. [538, 545]
[473] S. M. Cioaba. On the extreme eigenvalues of regular graphs. J. Combin. TheorySer. B, 96(3):367–373, 2006. [538, 545]
[474] S. M. Cioaba and M. R. Murty. Expander graphs and gaps between primes. ForumMath., 20(4):745–756, 2008. [544, 545]
Miscellaneous applications 727
[475] J. A. Cipra. Waring’s number in a finite field. Integers, 9:A34, 435–440, 2009. [140,169, 170]
[476] J. A. Cipra, T. Cochrane, and C. Pinner. Heilbronn’s conjecture on Waring’s number(mod p). J. Number Theory, 125(2):289–297, 2007. [168, 169, 170]
[477] M. Cipu. Dickson polynomials that are permutations. Serdica Math. J., 30(2-3):177–194, 2004. [182, 185]
[478] M. Cipu and S. D. Cohen. Dickson polynomial permutations. In Finite fields andapplications, volume 461 of Contemp. Math., pages 79–90. Amer. Math. Soc.,Providence, RI, 2008. [182, 185]
[479] T. Cochrane, J. Coffelt, and C. Pinner. A further refinement of Mordell’s bound onexponential sums. Acta Arith., 116(1):35–41, 2005. [132]
[480] T. Cochrane, M.-C. Liu, and Z. Zheng. Upper bounds on n-dimensional Klooster-man sums. J. Number Theory, 106(2):259–274, 2004. [117, 118]
[481] T. Cochrane and C. Pinner. Sum-product estimates applied to Waring’s problemmod p. Integers, 8:A46, 18, 2008. [134, 169, 170]
[482] T. Cochrane and Z. Zheng. A survey on pure and mixed exponential sums moduloprime powers. In Number theory for the millennium, I (Urbana, IL, 2000),pages 273–300. A K Peters, Natick, MA, 2002. [117, 118]
[483] H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren,editors. Handbook of Elliptic and Hyperelliptic Curve Cryptography. DiscreteMathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, BocaRaton, FL, 2006. [19, 20, 354, 355, 356, 357, 358, 359, 360, 666, 670]
[484] S. Cohen and H. Niederreiter, editors. Finite fields and applications, volume 233of London Mathematical Society Lecture Note Series, Cambridge, 1996. Cam-bridge University Press. [20]
[485] S. D. Cohen. The distribution of irreducible polynomials in several indeterminatesover a finite field. Proc. Edinburgh Math. Soc. (2), 16:1–17, 1968/1969. [51,52, 55]
[486] S. D. Cohen. On irreducible polynomials of certain types in finite fields. Proc.Cambridge Philos. Soc., 66:335–344, 1969. [28, 29, 30, 31, 34]
[487] S. D. Cohen. The distribution of polynomials over finite fields. Acta Arith., 17:255–271, 1970. [40, 184, 185, 190, 192, 193]
[488] S. D. Cohen. Some arithmetical functions in finite fields. Glasgow Math. J., 11:21–36, 1970. [50, 53, 55]
[489] S. D. Cohen. Uniform distribution of polynomials over finite fields. J. London Math.Soc. (2), 6:93–102, 1972. [47, 49]
[490] S. D. Cohen. The reducibility theorem for linearised polynomials over finite fields.Bull. Austral. Math. Soc., 40(3):407–412, 1989. [35]
[491] S. D. Cohen. Windmill polynomials over fields of characteristic two. Monatsh.Math., 107(4):291–301, 1989. [37, 38, 58, 59]
[492] S. D. Cohen. Primitive elements and polynomials with arbitrary trace. DiscreteMath., 83(1):1–7, 1990. [62, 65]
[493] S. D. Cohen. Proof of a conjecture of Chowla and Zassenhaus on permutationpolynomials. Canad. Math. Bull., 33(2):230–234, 1990. [184, 185]
[494] S. D. Cohen. Permutation polynomials and primitive permutation groups. Arch.Math. (Basel), 57(5):417–423, 1991. [174]
[495] S. D. Cohen. The explicit construction of irreducible polynomials over finite fields.
[496] S. D. Cohen. Dickson polynomials of the second kind that are permutations. Canad.J. Math., 46(2):225–238, 1994. [182, 185]
[497] S. D. Cohen. Dickson permutations. In Number-theoretic and algebraic methods incomputer science (Moscow, 1993), pages 29–51. World Sci. Publ., River Edge,NJ, 1995. [182, 185]
[498] S. D. Cohen. Permutation group theory and permutation polynomials. In Algebrasand combinatorics (Hong Kong, 1997), pages 133–146. Springer, Singapore,1999. [172, 185]
[499] S. D. Cohen. Gauss sums and a sieve for generators of Galois fields. Publ. Math.Debrecen, 56(3-4):293–312, 2000. Dedicated to Professor Kalman GyHory onthe occasion of his 60th birthday. [58, 59, 62, 63, 65]
[500] S. D. Cohen. Kloosterman sums and primitive elements in Galois fields. Acta Arith.,94(2):173–201, 2000. [62, 65]
[501] S. D. Cohen. Primitive polynomials over small fields. In Finite fields and applica-tions, volume 2948 of Lecture Notes in Comput. Sci., pages 197–214. Springer,Berlin, 2004. [62, 65]
[502] S. D. Cohen. Explicit theorems on generator polynomials. Finite Fields Appl.,11(3):337–357, 2005. [31, 33, 34, 46, 49]
[503] S. D. Cohen. Primitive polynomials with a prescribed coefficient. Finite FieldsAppl., 12(3):425–491, 2006. [61, 62, 65]
[504] S. D. Cohen and M. D. Fried. Lenstra’s proof of the Carlitz-Wan conjecture onexceptional polynomials: an elementary version. Finite Fields Appl., 1(3):372–375, 1995. [174, 185]
[505] S. D. Cohen and M. J. Ganley. Commutative semifields, two-dimensional over theirmiddle nuclei. J. Algebra, 75(2):373–385, 1982. [228, 229, 233, 234]
[506] S. D. Cohen and D. Hachenberger. Primitive normal bases with prescribed trace.Appl. Algebra Engrg. Comm. Comput., 9(5):383–403, 1999. [57, 59]
[507] S. D. Cohen and D. Hachenberger. Primitivity, freeness, norm and trace. DiscreteMath., 214(1-3):135–144, 2000. [62, 63, 65]
[508] S. D. Cohen and S. Huczynska. Primitive free quartics with specified norm andtrace. Acta Arith., 109(4):359–385, 2003. [58, 59, 62, 63, 65]
[509] S. D. Cohen and S. Huczynska. The primitive normal basis theorem—without acomputer. J. London Math. Soc. (2), 67(1):41–56, 2003. [63, 65]
[510] S. D. Cohen and S. Huczynska. The strong primitive normal basis theorem. ActaArith., 143(4):299–332, 2010. [64, 65]
[511] S. D. Cohen and C. King. The three fixed coefficient primitive polynomial theorem.JP J. Algebra Number Theory Appl., 4(1):79–87, 2004. [62, 65]
[512] S. D. Cohen and R. W. Matthews. A class of exceptional polynomials. Trans. Amer.Math. Soc., 345(2):897–909, 1994. [254, 255]
[513] S. D. Cohen and D. Mills. Primitive polynomials with first and second coefficientsprescribed. Finite Fields Appl., 9(3):334–350, 2003. [62]
[514] S. D. Cohen, G. L. Mullen, and P. J.-S. Shiue. The difference between permutationpolynomials over finite fields. Proc. Amer. Math. Soc., 123(7):2011–2015, 1995.[184, 185]
[515] S. D. Cohen and M. Presern. Primitive finite field elements with prescribed trace.Southeast Asian Bull. Math., 29(2):283–300, 2005. [62, 65]
Miscellaneous applications 729
[516] S. D. Cohen and M. Presern. Primitive polynomials with prescribed second coeffi-cient. Glasg. Math. J., 48(2):281–307, 2006. [62]
[517] S. D. Cohen and M. Presern. The Hansen-Mullen primitive conjecture: completionof proof. In Number theory and polynomials, volume 352 of London Math. Soc.Lecture Note Ser., pages 89–120. Cambridge Univ. Press, Cambridge, 2008.[62, 65]
[518] C. J. Colbourn. Covering arrays from cyclotomy. Des. Codes Cryptogr., 55(2-3):201–219, 2010. [549]
[519] C. J. Colbourn. Covering arrays and hash families. In Information Security andRelated Combinatorics, NATO Peace and Information Security, pages 99–136.IOS Press, 2011. [549]
[520] C. J. Colbourn and J. H. Dinitz. The CRC Handbook of Combinatorial Designs.CRC Press, 1996. [20, 264]
[521] C. J. Colbourn and J. H. Dinitz, editors. Handbook of combinatorial designs. Dis-crete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC,Boca Raton, FL, second edition, 2007. [20, 462, 463, 466, 467, 476, 484, 485,486, 512, 519]
[522] C. J. Colbourn and J. H. Dinitz, editors. Handbook of Combinatorial Designs.CRC/Chapman and Hall, Boca Raton FL, second edition, 2007. [20, 509, 511,556, 558]
[523] C. J. Colbourn and A. C. H. Ling. Linear hash families and forbidden configurations.Des. Codes Cryptogr., 52(1):25–55, 2009. [552]
[524] C. J. Colbourn and A. Rosa. Triple systems. Oxford Mathematical Monographs.The Clarendon Press Oxford University Press, New York, 1999. [503, 507]
[525] Computational Algebra Group, University of Sydney. The MAGMA computationalalgebra system for algebra, number theory and geometry. http://magma.maths.usyd.edu.au/magma/, 2005. [664]
[526] A. Conflitti. On elements of high order in finite fields. In Cryptography and com-putational number theory (Singapore, 1999), volume 20 of Progr. Comput. Sci.Appl. Logic, pages 11–14. Birkhauser, Basel, 2001. [69, 70]
[527] K. Conrad. Jacobi sums and Stickelberger’s congruence. Enseign. Math. (2), 41(1-2):141–153, 1995. [109, 118]
[528] K. Conrad. On Weil’s proof of the bound for Kloosterman sums. J. Number Theory,97(2):439–446, 2002. [111, 112, 118]
[529] S. Contini and I. E. Shparlinski. On stern’s attack against secret truncated linearcongruential generators. 3574:52–60, 2005. [283, 289]
[530] D. Coppersmith. Solving homogeneous linear equations over GF(2) via block Wiede-mann algorithm. Math. Comp., 62(205):333–350, 1994. [436]
[531] D. Coppersmith, J. Stern, and S. Vaudenay. The security of the birational per-mutation signature schemes. J. Cryptology, 10(3):207–221, 1997. [652, 658,662]
[532] D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progres-sions. J. Symbolic Comput., 9(3):251–280, 1990. [425, 436]
[533] R. Coulter, M. Henderson, and R. Matthews. A note on constructing permutationpolynomials. Finite Fields Appl., 15(5):553–557, 2009. [180, 185]
[534] R. S. Coulter. The classification of planar monomials over fields of prime squareorder. Proc. Amer. Math. Soc., 134(11):3373–3378 (electronic), 2006. [232,234]
730 Handbook of Finite Fields
[535] R. S. Coulter and M. Henderson. The compositional inverse of a class of permutationpolynomials over a finite field. Bull. Austral. Math. Soc., 65(3):521–526, 2002.[184, 185]
[536] R. S. Coulter and M. Henderson. Commutative presemifields and semifields. Adv.Math., 217(1):282–304, 2008. [229, 233, 234]
[537] R. S. Coulter, M. Henderson, and P. Kosick. Planar polynomials for commutativesemifields with specified nuclei. Des. Codes Cryptogr., 44(1-3):275–286, 2007.[228, 229, 233, 234]
[538] R. S. Coulter and P. Kosick. Commutative semifields of order 243 and 3125. InFinite fields: theory and applications, volume 518 of Contemp. Math., pages129–136. Amer. Math. Soc., Providence, RI, 2010. [227, 229]
[539] R. S. Coulter and F. Lazebnik. On the classification of planar monomials over fieldsof square order. submitted. [229, 232, 233]
[540] R. S. Coulter and R. W. Matthews. Planar functions and planes of Lenz-Barlotticlass II. Des. Codes Cryptogr., 10(2):167–184, 1997. [221, 224, 231, 232, 234]
[541] R. S. Coulter and R. W. Matthews. On the permutation behaviour of Dicksonpolynomials of the second kind. Finite Fields Appl., 8(4):519–530, 2002. [182,185]
[542] R. S. Coulter and R. W. Matthews. On the number of distinct values of a classof functions over a finite field. Finite Fields Appl., 17(3):220–224, 2011. [232,234]
[543] N. Courtois, L. Goubin, W. Meier, and J.-D. Tacier. Solving underdefined systemsof multivariate quadratic equations. In 2002, volume 2274 of Lecture Notes inComputer Science, pages 211–227. David Naccache and Pascal Paillier, editors,2002. [663]
[544] N. Courtois, L. Goubin, and J. Patarin. SFLASH: Primitive specification (secondrevised version), 2002. https://www.cosic.esat.kuleuven.be/nessie, Sub-missions, Sflash, 11 pages. [656]
[545] N. Courtois, A. Klimov, J. Patarin, and A. Shamir. Efficient algorithms for solv-ing overdefined systems of multivariate polynomial equations. In Advances incryptology—EUROCRYPT 2000 (Bruges), volume 1807 of Lecture Notes inComput. Sci., pages 392–407. Springer, Berlin, 2000. [662, 664]
[546] N. T. Courtois. Fast algebraic attacks on stream ciphers with linear feedback.In Advances in cryptology—CRYPTO 2003, volume 2729 of Lecture Notes inComput. Sci., pages 176–194. Springer, Berlin, 2003. [202, 204]
[547] N. T. Courtois. Algebraic attacks over GF(2k), application to HFE Challenge 2and Sflash-v2. In Public key cryptography—PKC 2004, volume 2947 of LectureNotes in Comput. Sci., pages 201–217. Springer, Berlin, 2004. [664]
[548] N. T. Courtois, M. Daum, and P. Felke. On the security of HFE, HFEv- andQuartz. In Public key cryptography—PKC 2003, volume 2567 of Lecture Notesin Comput. Sci., pages 337–350. Springer, Berlin, 2002. [654, 662]
[549] N. T. Courtois and W. Meier. Algebraic attacks on stream ciphers with linearfeedback. In Advances in cryptology—EUROCRYPT 2003, volume 2656 ofLecture Notes in Comput. Sci., pages 345–359. Springer, Berlin, 2003. [201,204]
[550] N. T. Courtois and J. Patarin. About the XL algorithm over GF(2). In Topicsin cryptology—CT-RSA 2003, volume 2612 of Lecture Notes in Comput. Sci.,pages 141–157. Springer, Berlin, 2003. [664]
Miscellaneous applications 731
[551] N. T. Courtois and J. Pieprzyk. Cryptanalysis of block ciphers with overdefinedsystems of equations. In Advances in cryptology—ASIACRYPT 2002, volume2501 of Lecture Notes in Comput. Sci., pages 267–287. Springer, Berlin, 2002.[664, 665]
[552] J. Couveignes and R. Lercier. Fast construction of irreducible polynomials over finitefields. Israel Journal of Mathematics, 2011. To appear. ArXiv:0905.1642v2.[297, 299]
[553] J.-M. Couveignes and T. Henocq. Action of modular correspondences around CMpoints. In C. Fieker and D. R. Kohel, editors, Algorithmic Number Theory —ANTS-V, volume 2369 of Lecture Notes in Computer Science, pages 234–243,Berlin, 2002. Springer-Verlag. [669]
[554] J.-M. Couveignes and J.-G. Kammerer. The geometry of flex tangents to a cubiccurve and its parameterizations. Preprint ArXiv 1101.3630v1, 2011. [679]
[555] D. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms. UndergraduateTexts in Mathematics. Springer, New York, third edition, 2007. An introduc-tion to computational algebraic geometry and commutative algebra. [684, 692]
[556] R. Crandall and C. Pomerance. Prime numbers: a computational perspective. 2ndedition. Springer, New York, 2005. [410, 414]
[557] R. M. Crew. Etale p-covers in characteristic p. Compositio Math., 52(1):31–45,1984. [401, 402]
[558] C. Culbert and G. L. Ebert. Circle geometry and three-dimensional subregulartranslation planes. Innov. Incidence Geom., 1:3–18, 2005. [480, 486]
[559] T. W. Cusick. Value sets of some polynomials over finite fields GF(22m). SIAM J.Comput., 27(1):120–131 (electronic), 1998. [191, 192]
[560] T. W. Cusick. Polynomials over base 2 finite fields with evenly distributed values.Finite Fields Appl., 11(2):278–291, 2005. [191, 192]
[561] T. W. Cusick, C. Ding, and A. Renvall. Stream ciphers and number theory, vol-ume 66 of North-Holland Mathematical Library. Elsevier Science B.V., Ams-terdam, revised edition, 2004. [19, 20, 272, 273, 279, 281]
[562] T. W. Cusick and P. Muller. Wan’s bound for value sets of polynomials. In Finitefields and applications (Glasgow, 1995), volume 233 of London Math. Soc.Lecture Note Ser., pages 69–72. Cambridge Univ. Press, Cambridge, 1996. [189,191, 192]
[563] S. Czapor, K. Geddes, and G. Labahn. Algorithms for Computer Algebra. KluwerAcademic Publishers, 1992. [20, 301, 311]
[564] E. D. D. Bernstein, J. Buchmann. Post-quantum cryptography. Springer, 2009.Chapter: Multivariate public key cryptography by J. Ding and B. Yang. [648]
[565] J. Daemen and V. Rijmen. The design of Rijndael: AES – the Advanced EncryptionStandard. Springer-Verlag, 2002. [19, 20, 635, 644, 645, 647]
[566] X. Dahan and J.-P. Tillich. Ramanujan graphs of very large girth based on octo-nions, 2010. [540, 545]
[567] Z. Dai. Multi-continued fraction algorithms and their applications to sequences. InSequences and their applications—SETA 2006, volume 4086 of Lecture Notesin Comput. Sci., pages 17–33. Springer, Berlin, 2006. [275, 281]
[568] Z. Dai and X. Feng. Classification and counting on multi-continued fractions and itsapplication to multi-sequences. Sci. China Ser. F, 50(3):351–358, 2007. [275,281]
[569] Z. Dai, K. Wang, and D. Ye. Multi-continued fraction algorithm on multi-formal
[570] Z. Dai and J. Yang. Multi-continued fraction algorithm and generalized B-M algo-rithm over Fq. Finite Fields Appl., 12(3):379–402, 2006. [275, 281]
[571] A. Danilevsky. The numerical solution of the secular equation. Matem. sbornik,44(2):169–171, 1937. In Russian. [294, 299]
[572] P. Das. The number of permutation polynomials of a given degree over a finite field.Finite Fields Appl., 8(4):478–490, 2002. [175, 185]
[573] P. Das. The number of polynomials of a given degree over a finite field with valuesets of a given cardinality. Finite Fields Appl., 9(2):168–174, 2003. [191, 192]
[574] P. Das. Value sets of polynomials and the Cauchy Davenport theorem. Finite FieldsAppl., 10(1):113–122, 2004. [191, 192]
[575] P. Das and G. L. Mullen. Value sets of polynomials over finite fields. In Finite fieldswith applications to coding theory, cryptography and related areas (Oaxaca,2001), pages 80–85. Springer, Berlin, 2002. [190, 192]
[576] H. Davenport. Bases for finite fields. J. London Math. Soc., 43:21–39, 1968. [92]
[577] H. Davenport and D. J. Lewis. Notes on congruences. I. Quart. J. Math. OxfordSer. (2), 14:51–60, 1963. [194, 245, 255]
[578] J. H. Davenport, Y. Siret, and E. Tournier. Calcul formel : systemes et algorithmesde manipulations algebriques. Masson, Paris, France, 1987. [301, 311]
[579] J. H. Davenport and B. M. Trager. Factorization over finitely generated fields.In SYMSAC’81: Proceedings of the fourth ACM symposium on Symbolic andalgebraic computation, pages 200–205. ACM Press, 1981. [306, 311]
[580] G. Davidoff, P. Sarnak, and A. Valette. Elementary number theory, group theory,and Ramanujan graphs, volume 55 of London Mathematical Society StudentTexts. Cambridge University Press, Cambridge, 2003. [545]
[581] J. A. Davis. Difference sets in abelian 2-groups. J. Combin. Theory Ser. A,57(2):262–286, 1991. [517, 519]
[582] J. A. Davis and J. Jedwab. A unifying construction for difference sets. J. Combin.Theory Ser. A, 80(1):13–78, 1997. [518, 519]
[583] J. A. Davis and J. Jedwab. Peak-to-mean power control in OFDM, Golay com-plementary sequences, and Reed-Muller codes. IEEE Trans. Inform. Theory,45(7):2397–2417, 1999. [696, 701]
[584] E. Dawson and L. Simpson. Analysis and design issues for synchronous streamciphers. In Coding theory and cryptology (Singapore, 2001), volume 1 of Lect.Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 49–90. World Sci. Publ.,River Edge, NJ, 2002. [271, 281]
[585] J. De Beule and L. Storme. Current Research Topics in Galois Geometry. NovaAcademic Publishers, New York, 2011. [20]
[586] J. De Beule and L. Storme. Current research topics in Galois geometry. NOVAAcademic Publishers, Inc., New York, 2012. [20, 485, 486]
[587] P. de la Harpe and A. Musitelli. Expanding graphs, Ramanujan graphs, and 1-factorperturbations. Bull. Belg. Math. Soc. Simon Stevin, 13(4):673–680, 2006. [544,545]
[588] M. J. de Resmini and N. Hamilton. Hyperovals and unitals in Figueroa planes.European J. Combin., 19(2):215–220, 1998. [484, 486]
[589] P. Deligne. Les constantes des equations fonctionnelles des fonctions L. In Modularfunctions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp,
[590] P. Deligne. La conjecture de Weil. I. Inst. Hautes Etudes Sci. Publ. Math., (43):273–307, 1974. [120, 127, 153, 155, 158, 384, 387, 390, 393]
[591] P. Deligne. Applications de la formule des traces aux sommes trigonometriques,rm in Cohomologie etale. Lecture Notes in Mathematics, Vol. 569. Springer-Verlag, Berlin, 1977. Seminaire de Geometrie Algebrique du Bois-Marie SGA41øer2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie etJ. L. Verdier. [126]
[592] P. Deligne. Cohomologie etale. Lecture Notes in Mathematics, Vol. 569. Springer-Verlag, Berlin, 1977. Seminaire de Geometrie Algebrique du Bois-Marie SGA41øer2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie etJ. L. Verdier. [20, 385, 386, 389, 390, 392, 393]
[593] P. Deligne. La conjecture de Weil. II. Inst. Hautes Etudes Sci. Publ. Math., (52):137–252, 1980. [387, 390, 393, 394]
[594] P. Delsarte. An algebraic approach to the association schemes of coding theory.Philips Res. Rep. Suppl., (10):vi+97, 1973. [204]
[595] P. Delsarte. Four fundamental parameters of a code and their combinatorial signif-icance. Information and Control, 23:407–438, 1973. [520, 531, 563, 564, 565,573, 591, 602]
[596] P. Delsarte. On subfield subcodes of modified Reed-Solomon codes. IEEE Trans.Information Theory, IT-21(5):575–576, 1975. [569, 584, 602]
[597] P. Delsarte. Bilinear forms over a finite field, with applications to coding theory. J.Combin. Theory Ser. A, 25(3):226–241, 1978. [699, 701]
[598] P. Delsarte and J.-M. Goethals. Alternating bilinear forms over GF (q). J. Combi-natorial Theory Ser. A, 19:26–50, 1975. [601, 602]
[599] P. Delsarte, J.-M. Goethals, and F. J. MacWilliams. On generalized Reed-Mullercodes and their relatives. Information and Control, 16:403–442, 1970. [586,602]
[600] P. Delsarte and V. I. Levenshtein. Association schemes and coding theory. IEEETrans. Inform. Theory, 44(6):2477–2504, 1998. Information theory: 1948–1998.[591, 602]
[601] E. D. Demaine, M. L. Demaine, and T. Rodgers, editors. A lifetime of puzzles. AK Peters Ltd., Wellesley, MA, 2008. A collection of puzzles in honor of MartinGardner’s 90th birthday. [531, 734]
[602] P. Dembowski. Finite geometries. Ergebnisse der Mathematik und ihrer Grenzge-biete, Band 44. Springer-Verlag, Berlin, 1968. [16, 20, 225, 229, 476, 479, 486,501, 503]
[603] P. Dembowski and T. G. Ostrom. Planes of order n with collineation groups oforder n2. Math. Z., 103:239–258, 1968. [206, 213, 231, 232, 234]
[604] U. Dempwolff. Semifield planes of order 81. J. Geom., 89(1-2):1–16, 2008. [227,229]
[605] U. Dempwolff and M. Roder. On finite projective planes defined by planar mono-mials. Innov. Incidence Geom., 4:103–108, 2006. [231, 232, 234]
[606] J. Denef and F. Loeser. Weights of exponential sums, intersection cohomology, andNewton polyhedra. Invent. Math., 106(2):275–294, 1991. [122, 127, 154, 158,391, 393]
[607] J. Denef and F. Loeser. Character sums associated to finite Coxeter groups. Trans.
[608] J. Denef and F. Loeser. Definable sets, motives and p-adic integrals. J. Amer. Math.Soc., 14(2):429–469 (electronic), 2001. [254, 255]
[609] J. Denef and F. Vercauteren. An extension of Kedlaya’s algorithm to Artin-Schreiercurves in characteristic 2. In Algorithmic number theory (Sydney, 2002), vol-ume 2369 of Lecture Notes in Comput. Sci., pages 308–323. Springer, Berlin,2002. [358, 360]
[610] J. Denef and F. Vercauteren. Counting points on Cab curves using Monsky-Washnitzer cohomology. Finite Fields Appl., 12(1):78–102, 2006. [406]
[611] J. Denef and F. Vercauteren. An extension of Kedlaya’s algorithm to hyperellipticcurves in characteristic 2. J. Cryptology, 19(1):1–25, 2006. [358, 360, 406]
[612] J. Denes and A. D. Keedwell. Latin squares and their applications. Academic Press,New York, 1974. [467]
[613] J. Denes and A. D. Keedwell. Latin squares, volume 46 of Annals of Discrete Math-ematics. North-Holland Publishing Co., Amsterdam, 1991. New developmentsin the theory and applications, With contributions by G. B. Belyavskaya, A.E. Brouwer, T. Evans, K. Heinrich, C. C. Lindner and D. A. Preece, With aforeword by Paul ErdHos. [20, 467]
[614] R. H. F. Denniston. Some maximal arcs in finite projective planes. J. CombinatorialTheory, 6:317–319, 1969. [484, 486]
[615] R. H. F. Denniston. Uniqueness of the inverse plane of order 5. Manuscripta Math.,8:11–19, 1973. [501]
[616] R. H. F. Denniston. Uniqueness of the inversive plane of order 7. ManuscriptaMath., 8:21–26, 1973. [501]
[617] a. C.-Y. S. Derrcik Hart, Liangpan Li. Fourier analysis and expanding phenomenain finite fields. Proc. Amer. Math. Soc. [130]
[618] J.-M. Deshouillers, G. Effinger, H. te Riele, and D. Zinoviev. A complete Vinogradov3-primes theorem under the Riemann hypothesis. Electron. Res. Announc.Amer. Math. Soc., 3:99–104, 1997. [411, 414]
[619] M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkorper.Abh. Math. Sem. Hansischen Univ., 14:197–272, 1941. [342, 351]
[620] M. Dewar, L. Moura, D. Panario, B. Stevens, and Q. Wang. Division of trinomialsby pentanomials and orthogonal arrays. Des. Codes Cryptogr., 45(1):1–17,2007. [528, 529, 531]
[621] A. D´iaz and E. Kaltofen. FoxBox a system for manipulating symbolic objects inblack box representation. In ISSAC ’98: Proceedings of the 1998 InternationalSymposium on Symbolic and Algebraic Computation, pages 30–37, 1998. [311]
[622] J. W. Di Paola. On minimum blocking coalitions in small projective plane games.SIAM J. Appl. Math., 17:378–392, 1969. [472, 475]
[623] P. Diaconis and R. Graham. Products of Universal Cycles, pages 35–55. In Demaineet al. [601], 2008. A collection of puzzles in honor of Martin Gardner’s 90thbirthday. [531]
[624] P. Diaconis and R. Graham. Magical Mathematics: The Mathematical Ideas thatAnimate Great Magic Tricks. Princeton University Press, November 2011.[531]
[625] P. Diaconis and M. Shahshahani. Generating a random permutation with randomtranspositions. Z. Wahrsch. Verw. Gebiete, 57(2):159–179, 1981. [536, 537,545]
Miscellaneous applications 735
[626] J. Dick. Walsh spaces containing smooth functions and quasi-Monte Carlo rules ofarbitrary high order. SIAM J. Numer. Anal., 46:1519–1553, 2008. [376, 381,383]
[627] J. Dick, P. Kritzer, G. Leobacher, and F. Pillichshammer. Constructions of generalpolynomial lattice rules based on the weighted star discrepancy. Finite FieldsAppl., 13:1045–1070, 2007. [377, 383]
[628] J. Dick and H. Niederreiter. On the exact t-value of Niederreiter and Sobol’ se-quences. J. Complexity, 24:572–581, 2008. [381, 383]
[629] J. Dick and H. Niederreiter. Duality for digital sequences. J. Complexity, 25:406–414, 2009. [381, 383]
[630] J. Dick and F. Pillichshammer. Digital nets and sequences: discrepancy theory andquasi-Monte Carlo integration. Cambridge University Press, Cambridge, 2010.[373, 376, 377, 378, 381, 383]
[631] L. E. Dickson. The analytic representation of substitutions on a power of a primenumber of letters with a discussion of the linear group. Ann. of Math., 11(1-6):65–120, 1896/97. [172, 185]
[632] L. E. Dickson. On finite algebras. In Gesellschaften der Wissenschaften zuGottingen, pages 358–393. 1905. [227, 229]
[633] L. E. Dickson. Criteria for the irreducibility of functions in a finite field. Bull.Amer. Math. Soc., 13(1):1–8, 1906. [36, 38, 41]
[634] L. E. Dickson. On commutative linear algebras in which division is always uniquelypossible. Trans. Amer. Math. Soc., 7(4):514–522, 1906. [227, 229, 233, 234]
[635] L. E. Dickson. Linear groups: With an exposition of the Galois field theory. withan introduction by W. Magnus. Dover Publications Inc., New York, 1958. [19,20, 39, 40]
[636] C. Diem. The GHS attack in odd characteristic. Journal of the Ramanujan Math-ematical Society, 18(1):1–32, 2003. [668]
[637] C. Diem. The XL-algorithm and a conjecture from commutative algebra. In Ad-vances in cryptology—ASIACRYPT 2004, volume 3329 of Lecture Notes inComput. Sci., pages 323–337. Springer, Berlin, 2004. [665]
[638] J. Dieudonne. Sur les groupes classiques. Actualites Sci. Ind., no. 1040 = Publ. Inst.Math. Univ. Strasbourg (N.S.) no. 1 (1945). Hermann et Cie., Paris, 1948. [439,440, 441, 442, 443, 444, 445, 446, 447]
[639] J. A. Dieudonne. La geometrie des groupes classiques. Springer-Verlag, Berlin, 1971.Troisieme edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band5. [438, 443, 447]
[640] W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Trans. Infor-mation Theory, IT-22(6):644–654, 1976. [147, 629, 634]
[641] W. Diffie and M. E. Hellman. New directions in cryptography. In Secure communi-cations and asymmetric cryptosystems, volume 69 of AAAS Sel. Sympos. Ser.,pages 143–180. Westview, Boulder, CO, 1982. [649]
[642] J. F. Dillon. ELEMENTARY HADAMARD DIFFERENCE-SETS. ProQuest LLC,Ann Arbor, MI, 1974. Thesis (Ph.D.)–University of Maryland, College Park.[221, 223, 224]
[643] J. F. Dillon. Multiplicative difference sets via additive characters. Des. CodesCryptogr., 17(1-3):225–235, 1999. [212, 213, 515, 519]
[644] J. F. Dillon. Geometry, codes and difference sets: exceptional connections. In Codesand designs (Columbus, OH, 2000), volume 10 of Ohio State Univ. Math. Res.
736 Handbook of Finite Fields
Inst. Publ., pages 73–85. de Gruyter, Berlin, 2002. [212, 213]
[645] J. F. Dillon and H. Dobbertin. New cyclic difference sets with Singer parameters.Finite Fields Appl., 10(3):342–389, 2004. [212, 213, 219, 224, 265, 515, 519,639, 640, 647]
[646] J. F. Dillon and G. McGuire. Near bent functions on a hyperplane. Finite FieldsAppl., 14(3):715–720, 2008. [261, 262]
[647] E. Dimitrova, L. D. GarcIa-Puente, F. Hinkelmann, A. S. Jarrah, R. Lauben-bacher, B. Stigler, M. Stillman, and P. Vera-Licona. Polynome. Availableat http://polymath.vbi.vt.edu/polynome/, 2010. [690]
[648] E. Dimitrova, L. D. Garc`ia-Puente, F. Hinkelmann, A. S. Jarrah, R. Laubenbacher,B. Stigler, M. Stillman, and P. Vera-Licona. Parameter estimation for booleanmodels of biological networks. Theoretical Computer Science, 412(26):2816 –2826, 2011. Foundations of Formal Reconstruction of Biochemical Networks.[689, 692]
[649] E. S. Dimitrova, A. S. Jarrah, R. Laubenbacher, and B. Stigler. A Grobner fanmethod for biochemical network modeling. In ISSAC 2007, pages 122–126.ACM, New York, 2007. [689, 692]
[650] C. Ding, T. Helleseth, and H. Niederreiter, editors. Sequences and their applications,Springer Series in Discrete Mathematics and Theoretical Computer Science,London, 1999. Springer-Verlag London Ltd. [20]
[651] C. Ding, D. Pei, and A. Salomaa. Chinese remainder theorem. World ScientificPublishing Co. Inc., River Edge, NJ, 1996. Applications in computing, coding,cryptography. [185]
[652] C. Ding, Z. Wang, and Q. Xiang. Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3, 32h+1). J. Combin. Theory Ser. A,114(5):867–887, 2007. [185, 231, 232, 234, 516, 519]
[653] C. Ding, Q. Xiang, J. Yuan, and P. Yuan. Explicit classes of permutation polyno-mials of F33m . Sci. China Ser. A, 52(4):639–647, 2009. [182, 185]
[654] C. Ding, G. Xiao, and W. Shan. The stability theory of stream ciphers, volume 561of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1991. [271,275, 281]
[655] C. Ding and J. Yuan. A family of skew Hadamard difference sets. J. Combin.Theory Ser. A, 113(7):1526–1535, 2006. [185, 231, 234, 516, 519]
[656] C. Ding and P. Yuan. Permutation polynomials over finite fields from a powerfullemma. Finite Fields Appl., 2011. [177, 180, 181, 185]
[657] C. S. Ding, H. Niederreiter, and C. P. Xing. Some new codes from algebraic curves.IEEE Trans. Inform. Theory, 46:2638–2642, 2000. [607, 612]
[658] J. Ding. A new variant of the Matsumoto-Imai cryptosystem through perturba-tion. In Public key cryptography—PKC 2004, volume 2947 of Lecture Notes inComput. Sci., pages 305–318. Springer, Berlin, 2004. [656]
[659] J. Ding. Mutants and its impact on polynomial solving strategies and algorithms.Privately distributed research note, University of Cincinnati and TechnicalUniversity of Darmstadt, 2006. [664]
[660] J. Ding. Inverting square systems algebraically is exponential. Cryptology ePrintArchive, Report 2011/275, 2011. http://eprint.iacr.org/. [654, 658, 664]
[661] J. Ding, J. Buchmann, M. S. E. Mohamed, W. S. A. M. Mohamed, and R.-P. Wein-mann. Mutant xl. First International Conference on Symbolic Computationand Cryptography – SCC, 2008. [664]
Miscellaneous applications 737
[662] J. Ding, V. Dubois, B.-Y. Yang, O. C.-H. Chen, and C.-M. Cheng. Could SFLASHbe repaired? In Automata, languages and programming. Part II, volume 5126of Lecture Notes in Comput. Sci., pages 691–701. Springer, Berlin, 2008. [656,657, 662]
[663] J. Ding and J. E. Gower. Inoculating multivariate schemes against differentialattacks. In Public key cryptography—PKC 2006, volume 3958 of Lecture Notesin Comput. Sci., pages 290–301. Springer, Berlin, 2006. [658, 660]
[664] J. Ding, J. E. Gower, and D. S. Schmidt. Multivariate public key cryptosystems,volume 25 of Advances in Information Security. Springer, New York, 2006.[648]
[665] J. Ding and T. Hodges. Cryptanalysis of an implementation scheme of the tamedtransformation method cryptosystem. J. Algebra Appl., 3(3):273–282, 2004.[658, 659]
[666] J. Ding and T. Hodges. Inverting the hfe systems is quasipolynomial for all fields.Accept for Crypto 2011, Febuary 2011. [654, 658, 664]
[667] J. Ding and D. Schmidt. A common defect of the TTM cryptosystem. In Proceed-ings of the technical track of the ACNS’03, ICISA Press, pages 68–78, 2003.http://eprint.iacr.org/2003/085. [658, 659]
[668] J. Ding and D. Schmidt. The new implementation schemes of the TTM cryptosystemare not secure. In Coding, cryptography and combinatorics, volume 23 of Progr.Comput. Sci. Appl. Logic, pages 113–127. Birkhauser, Basel, 2004. [658, 659]
[669] J. Ding and D. Schmidt. Cryptanalysis of HFEv and internal perturbation of HFE.In Public key cryptography—PKC 2005, volume 3386 of Lecture Notes in Com-put. Sci., pages 288–301. Springer, Berlin, 2005. [657]
[670] J. Ding and D. Schmidt. Rainbow, a new multivariable polynomial signature scheme.In Conference on Applied Cryptography and Network Security — ACNS 2005,volume 3531 of LNCS, pages 164–175. Springer, 2005. [655]
[671] J. Ding, D. Schmidt, and F. Werner. Algebraic attack on hfe revisited. In ISC 2008,Lecture Notes in Computer Science. Springer, 2007. [654, 658]
[672] J. Ding, D. Schmidt, and Z. Yin. Cryptanalysis of the new tts scheme in ches 2004.Int. J. Inf. Sec., 5(4):231–240, 2006. [655]
[673] J. Ding, C. Wolf, and B.-Y. Yang. l-invertible cycles for Multivariate Quadratic(MQ) public key cryptography. In Public key cryptography—PKC 2007, vol-ume 4450 of Lecture Notes in Comput. Sci., pages 266–281. Springer, Berlin,2007. [658]
[674] J. Ding and B.-Y. Yang. Multivariate polynomials for hashing. In Inscrypt, LNCS.Springer, 2007. to appear, cf. http://eprint.iacr.org/2007/137. [665]
[675] J. Ding, B.-Y. Yang, C.-H. O. Chen, M.-S. Chen, and C.-M. Cheng. New differential-algebraic attacks and reparametrization of rainbow. In Applied Cryptographyand Network Security, volume 5037 of LNCS, pages 242–257. Springer, 2008.cf. http://eprint.iacr.org/2008/108. [655, 656, 662, 663]
[676] J. Ding and Z. Yin. Cryptanalysis of TTS and Tame–like signature schemes. InThird International Workshop on Applied Public Key Infrastructures, 2004.[658]
[677] J. H. Dinitz. New lower bounds for the number of pairwise orthogonal symmetricLatin squares. In Proceedings of the Tenth Southeastern Conference on Combi-natorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton,Fla., 1979), Congress. Numer., XXIII–XXIV, pages 393–398, Winnipeg, Man.,
738 Handbook of Finite Fields
1979. Utilitas Math. [553]
[678] J. H. Dinitz and D. R. Stinson. The construction and uses of frames. Ars Combin.,10:31–53, 1980. [554]
[679] J. H. Dinitz and D. R. Stinson. Room squares and related designs. In Contempo-rary design theory, Wiley-Intersci. Ser. Discrete Math. Optim., pages 137–204.Wiley, New York, 1992. [555]
[680] J. H. Dinitz and G. S. Warrington. The spectra of certain classes of Room frames:the last cases. Electron. J. Combin., 17(1):Research Paper 74, 13, 2010. [556]
[681] V. Dmytrenko, F. Lazebnik, and J. Williford. On monomial graphs of girth eight.Finite Fields Appl., 13(4):828–842, 2007. [185]
[682] H. Dobbertin. Almost perfect nonlinear power functions on GF(2n): the Niho case.Inform. and Comput., 151(1-2):57–72, 1999. [185, 213]
[683] H. Dobbertin. Almost perfect nonlinear power functions on GF(2n): the Welch case.IEEE Trans. Inform. Theory, 45(4):1271–1275, 1999. [185, 213]
[684] H. Dobbertin. Kasami power functions, permutation polynomials and cyclic differ-ence sets. In Difference sets, sequences and their correlation properties (BadWindsheim, 1998), volume 542 of NATO Adv. Sci. Inst. Ser. C Math. Phys.Sci., pages 133–158. Kluwer Acad. Publ., Dordrecht, 1999. [515, 519]
[685] H. Dobbertin. Almost perfect nonlinear power functions on GF(2n): a new casefor n divisible by 5. In Finite fields and applications (Augsburg, 1999), pages113–121. Springer, Berlin, 2001. [183, 185, 213]
[686] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, and P. Gaborit. Con-struction of bent functions via Niho power functions. J. Combin. Theory Ser.A, 113(5):779–798, 2006. [220, 224]
[687] H. Dobbertin, D. Mills, E. N. Muller, A. Pott, and W. Willems. APN functions inodd characteristic. Discrete Math., 267(1-3):95–112, 2003. Combinatorics 2000(Gaeta). [208, 213]
[688] G. Dolinar, A. E. Guterman, B. Kuzma, and M. Orel. On the Polya permanentproblem over finite fields. European J. Combin., 32(1):116–132, 2011. [424]
[689] G. Dorfer and H. Maharaj. Generalized AG codes and generalized duality. FiniteFields Appl., 9:194–210, 2003. [608, 612]
[690] G. Dorfer, W. Meidl, and A. Winterhof. Counting functions and expected valuesfor the lattice profile at n. Finite Fields Appl., 10(4):636–652, 2004. [280, 281]
[691] G. Dorfer and A. Winterhof. Lattice structure and linear complexity profile ofnonlinear pseudorandom number generators. Appl. Algebra Engrg. Comm.Comput., 13(6):499–508, 2003. [280, 281]
[692] J. M. Dover. A family of non-Buekenhout unitals in the Hall planes. In Mostlyfinite geometries (Iowa City, IA, 1996), volume 190 of Lecture Notes in Pureand Appl. Math., pages 197–205. Dekker, New York, 1997. [484, 486]
[693] K. Drakakis, R. Gow, and G. McGuire. APN permutations on Zn and Costas arrays.Discrete Appl. Math., 157(15):3320–3326, 2009. [185]
[694] K. Drakakis, F. Iorio, and S. Rickard. The enumeration of costas arrays of order 28and its consequences. Adv. Math. Commun., 5(1):69–86, 2011. [547]
[695] V. G. Drinfeld. Elliptic modules. Mat. Sb. (N.S.), 94(136):594–627, 656, 1974. [448,450, 452, 456, 457]
[696] V. G. Drinfeld. Elliptic modules. II. Mat. Sb. (N.S.), 102(144)(2):182–194, 325,1977. [448, 450, 451]
Miscellaneous applications 739
[697] M. Drmota and R. F. Tichy. Sequences, discrepancies and applications, volume 1651of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1997. [19, 20, 139]
[698] V. Dubois, P.-A. Fouque, A. Shamir, and J. Stern. Practical cryptanalysis ofSFLASH. In Advances in cryptology—CRYPTO 2007, volume 4622 of Lec-ture Notes in Comput. Sci., pages 1–12. Springer, Berlin, 2007. [656, 657,662]
[699] V. Dubois, P.-A. Fouque, and J. Stern. Cryptanalysis of SFLASH with slightlymodified parameters. In Advances in cryptology—EUROCRYPT 2007, volume4515 of Lecture Notes in Comput. Sci., pages 264–275. Springer, Berlin, 2007.[656, 657, 661]
[700] V. Dubois and N. Gama. The degree of regularity of HFE systems. In Advancesin cryptology—ASIACRYPT 2010, volume 6477 of Lecture Notes in Comput.Sci., pages 557–576. Springer, Berlin, 2010. [664]
[701] I. S. Duff, A. M. Erisman, and J. K. Reid. Direct methods for sparse matrices.Monographs on Numerical Analysis. The Clarendon Press Oxford UniversityPress, New York, second edition, 1989. Oxford Science Publications. [434, 436]
[702] W. Duke. On multiple Salie sums. Proc. Amer. Math. Soc., 114(3):623–625, 1992.[111, 118]
[704] J.-G. Dumas, L. Fousse, and B. Salvy. Simultaneous modular reduction and kro-necker substitution for small finite fields. Journal of Symbolic Computation,46(7):823 – 840, 2011. Special Issue in Honour of Keith Geddes on his 60thBirthday. [426, 427, 436]
[705] J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. InProceedings of the 2002 International Symposium on Symbolic and AlgebraicComputation, pages 63–74 (electronic), New York, 2002. ACM. [427, 436]
[706] J.-G. Dumas, P. Giorgi, and C. Pernet. Dense linear algebra over word-size primefields: the FFLAS and FFPACK packages. ACM Trans. Math. Software,35(3):Art. 19, 35, 2008. [427, 428, 436]
[707] J.-G. Dumas and G. Villard. Computing the rank of sparse matrices over finitefields. In V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, editors, CASC2002, Proceedings of the fifth International Workshop on Computer Algebrain Scientific Computing, Yalta, Ukraine, pages 47–62. Technische UniversitatMunchen, Germany, Sept. 2002. [432, 433, 434, 435, 436]
[708] A. Duran, B. Saunders, and Z. Wan. Hybrid algorithms for rank of sparse matrices.In R. Mathias and H. Woerdeman, editors, SIAM Conference on Applied LinearAlgebra, Williamsburg, VA, USA, July 2003. [435, 436]
[709] I. Duursma and K.-H. Mak. On lower bounds for the Ihara constants A(2) andA(3). arXiv:1102.4127v2[math.NT], 2011. [367, 368, 372]
[710] B. Dwork. On the rationality of the zeta function of an algebraic variety. Amer. J.Math., 82:631–648, 1960. [120, 127, 394]
[712] B. Dwork. Bessel functions as p-adic functions of the argument. Duke Math. J.,41:711–738, 1974. [394, 402]
[713] B. M. Dwork. On the zeta function of a hypersurface. III. Ann. of Math. (2),83:457–519, 1966. [254, 255]
740 Handbook of Finite Fields
[714] W. Eberly, M. Giesbrecht, P. Giorgi, A. Storjohann, and G. Villard. Faster inversionand other black box matrix computations using efficient block projections. InISSAC 2007, pages 143–150. ACM, New York, 2007. [436]
[715] W. Eberly and E. Kaltofen. On randomized Lanczos algorithms. In Proceedingsof the 1997 International Symposium on Symbolic and Algebraic Computation(Kihei, HI), pages 176–183 (electronic), New York, 1997. ACM. [432, 433, 436]
[716] G. L. Ebert. Partitioning projective geometries into caps. Canad. J. Math.,37(6):1163–1175, 1985. [480, 486]
[717] G. L. Ebert. Nests, covers, and translation planes. Ars Combin., 25(C):213–233,1988. Eleventh British Combinatorial Conference (London, 1987). [480, 486]
[718] G. L. Ebert. Spreads admitting regular elliptic covers. European J. Combin.,10(4):319–330, 1989. [480, 486]
[719] G. L. Ebert. Partitioning problems and flag-transitive planes. Rend. Circ. Mat.Palermo (2) Suppl., (53):27–44, 1998. Combinatorics ’98 (Mondello). [481,486]
[720] G. L. Ebert, G. Marino, O. Polverino, and R. Trombetti. Infinite families of newsemifields. Combinatorica, 29(6):637–663, 2009. [228, 229]
[721] Y. Edel, G. Kyureghyan, and A. Pott. A new APN function which is not equivalentto a power mapping. IEEE Trans. Inform. Theory, 52(2):744–747, 2006. [209,213]
[722] Y. Edel and A. Pott. A new almost perfect nonlinear function which is not quadratic.Adv. Math. Commun., 3(1):59–81, 2009. [208, 213]
[723] G. A. Edgar and C. Miller. Borel subrings of the reals. Proc. Amer. Math. Soc.,131(4):1121–1129 (electronic), 2003. [128]
[724] G. Effinger. A Goldbach theorem for polynomials of low degree over odd finite fields.Acta Arith., 42(4):329–365, 1983. [412, 414]
[725] G. Effinger. A Goldbach 3-primes theorem for polynomials of low degree over finitefields of characteristic 2. J. Number Theory, 29(3):345–363, 1988. [412, 414]
[726] G. Effinger. Toward a complete twin primes theorem for polynomials over finitefields. In Finite Fields and Applications, volume 461 of Contemp. Math., pages103–110. Amer. Math. Soc., Providence, 2008. [410, 414]
[727] G. Effinger and D. Hayes. A complete solution to the polynomial 3-primes problem.Bull. Amer. Math. Soc., 24(2):363–369, 1991. [412, 414]
[728] G. Effinger and D. R. Hayes. Additive number theory of polynomials over a finitefield. Oxford Mathematical Monographs. Oxford University Press, New York,1991. [19, 20, 411, 412, 413, 414]
[729] G. Effinger, K. Hick, and G. L. Mullen. Twin irreducible polynomials over finitefields. In Finite fields with applications to coding theory, cryptography andrelated areas, pages 94–111. Springer, Berlin, 2002. [410, 414]
[730] G. Effinger, K. Hick, and G. L. Mullen. Integers and polynomials: comparing theclose cousins Z and Fq[x]. Math. Intelligencer, 27(2):26–34, 2005. [407, 414]
[731] M. Einsiedler and T. Ward. Ergodic theory with a view towards number theory,volume 259 of Graduate Texts in Mathematics. Springer-Verlag London Ltd.,London, 2011. [282, 283, 289]
[732] T. ElGamal. A public key cryptosystem and a signature scheme based on discretelogarithms. IEEE Trans. Inform. Theory, 31(4):469–472, 1985. [629, 634]
[733] S. Eliahou, M. Kervaire, and B. Saffari. On Golay polynomial pairs. Adv. in Appl.Math., 12(3):235–292, 1991. [696, 701]
Miscellaneous applications 741
[734] N. D. Elkies. The existence of infinitely many supersingular primes for every ellipticcurve over Q. Invent. Math., 89(3):561–567, 1987. [349, 351]
[735] N. D. Elkies. Distribution of supersingular primes. Asterisque, (198-200):127–132(1992), 1991. Journees Arithmetiques, 1989 (Luminy, 1989). [349, 351]
[736] N. D. Elkies. Elliptic and modular curves over finite fields and related computa-tional issues. In Computational perspectives on number theory (Chicago, IL,1995), volume 7 of AMS/IP Stud. Adv. Math., pages 21–76. Amer. Math. Soc.,Providence, RI, 1998. [669]
[737] N. D. Elkies. Explicit modular towers. Proceedings of the 35th Allerton conferenceon communication, control and computing, pages 23–32, 1998. [368, 372]
[738] N. D. Elkies. Explicit towers of Drinfeld modular curves. In European Congressof Mathematics, Vol. II (Barcelona, 2000), volume 202 of Progr. Math., pages189–198. Birkhauser, Basel, 2001. [368, 372]
[739] N. D. Elkies, E. W. Howe, A. Kresch, B. Poonen, J. L. Wetherell, and M. E. Zieve.Curves of every genus with many points. II. Asymptotically good families.Duke Math. J., 122(2):399–422, 2004. [367]
[741] B. Elspas. The theory of autonomous linear sequential networks. In Linear Sequen-tial Switching Circuits, pages 21–61. Holden-Day, San Francisco, Calif., 1965.[692]
[742] H. Enderling, M. Chaplain, and P. Hahnfeldt. Quantitative modeling of tu-mor dynamics and radiotherapy. Acta Biotheoretica, 58:341–353, 2010.10.1007/s10441-010-9111-z. [689, 692]
[743] A. Enge. Computing discrete logarithms in high-genus hyperelliptic Jacobians inprovably subexponential time. Math. Comp., 71(238):729–742 (electronic),2002. [360]
[744] A. Enge. The complexity of class polynomial computation via floating point ap-proximations. Mathematics of Computation, 78(266):1089–1107, 2009. [669]
[745] A. Enge. Computing modular polynomials in quasi-linear time. Mathematics ofComputation, 78(267):1809–1824, 2009. [670]
[746] A. Enge and P. Gaudry. A general framework for subexponential discrete logarithmalgorithms. Acta Arith., 102(1):83–103, 2002. [360]
[747] S. S. Erdem, T. Yanik, and C. K. Koc. Polynomial basis multiplication over GF(2m).Acta Appl. Math., 93(1-3):33–55, 2006. [79]
[748] S. Erickson, M. J. Jacobson, Jr., N. Shang, S. Shen, and A. Stein. Explicit formulasfor real hyperelliptic curves of genus 2 in affine representation. In Arithmeticof finite fields, volume 4547 of Lecture Notes in Comput. Sci., pages 202–218.Springer, Berlin, 2007. [357, 360]
[749] S. Erickson, M. J. Jacobson, Jr., and A. Stein. Explicit formulas for real hyper-elliptic curves of genus 2 in affine representation. to appear in Advances inMathematics of Communication, 2011. [356, 357, 360]
[750] T. eSTREAM Project. [636, 638, 639, 647]
[751] J. Ethier and G. L. Mullen. Strong forms of orthogonality for sets of frequencyhypercubes. Preprint, 2011. [466]
[752] J. Ethier and G. L. Mullen. Strong forms of orthogonality for sets of hypercubes.Preprint, 2011. [466]
[753] A. B. Evans. Maximal sets of mutually orthogonal Latin squares. II. European J.Combin., 13(5):345–350, 1992. [184, 185]
742 Handbook of Finite Fields
[754] A. B. Evans. Orthomorphism graphs of groups, volume 1535 of Lecture Notes inMathematics. Springer-Verlag, Berlin, 1992. [135, 184, 185]
[755] R. Evans. Residuacity of primes. Rocky Mountain J. Math., 19(4):1069–1081, 1989.[98, 118]
[756] R. Evans. Character sums as orthogonal eigenfunctions of adjacency operatorsfor Cayley graphs. In Finite fields: theory, applications, and algorithms (LasVegas, NV, 1993), volume 168 of Contemp. Math., pages 33–50. Amer. Math.Soc., Providence, RI, 1994. [112, 118]
[757] R. Evans. Congruences for Jacobi sums. J. Number Theory, 71(1):109–120, 1998.[104, 118]
[758] R. Evans. Gauss sums and Kloosterman sums over residue rings of algebraic integers.Trans. Amer. Math. Soc., 353(11):4429–4445 (electronic), 2001. [117, 118]
[759] R. Evans. Gauss sums of orders six and twelve. Canad. Math. Bull., 44(1):22–26,2001. [107, 108, 118]
[760] R. Evans. Twisted hyper-Kloosterman sums over finite rings of integers. In Numbertheory for the millennium, I (Urbana, IL, 2000), pages 429–448. A K Peters,Natick, MA, 2002. [112, 117, 118]
[761] R. Evans. Hypergeometric 3F2(1/4) evaluations over finite fields and Hecke eigen-forms. Proc. Amer. Math. Soc., 138(2):517–531, 2010. [115, 118]
[762] R. Evans. Seventh power moments of Kloosterman sums. Israel J. Math., 175:349–362, 2010. [114, 115, 118]
[763] R. Evans and J. Greene. Clausen’s theorem and hypergeometric functions over finitefields. Finite Fields Appl., 15(1):97–109, 2009. [103, 118]
[764] R. Evans and J. Greene. Evaluations of hypergeometric functions over finite fields.Hiroshima Math. J., 39(2):217–235, 2009. [103, 118]
[765] R. Evans, H. D. L. Hollmann, C. Krattenthaler, and Q. Xiang. Gauss sums, Jacobisums, and p-ranks of cyclic difference sets. J. Combin. Theory Ser. A, 87(1):74–119, 1999. [109, 118]
[766] R. J. Evans. Identities for products of Gauss sums over finite fields. Enseign. Math.(2), 27(3-4):197–209 (1982), 1981. [103, 118]
[767] R. J. Evans. Pure Gauss sums over finite fields. Mathematika, 28(2):239–248 (1982),1981. [102, 118]
[768] R. J. Evans. Period polynomials for generalized cyclotomic periods. ManuscriptaMath., 40(2-3):217–243, 1982. [117, 118]
[769] R. J. Evans. Character sum analogues of constant term identities for root systems.Israel J. Math., 46(3):189–196, 1983. [103, 118]
[770] R. J. Evans. The evaluation of Selberg character sums. Enseign. Math. (2), 37(3-4):235–248, 1991. [103, 118]
[771] R. J. Evans. Selberg-Jack character sums of dimension 2. J. Number Theory,54(1):1–11, 1995. [103, 118]
[772] R. J. Evans, J. Greene, and H. Niederreiter. Linearized polynomials and permutationpolynomials of finite fields. Michigan Math. J., 39(3):405–413, 1992. [184, 185]
[773] S. A. Evdokimov. Efficient factorization of polynomials over finite fields and thegeneralized Riemann hypothesis. Translation of Zapiski Nauchnyck SeminarovLeningradskgo Otdeleniya Mat. Inst. V.A. Steklova Akad. Nauk SSSR (LOMI),volume 176, 1989, pp. 104–117, 1986. [298, 299]
[774] G. Everest and T. Ward. Heights of polynomials and entropy in algebraic dynamics.
Miscellaneous applications 743
Universitext. Springer-Verlag London Ltd., London, 1999. [282, 283, 289]
[775] J.-H. Evertse. Linear equations with unknowns from a multiplicative group whosesolutions lie in a small number of subspaces. Indag. Math. (N.S.), 15(3):347–355, 2004. [255]
[776] C. Faber and G. van der Geer. Complete subvarieties of moduli spaces and thePrym map. J. Reine Angew. Math., 573:117–137, 2004. [401, 402]
[777] C. C. Faith. Extensions of normal bases and completely basic fields. Trans. Amer.Math. Soc., 85:406–427, 1957. [83, 84]
[778] G. Faltings. Finiteness theorems for abelian varieties over number fields. In Arith-metic geometry (Storrs, Conn., 1984), pages 9–27. Springer, New York, 1986.Translated from the German original [Invent. Math. 73 (1983), no. 3, 349–366;ibid. 75 (1984), no. 2, 381; MR 85g:11026ab] by Edward Shipz. [344, 351]
[779] S. Fan. Primitive normal polynomials with the last half coefficients prescribed.Finite Fields Appl., 15(5):604–614, 2009. [64, 65]
[780] S. Fan and W. Han. Character sums over Galois rings and primitive polynomialsover finite fields. Finite Fields Appl., 10(1):36–52, 2004. [61, 62, 65]
[781] S. Fan and W. Han. p-adic formal series and Cohen’s problem. Glasg. Math. J.,46(1):47–61, 2004. [61, 65]
[782] S. Fan and W. Han. p-adic formal series and primitive polynomials over finite fields.Proc. Amer. Math. Soc., 132(1):15–31 (electronic), 2004. [61, 62, 65]
[783] S. Fan and W. Han. Primitive polynomial with three coefficients prescribed. FiniteFields Appl., 10(4):506–521, 2004. [62, 65]
[784] S. Fan, W. Han, and K. Feng. Primitive normal polynomials with multiple coeffi-cients prescribed: an asymptotic result. Finite Fields Appl., 13(4):1029–1044,2007. [61, 64, 65]
[785] S. Fan, W. Han, K. Feng, and X. Zhang. Primitive normal polynomials with thefirst two coefficients prescribed: a revised p-adic method. Finite Fields Appl.,13(3):577–604, 2007. [64, 65]
[786] S. Fan and X. Wang. Primitive normal polynomials with a prescribed coefficient.Finite Fields Appl., 15(6):682–730, 2009. [63, 65]
[787] S. Fan and X. Wang. Primitive normal polynomials with the specified last twocoefficients. Discrete Math., 309(13):4502–4513, 2009. [62, 64, 65]
[788] R. R. Farashahi. Hashing into Hessian curves. To appear in Africacrypt, 2011. [679]
[789] J.-C. Faugere. A new efficient algorithm for computing Grobner bases (F4). J. PureAppl. Algebra, 139(1-3):61–88, 1999. Effective methods in algebraic geometry(Saint-Malo, 1998). [664]
[790] J.-C. Faugere. A new efficient algorithm for computing Grobner bases withoutreduction to zero (F5). In Proceedings of the 2002 International Symposiumon Symbolic and Algebraic Computation, pages 75–83 (electronic), New York,2002. ACM. [664]
[791] J.-C. Faugere and A. Joux. Algebraic cryptanalysis of hidden field equation (HFE)cryptosystems using Grobner bases. In Advances in cryptology—CRYPTO2003, volume 2729 of Lecture Notes in Comput. Sci., pages 44–60. Springer,Berlin, 2003. [654, 662, 664]
[792] J.-C. Faugere and S. Lachartre. Parallel gaussian elimination for grobner bases com-putations in finite fields. In M. M. Maza and J.-L. Roch, editors, PASCO 2010,Proceedings of the 4th International Workshop on Parallel Symbolic Computa-tion, Grenoble, France, pages 89–97. ACM, July 2010. [435, 436]
744 Handbook of Finite Fields
[793] J.-C. Faugere and L. Perret. Polynomial equivalence problems: algorithmic andtheoretical aspects. In Advances in cryptology—EUROCRYPT 2006, volume4004 of Lecture Notes in Comput. Sci., pages 30–47. Springer, Berlin, 2006.[651]
[794] H. Faure. Discrepance de suites associees a un systeme de numeration (en dimensions). Acta Arith., 41:337–351, 1982. [381, 383]
[795] H. Feistel, W. Notz, and J. Smith. Some cryptographic techniques for machine-to-machine data communications. Proceedings of the IEEE, 63(11):1545–1554,1975. [627, 634]
[796] H. Fell and W. Diffie. Analysis of a public key approach based on polynomialsubstitution. In Advances in cryptology—CRYPTO ’85 (Santa Barbara, Calif.,1985), volume 218 of Lecture Notes in Comput. Sci., pages 340–349. Springer,Berlin, 1986. [649, 651, 652]
[797] G. Fellegara. Gli ovaloidi in uno spazio tridimensionale di Galois di ordine 8. AttiAccad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 32:170–176, 1962. [501]
[798] B. Felszeghy. On the solvability of some special equations over finite fields. Publ.Math. Debrecen, 68(1-2):15–23, 2006. [167, 170]
[799] G. L. Feng and K. K. Tzeng. A generalization of the Berlekamp-Massey algorithmfor multisequence shift-register synthesis with applications to decoding cycliccodes. IEEE Trans. Inform. Theory, 37(5):1274–1287, 1991. [275, 281]
[800] K. Feng and J. Luo. Value distributions of exponential sums from perfect nonlinearfunctions and their applications. IEEE Trans. Inform. Theory, 53(9):3035–3041, 2007. [221, 224]
[801] K. Feng and J. Luo. Weight distribution of some reducible cyclic codes. FiniteFields Appl., 14(2):390–409, 2008. [163]
[802] K. Feng, H. Niederreiter, and C. Xing, editors. Coding, cryptography and com-binatorics, volume 23 of Progress in Computer Science and Applied Logic.Birkhauser Verlag, Basel, 2004. [20]
[803] X. Feng and Z. Dai. Expected value of the linear complexity of two-dimensionalbinary sequences. In Sequences and their applications—SETA 2004, volume3486 of Lecture Notes in Comput. Sci., pages 113–128. Springer, Berlin, 2005.[276, 281]
[804] F. Fiedler, K. H. Leung, and Q. Xiang. On Mathon’s construction of maximal arcsin Desarguesian planes. Adv. Geom., (suppl.):S119–S139, 2003. Special issuededicated to Adriano Barlotti. [485, 486]
[805] N. J. Fine and I. N. Herstein. The probability that a matrix be nilpotent. IllinoisJ. Math., 2:499–504, 1958. [417, 424]
[806] FIPS 180-3. Secure hash standard (SHS). Federal Information Processing StandardsPublication 180-3, National Institute of Standards and Technology, 2008. [629,634]
[807] FIPS 186-3. Digital signature standard (DSS). Federal Information ProcessingStandards Publication 186-3, National Institute of Standards and Technology,2009. [630, 634]
[808] FIPS 46-3. Data encryption standard (DES). Federal Information Processing Stan-dards Publication 46-3, National Institute of Standards and Technology, 1999.[627, 634]
[809] S. Fischer and W. Meier. Algebraic immunity of s-boxes and augmented functions.In Proceedings of Fast Software Encryption 2007, volume 4593 of Lecture Notes
Miscellaneous applications 745
in Comput. Sci., pages 366–381. 2007. [202, 204]
[810] S. D. Fisher. Classroom Notes: Matrices over a Finite Field. Amer. Math. Monthly,73(6):639–641, 1966. [415]
[811] R. W. Fitzgerald. A characterization of primitive polynomials over finite fields.Finite Fields Appl., 9(1):117–121, 2003. [57, 59]
[812] R. W. Fitzgerald. Highly degenerate quadratic forms over finite fields of character-istic 2. Finite Fields Appl., 11(2):165–181, 2005. [163]
[813] R. W. Fitzgerald. Highly degenerate quadratic forms over F2. Finite Fields Appl.,13(4):778–792, 2007. [161, 163]
[814] R. W. Fitzgerald. Invariants of trace forms over finite fields of characteristic 2.Finite Fields Appl., 15(2):261–275, 2009. [162, 163]
[815] R. W. Fitzgerald. Trace forms over finite fields of characteristic 2 with prescribedinvariants. Finite Fields Appl., 15(1):69–81, 2009. [162, 163]
[816] R. W. Fitzgerald and J. L. Yucas. Irreducible polynomials over GF(2) with threeprescribed coefficients. Finite Fields Appl., 9(3):286–299, 2003. [27, 30, 48, 49]
[817] R. W. Fitzgerald and J. L. Yucas. Pencils of quadratic forms over finite fields.Discrete Math., 283(1-3):71–79, 2004. [163]
[818] R. W. Fitzgerald and J. L. Yucas. Sums of Gauss sums and weights of irreduciblecodes. Finite Fields Appl., 11(1):89–110, 2005. [98, 118]
[819] R. W. Fitzgerald and J. L. Yucas. Generalized reciprocals, factors of Dickson poly-nomials and generalized cyclotomic polynomials over finite fields. Finite FieldsAppl., 13(3):492–515, 2007. [236, 238, 239, 242]
[820] P. Flajolet and A. M. Odlyzko. Random mapping statistics. In EUROCRYPT,pages 329–354, 1989. [638, 647]
[821] J. J. Flynn. Near-exceptionality over finite fields. PhD dissertation, University ofCalifornia, Berkeley, Department of Mathematics, 2001. [189, 192]
[822] S. Fomin and A. Zelevinsky. The Laurent phenomenon. Adv. in Appl. Math.,28(2):119–144, 2002. [282, 289]
[823] F. Fontein. Groups from cyclic infrastructures and Pohlig-Hellman in certain in-frastructures. Adv. Math. Commun., 2(3):293–307, 2008. [360]
[824] G. D. Forney, Jr. On decoding BCH codes. IEEE Trans. Information Theory,IT-11:549–557, 1965. [593, 601, 602]
[825] G. D. Forney, Jr. Generalized minimum distance decoding. IEEE Trans. Informa-tion Theory, IT-12:125–131, 1966. [596, 597, 601, 602]
[826] G. D. Forney, Jr., N. J. A. Sloane, and M. D. Trott. The Nordstrom-Robinson codeis the binary image of the octacode. In Coding and quantization (Piscataway,NJ, 1992), volume 14 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci.,pages 19–26. Amer. Math. Soc., Providence, RI, 1993. [601, 602]
[827] P.-A. Fouque, L. Granboulan, and J. Stern. Differential cryptanalysis for multivari-ate schemes. In Advances in cryptology—EUROCRYPT 2005, volume 3494 ofLecture Notes in Comput. Sci., pages 341–353. Springer, Berlin, 2005. [656,660]
[828] P.-A. Fouque, G. Macario-Rat, L. Perret, and J. Stern. Total break of the l-ICsignature scheme. In Public key cryptography—PKC 2008, volume 4939 ofLecture Notes in Comput. Sci., pages 1–17. Springer, Berlin, 2008. [657]
[829] D. M. Freeman. Converting pairing-based cryptosystems from composite-ordergroups to prime-order groups. In H. Gilbert, editor, Advances in Cryptology
746 Handbook of Finite Fields
— EUROCRYPT 2010, volume 6110 of Lecture Notes in Computer Science,pages 44–61, Berlin, 2010. Springer-Verlag. [670, 674]
[830] J. W. Freeman. Reguli and pseudoreguli in PG(3, s2). Geom. Dedicata, 9(3):267–280, 1980. [482, 486]
[831] T. S. Freeman, G. Imirzian, E. Kaltofen, and Lakshman Yagati. Dagwood: Asystem for manipulating polynomials given by straight-line programs. ACMTrans. Math. Software, 14(3):218–240, 1988. [309, 311]
[832] D. Freemann, M. Scott, and E. Teske. A taxonomy of pairing-friendly elliptic curves.Journal of Cryptology, 23(2):224–280, 2010. [675, 676]
[833] G. Frey. Applications of arithmetical geometry to cryptographic constructions.In D. Jungnickel and H. Niederreiter, editors, Finite Fields and Applications— Proceedings of The Fifth International Conference on Finite Fields andApplications Fq5 , held at the University of Augsburg, Germany, August 2–6,1999, pages 128–161, Berlin, 2001. Springer-Verlag. [668]
[834] G. Frey and T. Lange. Varieties over special fields. In Handbook of elliptic andhyperelliptic curve cryptography, Discrete Math. Appl. (Boca Raton), pages87–113. Chapman & Hall/CRC, Boca Raton, FL, 2006. [19, 20]
[835] G. Frey, M. Muller, and H.-G. Ruck. The Tate pairing and the discrete loga-rithm applied to elliptic curve cryptosystems. IEEE Trans. Inform. Theory,45(5):1717–1719, 1999. [360]
[836] G. Frey, M. Perret, and H. Stichtenoth. On the different of abelian extensions ofglobal fields. In Coding theory and algebraic geometry (Luminy, 1991), volume1518 of Lecture Notes in Math., pages 26–32. Springer, Berlin, 1992. [372]
[837] M. Fried. On a conjecture of Schur. Michigan Math. J., 17:41–55, 1970. [183, 236,246, 247, 255]
[838] M. Fried. The field of definition of function fields and a problem in the reducibilityof polynomials in two variables. Illinois J. Math., 17:128–146, 1973. [253, 255]
[839] M. Fried. On a theorem of Ritt and related Diophantine problems. J. Reine Angew.Math., 264:40–55, 1973. [248, 255]
[840] M. Fried. On a theorem of MacCluer. Acta Arith., 25:121–126, 1973/74. [245, 246,255]
[841] M. Fried. On Hilbert’s irreducibility theorem. J. Number Theory, 6:211–231, 1974.[247, 249, 252, 255]
[842] M. Fried. Fields of definition of function fields and Hurwitz families—groups asGalois groups. Comm. Algebra, 5(1):17–82, 1977. [245, 255]
[843] M. Fried. Galois groups and complex multiplication. Trans. Amer. Math. Soc.,235:141–163, 1978. [251, 252, 255]
[844] M. Fried and R. Lidl. On Dickson polynomials and Redei functions. In Contributionsto general algebra, 5 (Salzburg, 1986), pages 139–149. Holder-Pichler-Tempsky,Vienna, 1987. [236, 242]
[845] M. Fried and G. Sacerdote. Solving Diophantine problems over all residue classfields of a number field and all finite fields. Ann. of Math. (2), 104(2):203–233,1976. [254, 255]
[846] M. D. Fried. The place of exceptional covers among all Diophantine relations. FiniteFields Appl., 11(3):367–433, 2005. [245, 246, 247, 248, 249, 250, 251, 252, 253,254, 255]
[847] M. D. Fried. Variables separated equations: Strikingly different roles for the branchcycle lemma and the finite simple group classification. Science China Mathe-
[848] M. D. Fried, R. Guralnick, and J. Saxl. Schur covers and Carlitz’s conjecture. IsraelJ. Math., 82(1-3):157–225, 1993. [173, 174, 185, 193, 194, 247, 254, 255]
[849] M. D. Fried and M. Jarden. Field arithmetic, volume 11 of Ergebnisse der Math-ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas(3)]. Springer-Verlag, Berlin, 1986. [19, 20, 247, 252, 254, 255]
[850] M. D. Fried and M. Jarden. Field arithmetic, volume 11 of Ergebnisse der Mathe-matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathe-matics [Results in Mathematics and Related Areas. 3rd Series. A Series of Mod-ern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, 2005. [19,20]
[851] M. D. Fried and M. Jarden. Field arithmetic, volume 11 of Ergebnisse der Mathe-matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Math-ematics [Results in Mathematics and Related Areas. 3rd Series. A Series ofModern Surveys in Mathematics]. Springer-Verlag, Berlin, third edition, 2008.Revised by Jarden. [19, 20]
[852] M. D. Fried and R. E. MacRae. On curves with separated variables. Math. Ann.,180:220–226, 1969. [253, 255]
[853] M. D. Fried and R. E. MacRae. On the invariance of chains of fields. Illinois J.Math., 13:165–171, 1969. [246, 255]
[854] E. Friedman and L. C. Washington. On the distribution of divisor class groups ofcurves over a finite field. In Theorie des nombres (Quebec, PQ, 1987), pages227–239. de Gruyter, Berlin, 1989. [355, 360]
[855] J. Friedman. On the second eigenvalue and random walks in random d-regulargraphs. Combinatorica, 11(4):331–362, 1991. [545]
[856] J. Friedman. Some geometric aspects of graphs and their eigenfunctions. DukeMath. J., 69(3):487–525, 1993. [537, 538, 543, 545]
[857] J. Friedman. A proof of Alon’s second eigenvalue conjecture and related problems.Mem. Amer. Math. Soc., 195(910):viii+100, 2008. [545]
[858] C. Friesen. A special case of Cohen-Lenstra heuristics in function fields. In Numbertheory (Ottawa, ON, 1996), volume 19 of CRM Proc. Lecture Notes, pages99–105. Amer. Math. Soc., Providence, RI, 1999. [355, 360]
[859] C. Friesen. Class group frequencies of real quadratic function fields: the degree 4case. Math. Comp., 69(231):1213–1228, 2000. [355, 360]
[860] C. Friesen. Bounds for frequencies of class groups of real quadratic genus 1 functionfields. Acta Arith., 96(4):313–331, 2001. [355, 360]
[861] S. Frisch. When are weak permutation polynomials strong? Finite Fields Appl.,1(4):437–439, 1995. [188]
[862] D. Fu and J. Solinas. IKE and IKEv2 authentication using the elliptic curve digitalsignature algorithm (ECDSA). RFC 4754, Internet Engineering Task Force,2007. http://www.ietf.org/rfc/rfc4754.txt. [667]
[863] F.-W. Fu, H. Niederreiter, and F. Ozbudak. On the joint linear complexity of linearrecurring multisequences. In Coding and cryptology, volume 4 of Ser. CodingTheory Cryptol., pages 125–142. World Sci. Publ., Hackensack, NJ, 2008. [270,276, 281]
[864] F.-W. Fu, H. Niederreiter, and F. Ozbudak. Joint linear complexity of arbitrarymultisequences consisting of linear recurring sequences. Finite Fields Appl.,15(4):475–496, 2009. [276, 281]
748 Handbook of Finite Fields
[865] F.-W. Fu, H. Niederreiter, and F. Ozbudak. Joint linear complexity of multise-quences consisting of linear recurring sequences. Cryptogr. Commun., 1(1):3–29, 2009. [276, 281]
[866] F.-W. Fu, H. Niederreiter, and M. Su. The expectation and variance of the joint lin-ear complexity of random periodic multisequences. J. Complexity, 21(6):804–822, 2005. [276, 281]
[867] L. Fu. Weights of twisted exponential sums. Math. Z., 262(2):449–472, 2009. [125,127]
[868] L. Fu and C. Liu. Equidistribution of Gauss sums and Kloosterman sums. Math.Z., 249(2):269–281, 2005. [97, 118]
[869] L. Fu and D. Wan. Moment L-functions, partial L-functions and partial exponentialsums. Math. Ann., 328(1-2):193–228, 2004. [126, 127, 156, 158]
[870] L. Fu and D. Wan. Mirror congruence for rational points on Calabi-Yau varieties.Asian J. Math., 10(1):1–10, 2006. [158]
[871] R. Fuhrmann, A. Garcia, and F. Torres. On maximal curves. J. Number Theory,67(1):29–51, 1997. [366, 367]
[872] R. Fuhrmann and F. Torres. The genus of curves over finite fields with many rationalpoints. Manuscripta Math., 89(1):103–106, 1996. [366, 367]
[873] R. Fuji-Hara, K. Momihara, and M. Yamada. Perfect difference systems of sets andJacobi sums. Discrete Math., 309(12):3954–3961, 2009. [100, 118]
[874] W. Fulton. Algebraic curves. Advanced Book Classics. Addison-Wesley PublishingCompany Advanced Book Program, Redwood City, CA, 1989. An introductionto algebraic geometry, Notes written with the collaboration of Richard Weiss,Reprint of 1969 original. [317, 332, 333, 367]
[875] E. M. Gabidulin. Theory of codes with maximum rank distance. Problemy PeredachiInformatsii, 21(1):3–16, 1985. [699, 701]
[876] A. Gacs. A remark on blocking sets of almost Redei type. J. Geom., 60(1-2):65–73,1997. [471, 475]
[877] A. Gacs. On a generalization of Redei’s theorem. Combinatorica, 23(4):585–598,2003. [471, 475]
[878] A. Gacs, L. Lovasz, and T. SzHonyi. Directions in AG(2, p2). Innov. IncidenceGeom., 6/7:189–201, 2007/08. [471, 475]
[879] A. Gacs, P. Sziklai, and T. SzHonyi. Two remarks on blocking sets and nuclei inplanes of prime order. Des. Codes Cryptogr., 10(1):29–39, 1997. [471, 475]
[880] S. D. Galbraith. Supersingular curves in cryptography. In Advances in cryptology—ASIACRYPT 2001 (Gold Coast), volume 2248 of Lecture Notes in Comput.Sci., pages 495–513. Springer, Berlin, 2001. [359, 360]
[881] S. D. Galbraith, M. Harrison, and D. J. Mireles Morales. Efficient hyperellipticarithmetic using balanced representation for divisors. In Algorithmic numbertheory, volume 5011 of Lecture Notes in Comput. Sci., pages 342–356. Springer,Berlin, 2008. [356, 360]
[882] S. D. Galbraith, F. Hess, and N. P. Smart. Extending the GHS Weil descent at-tack. In L. Knudsen, editor, Advances in Cryptology — EUROCRYPT 2002,volume 2332 of Lecture Notes in Computer Science, pages 29–44, Berlin, 2002.Springer-Verlag. [668]
[883] S. D. Galbraith and K. G. Paterson, editors. Pairing-Based Cryptography — Pair-ing 2008, volume 5209 of Lecture Notes in Computer Science, Berlin, 2008.Springer-Verlag. [670]
Miscellaneous applications 749
[884] S. D. Galbraith and N. P. Smart. A cryptographic application of Weil descent. InM. Walker, editor, Cryptography and Coding, volume 1746 of Lecture Notes inComputer Science, pages 191–200, Berlin, 1999. Springer-Verlag. [668]
[885] Z. Galil, R. Kannan, and E. Szemeredi. On nontrivial separators for k-page graphsand simulations by nondeterministic one-tape Turing machines. J. Comput.System Sci., 38(1):134–149, 1989. 18th Annual ACM Symposium on Theoryof Computing (Berkeley, CA, 1986). [148]
[886] R. G. Gallager. A simple derivation of the coding theorem and some applications.IEEE Trans. Information Theory, IT-11:3–18, 1965. [560, 561, 602]
[887] R. Gallant, R. Lambert, and S. Vanstone. Improving the parallelized Pollardlambda search on binary anomalous curves. Mathematics of Computation,69(232):1699–1705, 2000. [668]
[888] L. H. Gallardo and L. N. Vaserstein. The strict waring problem for polynomialrings. J. Number Theory, 128(12):2963–2972, 2008. [413, 414]
[889] E. Galois. Sur la theorie des nombres. Bulletin des Sciences mathematiquesXIII, pages 428–435, 1830. Reprinted in Ecrits et Memoires Matheematiquesd’Evariste Galois, pp. 112-128. [296, 299]
[890] R. A. Games and A. H. Chan. A fast algorithm for determining the complexity of abinary sequence with period 2n. IEEE Trans. Inform. Theory, 29(1):144–146,1983. [274, 281]
[891] M. J. Ganley. Central weak nucleus semifields. European J. Combin., 2(4):339–347,1981. [228, 229, 233, 234]
[892] S. Gao. Normal bases over finite fields. ProQuest LLC, Ann Arbor, MI, 1993. Thesis(Ph.D.)–University of Waterloo (Canada). [31, 32, 33, 34, 72, 79]
[893] S. Gao. Elements of provable high orders in finite fields. Proc. Amer. Math. Soc.,127(6):1615–1623, 1999. [69, 70]
[894] S. Gao. Absolute irreducibility of polynomials via Newton polytopes. J. Algebra,237(2):501–520, 2001. [307, 311]
[895] S. Gao. Factoring multivariate polynomials via partial differential equations. Math.Comp., 72(242):801–822, 2003. [304, 305, 311]
[896] S. Gao, J. Howell, and D. Panario. Irreducible polynomials of given forms. In Finitefields: theory, applications, and algorithms (Waterloo, ON, 1997), volume 225of Contemp. Math., pages 43–54. Amer. Math. Soc., Providence, RI, 1999. [58,59]
[897] S. Gao, E. Kaltofen, and A. Lauder. Deterministic distinct degree factorization forpolynomials over finite fields. J. Symbolic Comput., 38(6):1461–1470, 2004.[306, 311]
[898] S. Gao and A. G. B. Lauder. Hensel lifting and bivariate polynomial factorisationover finite fields. Math. Comp., 71(240):1663–1676, 2002. [304, 311]
[899] S. Gao and D. Panario. Tests and constructions of irreducible polynomials overfinite fields. In Foundations of Computational Mathematics, pages 346–361,1997. [295, 296, 299]
[900] M. Z. Garaev. Double exponential sums related to Diffie-Hellman distributions. Int.Math. Res. Not., (17):1005–1014, 2005. [148]
[901] M. Z. Garaev. An explicit sum-product estimate in Fp. Int. Math. Res. Not. IMRN,(11):Art. ID rnm035, 11, 2007. [129]
[902] M. Z. Garaev. A quantified version of Bourgain’s sum-product estimate in Fp forsubsets of incomparable sizes. Electron. J. Combin., 15(1):Research paper 58,
750 Handbook of Finite Fields
8, 2008. [129]
[903] M. Z. Garaev. The sum-product estimate for large subsets of prime fields. Proc.Amer. Math. Soc., 136(8):2735–2739, 2008. [129]
[904] M. Z. Garaev. Sums and products of sets and estimates for rational trigonometricsums in fields of prime order. Uspekhi Mat. Nauk, 65(4(394)):5–66, 2010. [129,131]
[905] M. Z. Garaev and V. C. Garcia. Waring type congruences involving factorialsmodulo a prime. Arch. Math. (Basel), 88(1):35–41, 2007. [170]
[906] M. Z. Garaev, F. Luca, I. E. Shparlinski, and A. Winterhof. On the lower boundof the linear complexity over Fp of Sidelnikov sequences. IEEE Trans. Inform.Theory, 52(7):3299–3304, 2006. [279, 281]
[907] A. Garcia, M. Q. Kawakita, and S. Miura. On certain subcovers of the Hermitiancurve. Comm. Algebra, 34(3):973–982, 2006. [166, 170]
[908] A. Garc´ia and H. Stichtenoth. A tower of Artin-Schreier extensions of functionfields attaining the Drinfeld-Vladut bound. Invent. Math., 121(1):211–222,1995. [368, 371, 372]
[909] A. Garcia and H. Stichtenoth. On the asymptotic behaviour of some towers offunction fields over finite fields. J. Number Theory, 61(2):248–273, 1996. [371,372]
[910] A. Garcia and H. Stichtenoth. On the Galois closure of towers. In Recent trendsin coding theory and its applications, volume 41 of AMS/IP Stud. Adv. Math.,pages 83–92. Amer. Math. Soc., Providence, RI, 2007. [372]
[911] A. Garcia, H. Stichtenoth, and H.-G. Ruck. On tame towers over finite fields. J.Reine Angew. Math., 557:53–80, 2003. [371, 372]
[912] A. Garcia, H. Stichtenoth, and C.-P. Xing. On subfields of the Hermitian functionfield. Compositio Math., 120(2):137–170, 2000. [366, 367]
[913] A. Garc´ia and J. F. Voloch. Fermat curves over finite fields. J. Number Theory,30(3):345–356, 1988. [169, 170]
[914] M. Garc´ia-Armas, S. R. Ghorpade, and S. Ram. Relatively prime polynomialsand nonsingular Hankel matrices over finite fields. J. Combin. Theory Ser. A,118(3):819–828, 2011. [423, 424]
[915] F. Gardeyn. A Galois criterion for good reduction of τ -sheaves. J. Number Theory,97(2):447–471, 2002. [454]
[916] T. Garefalakis. Irreducible polynomials with consecutive zero coefficients. FiniteFields Appl., 14(1):201–208, 2008. [46, 49]
[917] T. Garefalakis. Self-irreducible polynomials with prescribed coefficients. FiniteFields Appl., 17(?), 2011. [47, 49]
[918] G. Garg, T. Helleseth, and P. Kumar. Recent advances in low-correlation sequences.New Directions in Wireless Communications Research, 2009. [264]
[919] J. von zur Gathen. Factoring sparse multivariate polynomials. In 24th AnnualIEEE Symposium on Foundations of Computer Science, pages 172–179, LosAlamitos, CA, USA, 1983. IEEE Computer Society. [310, 311]
[920] J. von zur Gathen. Hensel and Newton methods in valuation rings. Math. Comp.,42(166):637–661, 1984. [304, 311]
[921] J. von zur Gathen. Irreducibility of multivariate polynomials. J. Comput. SystemSci., 31(2):225–264, 1985. Special issue: Twenty-fourth annual symposium onthe foundations of computer science (Tucson, Ariz., 1983). [305, 309, 311]
Miscellaneous applications 751
[922] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge Univer-sity Press, Cambridge, New York, Melbourne, 2003. [20, 54, 55, 295, 296, 299,304, 306, 311]
[923] J. von zur Gathen and E. Kaltofen. Factoring multivariate polynomials over finitefields. Math. Comp., 45:251–261, 1985. [305, 311]
[924] J. von zur Gather and E. Kaltofen. Factoring sparse multivariate polynomials. J.Comput. System Sci., 31:265–287, 1985. [309, 310, 311]
[925] P. Gaudry. An algorithm for solving the discrete log problem on hyperelliptic curves.In Advances in cryptology—EUROCRYPT 2000 (Bruges), volume 1807 of Lec-ture Notes in Comput. Sci., pages 19–34. Springer, Berlin, 2000. [360]
[926] P. Gaudry. Index calculus for abelian varieties of small dimension and the el-liptic curve discrete logarithm problem. Journal of Symbolic Computation,44(12):1690–1702, 2009. [668]
[927] P. Gaudry and N. Gurel. Counting points in medium characteristic using Kedlaya’salgorithm. Experiment. Math., 12(4):395–402, 2003. [406]
[928] P. Gaudry and R. Harley. Counting points on hyperelliptic curves over finite fields.In Algorithmic number theory (Leiden, 2000), volume 1838 of Lecture Notes inComput. Sci., pages 313–332. Springer, Berlin, 2000. [358, 360]
[929] P. Gaudry, F. Hess, and N. P. Smart. Constructive and destructive facets of Weildescent on elliptic curves. Journal of Cryptology, 15(1):19–46, 2002. [668]
[930] P. Gaudry and F. Morain. Fast algorithms for computing the eigenvalue in theSchoof–Elkies–Atkin algorithm. In J.-G. Dumas, editor, Proceedings of the2006 International Symposium on Symbolic and Algebraic Computations —ISSAC MMVI, pages 109–115, New York, 2006. ACM Press. [670]
[931] P. Gaudry, E. Thome, N. Theriault, and C. Diem. A double large prime variationfor small genus hyperelliptic index calculus. Math. Comp., 76(257):475–492(electronic), 2007. [360]
[932] G. Ge and L. Zhu. Authentication perpendicular arrays APA1(2, 5, v). J. Combin.Des., 4(5):365–375, 1996. [551]
[933] W. Geiselmann and D. Gollmann. Self-dual bases in Fqn . Des. Codes Cryptogr.,3(4):333–345, 1993. [73, 79]
[934] W. Geiselmann, W. Meier, and R. Steinwandt. An attack on the isomorphisms ofpolynomials problem with one secret. Int. Journal of Information Security,2(1):59–64, 2003. [651]
[935] E.-U. Gekeler. On the coefficients of Drinfel’d modular forms. Invent. Math.,93(3):667–700, 1988. [457]
[936] M. Genma, M. Mishima, and M. Jimbo. Cyclic resolvability of cyclic Steiner 2-designs. J. Combin. Des., 5(3):177–187, 1997. [507]
[937] S. R. Ghorpade, S. U. Hasan, and M. Kumari. Primitive polynomials, Singer cy-cles and word-oriented linear feedback shift registers. Des. Codes Cryptogr.,58(2):123–134, 2011. [416, 424]
[938] S. R. Ghorpade and G. Lachaud. Etale cohomology, Lefschetz theorems and numberof points of singular varieties over finite fields. Mosc. Math. J., 2(3):589–631,2002. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. [153,158]
[939] P. Gianni and B. Trager. Square-free algorithms in positive characteristic. Appl.Alg. Eng. Comm. Comp., 7(1):1–14, 1996. [303, 311]
[940] P. Giorgi, C.-P. Jeannerod, and G. Villard. On the complexity of polynomial matrix
752 Handbook of Finite Fields
computations. In Proceedings of the 2003 International Symposium on Sym-bolic and Algebraic Computation, pages 135–142 (electronic), New York, 2003.ACM. [436]
[941] D. Giry and J.-J. Quisquater. Bluekrypt cryptographic key length recommendation,2011. v26.0, April 18, http://www.keylength.com/. [666]
[942] M. Giulietti, J. W. P. Hirschfeld, G. Korchmaros, and F. Torres. Curves covered bythe Hermitian curve. Finite Fields Appl., 12(4):539–564, 2006. [166, 170]
[943] M. Giulietti and G. Korchmaros. A new family of maximal curves over a finite field.Math. Ann., 343(1):229–245, 2009. [366, 367]
[944] M. Giulietti, G. Korchmaros, and F. Torres. Quotient curves of the Suzuki curve.Acta Arith., 122(3):245–274, 2006. [366, 367]
[945] D. Glass and R. Pries. Hyperelliptic curves with prescribed p-torsion. ManuscriptaMath., 117(3):299–317, 2005. [401, 402]
[946] A. Glibichuk and M. Rudnev. On additive properties of product sets in an arbitraryfinite field. J. Anal. Math., 108:159–170, 2009. [134]
[947] A. A. Glibichuk. Sums of powers of subsets of an arbitrary finite field. Izv. RAN.Ser. Mat., (75):35–68, 2011. [140]
[948] A. A. Glibichuk and S. V. Konyagin. Additive properties of product sets in fieldsof prime order. In Additive combinatorics, volume 43 of CRM Proc. LectureNotes, pages 279–286. Amer. Math. Soc., Providence, RI, 2007. [129]
[949] D. Gligoroski, S. Markovski, and S. J. Knapskog. Multivariate quadratic trapdoorfunctions based on multivariate quadratic quasigroup. In Proceedings of TheAmerican Conference on Applied Mathematics, (MATH08), Cambridge, Mas-sachusetts, USA, March 2008. [658]
[950] D. Gluck. A note on permutation polynomials and finite geometries. Discrete Math.,80(1):97–100, 1990. [232, 234]
[951] C. Godsil and G. Royle. Algebraic graph theory, volume 207 of Graduate Texts inMathematics. Springer-Verlag, New York, 2001. [534, 545]
[952] J.-M. Goethals. Nonlinear codes defined by quadratic forms over GF(2). Informationand Control, 31(1):43–74, 1976. [601, 602]
[953] J. S. Golan. Semirings and their applications. Kluwer Academic Publishers, Dor-drecht, 1999. Updated and expanded version of it The theory of semirings,with applications to mathematics and theoretical computer science [LongmanSci. Tech., Harlow, 1992; MR1163371 (93b:16085)]. [16, 20]
[954] M. Golay. Notes on digital coding. Proc. IRE, 37:657, 1949. [583, 601, 602]
[955] Golay, M.J.E. Static multislit spectrometry and its application to the panoramicdisplay of infrared spectra. J. Opt. Soc. Amer., 41:468–472, 1951. [695, 701]
[956] R. Gold. Maximal recursive sequences with 3-valued recursive crosscorrelation func-tions. IEEE Trans. Inform. Theory, 14:154–156, 1968. [183, 185]
[957] D. M. Goldschmidt. Algebraic functions and projective curves, volume 215 of Grad-uate Texts in Mathematics. Springer-Verlag, New York, 2003. [317, 333]
[958] D. Gollmann. Design of algorithms in cryptography. (Algorithmenentwurf inder Kryptographie.). Aspekte Komplexer Systeme. 1. Mannheim: B.I. Wis-senschaftsverlag. viii, 158 p. 68.00; oS 531.00; sFr 68.00 /hc , 1994. [73, 75,79]
[959] F. Gologlu, G. McGuire, and R. Moloney. Binary Kloosterman sums using Stickel-berger’s theorem and the Gross-Koblitz formula. Acta Arith., 148(3):269–279,2011. [111, 118]
Miscellaneous applications 753
[960] S. Golomb and G. Gong. Signal Design for Good Correlation: For Wireless Com-munication, Cryptography, and Radar. Cambridge University Press, 2004. [19,20, 640, 647]
[961] S. W. Golomb. Shift register sequences. With portions co-authored by Lloyd R.Welch, Richard M. Goldstein, and Alfred W. Hales. Holden-Day Inc., SanFrancisco, Calif., 1967. [38]
[962] S. W. Golomb. Periodic binary sequences: solved and unsolved problems. In Se-quences, subsequences, and consequences, volume 4893 of Lecture Notes inComput. Sci., pages 1–8. Springer, Berlin, 2007. [66, 68]
[963] S. W. Golomb and G. Gong. Signal design for good correlation. Cambridge Univer-sity Press, Cambridge, 2005. For wireless communication, cryptography, andradar. [19, 20, 137, 515, 519]
[964] S. W. Golomb and G. Gong. Signal Design for Good Correlation: For WirelessCommunication, Cryptography, and Radar. Cambridge University Press, 2005.[19, 20, 264]
[965] S. W. Golomb and O. Moreno. On periodicity properties of Costas arrays and aconjecture on permutation polynomials. IEEE Trans. Inform. Theory, 42(6,part 2):2252–2253, 1996. [185]
[966] S. W. Golomb, M. G. Parker, A. Pott, and A. Winterhof, editors. Sequences andtheir applications—SETA 2008, volume 5203 of Lecture Notes in ComputerScience, Berlin, 2008. Springer. [20]
[967] D. Gomez, J. Gutierrez, and A. Ibeas. Attacking the Pollard generator. IEEETrans. Inform. Theory, 52(12):5518–5523, 2006. [283, 289]
[968] D. Gomez and A. P. Nicolas. An estimate on the number of stable quadratic poly-nomials. Finite Fields Appl., 16(6):401–405, 2010. [143, 287, 288, 289]
[969] D. Gomez and A. Winterhof. Waring’s problem in finite fields with Dickson polyno-mials. In Finite fields: theory and applications, volume 518 of Contemp. Math.,pages 185–192. Amer. Math. Soc., Providence, RI, 2010. [169]
[970] J. Gomez-Calderon. On the cardinality of value set of polynomials with coefficientsin a finite field. Proc. Japan Acad. Ser. A Math. Sci., 68(10):338–340, 1992.[192]
[971] J. Gomez-Calderon and D. J. Madden. Polynomials with small value set over finitefields. J. Number Theory, 28(2):167–188, 1988. [189, 192]
[972] G. Gong, T. Helleseth, H.-Y. Song, and K. Yang, editors. Sequences and theirapplications—SETA 2006, volume 4086 of Lecture Notes in Computer Science,Berlin, 2006. Springer. [20]
[973] G. Gong and A. M. Youssef. Cryptographic properties of the welch-gong trans-formation sequence generators. IEEE Transactions on Information Theory,48(11):2837–2846, 2002. [640, 647]
[974] P. Gopalan, V. Guruswami, and R. J. Lipton. Algorithms for modular countingof roots of multivariate polynomials. In LATIN 2006: Theoretical informatics,volume 3887 of Lecture Notes in Comput. Sci., pages 544–555. Springer, Berlin,2006. [403, 404, 406]
[975] V. D. Goppa. A new class of linear correcting codes. Problemy Peredaci Informacii,6(3):24–30, 1970. [584, 601, 602]
[976] V. D. Goppa. Rational representation of codes and (L, g)-codes. Problemy PeredaciInformacii, 7(3):41–49, 1971. [584, 601, 602]
[977] V. D. Goppa. Codes that are associated with divisors (Russian). Problemy Peredaci
754 Handbook of Finite Fields
Informacii, 13:33–39, 1977. [603, 612]
[978] V. D. Goppa. Codes on algebraic curves (Russian). Dokl. Akad. Nauk SSSR,259:1289–1290, 1981. [603, 612]
[979] V. D. Goppa. Algebraic-geometric codes (Russian). Izv. Akad. Nauk SSSR Ser.Mat., 46:762–781, 1982. [603, 612]
[980] B. Gordon, W. H. Mills, and L. R. Welch. Some new difference sets. Canad. J.Math., 14:614–625, 1962. [265, 515, 519]
[981] D. M. Gordon. The prime power conjecture is true for n < 2, 000, 000. Electron. J.Combin., 1:Research Paper 6, approx. 7 pp. (electronic), 1994. [514, 519]
[982] D. Gorenstein and N. Zierler. A class of error-correcting codes in pm symbols. J.Soc. Indust. Appl. Math., 9:207–214, 1961. [578, 591, 592, 601, 602]
[983] D. Goss. π-adic Eisenstein series for function fields. Compositio Math., 41(1):3–38,1980. [457]
[984] D. Goss. Basic structures of function field arithmetic, volume 35 of Ergebnisse derMathematik und ihrer Grenzgebiete (3) [Results in Mathematics and RelatedAreas (3)]. Springer-Verlag, Berlin, 1996. [19, 20, 448, 449, 451, 453, 454, 455]
[985] D. Goss. Applications of non-Archimedean integration to the L-series of τ -sheaves.J. Number Theory, 110(1):83–113, 2005. [454]
[986] D. Goss. ζ-phenomenology. In Noncommutative Geometry, Arithmetic, and RelatedTopics: Proceedings of the Twenty-First Meeting of the Japan-U.S. Mathemat-ics Institute. The Johns Hopkins University Press, Baltimore, MD, 2011. [455,456]
[987] K. Goto and R. van de Geijn. High-performance implementation of the level-3BLAS. ACM Trans. Math. Software, 35(1):Art. 4, 14, 2009. [427, 436]
[988] L. Goubin and N. T. Courtois. Cryptanalysis of the TTM cryptosystem. In Advancesin cryptology—ASIACRYPT 2000 (Kyoto), volume 1976 of Lecture Notes inComput. Sci., pages 44–57. Springer, Berlin, 2000. [652, 656, 657, 662]
[989] A. Gouget and J. Patarin. Probabilistic multivariate cryptography. In P. Q. Nguyen,editor, VIETCRYPT, volume 4341 of Lecture Notes in Computer Science,pages 1–18. Springer, 2006. [654]
[990] P. Goutet. An explicit factorisation of the zeta functions of Dwork hypersurfaces.Acta Arith., 144(3):241–261, 2010. [98, 118]
[991] P. Goutet. On the zeta function of a family of quintics. J. Number Theory,130(3):478–492, 2010. [98, 118]
[992] P. Goutet. Isotypic decomposition of the cohomology and factorization of the zetafunctions of dwork hypersurfaces. Finite Fields Appl., 17(2):113–137, 2011.[386, 393]
[993] W. T. Gowers. A new proof of Szemeredi’s theorem. Geom. Funct. Anal., 11(3):465–588, 2001. [130]
[994] B. Grammaticos, R. G. Halburd, A. Ramani, and C.-M. Viallet. How to detect theintegrability of discrete systems. J. Phys. A, 42(45):454002, 30, 2009. [282,289]
[995] L. Granboulan, A. Joux, and J. Stern. Inverting HFE is quasipolynomial. In Ad-vances in cryptology—CRYPTO 2006, volume 4117 of Lecture Notes in Com-put. Sci., pages 345–356. Springer, Berlin, 2006. [664]
[996] R. M. Gray. Toeplitz and circulant matrices: a review. 2005. [422, 424]
[997] D. R. Grayson and M. E. Stillman. Macaulay2, a software system for research
Miscellaneous applications 755
in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/,1992. [689, 692]
[998] M. Greig. Some balanced incomplete block design constructions. In Proceedingsof the Twenty-first Southeastern Conference on Combinatorics, Graph Theory,and Computing (Boca Raton, FL, 1990), volume 77, pages 121–134, 1990. [507]
[999] M. Greig. Some group divisible design constructions. J. Combin. Math. Combin.Comput., 27:33–52, 1998. [509]
[1000] F. Griffin, H. Niederreiter, and I. E. Shparlinski. On the distribution of nonlinearrecursive congruential pseudorandom numbers of higher orders. In Appliedalgebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999),volume 1719 of Lecture Notes in Comput. Sci., pages 87–93. Springer, Berlin,1999. [283, 285, 289]
[1001] F. Griffin and I. E. Shparlinski. On the linear complexity profile of the powergenerator. IEEE Trans. Inform. Theory, 46(6):2159–2162, 2000. [278, 281]
[1002] K. C. Gupta and S. Maitra. Multiples of primitive polynomials over GF(2). InProgress in cryptology—INDOCRYPT 2001 (Chennai), volume 2247 of LectureNotes in Comput. Sci., pages 62–72. Springer, Berlin, 2001. [523, 531]
[1003] S. Gurak. Gauss and Eisenstein sums of order twelve. Canad. Math. Bull.,46(3):344–355, 2003. [108, 118]
[1004] S. Gurak. Gauss sums for prime powers in p-adic fields. Acta Arith., 142(1):11–39,2010. [117, 118]
[1005] S. Gurak. Jacobi sums and irreducible polynomials with prescribed trace and re-stricted norm. In Finite fields: theory and applications, volume 518 of Contemp.Math., pages 193–208. Amer. Math. Soc., Providence, RI, 2010. [100, 118]
[1006] S. J. Gurak. Kloosterman sums for prime powers in p-adic fields. J. Theor. NombresBordeaux, 21(1):175–201, 2009. [117, 118]
[1007] R. Guralnick and D. Wan. Bounds for fixed point free elements in a transitive groupand applications to curves over finite fields. Israel J. Math., 101:255–287, 1997.[189, 192]
[1008] R. M. Guralnick. Rational maps and images of rational points of curves over fi-nite fields. In Proceedings of the All Ireland Algebra Days, 2001 (Belfast),number 50, pages 71–95, 2003. [189, 192]
[1009] R. M. Guralnick and P. Muller. Exceptional polynomials of affine type. J. Algebra,194(2):429–454, 1997. [194]
[1010] R. M. Guralnick, P. Muller, and J. Saxl. The rational function analogue of a questionof Schur and exceptionality of permutation representations. Mem. Amer. Math.Soc., 162(773):viii+79, 2003. [251, 253, 255]
[1011] R. M. Guralnick, P. Muller, and M. E. Zieve. Exceptional polynomials of affinetype, revisited. preprint. [194]
[1012] R. M. Guralnick, J. Rosenberg, and M. E. Zieve. A new family of exceptionalpolynomials in characteristic two. Ann. of Math. (2), 172(2):1361–1390, 2010.[194]
[1013] R. M. Guralnick, T. J. Tucker, and M. E. Zieve. Exceptional covers and bijectionson rational points. Int. Math. Res. Not. IMRN, (1):Art. ID rnm004, 20, 2007.[195]
[1014] R. M. Guralnick and M. E. Zieve. Polynomials with PSL(2) monodromy. Ann. ofMath. (2), 172(2):1315–1359, 2010. [194]
[1015] V. Guruswami and A. C. Patthak. Correlated algebraic-geometric codes: improved
756 Handbook of Finite Fields
list decoding over bounded alphabets. Math. Comp., 77:447–473, 2008. [605,612]
[1016] V. Guruswami and A. Rudra. Limits to list decoding Reed-Solomon codes. IEEETrans. Inform. Theory, 52(8):3642–3649, 2006. [599, 602]
[1017] V. Guruswami and M. Sudan. Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Trans. Inform. Theory, 45(6):1757–1767, 1999. [599,602]
[1018] F. G. Gustavson. Analysis of the Berlekamp-Massey linear feedback shift-registersynthesis algorithm. IBM J. Res. Develop., 20(3):204–212, 1976. [275, 281]
[1019] J. Gutierrez and D. Gomez-Perez. Iterations of multivariate polynomials and dis-crepancy of pseudorandom numbers. In Applied algebra, algebraic algorithmsand error-correcting codes (Melbourne, 2001), volume 2227 of Lecture Notes inComput. Sci., pages 192–199. Springer, Berlin, 2001. [283, 285, 289]
[1020] J. Gutierrez and A. Ibeas. Inferring sequences produced by a linear congruentialgenerator on elliptic curves missing high-order bits. Des. Codes Cryptogr.,45(2):199–212, 2007. [283, 289]
[1021] J. Gutierrez, I. E. Shparlinski, and A. Winterhof. On the linear and nonlinearcomplexity profile of nonlinear pseudorandom number-generators. IEEE Trans.Inform. Theory, 49(1):60–64, 2003. [277, 278, 281]
[1022] K. Gyarmati and A. Sarkozy. Equations in finite fields with restricted solution sets.I. Character sums. Acta Math. Hungar., 118(1-2):129–148, 2008. [149]
[1023] K. Gyarmati and A. Sarkozy. Equations in finite fields with restricted solution sets.II. (Algebraic equations). Acta Math. Hungar., 119(3):259–280, 2008. [149]
[1024] D. Hachenberger. On completely free elements in finite fields. Des. Codes Cryptogr.,4(2):129–143, 1994. [83]
[1025] D. Hachenberger. Explicit iterative constructions of normal bases and completelyfree elements in finite fields. Finite Fields Appl., 2(1):1–20, 1996. [83]
[1026] D. Hachenberger. Normal bases and completely free elements in prime power ex-tensions over finite fields. Finite Fields Appl., 2(1):21–34, 1996. [83]
[1027] D. Hachenberger. Finite fields. The Kluwer International Series in Engineeringand Computer Science, 390. Kluwer Academic Publishers, Boston, MA, 1997.Normal bases and completely free elements. [19, 20, 82, 83, 84, 85, 86, 87, 88,89, 90, 94, 95]
[1028] D. Hachenberger. A decomposition theory for cyclotomic modules under the com-plete point of view. J. Algebra, 237(2):470–486, 2001. [82, 85, 86, 87, 88]
[1029] D. Hachenberger. Primitive complete normal bases for regular extensions. Glasg.Math. J., 43(3):383–398, 2001. [64, 65, 82, 84, 88, 89, 92]
[1030] D. Hachenberger. Universal generators for primary closures of Galois fields. In Finitefields and applications (Augsburg, 1999), pages 208–223. Springer, Berlin, 2001.[94]
[1031] D. Hachenberger. Primitive complete normal bases: existence in certain 2-powerextensions and lower bounds. Discrete Math., 310(22):3246–3250, 2010. [64,65, 92]
[1032] D. Hachenberger, H. Niederreiter, and C. P. Xing. Function-field codes. Appl.Algebra Engrg. Comm. Comput., 19:201–211, 2008. [608, 612]
[1033] C. D. Haessig. L-functions of symmetric powers of cubic exponential sums. J. ReineAngew. Math., 631:1–57, 2009. [394, 402]
[1034] A. W. Hales and D. W. Newhart. Swan’s theorem for binary tetranomials. Finite
Miscellaneous applications 757
Fields Appl., 12(2):301–311, 2006. [37, 38]
[1035] C. Hall. l-functions of twisted Legendre curves. J. Number Theory, 119(1):128–147,2006. [410, 414]
[1036] K. H. Ham and G. L. Mullen. Distribution of irreducible polynomials of smalldegrees over finite fields. Math. Comp., 67(221):337–341, 1998. [44, 49]
[1037] N. Hamilton and R. Mathon. More maximal arcs in Desarguesian projective planesand their geometric structure. Adv. Geom., 3(3):251–261, 2003. [485, 486]
[1038] N. Hamilton and R. Mathon. On the spectrum of non-Denniston maximal arcs inPG(2, 2h). European J. Combin., 25(3):415–421, 2004. [485, 486]
[1039] R. W. Hamming. Error detecting and error correcting codes. Bell System Tech. J.,29:147–160, 1950. [583, 601, 602]
[1040] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole.The Z4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEETrans. Inform. Theory, 40(2):301–319, 1994. [17, 18, 19, 599, 601, 602]
[1041] W. Han. The distribution of the coefficients of primitive polynomials over finitefields. In Cryptography and computational number theory (Singapore, 1999),volume 20 of Progr. Comput. Sci. Appl. Logic, pages 43–57. Birkhauser, Basel,2001. [62, 65]
[1042] W. B. Han. The coefficients of primitive polynomials over finite fields. Math. Comp.,65(213):331–340, 1996. [62, 65]
[1043] D. Hankerson, A. Menezes, and S. Vanstone. Guide to elliptic curve cryptography.Springer Professional Computing. Springer-Verlag, New York, 2004. [19, 20,360]
[1044] J. P. Hansen and J. P. Pedersen. Automorphism groups of Ree type, Deligne-Lusztigcurves and function fields. J. Reine Angew. Math., 440:99–109, 1993. [365, 367]
[1045] S. H. Hansen. Error-correcting codes from higher-dimensional varieties. FiniteFields Appl., 7:530–552, 2001. [605, 612]
[1046] T. Hansen and G. L. Mullen. Primitive polynomials over finite fields. Math. Comp.,59(200):639–643, S47–S50, 1992. [44, 49, 57, 59, 61, 65, 66, 68]
[1047] B. Hanson, D. Panario, and D. Thomson. Swan-like results for binomials and trino-mials over finite fields of odd characteristic. To appear in Designs, Codes andCryptography, 2012. [38]
[1048] G. Hardy and L. J.E. Some problems of ’partitio numerorum’:iv. the singular seriesin waring’s problem and the value of the number g(k). Math. Z., 12(1):161–188,1922. [412, 414]
[1049] G. Hardy and J. Littlewood. Some problems of ‘Partitio numerorum’; III: On theexpression of a number as a sum of primes. Acta Math., 44(1):1–70, 1923. [411,414]
[1050] G. Hardy and E. Wright. An introduction to the theory of numbers. Oxford Uni-versity Press, Oxford, 2008. [409, 414]
[1051] R. Harley. Asymptotically optimal p-adic point-counting, Dec. 2002. Posting to theNumber Theory List, available at http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0212&L=NMBRTHRY&P=R1277. [670]
[1052] N. V. Harrach and C. Mengyan. Minimal blocking sets in PG(2, q) arising froma generalized construction of Megyesi. Innov. Incidence Geom., 6/7:211–226,2007/08. [471, 475]
[1053] D. Hart, A. Iosevich, and J. Solymosi. Sum-product estimates in finite fields viaKloosterman sums. Int. Math. Res. Not. IMRN, (5):Art. ID rnm007, 14, 2007.
758 Handbook of Finite Fields
[129]
[1054] R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. GraduateTexts in Mathematics, No. 52. [244, 247, 255, 332, 333, 384, 393]
[1055] D. Harvey. Kedlaya’s algorithm in larger characteristic. Int. Math. Res. Not. IMRN,(22):Art. ID rnm095, 29, 2007. [404, 406, 670]
[1056] M. A. Hasan and T. Helleseth, editors. Arithmetic of finite fields, volume 6087 ofLecture Notes in Computer Science, Berlin, 2010. Springer. [20]
[1057] S. Hasegawa and T. Kaneko. An attacking method for a public key cryptosystembased on the difficulty of solving a system of non-linear equations. In Proc.10th Symposium on Information Theory and Its applications, pages JA5–3,1987. [652]
[1058] K.-i. Hashimoto. Zeta functions of finite graphs and representations of p-adic groups.In Automorphic forms and geometry of arithmetic varieties, volume 15 of Adv.Stud. Pure Math., pages 211–280. Academic Press, Boston, MA, 1989. [545]
[1059] H. Hasse. Theorie der relativ-zyklischen algebraischen funktionenkrper, insbeson-dere bei endlichen konstantkrper. J. Reine Angew. Math., 172:37–54, 1934.[119]
[1060] P. Hawkes and G. G. Rose. Exploiting multiples of the connection polynomialin word-oriented stream ciphers. In Advances in cryptology—ASIACRYPT2000 (Kyoto), volume 1976 of Lecture Notes in Comput. Sci., pages 303–316.Springer, Berlin, 2000. [271, 281]
[1061] P. Hawkes and G. G. Rose. Rewriting variables: the complexity of fast algebraicattacks on stream ciphers. In Advances in cryptology—CRYPTO 2004, volume3152 of Lecture Notes in Comput. Sci., pages 390–406. Springer, Berlin, 2004.[202, 204]
[1062] D. R. Hayes. The distribution of irreducibles in GF[q, x]. Trans. Amer. Math. Soc.,117:101–127, 1965. [43, 49, 409, 414]
[1063] D. R. Hayes. The expression of a polynomial as a sum of three irreducibles. ActaArith., 11:461–488, 1966. [411, 414]
[1064] D. R. Hayes. Explicit class field theory for rational function fields. Trans. Amer.Math. Soc., 189:77–91, 1974. [450]
[1065] D. R. Hayes. Explicit class field theory in global function fields. In Studies in algebraand number theory, volume 6 of Adv. in Math. Suppl. Stud., pages 173–217.Academic Press, New York, 1979. [450]
[1066] D. R. Hayes. A brief introduction to Drinfeld modules. In The arithmetic of functionfields (Columbus, OH, 1991), volume 2 of Ohio State Univ. Math. Res. Inst.Publ., pages 1–32. de Gruyter, Berlin, 1992. [448]
[1067] D. R. Heath-Brown. Arithmetic applications of Kloosterman sums. Nieuw Arch.Wiskd. (5), 1(4):380–384, 2000. [111, 118]
[1068] D. R. Heath-Brown and S. Konyagin. New bounds for Gauss sums derived fromkth powers, and for Heilbronn’s exponential sum. Q. J. Math., 51(2):221–235,2000. [98, 118, 137, 141]
[1069] D. R. Heath-Brown and S. J. Patterson. The distribution of Kummer sums at primearguments. J. Reine Angew. Math., 310:111–130, 1979. [107, 118]
[1070] A. Hedayat, D. Raghavarao, and E. Seiden. Further contributions to the theory ofF -squares design. Ann. Statist., 3:712–716, 1975. [465, 467]
[1071] A. S. Hedayat, N. J. A. Sloane, and J. Stufken. Orthogonal arrays. Springer Seriesin Statistics. Springer-Verlag, New York, 1999. Theory and applications, With
Miscellaneous applications 759
a foreword by C. R. Rao. [520, 531]
[1072] A. Hefez. On the value sets of special polynomials over finite fields. Finite FieldsAppl., 2(4):337–347, 1996. [191, 192]
[1073] L. Heffter. Ueber Tripelsysteme. Math. Ann., 49(1):101–112, 1897. [504]
[1074] H. Heilbronn. Lecture notes on additive number theory mod p. California Instituteof Technology, 1964. [169, 170]
[1075] R. Heindl. New directions in multivariate public key cryptography. PhD. The-sis, Clemson University, 2009. http://etd.lib.clemson.edu/documents/
1247508584/. [658]
[1076] J. Heintz and M. Sieveking. Absolute primality of polynomials is decidable inrandom polynomial time in the number of variables. In Automata, languagesand programming (Akko, 1981), volume 115 of Lecture Notes in Comput. Sci.,pages 16–28. Springer-Verlag, 1981. [305, 311]
[1077] H. A. Helfgott. Growth and generation in SL2(Z/pZ). Ann. of Math. (2),167(2):601–623, 2008. [134]
[1078] H. A. Helfgott. Growth in SL3(Z/pZ). J. Eur. Math. Soc. (JEMS), 13(3):761–851,2011. [134]
[1079] H. A. Helfgott and M. Rudnev. An explicit incidence theorem in Fp. Mathematika,57(1):135–145, 2011. [133]
[1080] H. A. Helfgott and A. Seress. On the diameter of permutation groups.arXiv:1109.3550. [134]
[1081] T. Helleseth. Some results about the cross-correlation function between two maximallinear sequences. Discrete Math., 16(3):209–232, 1976. [212, 213, 266]
[1082] T. Helleseth. On the covering radius of cyclic linear codes and arithmetic codes.Discrete Appl. Math., 11(2):157–173, 1985. [140]
[1083] T. Helleseth, H. D. L. Hollmann, A. Kholosha, Z. Wang, and Q. Xiang. Proofs oftwo conjectures on ternary weakly regular bent functions. IEEE Trans. Inform.Theory, 55(11):5272–5283, 2009. [223, 224]
[1084] T. Helleseth and A. Kholosha. Monomial and quadratic bent functions over thefinite fields of odd characteristic. IEEE Trans. Inform. Theory, 52(5):2018–2032, 2006. [218, 222, 223, 224]
[1085] T. Helleseth and A. Kholosha. On the dual of monomial quadratic p-ary bent func-tions. In Sequences, subsequences, and consequences, volume 4893 of LectureNotes in Comput. Sci., pages 50–61. Springer, Berlin, 2007. [222, 224]
[1086] T. Helleseth and A. Kholosha. New binomial bent functions over the finite fields ofodd characteristic. IEEE Trans. Inform. Theory, 56(9):4646–4652, Sept. 2010.[223, 224]
[1087] T. Helleseth and P. V. Kumar. Sequences with low correlation. In Handbook ofcoding theory, Vol. I, II, pages 1765–1853. North-Holland, Amsterdam, 1998.[163, 264, 266]
[1088] T. Helleseth and P. V. Kumar. Pseudonoise sequences. The Mobile CommunicationsHandbook, 1999. [264]
[1089] T. Helleseth, P. V. Kumar, and H. Martinsen. A new family of ternary sequenceswith ideal two-level autocorrelation function. Des. Codes Cryptogr., 23(2):157–166, 2001. [516, 519]
[1090] T. Helleseth, P. V. Kumar, and K. Yang, editors. Sequences and their applications,Discrete Mathematics and Theoretical Computer Science (London), London,2002. Springer-Verlag London Ltd. [20]
760 Handbook of Finite Fields
[1091] T. Helleseth, C. Rong, and D. Sandberg. New families of almost perfect nonlinearpower mappings. IEEE Trans. Inform. Theory, 45(2):474–485, 1999. [183, 185,208, 213]
[1092] T. Helleseth, D. Sarwate, H.-Y. Song, and K. Yang, editors. Sequences and TheirApplications—SETA 2004, volume 3486 of Lecture Notes in Computer Science,Berlin, 2005. Springer. [20]
[1093] T. Helleseth and V. Zinoviev. New Kloosterman sums identities over F2m for all m.Finite Fields Appl., 9(2):187–193, 2003. [182]
[1094] M. Henderson. A note on the permutation behaviour of the Dickson polynomials ofthe second kind. Bull. Austral. Math. Soc., 56(3):499–505, 1997. [182, 185]
[1095] M. Henderson and R. Matthews. Permutation properties of Chebyshev polynomialsof the second kind over a finite field. Finite Fields Appl., 1(1):115–125, 1995.[182, 185]
[1096] M. Henderson and R. Matthews. Dickson polynomials of the second kind which arepermutation polynomials over a finite field. New Zealand J. Math., 27(2):227–244, 1998. [182, 185]
[1097] B. Hendrickson and E. Rothberg. Improving the run time and quality of nesteddissection ordering. SIAM J. Sci. Comput., 20(2):468–489 (electronic), 1998.[434, 436]
[1098] C. Hering. Eine nicht-desarguessche zweifach transitive affine Ebene der Ordnung27. Abh. Math. Sem. Univ. Hamburg, 34:203–208, 1969/1970. [481, 486]
[1099] J. R. Heringa, H. W. J. Blote, and A. Compagner. New primitive trinomials ofMersenne-exponent degrees for random-number generation. Internat. J. Mod-ern Phys. C, 3(3):561–564, 1992. [66, 68]
[1100] R. A. Hernandez Toledo. Linear finite dynamical systems. Comm. Algebra,33(9):2977–2989, 2005. [692]
[1101] F. Hernando and G. McGuire. Proof of a conjecture on the sequence of exceptionalnumbers, classifying cyclic codes and APN functions. Journal of Algebra, 2011.To appear. [212, 213]
[1102] M. Herrmann and G. Leander. A practical key recovery attack on Basic T CHo. InPublic key cryptography—PKC 2009, volume 5443 of Lecture Notes in Comput.Sci., pages 411–424. Springer, Berlin, 2009. [521, 531]
[1103] F. Hess. Pairing lattices. In S. D. Galbraith and K. Paterson, editors, Pairing-BasedCryptography — Pairing 2008, volume 5209 of Lecture Notes in ComputerScience, pages 18–38, Berlin, 2008. Springer-Verlag. [673]
[1104] F. Hess and I. E. Shparlinski. On the linear complexity and multidimensional dis-tribution of congruential generators over elliptic curves. Des. Codes Cryptogr.,35(1):111–117, 2005. [279, 281]
[1105] F. Hess, N. P. Smart, and F. Vercauteren. The eta pairing revisited. IEEE Trans-actions on Information Theory, 52(10):4595–4602, 2006. [672, 673]
[1106] A. E. Heydtmann. Sudan-decoding generalized geometric Goppa codes. FiniteFields Appl., 9:267–285, 2003. [608, 612]
[1107] K. Hicks, G. Mullen, J. Yucas, and R. Zavislak. A polynomial analogue of the 3n+1problem. Amer. Math. Monthly, 115(7):615–622, 2008. [414]
[1108] J. Hietarinta and C. Viallet. Searching for integrable lattice maps using factoriza-tion. J. Phys. A, 40(42):12629–12643, 2007. [282, 283, 289]
[1109] D. Hilbert. Ueber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligenCoefficienten. J. Reine Angew. Math., 110, 1892. [305, 311]
Miscellaneous applications 761
[1110] D. Hilbert. Beweis fur die darstellbarkeit der ganzen kahlen durch eine feste anzahlnter potenzen (waringsches problem)(german). Math. Ann., 67(3):281–300,1909. [412, 414]
[1111] F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, A. Veliz-Cuba, G. Blekherman,and R. Laubenbacher. Adam: Analysis of analysis of dynamic algebraic models.Available at http:/adam.vbi.vt.edu/, 2010. [685, 688, 690, 692]
[1112] F. Hinkelmann and A. S. Jarrah. Inferring biologically relevant models: Nestedcanalyzing functions. under review, 2010. [692]
[1113] F. Hinkelmann and R. Laubenbacher. Boolean models of bistable biological systems.Discrete Contin. Dyn. Syst. Ser. S, 4(6):1443–1456, 2011. [685, 692]
[1114] F. Hinkelmann, D. Murrugarra, A. Jarrah, and R. Laubenbacher. A mathematicalframework for agent based models of complex biological networks. Bulletinof Mathematical Biology, pages 1–20, 2010. 10.1007/s11538-010-9582-8. [685,689, 692]
[1115] Y. Hiramine. A conjecture on affine planes of prime order. J. Combin. Theory Ser.A, 52(1):44–50, 1989. [232, 234]
[1116] Y. Hiramine. On planar functions. J. Algebra, 133(1):103–110, 1990. [230, 234]
[1117] Y. Hiramine, M. Matsumoto, and T. Oyama. On some extension of 1-spread sets.Osaka J. Math., 24(1):123–137, 1987. [228, 229, 479, 486]
[1118] J. W. P. Hirschfeld. Finite projective spaces of three dimensions. Oxford Mathe-matical Monographs. The Clarendon Press Oxford University Press, New York,1985. Oxford Science Publications. [20, 492, 500, 501]
[1119] J. W. P. Hirschfeld. Projective geometries over finite fields. Oxford MathematicalMonographs. The Clarendon Press Oxford University Press, New York, secondedition, 1998. [20, 476, 482, 484, 486, 487, 488, 489, 490, 491, 492, 493, 494,495, 496, 497, 501]
[1120] J. W. P. Hirschfeld, G. Korchmaros, and F. Torres. Algebraic curves over a finitefield. Princeton Series in Applied Mathematics. Princeton University Press,Princeton, NJ, 2008. [19, 20, 317, 333, 366, 367, 496, 498, 499, 501]
[1121] J. W. P. Hirschfeld and L. Storme. The packing problem in statistics, coding theoryand finite projective spaces. J. Statist. Plann. Inference, 72(1-2):355–380, 1998.R. C. Bose Memorial Conference (Fort Collins, CO, 1995). [496, 498]
[1122] J. W. P. Hirschfeld and L. Storme. The packing problem in statistics, coding theoryand finite projective spaces: update 2001. In Finite geometries, volume 3 ofDev. Math., pages 201–246. Kluwer Acad. Publ., Dordrecht, 2001. [486, 496,498]
[1123] J. W. P. Hirschfeld, L. Storme, J. A. Thas, and J. F. Voloch. A characterization ofHermitian curves. J. Geom., 41(1-2):72–78, 1991. [166, 170]
[1124] J. W. P. Hirschfeld and J. A. Thas. General Galois geometries. Oxford MathematicalMonographs. The Clarendon Press Oxford University Press, New York, 1991.Oxford Science Publications. [20, 478, 486, 496, 498, 501]
[1129] J. Hoffstein, J. Pipher, and J. H. Silverman. An introduction to mathematical cryp-tography. Undergraduate Texts in Mathematics. Springer, New York, 2008.[19, 20, 634]
[1130] T. Høholdt. Personal communication. 2011. [583, 602]
[1131] T. Høholdt and H. E. Jensen. Determination of the merit factor of legendre se-quences. IEEE Trans. Inform. Theory, 34(1):161–164, 1988. [269]
[1132] T. Høholdt and H. E. Jensen. Determination of the merit factor of Legendre se-quences. IEEE Trans. Inf. Theory, 34(1):161–164, 1988. [695, 701]
[1133] T. Høholdt and R. Pellikaan. On the decoding of algebraic-geometric codes. IEEETrans. Inform. Theory, 41:1589–1614, 1995. [605, 612]
[1134] H. D. L. Hollmann and Q. Xiang. A proof of the Welch and Niho conjectures oncross-correlations of binary m-sequences. Finite Fields Appl., 7(2):253–286,2001. [213]
[1135] H. D. L. Hollmann and Q. Xiang. A class of permutation polynomials of F2m relatedto Dickson polynomials. Finite Fields Appl., 11(1):111–122, 2005. [183]
[1136] S. Hong. Newton polygons of L functions associated with exponential sums ofpolynomials of degree four over finite fields. Finite Fields Appl., 7(1):205–237,2001. Dedicated to Professor Chao Ko on the occasion of his 90th birthday.[399, 402]
[1137] S. Hong. Newton polygons for L-functions of exponential sums of polynomials ofdegree six over finite fields. J. Number Theory, 97(2):368–396, 2002. [399, 402]
[1138] C. Hooley. On Artin’s conjecture. J. Reine Angew. Math., 225:209–220, 1967. [40]
[1139] C. Hooley. On exponential sums and certain of their applications. In Number theorydays, 1980 (Exeter, 1980), volume 56 of London Math. Soc. Lecture Note Ser.,pages 92–122. Cambridge Univ. Press, Cambridge, 1982. [121, 127]
[1140] C. Hooley. On the number of points on a complete intersection over a finite field. J.Number Theory, 38(3):338–358, 1991. With an appendix by Nicholas M. Katz.[153, 158]
[1141] S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications.Bull. Amer. Math. Soc. (N.S.), 43(4):439–561 (electronic), 2006. [532, 538,539, 545]
[1142] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press,Cambridge, 1985. [545]
[1143] A. Hoshi. Explicit lifts of quintic Jacobi sums and period polynomials for Fq. Proc.Japan Acad. Ser. A Math. Sci., 82(7):87–92, 2006. [98, 106, 118]
[1144] X. Hou and G. L. Mullen. Number of irreducible polynomials and pairs of relativelyprime polynomials in several variables over finite fields. Finite Fields Appl.,15:304–331, 2009. [50, 51, 52, 53, 55]
[1145] X.-D. Hou. p-ary and q-ary versions of certain results about bent functions andresilient functions. Finite Fields Appl., 10(4):566–582, 2004. [216, 224]
[1146] X.-D. Hou. A note on the proof of a theorem of Katz. Finite Fields Appl., 11(2):316–319, 2005. [157, 158]
[1147] X.-d. Hou. Affinity of permutations of Fn2 . Discrete Appl. Math., 154(2):313–325,2006. [185, 208, 213]
[1148] X.-d. Hou. Two classes of permutation polynomials over finite fields. J. Combin.
Miscellaneous applications 763
Theory Ser. A, 118(2):448–454, 2011. [183, 185]
[1149] X.-d. Hou and T. Ly. Necessary conditions for reversed Dickson polynomials to bepermutational. Finite Fields Appl., 16(6):436–448, 2010. [183, 185]
[1150] X.-d. Hou, G. L. Mullen, J. A. Sellers, and J. L. Yucas. Reversed Dickson poly-nomials over finite fields. Finite Fields Appl., 15(6):748–773, 2009. [182, 183,185]
[1151] X.-D. Hou and C. Sze. On certain diagonal equations over finite fields. Finite FieldsAppl., 15(6):633–643, 2009. [165, 170]
[1152] E. Howe and K. Lauter. Improved upper bounds for the number of points on curvesover finite fields. Ann. Inst. Fourier (Grenoble), 53(6):1677–1737, 2003. [364,367]
[1153] E. Howe, K. Lauter, C. Ritzenthaler, and G. van der Geer. manYPoints - table ofcurves with many points. http://www.manypoints.org/. [364, 367]
[1154] C.-N. Hsu. The distribution of irreducible polynomials in Fq[t]. J. Number Theory,61(1):85–96, 1996. [46, 49]
[1155] H. Hubrechts. Point counting in families of hyperelliptic curves in characteristic 2.LMS J. Comput. Math., 10:207–234, 2007. [359, 360]
[1156] H. Hubrechts. Point counting in families of hyperelliptic curves. Found. Comput.Math., 8(1):137–169, 2008. [359, 360, 406]
[1157] S. Huczynska and S. D. Cohen. Primitive free cubics with specified norm and trace.Trans. Amer. Math. Soc., 355(8):3099–3116 (electronic), 2003. [58, 59, 62, 63,65]
[1158] W. C. Huffman and V. Pless. Fundamentals of error-correcting codes. CambridgeUniversity Press, Cambridge, 2003. [19, 20, 561, 563, 572, 574, 577, 578, 581,582, 602]
[1159] D. R. Hughes. On t-designs and groups. Amer. J. Math., 87:761–778, 1965. [510]
[1160] D. R. Hughes and F. C. Piper. Projective planes. Springer-Verlag, New York, 1973.Graduate Texts in Mathematics, Vol. 6. [16, 20, 225, 229, 476, 486, 491]
[1161] T. W. Hungerford. Algebra, volume 73 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, first edition, 1980. Reprint of the 1974 original. [50,55]
[1162] N. E. Hurt. Exponential sums and coding theory: a review. Acta Appl. Math.,46(1):49–91, 1997. [111, 118]
[1163] D. Husemoller. Elliptic curves, volume 111 of Graduate Texts in Mathematics.Springer-Verlag, New York, second edition, 2004. With appendices by OttoForster, Ruth Lawrence and Stefan Theisen. [19, 20, 334, 351]
[1164] T. Icart. How to hash into elliptic curves. In S. Halevi, editor, Advances in Cryp-tology — CRYPTO 2009, volume 5677 of Lecture Notes in Computer Science,pages 303–316, Berlin, 2009. Springer-Verlag. [678]
[1165] IEEE. Standard specifications for public-key cryptography. Technical Report IEEEStd 1361-2000. IEEE Inc., 3 Park Ave., NY 10016-5997, USA. [36, 37, 38]
[1166] IEEE. Standard specifications for public key cryptography. Standard P1363-2000,Institute of Electrical and Electronics Engineering, 2000. Draft D13 availableat http://grouper.ieee.org/groups/1363/P1363/draft.html. [667]
[1167] Y. Ihara. On discrete subgroups of the two by two projective linear group overp-adic fields. J. Math. Soc. Japan, 18:219–235, 1966. [545]
[1168] Y. Ihara. Some remarks on the number of rational points of algebraic curves over
764 Handbook of Finite Fields
finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3):721–724 (1982),1981. [365, 367, 368, 372]
[1169] L. Illusie. Ordinarite des intersections completes generales. In The GrothendieckFestschrift, Vol. II, volume 87 of Progr. Math., pages 376–405. BirkhauserBoston, Boston, MA, 1990. [398, 402]
[1170] L. Illusie. Crystalline cohomology. In Motives (Seattle, WA, 1991), volume 55 ofProc. Sympos. Pure Math., pages 43–70. Amer. Math. Soc., Providence, RI,1994. [394]
[1171] K. Imamura. On self-complementary bases of GF (qn) over GF(q). Trans. IECEJapan, E, 66(12):717–721, 1983. [73, 74, 79]
[1172] H. Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studiesin Mathematics. American Mathematical Society, Providence, RI, 1997. [114,118]
[1173] H. Iwaniec and E. Kowalski. Analytic number theory, volume 53 of American Math-ematical Society Colloquium Publications. American Mathematical Society,Providence, RI, 2004. [97, 111, 113, 116, 117, 118]
[1174] F. Jacob and J. Monod. Genetic regulatory mechanisms in the synthesis of proteins†.Journal of Molecular Biology, 3(3):318–356, June 1961. [685, 692]
[1175] C. G. J. Jacobi. Uber die kreistheilung und ihre anwendung auf die zahlentheorie.Gesammelte Werke, 6:254–274, 1846. [15, 20]
[1176] M. Jacobson, Jr., A. Menezes, and A. Stein. Hyperelliptic curves and cryptography.In High primes and misdemeanours: lectures in honour of the 60th birthdayof Hugh Cowie Williams, volume 41 of Fields Inst. Commun., pages 255–282.Amer. Math. Soc., Providence, RI, 2004. [356, 360]
[1177] M. Jacobson, Jr., R. Scheidler, and A. Stein. Cryptographic aspects of real hyper-elliptic curves. Tatra Mt. Math. Publ., 47:31–65, 2010. [357, 360]
[1178] M. J. Jacobson, Jr., R. Scheidler, and A. Stein. Fast arithmetic on hyperellipticcurves via continued fraction expansions. In Advances in coding theory andcryptography, volume 3 of Ser. Coding Theory Cryptol., pages 200–243. WorldSci. Publ., Hackensack, NJ, 2007. [356, 357, 360]
[1179] N. Jacobson. Basic algebra. I. W. H. Freeman and Company, New York, secondedition, 1985. [437, 438, 440, 443, 444, 445, 447]
[1180] R. Jain. Error characteristics of fiber distributed data interface (fddi). Communi-cations, IEEE Transactions on, 38(8):1244 –1252, aug 1990. [524, 528, 531]
[1181] K. Jambunathan. On choice of connection-polynomials for LFSR-based streamciphers. In Progress in cryptology—INDOCRYPT 2000 (Calcutta), volume1977 of Lecture Notes in Comput. Sci., pages 9–18. Springer, Berlin, 2000.[522, 523, 531]
[1182] N. S. James and R. Lidl. Permutation polynomials on matrices. Linear AlgebraAppl., 96:181–190, 1987. [182, 185]
[1183] G. J. Janusz. Separable algebras over commutative rings. Trans. Amer. Math. Soc.,122:461–479, 1966. [17]
[1184] H. Janwa, G. M. McGuire, and R. M. Wilson. Double-error-correcting cyclic codesand absolutely irreducible polynomials over GF(2). J. Algebra, 178(2):665–676,1995. [212, 213]
[1185] H. Janwa and R. M. Wilson. Hyperplane sections of Fermat varieties in P3 in char. 2and some applications to cyclic codes. In Applied algebra, algebraic algorithmsand error-correcting codes (San Juan, PR, 1993), volume 673 of Lecture Notes
[1186] A. S. Jarrah and R. Laubenbacher. On the algebraic geometry of polynomial dy-namical systems. In Emerging applications of algebraic geometry, volume 149of IMA Vol. Math. Appl., pages 109–123. Springer, New York, 2009. [282, 289]
[1187] A. S. Jarrah, R. Laubenbacher, B. Stigler, and M. Stillman. Reverse-engineering ofpolynomial dynamical systems. Advances in Applied Mathematics, 39(4):477 –489, 2007. [689, 692]
[1188] A. S. Jarrah, R. Laubenbacher, and A. Veliz-Cuba. The dynamics of conjunctiveand disjunctive Boolean network models. Bull. Math. Biol., 72(6):1425–1447,2010. [692]
[1189] A. S. Jarrah, B. Raposa, and R. Laubenbacher. Nested canalyzing, unate cascade,and polynomial functions. Phys. D, 233(2):167–174, 2007. [691, 692]
[1190] C.-P. Jeannerod and C. Mouilleron. Computing specified generators of structuredmatrix inverses. In W. Koepf, editor, Symbolic and Algebraic Computation,International Symposium, ISSAC 2010, Munich, Germany, July 25-28, 2010,Proceedings, pages 281–288. ACM, 2010. [434, 436]
[1191] J. Jedwab. What can be used instead of a Barker sequence? In Finite fields andapplications, volume 461 of Contemp. Math., pages 153–178. Amer. Math. Soc.,Providence, RI, 2008. [694, 701]
[1192] J. Jedwab, D. J. Katz, and K.-U. Schmidt. Littlewood polynomials with small L4
norm. preprint, 2011. [695, 701]
[1193] E. Jensen and M. R. Murty. Artin’s conjecture for polynomials over finite fields.In Number Theory, Trends in Mathematics, pages 167–181. Birkhauser, Basel,2000. [411, 414]
[1194] J. M. Jensen, H. E. Jensen, and T. Høholdt. The merit factor of binary sequencesrelated to difference sets. IEEE Trans. Inform. Theory, 37(3, part 1):617–626,1991. [695, 701]
[1195] V. Jha and N. L. Johnson. An analog of the Albert-Knuth theorem on the ordersof finite semifields, and a complete solution to Cofman’s subplane problem.Algebras Groups Geom., 6(1):1–35, 1989. [228, 229]
[1196] V. Jha and N. L. Johnson. Nests of reguli and flocks of quadratic cones. SimonStevin, 63(3-4):311–338, 1989. [480, 486]
[1197] X. Jiang, J. Ding, and L. Hu. Kipnis-Shamir attack on HFE revisited. In Informationsecurity and cryptology, volume 4990 of Lecture Notes in Comput. Sci., pages399–411. Springer, Berlin, 2008. [662]
[1198] N. L. Johnson. Projective planes of prime order p that admit collineation groups oforder p2. J. Geom., 30(1):49–68, 1987. [232, 234]
[1199] N. L. Johnson. Nest replaceable translation planes. J. Geom., 36(1-2):49–62, 1989.[480, 486]
[1200] N. L. Johnson, V. Jha, and M. Biliotti. Handbook of finite translation planes, volume289 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC,Boca Raton, FL, 2007. [478, 486]
[1201] N. L. Johnson and R. Pomareda. Andre planes and nests of reguli. Geom. Dedicata,31(3):245–260, 1989. [480, 486]
[1202] N. L. Johnson and R. Pomareda. Mixed nests. J. Geom., 56(1-2):59–86, 1996. [480,486]
[1203] S. C. Johnson. Sparse polynomial arithmetic. ACM SIGSAM Bull., 8(3):63–71,1974. [301, 311]
766 Handbook of Finite Fields
[1204] R. Jones. Iterated Galois towers, their associated martingales, and the p-adic Man-delbrot set. Compos. Math., 143(5):1108–1126, 2007. [282, 287, 289]
[1205] R. Jones. The density of prime divisors in the arithmetic dynamics of quadraticpolynomials. J. Lond. Math. Soc. (2), 78(2):523–544, 2008. [282, 287, 289]
[1206] R. Jones and N. Boston. Settled polynomials over finite fields. Preprint, 2009. [287,289]
[1207] R. Jones and N. Boston. Settled polynomials over finite fields. Proc. Amer. Math.Soc., 2011. [142]
[1208] H. F. Jordan and D. C. M. Wood. On the distribution of sums of successive bitsof shift-register sequences. IEEE Trans. Computers, C-22:400–408, 1973. [521,531]
[1209] J.-P. Jouanolou. Theoremes de Bertini et applications, volume 42 of Progress inMathematics. Birkhauser Boston, 1983. [305, 311]
[1210] A. Joux. A one round protocol for tripartite Diffie-Hellman. J. Cryptology,17(4):263–276, 2004. [631, 634]
[1211] A. Joux and V. Vitse. Cover and decomposition index calculus on elliptic curvesmade practical — Application to a seemingly secure curve over Fp6 . Preprint,http://eprint.iacr.org/2011/020.pdf, 2011. [668]
[1212] M. Joye, A. Miyaji, and A. Otsuka, editors. Pairing-Based Cryptography — Pair-ing 2010, volume 6487 of Lecture Notes in Computer Science, Berlin, 2010.Springer-Verlag. [670]
[1214] D. Jungnickel, T. Beth, and W. Geiselmann. A note on orthogonal circulant matricesover finite fields. Arch. Math. (Basel), 62(2):126–133, 1994. [420, 424]
[1215] D. Jungnickel and M. J. de Resmini. Another case of the prime power conjecturefor finite projective planes. Adv. Geom., 2(3):215–218, 2002. [485, 486]
[1216] D. Jungnickel, A. J. Menezes, and S. A. Vanstone. On the number of self-dual basesof GF(qm) over GF(q). Proc. Amer. Math. Soc., 109(1):23–29, 1990. [74, 79]
[1217] D. Jungnickel and H. Niederreiter, editors. Finite fields and applications, Berlin,2001. Springer-Verlag. [20]
[1218] D. Jungnickel and S. A. Vanstone. On primitive polynomials over finite fields. J.Algebra, 124(2):337–353, 1989. [62, 65]
[1219] J. Justesen. A class of constructive asymptotically good algebraic codes. IEEETrans. Information Theory, IT-18:652–656, 1972. [589, 601, 602]
[1220] J. Justesen and T. Høholdt. A course in error-correcting codes. EMS Textbooksin Mathematics. European Mathematical Society (EMS), Zurich, 2004. [561,602]
[1221] V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests meansproving circuit lower bounds. Comput complexity, 13(1-2):1–46, 2004. [309,311]
[1222] T. Kaida, S. Uehara, and K. Imamura. An algorithm for the k-error linear complex-ity of sequences over GF(pm) with period pn, p a prime. Inform. and Comput.,151(1-2):134–147, 1999. [274, 281]
[1223] T. Kailath, S. Y. Kung, and M. Morf. Displacement ranks of a matrix. Bull. Amer.Math. Soc. (N.S.), 1(5):769–773, 1979. [434, 436]
Miscellaneous applications 767
[1224] E. Kaltofen. A polynomial reduction from multivariate to bivariate integral polyno-mial factorization. In Proceedings of the 14th Symposium on Theory of Com-puting, pages 261–266. ACM Press, 1982. [305, 306, 311]
[1225] E. Kaltofen. A polynomial-time reduction from bivariate to univariate integralpolynomial factorization. In Proc. 23rd Annual Symp. Foundations of Comp.Sci., pages 57–64. IEEE, 1982. [306, 311]
[1226] E. Kaltofen. Effective Hilbert irreducibility. Information and Control, 66:123–137,1985. [305, 311]
[1227] E. Kaltofen. Fast parallel absolute irreducibility testing. J. Symbolic Comput.,1(1):57–67, 1985. [305, 311]
[1228] E. Kaltofen. Sparse Hensel lifting. In Proceedings of EUROCAL ’85, Vol. 2 (Linz,1985), volume 204 of Lecture Notes in Comput. Sci., pages 4–17. Springer-Verlag, 1985. [305, 310, 311]
[1229] E. Kaltofen. Uniform closure properties of p-computable functions. In Proc. 18thAnnual ACM Symp. Theory Comput., pages 330–337. ACM, 1986. Also pub-lished as part of [1231] and [1232]. [309, 311]
[1230] E. Kaltofen. Deterministic irreducibility testing of polynomials over large finitefields. J. Symbolic Comput., 4:77–82, 1987. [306, 311]
[1231] E. Kaltofen. Greatest common divisors of polynomials given by straight-line pro-grams. J. ACM, 35(1):231–264, 1988. [767]
[1232] E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Mi-cali, editor, Randomness and Computation, volume 5 of Advances in ComputingResearch, pages 375–412. JAI Press Inc., Greenwhich, Connecticut, 1989. [309,311, 767]
[1233] E. Kaltofen. Polynomial factorization 1982-1986. In D. V. Chudnovsky and R. D.Jenks, editors, Computers in Mathematics, volume 125 of Lecture Notes inPure and Applied Mathematics, pages 285–309. Marcel Dekker, New York, N.Y., 1990. [306, 311]
[1234] E. Kaltofen. Polynomial factorization 1987-1991. In I. Simon, editor, Proc. LATIN’92, volume 583 of Lect. Notes Comput. Sci., pages 294–313. Springer-Verlag,1992. [306, 311]
[1235] E. Kaltofen. Asymptotically fast solution of toeplitz-like singular linear systems.In Proceedings of the international symposium on Symbolic and algebraic com-putation, ISSAC ’94, pages 297–304, New York, NY, USA, 1994. ACM. [434,436]
[1236] E. Kaltofen. Analysis of Coppersmith’s block Wiedemann algorithm for the parallelsolution of sparse linear systems. Math. Comp., 64(210):777–806, 1995. [436]
[1237] E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput.System Sci., 50(2):274–295, 1995. [305, 311]
[1238] E. Kaltofen. Polynomial factorization: a success story. In ISSAC ’03: Proceedingsof the 2003 international symposium on Symbolic and algebraic computation,pages 3–4. ACM Press, 2003. [306, 311]
[1239] E. Kaltofen and P. Koiran. On the complexity of factoring bivariate supersparse (la-cunary) polynomials. In ISSAC ’05: Proceedings of the 2005 International Sym-posium on Symbolic and Algebraic Computation, pages 208–215, 2005. [309,311]
[1240] E. Kaltofen and P. Koiran. Finding small degree factors of multivariate supersparse(lacunary) polynomials over algebraic number fields. In ISSAC ’06: Proceedings
768 Handbook of Finite Fields
of the 2006 International Symposium on Symbolic and Algebraic Computation,pages 162–168, 2006. [308, 311]
[1241] E. Kaltofen and W. Lee. Early termination in sparse interpolation algorithms. J.Symbolic Comput., 36(3–4):365–400, 2003. [311]
[1242] E. Kaltofen and V. Pan. Parallel solution of Toeplitz and Toeplitz-like linear systemsover fields of small positive characteristic. In First International Symposiumon Parallel Symbolic Computation—PASCO ’94 (Hagenberg/Linz, 1994), vol-ume 5 of Lecture Notes Ser. Comput., pages 225–233. World Sci. Publ., RiverEdge, NJ, 1994. [423, 424]
[1243] E. Kaltofen and B. D. Saunders. On Wiedemann’s method of solving sparse linearsystems. In Applied algebra, algebraic algorithms and error-correcting codes(New Orleans, LA, 1991), volume 539 of Lecture Notes in Comput. Sci., pages29–38. Springer, Berlin, 1991. [432, 436]
[1244] E. Kaltofen and V. Shoup. Subquadratic-time factoring of polynomials over finitefields. Mathematics of Computation, 67(223):1179–1198, July 1998. [295, 299]
[1245] E. Kaltofen and B. Trager. Computing with polynomials given by black boxesfor their evaluations: Greatest common divisors, factorization, separation ofnumerators and denominators. In Proc. 29th Annual Symp. Foundations ofComp. Sci., pages 296–305. IEEE, 1988. [310, 311]
[1246] E. Kaltofen and B. Trager. Computing with polynomials given by black boxesfor their evaluations: Greatest common divisors, factorization, separation ofnumerators and denominators. J. Symbolic Comput., 9(3):301–320, 1990. [310,311]
[1247] E. Kaltofen and G. Villard. On the complexity of computing determinants. Comput.Complexity, 13(3-4):91–130, 2004. [309, 311, 436]
[1248] N. Kamiya. On multisequence shift register synthesis and generalized-minimum-distance decoding of Reed-Solomon codes. Finite Fields Appl., 1(4):440–457,1995. [275, 281]
[1249] J.-G. Kammerer, R. Lercier, and G. Renault. Encoding points on hyperellipticcurves over finite fields in deterministic polynomial time. In M. Joye, A. Miyaji,and A. Otsuka, editors, Pairing-Based Cryptography — Pairing 2010, vol-ume 6487 of Lecture Notes in Computer Science, pages 278–297, Berlin, 2010.Springer-Verlag. [679]
[1250] W. M. Kantor. Two families of flag-transitive affine planes. Geom. Dedicata,41(2):191–200, 1992. [480, 481, 486]
[1251] W. M. Kantor. 2-transitive and flag-transitive designs. In Coding theory, designtheory, group theory (Burlington, VT, 1990), Wiley-Intersci. Publ., pages 13–30. Wiley, New York, 1993. [481, 486]
[1252] W. M. Kantor. Note on GMW designs. European J. Combin., 22(1):63–69, 2001.[515, 519]
[1253] W. M. Kantor. Commutative semifields and symplectic spreads. J. Algebra,270(1):96–114, 2003. [227, 229]
[1254] W. M. Kantor. Finite semifields. In Finite geometries, groups, and computation,pages 103–114. Walter de Gruyter GmbH & Co. KG, Berlin, 2006. [227, 229]
[1255] W. M. Kantor. HMO-planes. Adv. Geom., 9(1):31–43, 2009. [228, 229]
[1256] W. M. Kantor and R. A. Liebler. Semifields arising from irreducible semilineartransformations. J. Aust. Math. Soc., 85(3):333–339, 2008. [228, 229]
[1257] W. M. Kantor and C. Suetake. A note on some flag-transitive affine planes. J.
Miscellaneous applications 769
Combin. Theory Ser. A, 65(2):307–310, 1994. [481, 486]
[1258] W. M. Kantor and M. E. Williams. Symplectic semifield planes and Z4-linear codes.Trans. Amer. Math. Soc., 356(3):895–938, 2004. [227, 229]
[1259] G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioningirregular graphs. SIAM J. Sci. Comput., 20(1):359–392 (electronic), 1998. [434,436]
[1260] M. Kasahara and R. Sakai. A construction of public-key cryptosystem based onsingular simultaneous equations. In 2004, Jan. 27–30 2004. 6 pages. [655]
[1261] M. Kasahara and R. Sakai. A construction of public key cryptosystem for re-alizing ciphtertext of size 100 bit and digital signature scheme. IEICETrans. Fundamentals, E87-A(1):102–109, Jan. 2004. Electronic version: http://search.ieice.org/2004/files/e000a01.htm\#e87-a,1,102. [655]
[1262] T. Kasami. Weight distributions of Bose-Chaudhuri-Hocquenghem codes. In Combi-natorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina,Chapel Hill, N.C., 1967), pages 335–357. Univ. North Carolina Press, ChapelHill, N.C., 1969. [210, 213]
[1263] T. Kasami. The weight enumerators for several classes of subcodes of the 2nd orderbinary Reed-Muller codes. Information and Control, 18:369–394, 1971. [210,211, 213]
[1264] T. Kasami, S. Lin, and W. W. Peterson. Generalized Reed-Muller codes. Electron.Commun. Japan, 51(3):96–104, 1968. [586, 588, 602]
[1265] T. Kasami, S. Lin, and W. W. Peterson. Polynomial codes. IEEE Trans. Informa-tion Theory, IT-14:807–814, 1968. [588, 602]
[1266] T. Kasimi. The weight enumerators for several classes of subcodes of the secondorder binary reed-muller codes. Inform. and Control, 18:369–394, 1971. [183,185]
[1267] J. Katz and Y. Lindell. Introduction to modern cryptography. Chapman & Hall/CRCCryptography and Network Security. Chapman & Hall/CRC, Boca Raton, FL,2008. [19, 20, 634]
[1268] N. Katz and R. Livne. Sommes de Kloosterman et courbes elliptiques universelles encaracteristiques 2 et 3. C. R. Acad. Sci. Paris Ser. I Math., 309(11):723–726,1989. [223, 224]
[1269] N. H. Katz and C.-Y. Shen. Garaev’s inequality in finite fields not of prime order.Online J. Anal. Comb., (3):Art. 3, 6, 2008. [130]
[1270] N. M. Katz. On a theorem of Ax. Amer. J. Math., 93:485–499, 1971. [157, 158]
[1271] N. M. Katz. Slope filtration of F -crystals. In Journees de Geometrie Algebrique deRennes (Rennes, 1978), Vol. I, volume 63 of Asterisque, pages 113–163. Soc.Math. France, Paris, 1979. [398, 402]
[1272] N. M. Katz. Sommes exponentielles, volume 79 of Asterisque. Societe Mathematiquede France, Paris, 1980. Course taught at the University of Paris, Orsay, Fall1979, With a preface by Luc Illusie, Notes written by Gerard Laumon, Withan English summary. [113, 118]
[1273] N. M. Katz. Sommes exponentielles, volume 79 of Asterisque. Societe Mathematiquede France, Paris, 1980. Course taught at the University of Paris, Orsay, Fall1979, With a preface by Luc Illusie, Notes written by Gerard Laumon, Withan English summary. [122, 126, 127]
[1274] N. M. Katz. Gauss sums, Kloosterman sums, and monodromy groups, volume 116of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ,
770 Handbook of Finite Fields
1988. [19, 20]
[1275] N. M. Katz. Gauss sums, Kloosterman sums, and monodromy groups, volume 116of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ,1988. [97, 112, 113, 115, 118]
[1276] N. M. Katz. An estimate for character sums. J. Amer. Math. Soc., 2(2):197–200,1989. [126, 127, 147, 148]
[1277] N. M. Katz. Affine cohomological transforms, perversity, and monodromy. J. Amer.Math. Soc., 6(1):149–222, 1993. [125, 127]
[1278] N. M. Katz. Estimates for “singular” exponential sums. Internat. Math. Res.Notices, (16):875–899, 1999. [122, 127, 155, 158, 285, 289]
[1279] N. M. Katz. Frobenius-Schur indicator and the ubiquity of Brock-Granvillequadratic excess. Finite Fields Appl., 7(1):45–69, 2001. Dedicated to Pro-fessor Chao Ko on the occasion of his 90th birthday. [156, 158]
[1280] N. M. Katz. Sums of Betti numbers in arbitrary characteristic. Finite Fields Appl.,7(1):29–44, 2001. Dedicated to Professor Chao Ko on the occasion of his 90thbirthday. [388, 391, 393]
[1281] N. M. Katz. Estimates for nonsingular multiplicative character sums. Int. Math.Res. Not., (7):333–349, 2002. [123, 124, 127, 156, 158]
[1282] N. M. Katz. Moments, monodromy, and perversity: a Diophantine perspective, vol-ume 159 of Annals of Mathematics Studies. Princeton University Press, Prince-ton, NJ, 2005. [151, 158]
[1283] N. M. Katz. Estimates for nonsingular mixed character sums. Int. Math. Res. Not.IMRN, (19):Art. ID rnm069, 19, 2007. [125, 127]
[1284] N. M. Katz. Another look at the Dwork family. In Algebra, arithmetic, and geometry:in honor of Yu. I. Manin. Vol. II, volume 270 of Progr. Math., pages 89–126.Birkhauser Boston Inc., Boston, MA, 2009. [386, 393]
[1285] N. M. Katz. Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms. Annals of Mathematics Studies. Princeton UniversityPress, Princeton, NJ, 2012. [101, 114, 118]
[1286] N. M. Katz and G. Laumon. Transformation de Fourier et majoration de sommesexponentielles. Inst. Hautes Etudes Sci. Publ. Math., (62):361–418, 1985. [124,127]
[1287] N. M. Katz and Z. Zheng. On the uniform distribution of Gauss sums and Jacobisums. In Analytic number theory, Vol. 2 (Allerton Park, IL, 1995), volume139 of Progr. Math., pages 537–558. Birkhauser Boston, Boston, MA, 1996.[97, 101, 118]
[1288] S. Kauffman, C. Peterson, B. Samuelsson, and C. Troein. Genetic networks with can-alyzing boolean rules are always stable. Proceedings of the National Academyof Sciences of the United States of America, 101(49):17102–17107, 2004. [690]
[1289] S. A. Kauffman. Metabolic stability and epigenesis in randomly constructed geneticnets. Journal of Theoretical Biology, 22(3):437 – 467, 1969. [687, 692]
[1290] N. Kayal. Recognizing permutation functions in polynomial time. ECCC, TR05-008,2005. [173, 185, 311]
[1291] W. F. Ke and H. Kiechle. On the solutions of the equation xm + ym − zm = 1 in afinite field. Proc. Amer. Math. Soc., 123(5):1331–1339, 1995. [166, 170]
[1292] K. Kedlaya and C. Umans. Fast modular composition in any characteristic. InFoundations of Computer Science, 2008. FOCS ’08. IEEE 49th Annual IEEESymposium on, pages 146–155, 2008. [295, 297, 299]
Miscellaneous applications 771
[1293] K. S. Kedlaya. Counting points on hyperelliptic curves using Monsky-Washnitzercohomology. J. Ramanujan Math. Soc., 16(4):323–338, 2001. [358, 360, 406]
[1294] K. S. Kedlaya. Errata for: “Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology” [J. Ramanujan Math. Soc. 16 (2001), no. 4, 323–338;mr1877805]. J. Ramanujan Math. Soc., 18(4):417–418, 2003. Dedicated toProfessor K. S. Padmanabhan. [358, 360]
[1295] K. S. Kedlaya. Computing zeta functions via p-adic cohomology. In Algorithmicnumber theory, volume 3076 of Lecture Notes in Comput. Sci., pages 1–17.Springer, Berlin, 2004. [406]
[1296] D. Kelmer. Distribution of twisted Kloosterman sums modulo prime powers. Int.J. Number Theory, 6(2):271–280, 2010. [113, 118]
[1297] O. Kempthorne. A simple approach to confounding and fractional replication infactorial experiments. Biometrika, 34:255–272, 1947. [520, 531]
[1298] A. M. Kerdock. A class of low-rate nonlinear binary codes. Information and Control,20:182–187; ibid. 21 (1972), 395, 1972. [601, 602]
[1299] K. Khoo, G. Gong, and D. R. Stinson. New family of gold-like sequences. IEEEIntern. Symp. Inform. Theory, 2:181, 2002. [163]
[1300] D. S. Kim. Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums. Ann. Mat. Pura Appl. (4), 190(1):61–76, 2011.[111, 118]
[1301] J. H. Kim. Codes associated with Sp(4, q) and even-power moments of Kloostermansums. Bull. Aust. Math. Soc., 79(3):427–435, 2009. [114, 118]
[1302] R. Kim and W. Koepf. Parity of the number of irreducible factors for compositepolynomials. Finite Fields Appl., 16(3):137–143, 2010. [37, 38]
[1303] S.-H. Kim and J.-S. No. New families of binary sequences with low correlation.IEEE Trans. Inform. Theory, 49(11):3059–3065, 2003. [163]
[1304] A. Kipnis, J. Patarin, and L. Goubin. Unbalanced oil and vinegar signature schemes.In Advances in cryptology—EUROCRYPT ’99 (Prague), volume 1592 of Lec-ture Notes in Comput. Sci., pages 206–222. Springer, Berlin, 1999. [654, 663]
[1305] A. Kipnis and A. Shamir. Cryptanalysis of the oil and vinegar signature scheme.In Advances in cryptology—CRYPTO ’98 (Santa Barbara, CA, 1998), volume1462 of Lecture Notes in Comput. Sci., pages 257–266. Springer, Berlin, 1998.[663]
[1306] A. Kipnis and A. Shamir. Cryptanalysis of the HFE public key cryptosystem by re-linearization. In Advances in cryptology—CRYPTO ’99 (Santa Barbara, CA),volume 1666 of Lecture Notes in Comput. Sci., pages 19–30. Springer, Berlin,1999. [309, 311, 662]
[1307] T. Kiran and B. S. Rajan. Optimal rate-diversity tradeoff STBCs from codes overarbitrary finite fields. In IEEE Int. Conf. Commun., pages 453–457, May 2005.[700, 701]
[1308] T. P. Kirkman. On a problem in combinations. Cambridge and Dublin Math. J.,2:191–204, 1847. [503]
[1309] A. Klapper. Cross-correlations of geometric sequences in characteristic two. Des.Codes Cryptogr., 3(4):347–377, 1993. [162, 163]
[1310] A. Klapper. Cross-correlations of quadratic form sequences in odd characteristic.Des. Codes Cryptogr., 11(3):289–305, 1997. [162, 163]
[1311] A. Klapper, A. H. Chan, and M. Goresky. Cross-correlations of linearly and quadrat-ically related geometric sequences and GMW sequences. Discrete Appl. Math.,
772 Handbook of Finite Fields
46(1):1–20, 1993. [162, 163]
[1312] S. L. Kleiman. Bertini and his two fundamental theorems. Rend. Circ. Mat. Palermo(2) Suppl., 55:9–37, 1998. Studies in the history of modern mathematics, III.[305, 311]
[1313] E. Kleinfeld. Techniques for enumerating Veblen-Wedderburn systems. J. Assoc.Comput. Mach., 7:330–337, 1960. [227, 229]
[1314] R. Kloosterman. The zeta function of monomial deformations of Fermat hypersur-faces. Algebra Number Theory, 1(4):421–450, 2007. [394, 402]
[1315] A. A. Klyachko. Monodromy groups of polynomial mappings. In Studies in NumberTheory, volume 6, pages 82–91. 1975. [193, 194]
[1316] A. W. Knapp. Elliptic curves, volume 40 of Mathematical Notes. Princeton Univer-sity Press, Princeton, NJ, 1992. [19, 20, 334, 351]
[1317] N. Knarr and M. Stroppel. Polarities and unitals in the Coulter-Matthews planes.Des. Codes Cryptogr., 55(1):9–18, 2010. [231, 234]
[1318] D. E. Knuth. Finite semifields and projective planes. J. Algebra, 2:182–217, 1965.[225, 226, 228, 229]
[1319] N. Koblitz. p-adic variation of the zeta-function over families of varieties definedover finite fields. Compositio Math., 31(2):119–218, 1975. [400]
[1320] N. Koblitz. p-adic numbers, p-adic analysis, and zeta-functions. Springer-Verlag,New York, 1977. Graduate Texts in Mathematics, Vol. 58. [394, 402]
[1322] N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation,48(177):203–209, 1987. [666]
[1323] N. Koblitz. Hyperelliptic cryptosystems. J. Cryptology, 1(3):139–150, 1989. [630,634]
[1324] N. Koblitz. Introduction to elliptic curves and modular forms, volume 97 of GraduateTexts in Mathematics. Springer-Verlag, New York, second edition, 1993. [19,20, 334, 351]
[1325] N. Koblitz. Algebraic aspects of cryptography, volume 3 of Algorithms and Compu-tation in Mathematics. Springer-Verlag, Berlin, 1998. With an appendix byAlfred J. Menezes, Yi-Hong Wu and Robert J. Zuccherato. [19, 20]
[1326] W. Koepf and R. Kim. The parity of the number of irreducible factors for somepentanomials. Finite Fields Appl., 15(5):585–603, 2009. [37, 38]
[1327] J. F. Koksma. Some theorems on Diophantine inequalities. Scriptum no. 5. Math.Centrum Amsterdam, 1950. [285, 289]
[1328] K. Kononen. More exact solutions to Waring’s problem for finite fields. Acta Arith.,145(2):209–212, 2010. [169, 170]
[1329] K. Kononen, M. Moisio, M. Rinta-Aho, and K. Vaananen. Irreducible polynomialswith prescribed trace and restricted norm. JP J. Algebra Number Theory Appl.,11(2):223–248, 2008. [26, 48, 49, 100, 118]
[1330] K. Kononen, M. Rinta-Aho, and K. Vaananen. On the degree of a Kloostermansum as an algebraic integer. 2011. submitted. [111, 118]
[1331] K. P. Kononen, M. J. Rinta-aho, and K. O. Vaananen. On integer values of Kloost-erman sums. IEEE Trans. Inform. Theory, 56(8):4011–4013, 2010. [111, 112,118]
[1332] K. P. Kononen, M. J. Rinta-aho, and K. O. Vaananen. On integer values of Kloost-
Miscellaneous applications 773
erman sums. IEEE Trans. Inform. Theory, 56(8):4011–4013, Aug. 2010. [223,224]
[1333] S. Konyagin, T. Lange, and I. Shparlinski. Linear complexity of the discrete loga-rithm. Des. Codes Cryptogr., 28(2):135–146, 2003. [279, 281]
[1334] S. Konyagin and F. Pappalardi. Enumerating permutation polynomials over finitefields by degree. Finite Fields Appl., 8(4):548–553, 2002. [175, 185]
[1335] S. Konyagin and F. Pappalardi. Enumerating permutation polynomials over finitefields by degree. II. Finite Fields Appl., 12(1):26–37, 2006. [175, 185]
[1336] S. V. Konyagin. Estimates for Gaussian sums and Waring’s problem modulo aprime. Trudy Mat. Inst. Steklov., 198:111–124, 1992. [140, 141, 169, 170]
[1337] S. V. Konyagin. Estimates for trigonometric sums over subgroups and for Gausssums. In IV International Conference “Modern Problems of Number The-ory and its Applications”: Current Problems, Part III (Russian) (Tula, 2001),pages 86–114. Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow,2002. [98, 118]
[1338] P. Koopman. 32-bit cyclic redundancy codes for internet applications. In DependableSystems and Networks, 2002. DSN 2002. Proceedings. International Conferenceon, pages 459 – 468, 2002. [524, 528, 531]
[1339] P. Koopman and T. Chakravarty. Cyclic redundancy code (crc) polynomial selectionfor embedded networks. In Dependable Systems and Networks, 2004. [524, 531]
[1340] G. Korchmaros and T. SzHonyi. Fermat curves over finite fields and cyclic subsetsin high-dimensional projective spaces. Finite Fields Appl., 5(2):206–217, 1999.[166, 170]
[1341] P. Kosick. Commutative semifields of odd order and planar Dembowski-Ostrompolynomials. PhD thesis, Department of Mathematical Sciences, University ofDelaware, USA, 2010. [233, 234]
[1342] R. Kotter and F. R. Kschischang. Coding for errors and erasures in random networkcoding. IEEE Trans. Inform. Theory, 54(8):3579–3591, 2008. [701]
[1343] A. G. Kouchnirenko. Polyedres de Newton et nombres de Milnor. Invent. Math.,32(1):1–31, 1976. [397, 402]
[1344] R. G. Kraemer. Proof of a conjecture on Hadamard 2-groups. J. Combin. TheorySer. A, 63(1):1–10, 1993. [517, 519]
[1345] R. A. Kristiansen and M. G. Parker. Binary sequences with merit factor > 6.3.IEEE Trans. Inform. Theory, 50(12):3385–3389, 2004. [269]
[1346] M. Krivelevich and B. Sudakov. Pseudo-random graphs. In More sets, graphsand numbers, volume 15 of Bolyai Soc. Math. Stud., pages 199–262. Springer,Berlin, 2006. [534, 545]
[1347] W. Krull. Algebraische Theorie der Ringe. II. Math. Ann., 91(1-2):1–46, 1924. [17]
[1348] D. S. Kubert and S. Lichtenbaum. Jacobi-sum Hecke characters and Gauss-sumidentities. Compositio Math., 48(1):55–87, 1983. [103, 118]
[1349] R. Kubota. Waring’s problem for Fq[x]. Dissertationes Math. (Rozprawy Mat.),117:60pp, 1974. [413, 414]
[1350] T. Kumada, H. Leeb, Y. Kurita, and M. Matsumoto. New primitive t-nomials(t = 3, 5) over GF(2) whose degree is a Mersenne exponent. Math. Comp.,69(230):811–814, 2000. [66, 67, 68]
[1351] P. V. Kumar, R. A. Scholtz, and L. R. Welch. Generalized bent functions and theirproperties. J. Combin. Theory Ser. A, 40(1):90–107, 1985. [215, 216, 224]
774 Handbook of Finite Fields
[1352] V. A. Kurbatov and N. G. Starkov. The analytic representation of permutations.Sverdlovsk. Gos. Ped. Inst. Ucen. Zap., 31:151–158, 1965. [173, 185]
[1353] M. K. Kuregian. Recurrent methods of constructing irreducible polynomials overgf(2s)(Russian). J. Inform. Process. Cybernet EIK, 27(7):357–372, 1991. [31,32, 33, 34]
[1354] E. N. Kuz′min. Irreducible polynomials over a finite field and an analogue of Gausssums over a field of characteristic 2. Sibirsk. Mat. Zh., 32(6):100–108, 205,1991. [27, 47, 48, 49]
[1355] G. M. Kyureghyan. Crooked maps in F2n . Finite Fields Appl., 13(3):713–726, 2007.[211, 213]
[1356] G. M. Kyureghyan. Constructing permutations of finite fields via linear translators.J. Combin. Theory A, 118(3), 2011. [181, 184, 185]
[1357] G. M. Kyureghyan and A. Pott. Some theorems on planar mappings. In Arithmeticof finite fields, volume 5130 of Lecture Notes in Comput. Sci., pages 117–122.Springer, Berlin, 2008. [232, 234]
[1358] M. K. Kyuregyan. On the theory of the reducibility of polynomials over finite fields.Akad. Nauk Armyan. SSR Dokl., 86(1):17–22, 1988. [31, 33, 34]
[1359] M. K. Kyuregyan. Recurrent methods for constructing irreducible polynomials overGF(2s). Finite Fields Appl., 8(1):52–68, 2002. [31, 32, 33, 34, 238, 242]
[1360] M. K. Kyuregyan. Recurrent methods for constructing irreducible polynomials overFq of odd characteristics. Finite Fields Appl., 9(1):39–58, 2003. [31, 32, 33, 34]
[1361] M. K. Kyuregyan. Iterated constructions of irreducible polynomials over finite fieldswith linearly independent roots. Finite Fields Appl., 10(3):323–341, 2004. [31,32, 34]
[1362] M. K. Kyuregyan. Recurrent methods for constructing irreducible polynomials overFq of odd characteristics. II. Finite Fields Appl., 12(3):357–378, 2006. [31, 33,34]
[1363] G. Lachaud. Sommes d’Eisenstein et nombre de points de certaines courbesalgebriques sur les corps finis. C. R. Acad. Sci. Paris Ser. I Math., 305(16):729–732, 1987. [366, 367]
[1364] G. Lachaud. The parameters of projective Reed-Muller codes. Discrete Math.,81(2):217–221, 1990. [587, 602]
[1365] G. Lachaud and J. Wolfmann. The weights of the orthogonals of the extendedquadratic binary Goppa codes. IEEE Trans. Inform. Theory, 36(3):686–692,1990. [212, 213]
[1366] L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math.,147(1):1–241, 2002. [457]
[1367] L. Lafforgue. Chtoucas de Drinfeld, formule des traces d’Arthur-Selberg et cor-respondance de Langlands. In Proceedings of the International Congress ofMathematicians, Vol. I (Beijing, 2002), pages 383–400, Beijing, 2002. HigherEd. Press. [457]
[1368] J. C. Lagarias. Pseudorandom number generators in cryptography and numbertheory. In Cryptology and computational number theory (Boulder, CO, 1989),volume 42 of Proc. Sympos. Appl. Math., pages 115–143. Amer. Math. Soc.,Providence, RI, 1990. [283, 289]
[1369] Y. Laigle-Chapuy. A note on a class of quadratic permutations over F2n . In Appliedalgebra, algebraic algorithms and error-correcting codes, volume 4851 of LectureNotes in Comput. Sci., pages 130–137. Springer, Berlin, 2007. [180, 185]
Miscellaneous applications 775
[1370] Y. Laigle-Chapuy. Permutation polynomials and applications to coding theory.Finite Fields Appl., 13(1):58–70, 2007. [174, 179, 185]
[1371] D. Laksov. Linear recurring sequences over finite fields. Math. Scand., 16:181–196,1965. [523, 531]
[1372] C. Lam, M. Aagaard, and G. G. Hardware imple-mentations of multi-output welch-gong ciphers, 2011.http://www.cacr.math.uwaterloo.ca/techreports/2011/cacr2011-01.pdf.[638, 647]
[1373] C. W. H. Lam, G. Kolesova, and L. Thiel. A computer search for finite projectiveplanes of order 9. Discrete Math., 92(1-3):187–195, 1991. [476, 486]
[1374] C. W. H. Lam, L. Thiel, and S. Swiercz. The nonexistence of finite projective planesof order 10. Canad. J. Math., 41(6):1117–1123, 1989. [476, 486]
[1375] T. Y. Lam and K. H. Leung. Vanishing sums of mth roots of unity in finite fields.Finite Fields Appl., 2(4):422–438, 1996. [168, 170]
[1376] B. A. LaMacchia and A. M. Odlyzko. Solving large sparse linear systems over finitefields. Lecture Notes in Computer Science, 537:109–133, 1991. http://www.
[1377] R. Lambert. Computational aspects of discrete logarithms. PhD thesis, Universityof Waterloo, Ontario, Canada, 1996. http://www.cacr.math.uwaterloo.ca/techreports/2000/lambert-thesis.ps. [432, 436]
[1378] E. S. Lander. Symmetric designs: an algebraic approach, volume 74 of London Math-ematical Society Lecture Note Series. Cambridge University Press, Cambridge,1983. [512, 515, 519]
[1379] S. Lang. Elliptic curves: Diophantine analysis, volume 231 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences]. Springer-Verlag, Berlin, 1978. [19, 20, 334, 351]
[1380] S. Lang. Abelian varieties. Springer-Verlag, New York, 1983. Reprint of the 1959original. [121, 127]
[1381] S. Lang. Elliptic functions, volume 112 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1987. With an appendix by J. Tate. [19, 20,334, 351]
[1382] S. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag,New York, third edition, 2002. [326, 333, 372, 448, 449]
[1383] S. Lang and H. Trotter. Frobenius distributions in GL2-extensions. Springer-Verlag,Berlin, 1976. Distribution of Frobenius automorphisms in GL2-extensions ofthe rational numbers, Lecture Notes in Mathematics, Vol. 504. [20, 349, 351]
[1384] S. Lang and A. Weil. Number of points of varieties in finite fields. Amer. J. Math.,76:819–827, 1954. [152, 158]
[1385] V. Laohakosol and U. Pintoptang. A modification of Fitzgerald’s characterizationof primitive polynomials over a finite field. Finite Fields Appl., 14(1):85–91,2008. [57, 59]
[1386] G. Larcher and H. Niederreiter. Generalized (t, s)-sequences, Kronecker-type se-quences, and diophantine approximations of formal Laurent series. Trans.Amer. Math. Soc., 347:2051–2073, 1995. [379, 383]
[1387] R. Laubenbacher, A. Jarrah, H. Mortveit, and S. S. Ravi. Encyclopedia of Complex-ity and System Science, chapter A mathematical foundation for agent-basedcomputer simulation. Springer Verlag, New York, 2009. [685]
[1388] R. Laubenbacher and B. Stigler. A computational algebra approach to the re-
776 Handbook of Finite Fields
verse engineering of gene regulatory networks. Journal of Theoretical Biology,229(4):523 – 537, 2004. [282, 289, 689, 692]
[1389] A. G. B. Lauder. Computing zeta functions of Kummer curves via multiplicativecharacters. Found. Comput. Math., 3(3):273–295, 2003. [406]
[1390] A. G. B. Lauder. Counting solutions to equations in many variables over finitefields. Found. Comput. Math., 4(3):221–267, 2004. [404, 405, 406]
[1391] A. G. B. Lauder. Deformation theory and the computation of zeta functions. Proc.London Math. Soc. (3), 88(3):565–602, 2004. [405, 406]
[1392] A. G. B. Lauder and K. G. Paterson. Computing the error linear complexityspectrum of a binary sequence of period 2n. IEEE Trans. Inform. Theory,49(1):273–280, 2003. [274, 281]
[1393] A. G. B. Lauder and D. Wan. Computing zeta functions of Artin-Schreier curvesover finite fields. II. J. Complexity, 20(2-3):331–349, 2004. [359, 360, 406]
[1394] A. G. B. Lauder and D. Wan. Counting points on varieties over finite fields of smallcharacteristic. In Algorithmic number theory: lattices, number fields, curvesand cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages 579–612.Cambridge Univ. Press, Cambridge, 2008. [359, 360, 404, 405, 406]
[1395] G. Laumon. Majorations de sommes trigonometriques (d’apres P. Deligne et N.Katz). In The Euler-Poincare characteristic (French), volume 83 of Asterisque,pages 221–258. Soc. Math. France, Paris, 1981. [126]
[1396] G. Laumon. Transformation de Fourier, constantes d’equations fonctionnelles etconjecture de Weil. Inst. Hautes Etudes Sci. Publ. Math., (65):131–210, 1987.[392, 393]
[1397] G. Laumon. Exponential sums and l-adic cohomology: a survey. Israel J. Math.,120(part A):225–257, 2000. [126]
[1398] M. Lavrauw, L. Storme, and G. Van de Voorde. A proof of the linearity conjecture fork-blocking sets in PG(n, p3), p prime. J. Combin. Theory Ser. A, 118(3):808–818, 2011. [472, 475]
[1399] K. M. Lawrence. A combinatorial characterization of (t,m, s)-nets in base b. J.Combin. Des., 4:275–293, 1996. [374, 383]
[1400] K. M. Lawrence, A. Mahalanabis, G. L. Mullen, and W. C. Schmid. Constructionof digital (t,m, s)-nets from linear codes. In Finite fields and applications(Glasgow, 1995), volume 233 of London Math. Soc. Lecture Note Ser., pages189–208. Cambridge University Press, Cambridge, 1996. [378, 383]
[1401] C. F. Laywine and G. L. Mullen. Discrete mathematics using Latin squares. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley &Sons Inc., New York, 1998. A Wiley-Interscience Publication. [20, 463, 467]
[1402] C. F. Laywine, G. L. Mullen, and G. Whittle. d-dimensional hypercubes and theEuler and MacNeish conjectures. Monatsh. Math., 119(3):223–238, 1995. [464,465, 467]
[1403] D. Lazard. Grobner bases, Gaussian elimination and resolution of systems of alge-braic equations. In Computer algebra (London, 1983), volume 162 of LectureNotes in Comput. Sci., pages 146–156. Springer, Berlin, 1983. [664]
[1404] G. Leander and A. Kholosha. Bent functions with 2r Niho exponents. IEEE Trans.Inform. Theory, 52(12):5529–5532, 2006. [220, 224]
[1405] N. G. Leander. Monomial bent functions. IEEE Trans. Inform. Theory, 52(2):738–743, 2006. [219, 223, 224]
[1406] G. Lecerf. Sharp precision in Hensel lifting for bivariate polynomial factorization.
[1408] G. Lecerf. Fast separable factorization and applications. Appl. Alg. Eng. Comm.Comp., 19(2), 2008. [302, 303, 311]
[1409] G. Lecerf. New recombination algorithms for bivariate polynomial factorizationbased on Hensel lifting. Appl. Alg. Eng. Comm. Comp., 21(2):151–176, 2010.[303, 311]
[1410] A. M. Legendre. Recherches d’analyse indeterminee. Memoires Acad. Sci. Paris,pages 465–559, 1785. [146]
[1411] A. Lempel and H. Greenberger. Families of sequences with optimal Hamming cor-relation properties. IEEE Trans. Information Theory, IT-20:90–94, 1974. [698,701]
[1412] D. Lenskoi. On the arithmetic of polynomials over a finite field (russian). Volz.Mat. Sb., 4:155–159, 1966. [408, 414]
[1413] A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz. Factoring polynomials withrational coefficients. Math. Ann., 261(4):515–534, 1982. [306, 311]
[1414] A. K. Lenstra and E. R. Verheul. Selecting cryptographic key sizes (extended ab-stract). In H. Imai and Y. Zheng, editors, Public Key Cryptography — 3rdInternational Workshop on Practice and Theory in Public Key CryptosystemsPKC 2000, volume 1751 of Lecture Notes in Computer Science, pages 446–465,Berlin, 2000. Springer-Verlag. [666]
[1415] H. W. Lenstra, Jr. A normal basis theorem for infinite Galois extensions. Nederl.Akad. Wetensch. Indag. Math., 47(2):221–228, 1985. [93]
[1416] H. W. Lenstra, Jr. Finding small degree factors of lacunary polynomials. In K. Gy-Hory, H. Iwaniec, and J. Urbanowicz, editors, Number Theory in Progress, vol-ume 1 Diophantine Problems and Polynomials, pages 267–276. Stefan BanachInternat. Center, Walter de Gruyter Berlin/New York, 1999. Proc. Internat.Conf. Number Theory in Honor of the 60th Birthday of Andrzej Schinzel, Za-kopane, Poland June 30–July 9, 1997. [308, 311]
[1417] H. W. Lenstra, Jr. and R. J. Schoof. Primitive normal bases for finite fields. Math.Comp., 48(177):217–231, 1987. [63, 65, 92]
[1418] J. S. Leon, J. M. Masley, and V. Pless. Duadic codes. IEEE Trans. Inform. Theory,30(5):709–714, 1984. [581, 602]
[1419] R. Lercier and D. Lubicz. Counting points on elliptic curves over finite fields ofsmall characteristic in quasi quadratic time. In E. Biham, editor, Advances inCryptology — EUROCRYPT 2003, volume 2656 of Lecture Notes in ComputerScience, pages 360–373, Berlin, 2003. Springer-Verlag. [670]
[1420] R. Lercier and D. Lubicz. A quasi quadratic time algorithm for hyperelliptic curvepoint counting. Ramanujan J., 12(3):399–423, 2006. [358, 360, 406]
[1421] K. H. Leung, S. L. Ma, and B. Schmidt. Nonexistence of abelian differencesets: Lander’s conjecture for prime power orders. Trans. Amer. Math. Soc.,356(11):4343–4358 (electronic), 2004. [515, 519]
[1422] K. H. Leung, S. L. Ma, and B. Schmidt. New Hadamard matrices of order 4p2
obtained from Jacobi sums of order 16. J. Combin. Theory Ser. A, 113(5):822–838, 2006. [106, 118]
[1423] K. H. Leung, S. L. Ma, and B. Schmidt. On Lander’s conjecture for difference setswhose order is a power of 2 or 3. Des. Codes Cryptogr., 56(1):79–84, 2010.
778 Handbook of Finite Fields
[515, 516, 519]
[1424] K. H. Leung and B. Schmidt. The field descent method. Des. Codes Cryptogr.,36(2):171–188, 2005. [517, 519]
[1425] V. Levenshtein. Application of hadamard matrices to a problem of coding theorey.Problemy Kibernetiki, 5:123–136, 1961. [135]
[1426] F. Levy-dit Vehel and L. Perret. Polynomial equivalence problems and applicationsto multivariate cryptosystems. In Progress in cryptology—INDOCRYPT 2003,volume 2904 of Lecture Notes in Comput. Sci., pages 235–251. Springer, Berlin,2003. [651]
[1427] H. Li and H. J. Zhu. Zeta functions of totally ramified p-covers of the projectiveline. Rend. Sem. Mat. Univ. Padova, 113:203–225, 2005. [400, 402]
[1428] J. Li, D. B. Chandler, and Q. Xiang. Permutation polynomials of degree 6 or 7 overfinite fields of characteristic 2. Finite Fields Appl., 16(6):406–419, 2010. [172,185]
[1429] K.-Z. Li and F. Oort. Moduli of supersingular abelian varieties, volume 1680 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1998. [401, 402]
[1430] L. Li and O. Roche-Newton. An improved sum-product estimate for general finitefields. SIAM J. Discrete Math., 25:1285–1296, 2011. [130]
[1431] W.-C. W. Li. Character sums and abelian Ramanujan graphs. J. Number Theory,41(2):199–217, 1992. With an appendix by Ke Qin Feng and the author. [536,545]
[1432] W. C. W. Li. Number theory with applications, volume 7 of Series on UniversityMathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 1996. [19,20, 532, 536, 545]
[1433] W.-C. W. Li. Recent developments in automorphic forms and applications. InNumber theory for the millennium, II (Urbana, IL, 2000), pages 331–354. A KPeters, Natick, MA, 2002. [532, 545]
[1435] W.-C. W. Li and P. Sole. Spectra of regular graphs and hypergraphs and orthogonalpolynomials. European J. Combin., 17(5):461–477, 1996. [538, 545]
[1436] Y. Li, S. Ling, H. Niederreiter, H. Wang, C. Xing, and S. Zhang, editors. Codingand cryptology, volume 4 of Series on Coding Theory and Cryptology. WorldScientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. [20]
[1437] Y. Li and M. Wang. On EA-equivalence of certain permutations to power mappings.Des. Codes Cryptogr., 58(3):259–269, 2011. [182, 185]
[1438] R. Lidl and G. L. Mullen. Unsolved Problems: When Does a Polynomial Overa Finite Field Permute the Elements of the Field? Amer. Math. Monthly,95(3):243–246, 1988. [172, 185]
[1439] R. Lidl and G. L. Mullen. Cycle structure of Dickson permutation polynomials.Math. J. Okayama Univ., 33:1–11, 1991. [184, 185]
[1440] R. Lidl and G. L. Mullen. Unsolved Problems: When Does a Polynomial over aFinite Field Permute the Elements of the Field?, II. Amer. Math. Monthly,100(1):71–74, 1993. [172, 173, 185]
[1441] R. Lidl, G. L. Mullen, and G. Turnwald. Dickson polynomials, volume 65 of PitmanMonographs and Surveys in Pure and Applied Mathematics. Longman Scientific& Technical, Harlow, 1993. [19, 20, 185, 192, 235, 240, 241, 242, 246, 250, 255]
[1442] R. Lidl and H. Niederreiter. On orthogonal systems and permutation polynomials
Miscellaneous applications 779
in several variables. Acta Arith., 22:257–265, 1972/73. [186, 188]
[1443] R. Lidl and H. Niederreiter. Introduction to finite fields and their applications.Cambridge University Press, Cambridge, first edition, 1994. [3, 19, 20, 39]
[1444] R. Lidl and H. Niederreiter. Finite Fields, volume 20 of Encyclopedia of Mathematicsand its Applications. Cambridge University Press, Cambridge, second edition,1997. With a foreword by P. M. Cohn. [3, 14, 15, 19, 20, 31, 34, 35, 39, 56, 59,136, 137, 138, 140, 160, 163, 164, 170, 171, 172, 173, 183, 184, 188, 191, 205,213, 240, 261, 270, 273, 281, 296, 299, 424]
[1445] R. Lidl and C. Wells. Chebyshev polynomials in several variables. J. Reine Angew.Math., 255:104–111, 1972. [187, 188]
[1446] S. Lin. On a class of cyclic codes. In Error Correcting Codes (Proc. Sympos. Math.Res. Center, Madison, Wis., 1968), pages 131–148. John Wiley, New York,1968. [588, 596, 601, 602]
[1447] S. Lin and D. Costello. Error control coding. Prentice-Hall, Saddle River, NJ, secondedition, 2004. [19, 20, 561, 591, 602]
[1448] J. Lindholm. An analysis of the pseudo-randomness properties of subsequences oflong m -sequences. Information Theory, IEEE Transactions on, 14(4):569 –576, jul 1968. [521, 531]
[1449] S. Ling and C. Xing. Coding theory. Cambridge University Press, Cambridge, 2004.A first course. [19, 20, 561, 576, 580, 585, 602]
[1450] P. Lisonek and M. Moisio. On zeros of Kloosterman sums. Des. Codes Cryptogr.,59(1-3):223–230, 2011. [111, 118]
[1451] C. Liu. Twisted higher moments of Kloosterman sums. Proc. Amer. Math. Soc.,130(7):1887–1892 (electronic), 2002. [115, 118]
[1452] C. Liu. The L-functions of twisted Witt extensions. J. Number Theory, 125(2):267–284, 2007. [397, 402]
[1453] C. Liu and D. Wan. T -adic exponential sums over finite fields. Algebra NumberTheory, 3(5):489–509, 2009. [397, 402]
[1454] C. Liu and D. Wei. The L-functions of Witt coverings. Math. Z., 255(1):95–115,2007. [397, 402]
[1455] P. Loidreau. On the factiorization of trinomials over 3. INRIA rapport de recherche3918, 2000. [38]
[1456] D. Lorenzini. An invitation to arithmetic geometry, volume 9 of Graduate Studiesin Mathematics. American Mathematical Society, Providence, RI, 1996. [363,367]
[1457] S. R. Louboutin. Efficient computation of root numbers and class numbers ofparametrized families of real abelian number fields. Math. Comp., 76(257):455–473 (electronic), 2007. [100, 118]
[1458] L. Lovasz and A. Schrijver. Remarks on a theorem of Redei. Studia Sci. Math.Hungar., 16(3-4):449–454, 1983. [471, 475]
[1459] H.-f. Lu and P. V. Kumar. Rate-diversity tradeoff of space-time codes with fixedalphabet and optimal constructions of PSK modulation. IEEE Trans. Inform.Theory, 49(10):2747–2751, 2003. Special issue on space-time transmission, re-ception, coding and signal processing. [700, 701]
[1460] H.-F. Lu and P. V. Kumar. A unified construction of space-time codes with optimalrate-diversity tradeoff. IEEE Trans. Inform. Theory, 51(5):1709–1730, 2005.[700, 701]
[1461] Y. Lu and L. Zhu. On the existence of triplewhist tournaments TWh(v). J. Combin.
780 Handbook of Finite Fields
Des., 5(4):249–256, 1997. [558]
[1462] A. Lubotzky. Discrete groups, expanding graphs and invariant measures, volume125 of Progress in Mathematics. Birkhauser Verlag, Basel, 1994. With anappendix by Jonathan D. Rogawski. [532, 538, 545]
[1463] A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica,8(3):261–277, 1988. [540, 541, 542, 545]
[1464] G. Lunardon. Normal spreads. Geom. Dedicata, 75(3):245–261, 1999. [472, 475]
[1465] G. Lunardon and O. Polverino. Blocking sets of size qt + qt−1 + 1. J. Combin.Theory Ser. A, 90(1):148–158, 2000. [472, 475]
[1466] H. Luneburg. Uber projektive Ebenen, in denen jede Fahne von einer nicht-trivialenElation invariant gelassen wird. Abh. Math. Sem. Univ. Hamburg, 29:37–76,1965. [481, 486]
[1467] H. Luneburg. Translation planes. Springer-Verlag, Berlin, 1980. [479, 486]
[1468] J. Luo and K. Feng. On the weight distributions of two classes of cyclic codes. IEEETrans. Inform. Theory, 54(12):5332–5344, 2008. [163]
[1469] K. Ma and J. von zur Gathen. The computational complexity of recognizing per-mutation functions. Comput. Complexity, 5(1):76–97, 1995. [173, 185, 311]
[1470] K. Ma and J. von zur Gathen. Tests for permutation functions. Finite Fields Appl.,1(1):31–56, 1995. [173, 185]
[1471] F. S. Macaulay. The algebraic theory of modular systems. Cambridge MathematicalLibrary. Cambridge University Press, Cambridge, 1994. Revised reprint of the1916 original, With an introduction by Paul Roberts. [664]
[1472] C. R. MacCluer. On a conjecture of Davenport and Lewis concerning exceptionalpolynomials. Acta Arith, 12:289–299, 1966/1967. [245, 255]
[1473] H. F. MacNeish. Euler squares. Ann. of Math. (2), 23(3):221–227, 1922. [464, 467]
[1474] F. J. MacWilliams. Orthogonal circulant matrices over finite fields, and how to findthem. J. Combinatorial Theory Ser. A, 10:1–17, 1971. [420, 424]
[1475] F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Co., Amsterdam, 1977. North-Holland Mathematical Li-brary. [19, 20, 499]
[1476] F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. I.North-Holland Publishing Co., Amsterdam, 1977. North-Holland Mathemati-cal Library, Vol. 16. [19, 20, 143, 163, 211, 213, 561, 584, 591, 601, 602]
[1477] J. MacWilliams. Orthogonal matrices over finite fields. Amer. Math. Monthly,76:152–164, 1969. [420, 421, 422, 424]
[1478] S. Maitra, K. C. Gupta, and A. Venkateswarlu. Results on multiples of primitivepolynomials and their products over GF(2). Theoret. Comput. Sci., 341(1-3):311–343, 2005. [67, 68, 522, 524, 531]
[1479] C. Malvenuto and F. Pappalardi. Enumerating permutation polynomials. I. Per-mutations with non-maximal degree. Finite Fields Appl., 8(4):531–547, 2002.[176, 185]
[1480] C. Malvenuto and F. Pappalardi. Enumerating permutation polynomials. II. k-cycles with minimal degree. Finite Fields Appl., 10(1):72–96, 2004. [176, 185]
[1481] C. Malvenuto and F. Pappalardi. Corrigendum to: “Enumerating permutation poly-nomials. I. Permutations with non-maximal degree” [Finite Fields Appl. 8(2002), no. 4, 531–547; mr1933624]. Finite Fields Appl., 13(1):171–174, 2007.[176, 185]
Miscellaneous applications 781
[1482] F. Manganiello, E. Gorla, and J. Rosenthal. Spread codes and spread decodingin network coding. In Proc. Int. Symp. Inform. Theory, pages 881–885, July2008. [701]
[1483] J. I. Manin. The Hasse-Witt matrix of an algebraic curve. Izv. Akad. Nauk SSSRSer. Mat., 25:153–172, 1961. [401, 402]
[1484] Y. Mansury, M. Kimura, J. Lobo, and T. S. Deisboeck. Emerging patterns in tumorsystems: Simulating the dynamics of multicellular clusters with an agent-basedspatial agglomeration model. Journal of Theoretical Biology, 219(3):343 – 370,2002. [689, 692]
[1485] I. Mantin. Analysis of the Stream Cipher RC4. Master’s dissertation, The WeizmannInstitute of Science, Rehovot, 76100, Israel, 2001. [635, 637, 647]
[1486] J. E. Marcos. Specific permutation polynomials over finite fields. Finite FieldsAppl., 17(2):105–112, 2011. [177, 180, 181, 185]
[1487] D. A. Marcus. Number fields. Springer-Verlag, New York, 1977. Universitext. [700,701]
[1488] G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes andtheir applications in the construction of expanders and concentrators. ProblemyPeredachi Informatsii, 24(1):51–60, 1988. [540, 541, 542, 545]
[1489] W. J. Martin and D. R. Stinson. A generalized Rao bound for ordered orthogonalarrays and (t,m, s)-nets. Canad. Math. Bull., 42:359–370, 1999. [374, 383]
[1490] W. J. Martin and D. R. Stinson. Association schemes for ordered orthogonal arraysand (T,M, S)-nets. Canad. J. Math., 51:326–346, 1999. [374, 383]
[1491] W. J. Martin and T. I. Visentin. A dual Plotkin bound for (T,M, S)-nets. IEEETrans. Inform. Theory, 53:411–415, 2007. [374, 383]
[1492] J. L. Massey. Threshold decoding. Massachusetts Institute of Technology, ResearchLaboratory of Electronics, Tech. Rep. 410, Cambridge, Mass., 1963. [595, 602]
[1493] J. L. Massey. Shift-register synthesis and BCH decoding. IEEE Trans. InformationTheory, IT-15:122–127, 1969. [201, 204, 271, 281, 593, 601, 602]
[1494] J. L. Massey and S. Serconek. Linear complexity of periodic sequences: a generaltheory. In Advances in cryptology—CRYPTO ’96 (Santa Barbara, CA), volume1109 of Lecture Notes in Comput. Sci., pages 358–371. Springer, Berlin, 1996.[274, 281]
[1495] A. Masuda and D. Panario. Sequences of consecutive smooth polynomials over afinite field. Proc. Amer. Math. Soc., 135(5):1271–1277, 2007. [414]
[1496] A. Masuda, D. Panario, and Q. Wang. The number of permutation binomials overF4p+1 where p and 4p + 1 are primes. Electron. J. Combin., 13(1):ResearchPaper 65, 15 pp. (electronic), 2006. [173, 174, 179, 185]
[1497] A. M. Masuda and D. Panario. Topicos de Corpos Finitos com Aplicacoes emCriptografia e Teoria de Codigos. Publicacoes Matematicas do IMPA. [IMPAMathematical Publications]. Instituto Nacional de Matematica Pura e Apli-cada (IMPA), Rio de Janeiro, 2007. 26o Coloquio Brasileiro de Matematica.[26th Brazilian Mathematics Colloquium]. [3, 19, 20]
[1498] A. M. Masuda and M. E. Zieve. Nonexistence of permutation binomials of certainshapes. Electron. J. Combin., 14(1):Note 12, 5 pp. (electronic), 2007. [174,185]
[1499] A. M. Masuda and M. E. Zieve. Permutation binomials over finite fields. Trans.Amer. Math. Soc., 361(8):4169–4180, 2009. [174, 179, 185]
[1500] R. Mathon. Symmetric conference matrices of order pq2 + 1. Canad. J. Math.,
782 Handbook of Finite Fields
30(2):321–331, 1978. [548]
[1501] R. Mathon. New maximal arcs in Desarguesian planes. J. Combin. Theory Ser. A,97(2):353–368, 2002. [484, 486]
[1502] R. Mathon and G. F. Royle. The translation planes of order 49. Des. CodesCryptogr., 5(1):57–72, 1995. [226, 229]
[1503] M. Matsui. Linear cryptoanalysis method for des cipher. In EUROCRYPT, pages386–397. 1993. [205, 213]
[1504] T. Matsumoto and H. Imai. Public quadratic polynomial-tuples for efficientsignature-verification and message-encryption. In Advances in cryptology—EUROCRYPT ’88 (Davos, 1988), volume 330 of Lecture Notes in Comput.Sci., pages 419–453. Springer, Berlin, 1988. [649, 653]
[1505] T. Matsumoto, H. Imai, H. Harashima, and H. Miyakawa. A cryptographicallyuseful theorem on the connection between uni and multivariate polynomials.Transactions of the IECE of Japan, 68(3):139–146, Mar. 1985. [649, 652]
[1506] S. Mattarei. On a bound of Garcia and Voloch for the number of points of a Fermatcurve over a prime field. Finite Fields Appl., 13(4):773–777, 2007. [169, 170]
[1507] R. Matthews. Permutation polynomials over algebraic number fields. J. NumberTheory, 18(3):249–260, 1984. [246, 255]
[1508] R. Matthews. Some results on permutation polynomials over finite fields. Appl.Algebra Engrg. Comm. Comput., 3(1):63–65, 1992. [186, 188]
[1509] R. Matthews. Permutation properties of the polynomials 1 + x + · · · + xk over afinite field. Proc. Amer. Math. Soc., 120(1):47–51, 1994. [179, 185]
[1510] R. W. Matthews. Permutation polynomials in one and several variables, Ph.D.Thesis, University of Tasmania. PhD thesis, 1982. [182, 185]
[1511] R. W. Matthews. Permutation polynomials in one and several variables. PhD thesis,University of Tasmania, Hobart, Tasmania, Australia, 1990. [225, 229]
[1512] H. F. Mattson and G. Solomon. A new treatment of Bose-Chaudhuri codes. J. Soc.Indust. Appl. Math., 9:654–669, 1961. [578, 601, 602]
[1513] C. Mauduit, H. Niederreiter, and A. Sarkozy. On pseudorandom [0, 1) and binarysequences. Publ. Math. Debrecen, 71(3-4):305–324, 2007. [281]
[1514] C. Mauduit and A. Sarkozy. On finite pseudorandom binary sequences. I. Measureof pseudorandomness, the Legendre symbol. Acta Arith., 82(4):365–377, 1997.[146, 147, 280, 281, 694, 701]
[1515] U. M. Maurer and S. Wolf. The Diffie-Hellman protocol. Des. Codes Cryptogr.,19(2-3):147–171, 2000. Towards a quarter-century of public key cryptography.[629, 634]
[1516] J. P. May, D. Saunders, and Z. Wan. Efficient matrix rank computation withapplication to the study of strongly regular graphs. In ISSAC 2007, pages277–284. ACM, New York, 2007. [435, 436]
[1517] B. Mazur. Frobenius and the Hodge filtration (estimates). Ann. of Math. (2),98:58–95, 1973. [396, 402]
[1518] O. D. Mbodj. Quadratic Gauss sums. Finite Fields Appl., 4(4):347–361, 1998. [106,118]
[1519] K. McCann and K. S. Williams. The distribution of the residues of a quarticpolynomial. Glasgow Math. J., 8:67–88, 1967. [190, 192]
[1520] B. R. McDonald. Finite rings with identity. Marcel Dekker Inc., New York, 1974.Pure and Applied Mathematics, Vol. 28. [17, 18, 19]
Miscellaneous applications 783
[1521] R. J. McEliece. The theory of information and coding. Addison-Wesley PublishingCo., Reading, Mass.-London-Amsterdam, 1977. A mathematical framework forcommunication, With a foreword by Mark Kac, Encyclopedia of Mathematicsand its Applications, Vol. 3. [561, 584, 585, 594, 602]
[1522] R. J. McEliece. A public-key cryptosystem based on algebraic coding theory. DSNprogress report #42-44, Jet Propulsion Laboratory, Pasadena, California, 1978.[633, 634]
[1523] R. J. McEliece. Finite fields for computer scientists and engineers. The Kluwer In-ternational Series in Engineering and Computer Science, 23. Kluwer AcademicPublishers, Boston, MA, 1987. [3, 19, 20]
[1524] R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr., and L. R. Welch. New upperbounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEETrans. Information Theory, IT-23(2):157–166, 1977. [573, 574, 602]
[1525] R. L. McFarland. A family of difference sets in non-cyclic groups. J. CombinatorialTheory Ser. A, 15:1–10, 1973. [517, 519]
[1526] G. McGuire, G. L. Mullen, D. Panario, and I. E. Shparlinski, editors. Finite fields:theory and applications, volume 518 of Contemporary Mathematics, Provi-dence, RI, 2010. American Mathematical Society. [20]
[1527] B. D. McKay and I. M. Wanless. On the number of Latin squares. Ann. Comb.,9(3):335–344, 2005. [462, 467]
[1528] H. McKean and V. Moll. Elliptic curves. Cambridge University Press, Cambridge,1997. Function theory, geometry, arithmetic. [19, 20, 334, 351]
[1529] W. Meidl. Linear complexity and k-error linear complexity for pn-periodic se-quences. In Coding, cryptography and combinatorics, volume 23 of Progr.Comput. Sci. Appl. Logic, pages 227–235. Birkhauser, Basel, 2004. [274, 281]
[1530] W. Meidl. Reducing the calculation of the linear complexity of u2v-periodic binarysequences to Games-Chan algorithm. Des. Codes Cryptogr., 46(1):57–65, 2008.[274, 281]
[1531] W. Meidl and H. Niederreiter. Counting functions and expected values for thek-error linear complexity. Finite Fields Appl., 8(2):142–154, 2002. [276, 281]
[1532] W. Meidl and H. Niederreiter. Linear complexity, k-error linear complexity, and thediscrete Fourier transform. J. Complexity, 18(1):87–103, 2002. [276, 281]
[1533] W. Meidl and H. Niederreiter. On the expected value of the linear complexityand the k-error linear complexity of periodic sequences. IEEE Trans. Inform.Theory, 48(11):2817–2825, 2002. [276, 281]
[1534] W. Meidl and H. Niederreiter. The expected value of the joint linear complexity ofperiodic multisequences. J. Complexity, 19(1):61–72, 2003. [273, 274, 276, 281]
[1535] W. Meidl and H. Niederreiter. Periodic sequences with maximal linear complexityand large k-error linear complexity. Appl. Algebra Engrg. Comm. Comput.,14(4):273–286, 2003. [276, 281]
[1536] W. Meidl, H. Niederreiter, and A. Venkateswarlu. Error linear complexity measuresfor multisequences. J. Complexity, 23(2):169–192, 2007. [276, 281]
[1537] W. Meidl and F. Ozbudak. Linear complexity over Fq and over Fqm for linearrecurring sequences. Finite Fields Appl., 15(1):110–124, 2009. [270, 281]
[1538] W. Meidl and A. Winterhof. Lower bounds on the linear complexity of the discretelogarithm in finite fields. IEEE Trans. Inform. Theory, 47(7):2807–2811, 2001.[279, 281]
[1539] W. Meidl and A. Winterhof. Linear complexity and polynomial degree of a function
784 Handbook of Finite Fields
over a finite field. In Finite fields with applications to coding theory, cryptogra-phy and related areas (Oaxaca, 2001), pages 229–238. Springer, Berlin, 2002.[274, 281]
[1540] W. Meidl and A. Winterhof. On the linear complexity profile of explicit nonlinearpseudorandom numbers. Inform. Process. Lett., 85(1):13–18, 2003. [277, 281]
[1541] W. Meidl and A. Winterhof. On the autocorrelation of cyclotomic generators. InFinite fields and applications, volume 2948 of Lecture Notes in Comput. Sci.,pages 1–11. Springer, Berlin, 2004. [137]
[1542] W. Meidl and A. Winterhof. On the linear complexity profile of some new explicitinversive pseudorandom numbers. J. Complexity, 20(2-3):350–355, 2004. [277,281]
[1543] W. Meidl and A. Winterhof. On the joint linear complexity profile of explicitinversive multisequences. J. Complexity, 21(3):324–336, 2005. [277, 281]
[1544] W. Meidl and A. Winterhof. Some notes on the linear complexity of Sidel′nikov-Lempel-Cohn-Eastman sequences. Des. Codes Cryptogr., 38(2):159–178, 2006.[279, 281]
[1545] W. Meidl and A. Winterhof. On the linear complexity profile of nonlinear congru-ential pseudorandom number generators with Redei functions. Finite FieldsAppl., 13(3):628–634, 2007. [278, 281]
[1546] W. Meier and O. Staffelbach. Fast correlation attacks on stream ciphers. InD. Barstow, W. Brauer, P. Brinch Hansen, D. Gries, D. Luckham, C. Moler,A. Pnueli, G. Seegm??ller, J. Stoer, N. Wirth, and C. G??nther, editors, Ad-vances in Cryptology ??? EUROCRYPT ???88, volume 330 of Lecture Notesin Computer Science, pages 301–314. Springer Berlin / Heidelberg, 1988. [521,531]
[1547] W. Meier and O. Staffelbach. Fast correlation attacks on certain stream ciphers. J.Cryptology, 1(3):159–176, 1989. [201, 204]
[1548] Z. Mejias and J.-K. Accetta. Numero de waring en cuerpos finitos. Universidad dePuerto Rico, R´io Piedras, Informe Tecnico, 2011. [168, 170]
[1549] A. Menezes. Elliptic curve public key cryptosystems. The Kluwer InternationalSeries in Engineering and Computer Science, 234. Kluwer Academic Publishers,Boston, MA, 1993. With a foreword by Neal Koblitz, Communications andInformation Theory. [19, 20]
[1550] A. Menezes, I. Blake, X.-H. Gao, R. Mullin, S. Vanstone, and T. Yaghoobian. Ap-plications of Finite Fields. The Springer International Series in Engineeringand Computer Science, Vol. 199., Springer., 1993. [3, 19, 20, 31, 32, 33, 34, 39,40]
[1551] A. J. Menezes, T. Okamoto, and S. A. Vanstone. Reducing elliptic curve logarithmsto logarithms in a finite field. IEEE Trans. Inform. Theory, 39(5):1639–1646,1993. [351, 675]
[1552] A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of applied cryp-tography. CRC Press Series on Discrete Mathematics and its Applications.CRC Press, Boca Raton, FL, 1997. With a foreword by Ronald L. Rivest. [19,20, 634, 642, 647, 667]
[1553] G. Menichetti. On a Kaplansky conjecture concerning three-dimensional divisionalgebras over a finite field. J. Algebra, 47(2):400–410, 1977. [228, 229]
[1554] G. Menichetti. n-dimensional algebras over a field with a cyclic extension of degreen. Geom. Dedicata, 63(1):69–94, 1996. [228, 229]
Miscellaneous applications 785
[1555] P. Merkey and E. Posner. Optimum cyclic redundancy codes for noisy channels(corresp.). Information Theory, IEEE Transactions on, 30(6):865 – 867, nov1984. [521, 522, 524, 527, 528, 531]
[1556] S. Mesnager. A new class of bent and hyper-bent Boolean functions in polynomialforms. Des. Codes Cryptogr., 59(1-3):265–279, 2011. [111, 118]
[1557] J.-F. Mestre. Lettre adressee a Gaudry et Harley. Dec. 2000. [670]
[1558] J.-F. Mestre. Algorithmes pur compter des point de courbesen petite characteristique et en petit genres. Available athttp://www.math.jussieu.fr/∼mestre/, 2002. [358, 360]
[1559] H. Meyn. On the construction of irreducible self-reciprocal polynomials over finitefields. Appl. Algebra Engrg. Comm. Comput., 1(1):43–53, 1990. [28, 31, 32, 33,34, 238, 242]
[1560] P. Michel. Some recent applications of Kloostermania. In Physics and numbertheory, volume 10 of IRMA Lect. Math. Theor. Phys., pages 225–251. Eur.Math. Soc., Zurich, 2006. [113, 118]
[1561] T. Migler, K. E. Morrison, and M. Ogle. How much does a matrix of rank k weigh?Math. Mag., 79(4):262–271, 2006. [416, 424]
[1562] P. Mihailescu, F. Morain, and E. Schost. Computing the eigenvalue in the Schoof–Elkies–Atkin algorithm using abelian lifts. In C. W. Brown, editor, Proceedingsof the 2007 International Symposium on Symbolic and Algebraic Computation— ISSAC 2007, pages 285–292, New York, 2007. Association for ComputingMachinery. [670]
[1563] R. L. Miller. Necklaces, symmetries and self-reciprocal polynomials. Discrete Math.,22(1):25–33, 1978. [238, 242]
[1564] S. J. Miller and M. R. Murty. Effective equidistribution and the Sato-Tate law forfamilies of elliptic curves. J. Number Theory, 131(1):25–44, 2011. [341, 351]
[1565] V. S. Miller. Use of elliptic curves in cryptography. In Advances in cryptology—CRYPTO ’85 (Santa Barbara, Calif., 1985), volume 218 of Lecture Notes inComput. Sci., pages 417–426. Springer, Berlin, 1986. [630, 634]
[1566] V. S. Miller. Use of elliptic curves in cryptography. In H. C. Williams, editor, Ad-vances in Cryptology — CRYPTO ’85, volume 218 of Lecture Notes in Com-puter Science, pages 417–426, Berlin, 1986. Springer-Verlag. [666]
[1567] D. Mills. Factorizations of root-based polynomial compositions. Discrete Math.,240(1-3):161–173, 2001. [35, 38]
[1568] D. Mills. Existence of primitive polynomials with three coefficients prescribed. JPJ. Algebra Number Theory Appl., 4(1):1–22, 2004. [62, 65]
[1569] W. H. Mills. Polynomials with minimal value sets. Pacific J. Math., 14:225–241,1964. [189, 192]
[1570] J. S. Milne. Elliptic curves. BookSurge Publishers, Charleston, SC, 2006. [19, 20,334, 351]
[1571] R. Mines, F. Richman, and W. Ruitenburg. A course in constructive algebra. Uni-versitext. Springer-Verlag, 1988. [302, 311]
[1572] M. Minzlaff. Computing zeta functions of superelliptic curves in larger characteris-tic. Math. Comput. Sci., 3(2):209–224, 2010. [404, 406]
[1573] T. Moh. A public key system with signature and master key functions. Comm.Algebra, 27(5):2207–2222, 1999. [657]
[1574] M. S. E. Mohamed, D. Cabarcas, J. Ding, J. Buchmann, and S. Bulygin. MXL3:an efficient algorithm for computing Grobner bases of zero-dimensional ideals.
786 Handbook of Finite Fields
In Information security and cryptology—ICISC 2009, volume 5984 of LectureNotes in Comput. Sci., pages 87–100. Springer, Berlin, 2010. [664]
[1575] M. S. E. Mohamed, J. Ding, J. Buchmann, and F. Werner. Algebraic attack onthe mqq public key cryptosystem. In Cryptology and Network Security, 8thInternational Conference, CANS, pages 392–401, 2009. [658]
[1576] M. S. E. Mohamed, W. S. A. E. Mohamed, J. Ding, and J. Buchmann. Mxl2:Solving polynomial equations over gf(2) using an improved mutant strategy.In J. Buchmann and J. Ding, editors, PQCrypto, volume 5299 of Lecture Notesin Computer Science, pages 203–215. Springer, 2008. [664]
[1577] B. Mohar. A strengthening and a multipartite generalization of the Alon-Boppana-Serre theorem. Proc. Amer. Math. Soc., 138(11):3899–3909, 2010. [538, 539,545]
[1578] M. Moisio. On the number of rational points on some families of Fermat curvesover finite fields. Finite Fields Appl., 13(3):546–562, 2007. [165, 170]
[1579] M. Moisio. Kloosterman sums, elliptic curves, and irreducible polynomials withprescribed trace and norm. Acta Arith., 132(4):329–350, 2008. [44, 49, 223,224]
[1580] M. Moisio. On the moments of Kloosterman sums and fibre products of Kloostermancurves. Finite Fields Appl., 14(2):515–531, 2008. [114, 118]
[1581] M. Moisio and K. Ranto. Elliptic curves and explicit enumeration of irreduciblepolynomials with two coefficients prescribed. Finite Fields Appl., 14(3):798–815, 2008. [48, 49]
[1582] M. Moisio, K. Ranto, M. Rinta-Aho, and K. Vaananen. On the weight distributionof cyclic codes with one or two zeros. Adv. Appl. Discrete Math., 3(2):125–150,2009. [111, 118]
[1583] M. Moisio and D. Wan. On Katz’s bound for the number of elements with giventrace and norm. J. Reine Angew. Math., 638:69–74, 2010. [154, 158]
[1584] M. J. Moisio. The moments of a Kloosterman sum and the weight distribution of aZetterberg-type binary cyclic code. IEEE Trans. Inform. Theory, 53(2):843–847, 2007. [114, 118]
[1585] F. Moller. Exceptional polynomials with 2-transitive affine monodromy groups.Finite Fields Appl., 2012. to appear. [194]
[1586] R. A. Mollin and C. Small. On permutation polynomials over finite fields. Internat.J. Math. Math. Sci., 10(3):535–543, 1987. [179, 185]
[1587] R. Moloney. Divisibility Properties of Kloosterman Sums and Division Polynomialsfor Edwards Curves. PhD dissertation, University College Dublin, College ofEngineering, Mathematical and Physical Sciences, 2011. [111, 118]
[1588] M. Monagan and R. Pearce. Polynomial division using dynamic arrays, heaps, andpacked exponent vectors. In Proc. of CASC 2007, pages 295–315. Springer-Verlag, 2007. [301, 311]
[1589] M. Monagan and R. Pearce. Parallel sparse polynomial multiplication using heaps.In ISSAC ’09: Proceedings of the 2009 International Symposium on Symbolicand Algebraic Computation, pages 263–270, New York, NY, USA, 2009. ACMPress. [301, 311]
[1590] M. Monagan and R. Pearce. Sparse polynomial multiplication and division in Maple14. ACM Communications in Computer Algebra, 44(3/4), 2010. [301, 311]
[1591] T. Moon. Error correction coding: Mathematical methods and algorithms. JohnWiley and Sons, Hoboken, NJ, 2005. [561, 591, 602]
Miscellaneous applications 787
[1592] E. H. Moore. A two-fold generalization of Fermat’s theorem. Bull. Amer. Math.Soc., 2(7):189–199, 1896. [424]
[1593] D. J. M. Morales. An analysis of the infrastructure in real function fields. eprintarchive no. 2008/299, 2008. [360]
[1594] L. J. Mordell. On a sum analogous to a gauss sum. Quart. J. Math., 3:161–162,1932. [132]
[1595] C. Moreno. Algebraic curves over finite fields, volume 97 of Cambridge Tracts inMathematics. Cambridge University Press, Cambridge, 1991. [20]
[1596] O. Moreno. Discriminants and the irreducibility of a class of polynomials in a finitefield of arbitrary characteristic. J. Number Theory, 28(1):62–65, 1988. [35]
[1597] O. Moreno and F. N. Castro. On the calculation and estimation of Waring numberfor finite fields. In Arithmetic, geometry and coding theory (AGCT 2003),volume 11 of Semin. Congr., pages 29–40. Soc. Math. France, Paris, 2005.[168, 170]
[1598] O. Moreno and F. N. Castro. Optimal divisibility for certain diagonal equationsover finite fields. J. Ramanujan Math. Soc., 23(1):43–61, 2008. [167, 168, 170]
[1599] O. Moreno and C. J. Moreno. Improvements of the Chevalley-Warning and theAx-Katz theorems. Amer. J. Math., 117(1):241–244, 1995. [157, 158, 164, 167,170, 396, 402]
[1600] O. Moreno and I. Rubio. Cyclic decomposition of monomial permutations. In Pro-ceedings of the Twentieth Southeastern Conference on Combinatorics, GraphTheory, and Computing (Boca Raton, FL, 1989), volume 73, pages 147–158,1990. [184, 185]
[1601] O. Moreno, K. W. Shum, F. N. Castro, and P. V. Kumar. Tight bounds forChevalley-Warning-Ax-Katz type estimates, with improved applications. Proc.London Math. Soc. (3), 88(3):545–564, 2004. [396, 402]
[1602] M. Morf. Doubling algorithms for teoplitz and related equations. In Proc. 1980 Int’lConf. Acoustics Speech and Signal Processing, pages 954–959, Denver, Colo.,Apr. 1980. [434, 436]
[1603] I. H. Morgan. Construction of complete sets of mutually equiorthogonal frequencyhypercubes. Discrete Math., 186(1-3):237–251, 1998. [466, 467]
[1604] I. H. Morgan and G. L. Mullen. Primitive normal polynomials over finite fields.Math. Comp., 63(208):759–765, S19–S23, 1994. [57, 59]
[1605] I. H. Morgan and G. L. Mullen. Completely normal primitive basis generators offinite fields. Utilitas Math., 49:21–43, 1996. [58, 59, 64, 65, 92]
[1606] I. H. Morgan, G. L. Mullen, and M. Zivkovic. Almost weakly self-dual bases forfinite fields. Appl. Algebra Engrg. Comm. Comput., 8(1):25–31, 1997. [58, 59,77, 79]
[1607] J. P. Morgan. Nested designs. In Design and analysis of experiments, volume 13 ofHandbook of Statist., pages 939–976. North-Holland, Amsterdam, 1996. [507]
[1608] M. Morgenstern. Existence and explicit constructions of q + 1 regular Ramanujangraphs for every prime power q. J. Combin. Theory Ser. B, 62(1):44–62, 1994.[542, 545]
[1609] M. Morii and M. Kasahara. Generalized key-equation of remainder decoding algo-rithm for Reed-Solomon codes. IEEE Trans. Inform. Theory, 38(6):1801–1807,1992. [595, 602]
[1610] B. Morlaye. Equations diagonales non homogenes sur un corps fini. C. R. Acad.Sci. Paris Ser. A-B, 272:A1545–A1548, 1971. [165, 170]
788 Handbook of Finite Fields
[1611] K. E. Morrison. Integer sequences and matrices over finite fields. J. Integer Seq.,9(2):Article 06.2.1, 28 pp. (electronic), 2006. [417, 424]
[1612] E. Mortenson. Modularity of a certain Calabi-Yau threefold and combinatorialcongruences. Ramanujan J., 11(1):5–39, 2006. [98, 118]
[1613] M. J. Mossinghoff. Wieferich pairs and Barker sequences. Des. Codes Cryptogr.,53(3):149–163, 2009. [517, 519]
[1614] C. Mulcahy. Card colm. Mathematical Association of America Online. http:
//www.maa.org/columns/colm/cardcolm.html. [531]
[1615] G. Mullen and H. Stevens. Polynomial functions (modm). Acta Math. Hungar.,44(3-4):237–241, 1984. [185]
[1616] G. L. Mullen. Permutation polynomials in several variables over finite fields. ActaArith., 31(2):107–111, 1976. [186, 188]
[1617] G. L. Mullen. Polynomial representation of complete sets of mutually orthogonalfrequency squares of prime power order. Discrete Math., 69(1):79–84, 1988.[465, 467]
[1618] G. L. Mullen. Permutation polynomials and nonsingular feedback shift registersover finite fields. IEEE Trans. Inform. Theory, 35(4):900–902, 1989. [187, 188]
[1619] G. L. Mullen. Dickson polynomials over finite fields. Adv. in Math. (China),20(1):24–32, 1991. [182, 185]
[1620] G. L. Mullen. Permutation polynomials over finite fields. In Finite fields, coding the-ory, and advances in communications and computing (Las Vegas, NV, 1991),volume 141 of Lecture Notes in Pure and Appl. Math., pages 131–151. Dekker,New York, 1993. [172, 173, 174, 185]
[1621] G. L. Mullen. A candidate for the “next Fermat problem”. Math. Intelligencer,17(3):18–22, 1995. [463, 467]
[1622] G. L. Mullen. Permutation polynomials: a matrix analogue of Schur’s conjectureand a survey of recent results. Finite Fields Appl., 1(2):242–258, 1995. Specialissue dedicated to Leonard Carlitz. [172, 183, 185]
[1623] G. L. Mullen and C. Mummert. Finite fields and applications, volume 41 of StudentMathematical Library. American Mathematical Society, Providence, RI, 2007.[3, 19, 20]
[1624] G. L. Mullen, D. Panario, and I. E. Shparlinski, editors. Finite fields and appli-cations, volume 461 of Contemporary Mathematics. American MathematicalSociety, Providence, RI, 2008. Papers from the 8th International Conferenceheld in Melbourne, July 9–13, 2007. [20]
[1625] G. L. Mullen, A. Poli, and H. Stichtenoth, editors. Finite fields and applications,volume 2948 of Lecture Notes in Computer Science. Springer-Verlag, Berlin,2004. Revised papers from the 7th International Conference (Fq7) held inToulouse, May 5–9, 2003. [20]
[1626] G. L. Mullen and W. C. Schmid. An equivalence between (t,m, s)-nets and stronglyorthogonal hypercubes. J. Combin. Theory Ser. A, 76:164–174, 1996. [374, 383]
[1627] G. L. Mullen and P. J.-S. Shiue, editors. Finite fields, coding theory, and advancesin communications and computing, volume 141 of Lecture Notes in Pure andApplied Mathematics, New York, 1993. Marcel Dekker Inc. [20]
[1628] G. L. Mullen and P. J.-S. Shiue, editors. Finite fields: theory, applications, andalgorithms, volume 168 of Contemporary Mathematics, Providence, RI, 1994.American Mathematical Society. [20]
[1629] G. L. Mullen and I. Shparlinski. Open problems and conjectures in finite fields. In
Miscellaneous applications 789
Finite fields and applications (Glasgow, 1995), volume 233 of London Math.Soc. Lecture Note Ser., pages 243–268. Cambridge Univ. Press, Cambridge,1996. [41, 57, 58, 59, 66, 68]
[1630] G. L. Mullen, H. Stichtenoth, and H. Tapia-Recillas, editors. Finite fields with appli-cations to coding theory, cryptography and related areas, Berlin, 2002. Springer-Verlag. [20]
[1631] P. Muller. Primitive monodromy groups of polynomials. In Recent developmentsin the inverse Galois problem (Seattle, WA, 1993), volume 186 of Contemp.Math., pages 385–401. Amer. Math. Soc., Providence, RI, 1995. [253, 255]
[1632] P. Muller. A Weil-bound free proof of Schur’s conjecture. Finite Fields Appl.,3(1):25–32, 1997. [183, 193]
[1633] R. C. Mullin and G. L. Mullen, editors. Finite fields: theory, applications, andalgorithms, volume 225 of Contemporary Mathematics, Providence, RI, 1999.American Mathematical Society. [20]
[1634] R. C. Mullin and E. Nemeth. An existence theorem for room squares. Canad. Math.Bull., 12:493–497, 1969. [553]
[1635] R. C. Mullin, J. L. Yucas, and G. L. Mullen. A generalized counting and factoringmethod for polynomials over finite fields. J. Combin. Math. Combin. Comput.,72:121–143, 2010. [28, 29, 30]
[1636] D. Mumford. An algebro-geometric construction of commuting operators and ofsolutions to the Toda lattice equation, Korteweg deVries equation and relatednonlinear equation. In Proceedings of the International Symposium on Al-gebraic Geometry (Kyoto Univ., Kyoto, 1977), pages 115–153, Tokyo, 1978.Kinokuniya Book Store. [457]
[1637] D. Mumford. Algebraic geometry. I. Classics in Mathematics. Springer-Verlag,Berlin, 1995. Complex projective varieties, Reprint of the 1976 edition. [305,311]
[1638] D. Mumford. The red book of varieties and schemes, volume 1358 of Lecture Notesin Mathematics. Springer-Verlag, Berlin, expanded edition, 1999. Includes theMichigan lectures (1974) on curves and their Jacobians, With contributions byEnrico Arbarello. [243, 244, 245, 250, 255]
[1639] A. Munemasa. Orthogonal arrays, primitive trinomials, and shift-register sequences.Finite Fields Appl., 4(3):252–260, 1998. [59, 520, 524, 531]
[1640] A. Muratovic-Ribic. A note on the coefficients of inverse polynomials. Finite FieldsAppl., 13(4):977–980, 2007. [184, 185]
[1641] A. Muratovic-Ribic. Inverse of some classes of permutation binomials. J. Concr.Appl. Math., 7(1):47–53, 2009. [184, 185]
[1642] M. R. Murty. Problems in analytic number theory, volume 206 of Graduate Textsin Mathematics. Springer-Verlag, New York, 2001. Readings in Mathematics.[535, 545]
[1643] M. R. Murty. Ramanujan graphs. J. Ramanujan Math. Soc., 18(1):33–52, 2003.[532, 545]
[1644] M. R. Murty and K. Sinha. Effective equidistribution of eigenvalues of Hecke oper-ators. J. Number Theory, 129(3):681–714, 2009. [545]
[1645] D. R. Musser. Multivariate polynomial factorization. J. Assoc. Comput. Mach.,22:291–308, 1975. [304, 311]
[1646] M. Nagata. On automorphism group of k[x, y]. Kinokuniya Book-Store Co. Ltd.,Tokyo, 1972. Department of Mathematics, Kyoto University, Lectures in Math-
790 Handbook of Finite Fields
ematics, No. 5. [652]
[1647] S. Najib. Une generalisation de l’inegalite de Stein-Lorenzini. J. Algebra, 292:566–573, 2005. [53, 55]
[1648] A. Naldi, D. Thieffry, and C. Chaouiya. Decision diagrams for the representationand analysis of logical models of genetic networks. In CMSB’07: Proceedingsof the 2007 international conference on Computational methods in systemsbiology, pages 233–247, Berlin, Heidelberg, 2007. Springer-Verlag. [685]
[1649] Y. Nawaz and G. Gong. The wg stream cipher, 2005.http://www.cacr.math.uwaterloo.ca/techreports/2005/cacr2005-15.pdf.[635, 639, 640, 641, 647]
[1650] Nazarathy, M. and Newton, S.A. and Giffard, R.P. and Moberly, D.S. and Sischka,F. and Trutna, Jr., W.R. and Foster, S. Real-time long range complementarycorrelation optical time domain reflectometer. IEEE J. Lightwave Technology,7:24–38, 1989. [695, 701]
[1651] NESSIE: New European Schemes for Signatures, Integrity, and Encryption. Infor-mation Society Technologies programme of the European commission (IST-1999-12324). http://www.cryptonessie.org/. [656]
[1652] E. Netto. Zur Theorie der Tripelsysteme. Math. Ann., 42(1):143–152, 1893. [503]
[1653] D. K. Nguyen and B. Schmidt. Fast computation of Gauss sums and resolution ofthe root of unity ambiguity. Acta Arith., 140(3):205–232, 2009. [98, 118]
[1654] X. Nie, L. Hu, J. Li, C. Updegrove, and J. Ding. Breaking a new instance of ttmcryptosystems. In J. Zhou, M. Yung, and F. Bao, editors, ACNS, volume 3989of Lecture Notes in Computer Science, pages 210–225, 2006. [658]
[1655] H. Niederreiter. Permutation polynomials in several variables over finite fields. Proc.Japan Acad. 46 (1970), no. 10, suppl. to, 46(9):1001–1005, 1970. [187, 188]
[1656] H. Niederreiter. Orthogonal systems of polynomials in finite fields. Proc. Amer.Math. Soc., 28:415–422, 1971. [186, 187, 188]
[1657] H. Niederreiter. Permutation polynomials in several variables. Acta Sci. Math.(Szeged), 33:53–58, 1972. [187, 188]
[1658] H. Niederreiter. Low-discrepancy point sets. Monatsh. Math., 102:155–167, 1986.[375, 383]
[1659] H. Niederreiter. Continued fractions for formal power series, pseudorandom num-bers, and linear complexity of sequences. In Contributions to general algebra, 5(Salzburg, 1986), pages 221–233. Holder-Pichler-Tempsky, Vienna, 1987. [275,281]
[1660] H. Niederreiter. Point sets and sequences with small discrepancy. Monatsh. Math.,104:273–337, 1987. [373, 374, 375, 379, 381, 383]
[1661] H. Niederreiter. Low-discrepancy and low-dispersion sequences. J. Number Theory,30:51–70, 1988. [379, 381, 383]
[1662] H. Niederreiter. The probabilistic theory of linear complexity. In Advances incryptology—EUROCRYPT ’88 (Davos, 1988), volume 330 of Lecture Notes inComput. Sci., pages 191–209. Springer, Berlin, 1988. [274, 275, 281]
[1663] H. Niederreiter. Sequences with almost perfect linear complexity profile. In Advancesin cryptology—EUROCRYPT ’87, volume 304 of Lecture Notes in Comput.Sci., pages 37–51. Springer, Berlin, 1988. [273, 274, 275, 281]
[1664] H. Niederreiter. A combinatorial approach to probabilistic results on the linear-complexity profile of random sequences. J. Cryptology, 2(2):105–112, 1990.[275, 281]
Miscellaneous applications 791
[1665] H. Niederreiter. Keystream sequences with a good linear complexity profile for everystarting point. In Advances in cryptology—EUROCRYPT ’89 (Houthalen,1989), volume 434 of Lecture Notes in Comput. Sci., pages 523–532. Springer,Berlin, 1990. [275, 281]
[1666] H. Niederreiter. The distribution of values of Kloosterman sums. Arch. Math.(Basel), 56(3):270–277, 1991. [113, 118]
[1667] H. Niederreiter. The linear complexity profile and the jump complexity of keystreamsequences. In Advances in cryptology—EUROCRYPT ’90 (Aarhus, 1990), vol-ume 473 of Lecture Notes in Comput. Sci., pages 174–188. Springer, Berlin,1991. [274, 275, 281]
[1668] H. Niederreiter. Low-discrepancy point sets obtained by digital constructions overfinite fields. Czechoslovak Math. J., 42:143–166, 1992. [376, 377, 383]
[1669] H. Niederreiter. Random number generation and quasi-Monte Carlo methods, vol-ume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics.Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,1992. [139, 373, 375, 380, 383]
[1670] H. Niederreiter. Constructions of (t,m, s)-nets. In Monte Carlo and quasi-MonteCarlo methods 1998 (Claremont, CA), pages 70–85. Springer-Verlag, Berlin,2000. [378, 383]
[1671] H. Niederreiter, editor. Coding theory and cryptology, volume 1 of Lecture NotesSeries. Institute for Mathematical Sciences. National University of Singapore.World Scientific Publishing Co. Inc., River Edge, NJ, 2002. Expanded lecturenotes of the tutorials from the Inaugural Research Program of the Institute forMathematical Sciences held at the National University of Singapore, Singapore,July–December, 2001. [19, 20]
[1672] H. Niederreiter. Linear complexity and related complexity measures for sequences.In Progress in cryptology—INDOCRYPT 2003, volume 2904 of Lecture Notesin Comput. Sci., pages 1–17. Springer, Berlin, 2003. [275, 281]
[1673] H. Niederreiter. Periodic sequences with large k-error linear complexity. IEEETrans. Inform. Theory, 49(2):501–505, 2003. [276, 281]
[1674] H. Niederreiter. Digital nets and coding theory. In Coding, cryptography andcombinatorics, volume 23 of Progr. Comput. Sci. Appl. Logic, pages 247–257.Birkhauser, Basel, 2004. [377, 383]
[1675] H. Niederreiter. Constructions of (t,m, s)-nets and (t, s)-sequences. Finite FieldsAppl., 11:578–600, 2005. [378, 383]
[1676] H. Niederreiter. The probabilistic theory of the joint linear complexity of multi-sequences. In Sequences and their applications—SETA 2006, volume 4086 ofLecture Notes in Comput. Sci., pages 5–16. Springer, Berlin, 2006. [276, 281]
[1677] H. Niederreiter. Nets, (t, s)-sequences, and codes. In Monte Carlo and quasi-MonteCarlo methods 2006, pages 83–100. Springer-Verlag, Berlin, 2008. [376, 381,383]
[1678] H. Niederreiter. Quasi-Monte Carlo methods. In Encyclopedia of quantitative fi-nance, pages 1460–1472. John Wiley and Sons, Chichester, 2010. [373, 383]
[1679] H. Niederreiter and F. Ozbudak. Constructions of digital nets using global functionfields. Acta Arith., 105:279–302, 2002. [377, 383]
[1680] H. Niederreiter and F. Ozbudak. Constructive asymptotic codes with an improve-ment on the Tsfasman-Vladut-Zink and Xing bounds. In Coding, cryptographyand combinatorics, volume 23 of Progr. Comput. Sci. Appl. Logic, pages 259–
792 Handbook of Finite Fields
275. Birkhauser, Basel, 2004. [612]
[1681] H. Niederreiter and F. Ozbudak. Matrix-product constructions of digital nets. FiniteFields Appl., 10:464–479, 2004. [378, 383]
[1682] H. Niederreiter and F. Ozbudak. Further improvements on asymptotic bounds forcodes using distinguished divisors. Finite Fields Appl., 13:423–443, 2007. [612]
[1683] H. Niederreiter and F. Ozbudak. Improved asymptotic bounds for codes using distin-guished divisors of global function fields. SIAM J. Discrete Math., 21:865–899,2007. [612]
[1684] H. Niederreiter and F. Ozbudak. Low-discrepancy sequences using duality andglobal function fields. Acta Arith., 130:79–97, 2007. [382, 383]
[1685] H. Niederreiter and G. Pirsic. Duality for digital nets and its applications. ActaArith., 97:173–182, 2001. [376, 383]
[1686] H. Niederreiter and K. H. Robinson. Complete mappings of finite fields. J. Austral.Math. Soc. Ser. A, 33(2):197–212, 1982. [184, 185]
[1687] H. Niederreiter and I. E. Shparlinski. On the distribution and lattice structure ofnonlinear congruential pseudorandom numbers. Finite Fields Appl., 5(3):246–253, 1999. [285, 289]
[1688] H. Niederreiter and I. E. Shparlinski. On the distribution of inversive congruentialpseudorandom numbers in parts of the period. Math. Comp., 70(236):1569–1574 (electronic), 2001. [145, 285, 289]
[1689] H. Niederreiter and I. E. Shparlinski. Dynamical systems generated by rationalfunctions. In Applied algebra, algebraic algorithms and error-correcting codes(Toulouse, 2003), volume 2643 of Lecture Notes in Comput. Sci., pages 6–17.Springer, Berlin, 2003. [282, 283, 289]
[1690] H. Niederreiter and I. E. Shparlinski. Periodic sequences with maximal linear com-plexity and almost maximal k-error linear complexity. In Cryptography andcoding, volume 2898 of Lecture Notes in Comput. Sci., pages 183–189. Springer,Berlin, 2003. [276, 281]
[1691] H. Niederreiter and A. Venkateswarlu. Periodic multisequences with large errorlinear complexity. Des. Codes Cryptogr., 49(1-3):33–45, 2008. [276, 281]
[1692] H. Niederreiter and L.-P. Wang. Proof of a conjecture on the joint linear complex-ity profile of multisequences. In Progress in cryptology—INDOCRYPT 2005,volume 3797 of Lecture Notes in Comput. Sci., pages 13–22. Springer, Berlin,2005. [275, 281]
[1693] H. Niederreiter and L.-P. Wang. The asymptotic behavior of the joint linear com-plexity profile of multisequences. Monatsh. Math., 150(2):141–155, 2007. [275,276, 281]
[1694] H. Niederreiter and A. Winterhof. Cyclotomic r-orthomorphisms of finite fields.Discrete Mathematics, 295(1-3):161–171, 2005. [136]
[1695] H. Niederreiter and A. Winterhof. Cyclotomic R-orthomorphisms of finite fields.Discrete Math., 295(1-3):161–171, 2005. [177, 184, 185]
[1696] H. Niederreiter and A. Winterhof. Exponential sums for nonlinear recurring se-quences. Finite Fields Appl., 14(1):59–64, 2008. [285, 289]
[1697] H. Niederreiter and C. Xing. Rational points on curves over finite fields: theory andapplications, volume 285 of London Mathematical Society Lecture Note Series.Cambridge University Press, Cambridge, 2001. [20, 317, 333, 364, 367, 368,372, 458, 608, 609, 612]
[1698] H. Niederreiter and C. Xing. Algebraic geometry in coding theory and cryptography.
[1699] H. Niederreiter and C. P. Xing. Low-discrepancy sequences and global functionfields with many rational places. Finite Fields Appl., 2:241–273, 1996. [382,383]
[1700] H. Niederreiter and C. P. Xing. Quasirandom points and global function fields. In Fi-nite fields and applications (Glasgow, 1995), volume 233 of London Math. Soc.Lecture Note Ser., pages 269–296. Cambridge University Press, Cambridge,1996. [379, 383]
[1701] H. Niederreiter and C. P. Xing. Towers of global function fields with asymptoticallymany rational places and an improvement on the Gilbert-Varshamov bound.Math. Nachr., 195:171–186, 1998. [611, 612]
[1702] H. Niederreiter, C. P. Xing, and K. Y. Lam. A new construction of algebraic-geometry codes. Appl. Algebra Engrg. Comm. Comput., 9:373–381, 1999. [605,606, 612]
[1703] Y. Niho. Multi-valued cross-correlation functions between two maximal linear recur-sive sequences. PhD thesis, Univ. Southern California, 1972. [213]
[1704] Y. Niitsuma. Counting points of the curve y2 = x12 + a over a finite field. Tokyo J.Math., 31(1):59–94, 2008. [106, 118]
[1705] A. Nilli. On the second eigenvalue of a graph. Discrete Math., 91(2):207–210, 1991.[545]
[1706] A. Nilli. Tight estimates for eigenvalues of regular graphs. Electron. J. Combin.,11(1):Note 9, 4 pp. (electronic), 2004. [537, 538, 545]
[1707] NIST. Digital signature standard (DSS). Federal Information Processing StandardsPublication 186-3, National Institute of Standards and Technology, July 2009.[667, 669]
[1708] I. Niven. Fermat’s theorem for matrices. Duke Math. J., 15:823–826, 1948. [416,424]
[1709] J.-S. No, S. W. Golomb, G. Gong, H.-K. Lee, and P. Gaal. Binary pseudorandomsequences of period 2n − 1 with ideal autocorrelation. IEEE Transactions onInformation Theory, 44(2):814–817, 1998. [639, 640, 647]
[1710] W. Nobauer. On the length of cycles of polynomial permutations. In Contributionsto general algebra, 3 (Vienna, 1984), pages 265–274. Holder-Pichler-Tempsky,Vienna, 1985. [184, 185]
[1711] A. W. Nordstrom and J. P. Robinson. An optimum nonlinear code. Informationand Control, 11:613–616, 1967. [601, 602]
[1712] M. Noro and K. Yokoyama. Yet another practical implementation of polynomialfactorization over finite fields. In ISSAC ’02: Proceedings of the 2002 Inter-national Symposium on Symbolic and Algebraic Computation, pages 200–206.ACM Press, 2002. [306, 311]
[1713] Nowicki, A. and Secomski, W. and Litniewski, J. and Trots, I. and Lewin, P.A.On the application of signal compression using Golay’s codes sequences inultrasonic diagnostic. Arch. Acoustics, 28:313–324, 2003. [695, 701]
[1714] K. Nyberg. Perfect nonlinear S-boxes. In Advances in cryptology—EUROCRYPT’91 (Brighton, 1991), volume 547 of Lecture Notes in Comput. Sci., pages 378–386. Springer, Berlin, 1991. [206, 213]
[1715] K. Nyberg. Differentially uniform mappings for cryptography. In Advances incryptology—EUROCRYPT ’93 (Lofthus, 1993), volume 765 of Lecture Notes
[1716] K. Nyberg and L. R. Knudsen. Provable security against differential cryptanalysis.In Advances in cryptology—CRYPTO ’92 (Santa Barbara, CA, 1992), volume740 of Lecture Notes in Comput. Sci., pages 566–574. Springer, Berlin, 1993.[206, 207, 213]
[1717] A. P. Ogg. Abelian curves of small conductor. J. Reine Angew. Math., 226:204–215,1967. [252, 255]
[1718] A. P. Ogg. Rational points of finite order on elliptic curves. Invent. Math., 12:105–111, 1971. [251, 255]
[1719] E. Okamoto and K. Nakamura. Evaluation of public key cryptosystems proposedrecently. In Proc 1986’s Symposium of cryptography and information security,volume D1, 1986. [652]
[1720] C. M. O’Keefe and T. Penttila. Ovoids of PG(3, 16) are elliptic quadrics. J. Geom.,38(1-2):95–106, 1990. [501]
[1721] C. M. O’Keefe and T. Penttila. Ovoids of PG(3, 16) are elliptic quadrics. II. J.Geom., 44(1-2):140–159, 1992. [501]
[1722] C. M. O’Keefe, T. Penttila, and G. F. Royle. Classification of ovoids in PG(3, 32).J. Geom., 50(1-2):143–150, 1994. [501]
[1723] B. Omidi Koma, D. Panario, and Q. Wang. The number of irreducible polynomialsof degree n over Fq with given trace and constant terms. Discrete Math.,310(8):1282–1292, 2010. [45, 49]
[1724] R. Omrani, O. Moreno, and P. V. Kumar. Improved Johnson bounds for opticalorthogonal codes with λ > 1 and some optimal constructions. In Proc. Int.Symp. Inform. Theory, pages 259–263, September 2005. [697, 701]
[1725] H. Ong, C. Schnorr, and A. Shamir. Signatures through approximate representationsby quadratic forms. In Advances in cryptology, Crypto ’83, pages 117–131.Plenum Publ., 1984. [649, 651]
[1726] H. Ong, C.-P. Schnorr, and A. Shamir. Efficient signature schemes based on poly-nomial equations (preliminary version). In Advances in cryptology (Santa Bar-bara, Calif., 1984), volume 196 of Lecture Notes in Comput. Sci., pages 37–46.Springer, Berlin, 1985. [651]
[1727] F. Oort. Moduli of abelian varieties and Newton polygons. C. R. Acad. Sci. ParisSer. I Math., 312(5):385–389, 1991. [400, 402]
[1728] O. Ore. Contributions to the theory of finite fields. Trans. Amer. Math. Soc.,36(2):243–274, 1934. [35, 40]
[1729] A. Ostafe. Multivariate permutation polynomial systems and nonlinear pseudoran-dom number generators. Finite Fields Appl., 16(3):144–154, 2010. [188, 286,289]
[1730] A. Ostafe. Pseudorandom vector sequences derived from triangular polynomialsystems with constant multipliers. In Arithmetic of finite fields, volume 6087of Lecture Notes in Comput. Sci., pages 62–72. Springer, Berlin, 2010. [286,289]
[1731] A. Ostafe. Pseudorandom vector sequences of maximal period generated by poly-nomial dynamical systems. To appear in Designs, Codes and Cryptography,2011. [284, 287, 289]
[1732] A. Ostafe, E. Pelican, and I. E. Shparlinski. On pseudorandom numbers frommultivariate polynomial systems. Finite Fields Appl., 16(5):320–328, 2010.[283, 285, 289]
Miscellaneous applications 795
[1733] A. Ostafe and I. Shparlinski. On the waring problem with dickson polynomials infinite fields. Proc. Amer. Math. Soc., 8, 2011. [170]
[1734] A. Ostafe and I. E. Shparlinski. On the degree growth in some polynomial dynam-ical systems and nonlinear pseudorandom number generators. Math. Comp.,79(269):501–511, 2010. [284, 286, 289]
[1735] A. Ostafe and I. E. Shparlinski. On the length of critical orbits of stable quadraticpolynomials. Proc. Amer. Math. Soc., 138(8):2653–2656, 2010. [143, 287, 288,289]
[1736] A. Ostafe and I. E. Shparlinski. Pseudorandom numbers and hash functions fromiterations of multivariate polynomials. Cryptogr. Commun., 2(1):49–67, 2010.[284, 286, 289]
[1737] A. Ostafe and I. E. Shparlinski. On the Waring problem with Dickson polynomialsin finite fields. Proc. Amer. Math. Soc., 139(11):3815–3820, 2011. [134]
[1738] A. Ostafe, I. E. Shparlinski, and A. Winterhof. On the generalized joint linearcomplexity profile of a class of nonlinear pseudorandom multisequences. Adv.Math. Commun., 4(3):369–379, 2010. [286, 289]
[1739] A. Ostafe, I. E. Shparlinski, and A. Winterhof. Multiplicative character sums of aclass of nonlinear recurrence vector sequences. To appear in Intern. J. NumberTheory, 2011. [286, 289]
[1740] A. M. Ostrowski. Uber die Bedeutung der Theorie der konvexen Polyeder fur dieformale Algebra. Jahresber. Deutsch. Math.-Verein., 30(2):98–99, 1921. Talkgiven at Der Deutsche Mathematikertag vom 18–24 September 1921 in Jena.[307, 311, 795]
[1741] A. M. Ostrowski. On the significance of the theory of convex polyhedra for formalalgebra. ACM SIGSAM Bull., 33(1):5, 1999. Translated from [1740]. [307,311]
[1742] L. J. Paige. Neofields. Duke Math. J., 16:39–60, 1949. [16, 20]
[1743] R. Paley. On orthogonal matrices. J. Math. Phys., Mass. Inst. Techn., 12:311–320,1933. [135]
[1744] R. E. A. C. Paley. On orthogonal matrices. J. Math. Phys, 12:311–320, 1933. [547]
[1745] V. Y. Pan. Structured matrices and polynomials. Birkhauser Boston Inc., Boston,MA, 2001. Unified superfast algorithms. [434, 436]
[1746] D. Panario and A.Viola. Analysis of Rabin’s polynomial irreducibility test. In Proc.Latin American Theoretical Informatics Conference (LATIN), volume 1380 ofLecture Notes in Computer Science, Berlin, 1998. Springer-Verlag. [295, 299]
[1747] D. Panario, B. Pittel, B. Richmond, and A. Viola. Analysis of Rabin’s irreducibilitytest for polynomials over finite fields. Random Structures & Algorithms, 19(3-4):525–551, 2001. [295, 299]
[1748] D. Panario and B. Richmond. Analysis of Ben-Or’s polynomial irreducibility test.Random Structures and Algorithms, pages 439–456, 1998. [296, 299]
[1749] D. Panario, A. Sakzd, B. Stevens, and Q. Wang. Two new measures for permuta-tions: Ambiguity and deficiency. Preprint, 2011. [185]
[1750] D. Panario, O. Sosnovski, B. Stevens, and Q. Wang. Divisibility of polynomials overfinite fields and combinatorial applications. to appear in Des. Codes Cryptogr.[520, 529, 531]
[1751] D. Panario, B. Stevens, and Q. Wang. Ambiguity and deficiency in costas arrays andapn permutations. In LATIN 2010: Theoretical Informatics, volume 6034 ofLecture Notes in Computer Science, 2010, pages 397–406. Dekker, New York,
796 Handbook of Finite Fields
2010. [185]
[1752] D. Panario and D. Thomson. Efficient pth root computations in finite fields ofcharacteristic p. Des. Codes Cryptogr., 50(3):351–358, 2009. [39]
[1753] G. Panella. Caratterizzazione delle quadriche di uno spazio (tridimensionale) linearesopra un corpo finito. Boll. Un. Mat. Ital. (3), 10:507–513, 1955. [500]
[1754] Y. H. Park and J. B. Lee. Permutation polynomials and group permutation poly-nomials. Bull. Austral. Math. Soc., 63(1):67–74, 2001. [176, 177, 185]
[1755] F. Parvaresh and A. Vardy. Correcting errors beyond the Guruswami-Sudan radiusin polynomial time. In Proceedings of the 46th Annual IEEE Symposium onFoundations of Computer Science, 2005, pages 285–294, Oct. 2005. [599, 602]
[1756] E. Pasalic. On cryptographically significant mappings over GF(2n). In Arithmeticof finite fields, volume 5130 of Lecture Notes in Comput. Sci., pages 189–204.Springer, Berlin, 2008. [182, 185]
[1757] E. Pasalic and P. Charpin. Some results concerning cryptographically significantmappings over GF(2n). Des. Codes Cryptogr., 57(3):257–269, 2010. [182, 185]
[1758] J. Patarin. Cryptanalysis of the Matsumoto and Imai public key scheme of Eu-rocrypt ’88. In Advances in cryptology—CRYPTO ’95 (Santa Barbara, CA,1995), volume 963 of Lecture Notes in Comput. Sci., pages 248–261. Springer,Berlin, 1995. [653, 659]
[1759] J. Patarin. Asymmetric cryptography with a hidden monomial and a candidatealgorithm for ' 64 bits asymmetric signatures. In Advances in cryptology—CRYPTO ’96 (Santa Barbara, CA), volume 1109 of Lecture Notes in Comput.Sci., pages 45–60. Springer, Berlin, 1996. [650]
[1760] J. Patarin. Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP):two new families of asymmetric algorithms. In 1996, volume 1070 of LectureNotes in Computer Science, pages 33–48. Ueli Maurer, ed., 1996. ExtendedVersion: http://www.minrank.org/hfe.pdf. [651]
[1761] J. Patarin. The oil and vinegar signature scheme. Dagstuhl Workshop on Cryptog-raphy, September 1997, 1997. [654]
[1762] J. Patarin, N. Courtois, and L. Goubin. FLASH, a fast multivariate signature algo-rithm. In Topics in cryptology—CT-RSA 2001 (San Francisco, CA), volume2020 of Lecture Notes in Comput. Sci., pages 298–307. Springer, Berlin, 2001.[656]
[1763] J. Patarin, L. Goubin, and N. Courtois. C∗−+ and HM : Variations aroundtwo schemes of T. Matsumoto and H. Imai. In Asiacrypt 1998, volume1514 of LNCS, pages 35–49. Kazuo Ohta and Dingyi Pei, editors, Springer,1998. Extended Version: http://citeseer.nj.nec.com/patarin98plusmn.html. [656, 660]
[1764] J. Patarin, L. Goubin, and N. Courtois. Improved algorithms for Isomorphismsof Polynomials. In 1998, volume 1403 of Lecture Notes in Computer Sci-ence, pages 184–200. Kaisa Nyberg, ed., 1998. Extended Version: http:
//www.minrank.org/ip6long.ps. [651]
[1765] K. G. Paterson. Applications of exponential sums in communications theory [invitedpaper]. In Cryptography and coding (Cirencester, 1999), volume 1746 of LectureNotes in Comput. Sci., pages 1–24. Springer, Berlin, 1999. [143, 144]
[1766] S. Paulus and H.-G. Ruck. Real and imaginary quadratic representations of hyper-elliptic function fields. Math. Comp., 68(227):1233–1241, 1999. [356, 360]
[1767] S. E. Payne. Spreads, flocks, and generalized quadrangles. J. Geom., 33(1-2):113–
Miscellaneous applications 797
128, 1988. [480, 486]
[1768] F. Pellarin. Values of certain l-series in positive characteristic. 2011. [458]
[1769] A. Pellet. Sur les fonctions irreducibles suivant un module premier. C.R. Acad. Sci.Paris, 93:1065–1066, 1881. [31, 34]
[1770] A. E. Pellet. On irreducible functions to a prime modulus and a modular function.(Sur les fonctions irreductibles suivant un module premier et une fonctionmodulaire.). C. R. Acad. Sci. Paris., 70:328–330, 1870. [40]
[1771] A. E. Pellet. On the decomposition of an integral function into irreducible factorswith respect to a prime modulus. (Sur la decomposition d’une fonction entiereen facteurs irreductibles suivant un module premier.). C. R. Acad. Sci. Paris.,86:1071–1072., 1878. [35, 36, 38, 41]
[1772] R. Pellikaan, B.-Z. Shen, and G. J. M. van Wee. Which linear codes are algebraic-geometric? IEEE Trans. Inform. Theory, 37:583–602, 1991. [610, 612]
[1773] T. Penttila and G. F. Royle. Sets of type (m,n) in the affine and projective planesof order nine. Des. Codes Cryptogr., 6(3):229–245, 1995. [484, 486]
[1774] T. Penttila and B. Williams. Ovoids of parabolic spaces. Geom. Dedicata, 82(1-3):1–19, 2000. [233, 234]
[1775] G. I. Perel′muter. Estimate of a sum along an algebraic curve. Mat. Zametki,5:373–380, 1969. [125, 127]
[1776] C. Pernet and A. Storjohann. Faster algorithms for the characteristic polynomial.In ISSAC 2007, pages 307–314. ACM, New York, 2007. [431, 436]
[1777] L. Perret. A fast cryptanalysis of the isomorphism of polynomials with one secretproblem. In Advances in cryptology—EUROCRYPT 2005, volume 3494 ofLecture Notes in Comput. Sci., pages 354–370. Springer, Berlin, 2005. [651]
[1778] W. W. Peterson. Error-correcting codes. The M.I.T. Press, Cambridge, Mass., 1961.[561, 574, 592, 602]
[1779] W. W. Peterson and E. J. Weldon, Jr. Error-correcting codes. The M.I.T. Press,Cambridge, Mass.-London, second edition, 1972. [561, 573, 574, 581, 586, 588,591, 593, 596, 597, 602]
[1780] K. Petr. Uber die irreduzibilitat eines polynoms mit ganzzahligen koeffizienten nacheinem primzahlmodul. Casopis pro pestovan´i matematiky a fysiky, 66:85–94,1937. [294, 299]
[1781] D. Pierce and M. J. Kallaher. A note on planar functions and their planes. Bull.Inst. Combin. Appl., 42:53–75, 2004. [231, 234]
[1782] J. Pila. Frobenius maps of abelian varieties and finding roots of unity in finite fields.Math. Comp., 55(192):745–763, 1990. [404, 406]
[1783] G. Pirsic, J. Dick, and F. Pillichshammer. Cyclic digital nets, hyperplane nets, andmultivariate integration in Sobolev spaces. SIAM J. Numer. Anal., 44:385–411,2006. [378, 383]
[1784] N. L. Pitcher. Efficient point-counting on genus-2 hyperelliptic curves. ProQuestLLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Illinois at Chicago.[359, 360]
[1785] A. K. Pizer. Ramanujan graphs and Hecke operators. Bull. Amer. Math. Soc.(N.S.), 23(1):127–137, 1990. [545]
[1786] D. A. Plaisted. New NP-hard and NP-complete polynomial and integer divisibilityproblems. Theoret. Comput. Sci., 13:125–138, 1984. [309, 311]
[1787] V. Pless. Q-codes. J. Combin. Theory Ser. A, 43(2):258–276, 1986. [581, 602]
798 Handbook of Finite Fields
[1788] V. Pless. Duadic codes and generalizations. In Eurocode ’92 (Udine, 1992), volume339 of CISM Courses and Lectures, pages 3–15. Springer, Vienna, 1993. [581,602]
[1789] V. Pless. Introduction to the theory of error-correcting codes. Wiley-InterscienceSeries in Discrete Mathematics and Optimization. John Wiley & Sons Inc.,New York, third edition, 1998. A Wiley-Interscience Publication. [19, 20]
[1790] V. S. Pless, W. C. Huffman, and R. A. Brualdi, editors. Handbook of coding theory.Vol. I, II. North-Holland, Amsterdam, 1998. [19, 20, 561, 582, 583, 590, 591,602]
[1791] L. Poinsot. Reflexions sur les principes fondamentaux de la theorie des nombres.Journal de mathematiques pures et appliquees, 10:1–101, 1845. [39]
[1792] P. Polito and O. Polverino. Linear blocking sets in PG(2, q4). Australas. J. Combin.,26:41–48, 2002. [472, 475]
[1793] P. Pollack. An explicit approach to hypothesis H for polynomials over a finite field.In Anatomy of Integers, volume 46 of CRM Proc. Lecture Notes, pages 259–273.Amer. Math. Soc., Providence, 2008. [410, 414]
[1794] P. Pollack. A polynomial analogue of the twin primes conjecture. Proc. Amer.Math. Soc., 136(11):3775–3784, 2008. [410, 414]
[1795] P. Pollack. Simultaneous prime specializations of polynomials over finite fields. Proc.Lond. Math. Soc., 97(3):545–567, 2008. [410, 414]
[1796] P. Pollack. Revisiting gauss’s analogue of the prime number theorem for polynomialsover finite fields. Finite Fields Appl., 16(4):290–299, 2010. [408, 414]
[1797] J. M. Pollard. Monte Carlo methods for index computation (mod p). Math. Comp.,32(143):918–924, 1978. [629, 634]
[1798] J. M. Pollard and C.-P. Schnorr. An efficient solution of the congruence x2+ky2 = m(mod n). IEEE Trans. Inform. Theory, 33(5):702–709, 1987. [651]
[1799] O. Polverino. Small minimal blocking sets and complete k-arcs in PG(2, p3). DiscreteMath., 208/209:469–476, 1999. Combinatorics (Assisi, 1996). [474, 475]
[1800] O. Polverino. Small blocking sets in PG(2, p3). Des. Codes Cryptogr., 20(3):319–324,2000. [472, 474, 475]
[1801] O. Polverino and L. Storme. Small minimal blocking sets in PG(2, q3). EuropeanJ. Combin., 23(1):83–92, 2002. [474, 475]
[1802] B. Poonen. Local height functions and the Mordell-Weil theorem for Drinfeld mod-ules. Compositio Math., 97(3):349–368, 1995. [452]
[1803] A. G. Postnikov. Ergodic problems in the theory of congruences and of Diophantineapproximations. Proceedings of the Steklov Institute of Mathematics, No. 82(1966). Translated from the Russian by B. Volkmann. American MathematicalSociety, Providence, R.I., 1967. [282, 289]
[1804] A. Pott, Y. Tan, T. Feng, and S. Ling. Association schemes arising from bentfunctions. Des. Codes Cryptogr., 59(1–3):319–331, Apr. 2011. [219, 224]
[1805] B. Preneel et al. NESSIE security report. Technical Report D20-v2, New EuropeanSchemes for Signatures, Integrity, and Encryption, 2003. [666]
[1806] F. P. Preparata. A class of optimum nonlinear double-error-correcting codes. In-formation and Control, 13:378–400, 1968. [601, 602]
[1807] R. Pries and H. J. Zhu. p-rank stratification of artin-schreier curves. Ann. Inst.Fourier, to appear. [401, 402]
[1808] M. Ptashne. A genetic switch: Phage lambda and higher organisms. 1992. [687,
Miscellaneous applications 799
692]
[1809] S. Qi. On diagonal equations over finite fields. Finite Fields Appl., 3(2):175–179,1997. [165, 170]
[1810] G. Quenell. Spectral diameter estimates for k-regular graphs. Adv. Math.,106(1):122–148, 1994. [545]
[1811] M. Rabin. Probabilistic algorithms in finite fields. SIAM Journal on Computing,9(2):273–280, 1980. [295, 299]
[1812] R. Raghavendran. Finite associative rings. Compositio Math., 21:195–229, 1969.[17]
[1813] J. Rajsski and J. Tyszer. Primitive polynomials over gf(2) of degree upto 660 withuniformly distributed coefficients. J. Elect. Testing, 19(6):645–657, 2003. [67,68]
[1814] J. Ray and P. Koopman. Efficient high hamming distance crcs for embedded net-works. In Dependable Systems and Networks, 2006. DSN 2006. InternationalConference on, pages 3 –12, june 2006. [524, 531]
[1815] L. Redei. Luckenhafte Polynome uber endlichen Korpern. Birkhauser Verlag,Basel, 1970. Lehrbucher und Monographien aus dem Gebiete der exaktenWissenschaften, Mathematische Reihe, Band 42. [468, 475]
[1816] L. Redei. Lacunary polynomials over finite fields. North-Holland Publishing Co.,Amsterdam, 1973. Translated from the German by I. Foldes. [20, 468, 469,470, 475]
[1817] R. Ree. Proof of a conjecture of S. Chowla. J. Number Theory, 3:210–212, 1971.[40]
[1818] I. S. Reed and G. Solomon. Polynomial codes over certain finite fields. J. Soc.Indust. Appl. Math., 8:300–304, 1960. [579, 601, 602]
[1819] O. Reingold, S. Vadhan, and A. Wigderson. Entropy waves, the zig-zag graphproduct, and new constant-degree expanders. Ann. of Math. (2), 155(1):157–187, 2002. [542, 543, 545]
[1820] D. Ren, Q. Sun, and P. Yuan. Number of zeros of diagonal polynomials over finitefields. Finite Fields Appl., 7(1):197–204, 2001. Dedicated to Professor ChaoKo on the occasion of his 90th birthday. [166, 170]
[1821] G. Rhin. Repartition modulo 1 dans un corps de series formelles sur un corps fini.Dissertationes Math. (Rozprawy Mat.), 95:75, 1972. [46, 49]
[1822] C. Ritzenthaler. Optimal curves of genus 1,2 and 3. Publ. Math. Besancon (PMB),2011. [364, 367]
[1823] R. L. Rivest. Permutation polynomials modulo 2w. Finite Fields Appl., 7(2):287–292, 2001. [185]
[1824] R. L. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signaturesand public-key cryptosystems. Comm. ACM, 21(2):120–126, 1978. [628, 634]
[1825] A. M. Robert. The Gross-Koblitz formula revisited. Rend. Sem. Mat. Univ. Padova,105:157–170, 2001. [110, 118]
[1826] J. A. G. Roberts and F. Vivaldi. A combinatorial model for reversible rational mapsover finite fields. Nonlinearity, 22(8):1965–1982, 2009. [282, 289]
[1827] M. Roitman. On Zsigmondy primes. Proc. Amer. Math. Soc., 125(7):1913–1919,1997. [45, 49]
[1828] A. Rojas-Leon. Estimates for singular multiplicative character sums. Int. Math.Res. Not., (20):1221–1234, 2005. [123, 124, 127, 156, 158]
800 Handbook of Finite Fields
[1829] A. Rojas-Leon. Purity of exponential sums on An. Compos. Math., 142(2):295–306,2006. [121, 127]
[1830] A. Rojas-Leon and D. Wan. Moment zeta functions for toric Calabi-Yau hypersur-faces. Commun. Number Theory Phys., 1(3):539–578, 2007. [154, 156, 158]
[1831] A. Rojas-Leon and D. Wan. Improvements of the Weil bound for Artin-Schreiercurves. Math. Ann., 2011, to appear. [155, 158]
[1832] S. Rønjom and T. Helleseth. A new attack on the filter generator. IEEE Trans.Inform. Theory, 53(5):1752–1758, 2007. [201, 204]
[1833] L. Ronyai and T. SzHonyi. Planar functions over finite fields. Combinatorica,9(3):315–320, 1989. [232, 234]
[1834] L. A. Rosati. Unitals in Hughes planes. Geom. Dedicata, 27(3):295–299, 1988. [484,486]
[1835] M. Y. Rosenbloom and M. A. Tsfasman. Codes for the m-metric. Problems Inform.Transmission, 33:45–52, 1997. [375, 383]
[1836] R. Roth. Introduction to coding theory. Cambridge University Press, Cambridge,2006. [561, 580, 584, 591, 602]
[1837] R. M. Roth. Maximum-rank array codes and their application to crisscross errorcorrection. IEEE Trans. Inform. Theory, 37(2):328–336, 1991. [699, 701]
[1838] O. S. Rothaus. On “bent” functions. J. Combinatorial Theory Ser. A, 20(3):300–305, 1976. [216, 224]
[1839] I. F. Rua, E. F. Combarro, and J. Ranilla. Classification of semifields of order 64.J. Algebra, 322(11):4011–4029, 2009. [227, 229]
[1840] K. Rubin and A. Silverberg. Supersingular abelian varieties in cryptology. In Ad-vances in cryptology—CRYPTO 2002, volume 2442 of Lecture Notes in Com-put. Sci., pages 336–353. Springer, Berlin, 2002. [359]
[1841] I. M. Rubio and C. J. Corrada-Bravo. Cyclic decomposition of permutations of finitefields obtained using monomials. In Finite fields and applications, volume 2948of Lecture Notes in Comput. Sci., pages 254–261. Springer, Berlin, 2004. [184,185]
[1842] I. M. Rubio, G. L. Mullen, C. Corrada, and F. N. Castro. Dickson permutationpolynomials that decompose in cycles of the same length. In Finite fields andapplications, volume 461 of Contemp. Math., pages 229–239. Amer. Math. Soc.,Providence, RI, 2008. [184, 185]
[1843] H.-G. Ruck. A note on elliptic curves over finite fields. Math. Comp., 49(179):301–304, 1987. [342, 351]
[1844] H.-G. Ruck. A note on elliptic curves over finite fields. Mathematics of Computation,49(179):301–304, July 1987. [675]
[1845] H.-G. Ruck and H. Stichtenoth. A characterization of Hermitian function fields overfinite fields. J. Reine Angew. Math., 457:185–188, 1994. [166, 170, 366, 367]
[1846] M. Rudnev. An improved sumproduct inequality in fields of prime order. Int. Math.Res. Notices. [129]
[1847] A. Rudra. Limits to list decoding of random codes. IEEE Trans. InformationTheory, IT-57:1398–1408, 2011. [599, 602]
[1848] R. A. Rueppel. Analysis and design of stream ciphers. Communications and ControlEngineering Series. Springer-Verlag, Berlin, 1986. With a foreword by JamesL. Massey. [270, 271, 274, 275, 281]
[1849] R. A. Rueppel. Stream ciphers. In Contemporary cryptology, pages 65–134. IEEE,
Miscellaneous applications 801
New York, 1992. [270, 271, 273, 281]
[1850] W. M. Ruppert. Reduzibilitat ebener Kurven. J. Reine Angew. Math., 369:167–191,1986. [304, 305, 311]
[1851] W. M. Ruppert. Reducibility of polynomials f(x, y) modulo p. J. Number Theory,77(1):62–70, 1999. [304, 311]
[1852] J. J. Rushanan. Topics in integral matrices and abelian group codes: generalizedQ-codes. ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–CaliforniaInstitute of Technology. [581, 602]
[1853] F. Ruskey, C. R. Miers, and J. Sawada. The number of irreducible polynomialsand Lyndon words with given trace. SIAM J. Discrete Math., 14(2):240–245(electronic), 2001. [25, 30]
[1854] I. Z. Ruzsa. Essential components. Proc. London Math. Soc. (3), 54(1):38–56, 1987.[148]
[1855] W. E. Ryan and S. Lin. Channel codes. Cambridge University Press, Cambridge,2009. Classical and modern. [561, 602]
[1856] A. Sackmann, M. Heiner, and I. Koch. Application of petri net based analysistechniques to signal transduction pathways. BMC Bioinformatics, 7(1):482,2006. [685]
[1857] H. Sadjadpour, N. Sloane, M. Salehi, and G. Nebe. Interleaver design for turbocodes. Selected Areas in Communications, IEEE Journal on, 19(5):831 –837,may 2001. [521, 523, 531]
[1858] J. Saez-Rodriguez, L. G. Alexopoulos, J. Epperlein, R. Samaga, D. A. Lauffenburger,S. Klamt, and P. K. Sorger. Discrete logic modelling as a means to link proteinsignalling networks with functional analysis of mammalian signal transduction.Molecular Systems Biology, 5, Dec. 2009. [683, 692]
[1859] O. Sahin, H. Frohlich, C. Lobke, U. Korf, S. Burmester, M. Majety, J. Mattern,I. Schupp, C. Chaouiya, D. Thieffry, A. Poustka, S. Wiemann, T. Beissbarth,and D. Arlt. Modeling erbb receptor-regulated g1/s transition to find noveltargets for de novo trastuzumab resistance. BMC Systems Biology, 3(1):1,2009. [683, 692]
[1860] S. Sakata. n-dimensional Berlekamp-Massey algorithm for multiple arrays and con-struction of multivariate polynomials with preassigned zeros. In Applied alge-bra, algebraic algorithms and error-correcting codes (Rome, 1988), volume 357of Lecture Notes in Comput. Sci., pages 356–376. Springer, Berlin, 1989. [275,281]
[1861] S. Sakata. Extension of the Berlekamp-Massey algorithm to N dimensions. Inform.and Comput., 84(2):207–239, 1990. [275, 281]
[1862] A. Salagean. On the computation of the linear complexity and the k-error linearcomplexity of binary sequences with period a power of two. IEEE Trans.Inform. Theory, 51(3):1145–1150, 2005. [274, 281]
[1863] R. Sandler. The collineation groups of some finite projective planes. Portugal.Math., 21:189–199, 1962. [227, 229]
[1864] P. Sarnak. Some applications of modular forms, volume 99 of Cambridge Tracts inMathematics. Cambridge University Press, Cambridge, 1990. [533, 545]
[1865] P. Sarnak. Kloosterman, quadratic forms and modular forms. Nieuw Arch. Wiskd.(5), 1(4):385–389, 2000. [111, 118]
[1866] D. Sarwate and M. Pursley. Crosscorrelation properties of pseudorandom and re-lated sequences. Proceedings of the IEEE, 68(5):593–619, 1980. [264]
802 Handbook of Finite Fields
[1867] D. V. Sarwate. An upper bound on the aperiodic autocorrelation function for amaximal-length sequence. IEEE Trans. Inform. Theory, 30(4):685–687, 1984.[694, 701]
[1868] T. Sasaki, T. Saito, and T. Hilano. Analysis of approximate factorization algorithm.I. Japan J. Indust. Appl. Math., 9(3):351–368, 1992. [304, 311]
[1869] T. Sasaki and M. Sasaki. A unified method for multivariate polynomial factoriza-tions. Japan J. Indust. Appl. Math., 10(1):21–39, 1993. [304, 311]
[1870] T. Sasaki, M. Suzuki, M. Kolar, and M. Sasaki. Approximate factorization of multi-variate polynomials and absolute irreducibility testing. Japan J. Indust. Appl.Math., 8(3):357–375, 1991. [304, 311]
[1871] T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and itspoint counting. J. Ramanujan Math. Soc., 15(4):247–270, 2000. [406, 670]
[1872] T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete logalgorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul.,47(1):81–92, 1998. [351]
[1873] A. Scheerhorn. Trace- and norm-compatible extensions of finite fields. Appl. AlgebraEngrg. Comm. Comput., 3(3):199–209, 1992. [94]
[1874] A. Scheerhorn. Spur-kompatible Polynomfolgen uber endlichen Korpern. InSeminaire Lotharingien de Combinatoire (Thurnau, 1992), volume 1993/33 ofPrepubl. Inst. Rech. Math. Av., pages 73–79. Univ. Louis Pasteur, Strasbourg,1993. [94]
[1875] A. Scheerhorn. Iterated constructions of normal bases over finite fields. In Finitefields: theory, applications, and algorithms (Las Vegas, NV, 1993), volume 168of Contemp. Math., pages 309–325. Amer. Math. Soc., Providence, RI, 1994.[94, 238, 242]
[1876] A. Scheerhorn. Dickson polynomials and completely normal elements over finitefields. In Applications of finite fields (Egham, 1994), volume 59 of Inst. Math.Appl. Conf. Ser. New Ser., pages 47–55. Oxford Univ. Press, New York, 1996.[95]
[1877] A. Scheerhorn. Dickson polynomials, completely normal polynomials and the cyclicmodule structure of specific extensions of finite fields. Des. Codes Cryptogr.,9(2):193–202, 1996. [95]
[1878] A. Schinzel. Polynomials with special regard to reducibility, volume 77 of Encyclo-pedia of Mathematics and its Applications. Cambridge University Press, 2000.[304, 311]
[1879] B. Schmidt. Characters and cyclotomic fields in finite geometry, volume 1797 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. [512, 513, 519]
[1880] K. Schmidt. Dynamical systems of algebraic origin, volume 128 of Progress inMathematics. Birkhauser Verlag, Basel, 1995. [282, 289]
[1881] W. Schmidt. Equations over finite fields: an elementary approach. Kendrick Press,Heber City, UT, second edition, 2004. [19, 20]
[1882] W. M. Schmidt. Equations over finite fields. An elementary approach. LectureNotes in Mathematics, Vol. 536. Springer-Verlag, Berlin, 1976. [19, 20, 141,151, 152, 157, 158]
[1883] W. M. Schmidt. Construction and estimation of bases in function fields. J. NumberTheory, 39(2):181–224, 1991. [275, 281]
[1884] J. Scholten and H. J. Zhu. Families of supersingular curves in characteristic 2. Math.Res. Lett., 9(5-6):639–650, 2002. [402]
Miscellaneous applications 803
[1885] J. Scholten and H. J. Zhu. Hyperelliptic curves in characteristic 2. Int. Math. Res.Not., (17):905–917, 2002. [399, 402]
[1886] J. Scholten and H. J. Zhu. Slope estimates of Artin-Schreier curves. CompositioMath., 137(3):275–292, 2003. [399, 402]
[1887] R. A. Scholtz. The spread spectrum concept. IEEE Trans. Commun., COM-25(8):748–755, 1977. [695, 698, 701]
[1888] R. A. Scholtz and L. R. Welch. GMW sequences. IEEE Trans. Inform. Theory,30(3):548–553, 1984. [265]
[1889] R. Schoof. Elliptic curves over finite fields and the computation of square roots modp. Math. Comp., 44(170):483–494, 1985. [404, 406]
[1890] R. Schoof. Elliptic curves over finite fields and the computation of square roots modp. Mathematics of Computation, 44(170), Apr. 1985. [669]
[1891] R. Schoof. Algebraic curves over F2 with many rational points. J. Number Theory,41(1):6–14, 1992. [368, 372]
[1892] R. Schurer. A new lower bound on the t-parameter of (t, s)-sequences. In MonteCarlo and quasi-Monte Carlo methods 2006, pages 623–632. Springer-Verlag,Berlin, 2008. [379, 383]
[1893] M. P. Schutzenberger. A non-existence theorem for an infinite family of symmetricalblock designs. Ann. Eugenics, 14:286–287, 1949. [513, 519]
[1894] S. Schwarz. Contribution a la recluctibilite des polynomes dans la theorie des con-gruences. Vestnik Knalovske ceske spol. nauk., pages 1–7, 1939. [294, 299]
[1895] S. Schwarz. On the reducibility of polynomials over a finite field. Quart. J. Math.Oxford, 2(7):110–124, 1956. [294, 299]
[1896] B. Segre. Ovals in a finite projective plane. Canad. J. Math., 7:414–416, 1955. [497]
[1897] B. Segre. On complete caps and ovaloids in three-dimensional Galois spaces ofcharacteristic two. Acta Arith., 5:315–332 (1959), 1959. [500]
[1898] B. Segre. Introduction to Galois geometries. Atti Accad. Naz. Lincei Mem. Cl. Sci.Fis. Mat. Natur. Sez. I (8), 8:133–236, 1967. [497]
[1899] I. Semaev. Construction of polynomials, irreducible over a finite field, with linearlyindependent roots. Mat. Sbornik, 135(4):520–532, 1988. In Russian; Englishtranslation in Math. USSR-Sbornik, 63(2):507-519, 1989. [297, 299]
[1900] I. A. Semaev. Evaluation of discrete logarithms in a group of p-torsion points of anelliptic curve in characteristic p. Math. Comp., 67(221):353–356, 1998. [351]
[1901] G. Seroussi and A. Lempel. Factorization of symmetric matrices and trace-orthogonal bases in finite fields. SIAM J. Comput., 9(4):758–767, 1980. [73,79]
[1902] G. Seroussi and A. Lempel. On symmetric representations of finite fields. SIAM J.Algebraic Discrete Methods, 4(1):14–21, 1983. [418, 424]
[1903] J.-P. Serre. Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier,Grenoble, 6:1–42, 1955–1956. [449]
[1904] J.-P. Serre. Abelian l-adic representations and elliptic curves. McGill Universitylecture notes written with the collaboration of Willem Kuyk and John Labute.W. A. Benjamin, Inc., New York-Amsterdam, 1968. [251, 252, 255]
[1905] J.-P. Serre. Proprietes galoisiennes des points d’ordre fini des courbes elliptiques.Invent. Math., 15(4):259–331, 1972. [252, 255]
[1906] J.-P. Serre. A course in arithmetic. Springer-Verlag, New York, 1973. Translatedfrom the French, Graduate Texts in Mathematics, No. 7. [17]
804 Handbook of Finite Fields
[1907] J.-P. Serre. Majorations de sommes exponentielles. In Journees Arithmetiques deCaen (Univ. Caen, Caen, 1976), pages 111–126. Asterisque No. 41–42. Soc.Math. France, Paris, 1977. [126, 536, 545]
[1908] J.-P. Serre. Quelques applications du theoreme de densite de Chebotarev. Inst.Hautes Etudes Sci. Publ. Math., (54):323–401, 1981. [252, 255, 349, 351]
[1909] J.-P. Serre. Nombres de points des courbes algebriques sur Fq. In Seminar onnumber theory, 1982–1983 (Talence, 1982/1983), pages Exp. No. 22, 8. Univ.Bordeaux I, Talence, 1983. [364, 367]
[1910] J.-P. Serre. Sur le nombre des points rationnels d’une courbe algebrique sur uncorps fini. C. R. Acad. Sci. Paris Ser. I Math., 296(9):397–402, 1983. [364,367, 368, 372]
[1911] J.-P. Serre. Quel est le nombre maximum de points rationnels que peut avoir unecourbe algebrique de genre g sur un corps fini? Annuaire du College de France,84:397–402, 1984. [365, 367]
[1912] J.-P. Serre. Repartition asymptotique des valeurs propres de l’operateur de HeckeTp. J. Amer. Math. Soc., 10(1):75–102, 1997. [538, 545]
[1913] J.-P. Serre. On a theorem of Jordan. Bull. Amer. Math. Soc. (N.S.), 40(4):429–440(electronic), 2003. [252, 255]
[1914] J. A. Serret. Cours d’algebre supeeriure. Gauthier-Villars, Paris, Paris, 3rd edition,1866. [31, 34]
[1915] J. A. Serret. Memoire sur la theorie des congruences suivant un module premier etsuivant une fonction modularie irreducible. Mem. Acad. Sci., Inst. de France,1(35):617–688, 1866. [31, 34]
[1916] J.-A. Serret. Cours d’algebre superieure. Tome I. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Editions Jacques Gabay, Sceaux,1992. Reprint of the fourth (1877) edition. [39]
[1917] H. Shacham and B. Waters, editors. Pairing-Based Cryptography — Pairing 2009,volume 5671 of Lecture Notes in Computer Science, Berlin, 2009. Springer-Verlag. [670]
[1918] I. R. Shafarevich. Basic algebraic geometry. 1 Varieties in projective space. Springer-Verlag, second edition, 1994. [305, 311]
[1919] R. Shaheen and A. Winterhof. Permutations of finite fields for check digit systems.Des. Codes Cryptogr., 57(3):361–371, 2010. [185]
[1920] A. Shallue and C. E. van de Woestijne. Construction of rational points on ellipticcurves over finite fields. In F. Hess, S. Pauli, and M. Pohst, editors, AlgorithmicNumber Theory — ANTS-VII, volume 4076 of Lecture Notes in ComputerScience, pages 510–524, Berlin, 2006. Springer-Verlag. [678]
[1921] A. Shamir. Efficient signature schemes based on birational permutations. In 1993,volume 773 of Lecture Notes in Computer Science, pages 1–12. Douglas R.Stinson, ed., 1993. [652, 656, 658]
[1922] C. E. Shannon. A mathematical theory of communication. Bell System Tech. J.,27:379–423, 623–656, 1948. [561, 583, 602]
[1923] C. E. Shannon. Communication theory of secrecy systems. Bell System Tech. J.,28:656–715, 1949. [626, 634]
[1924] R. T. Sharifi. On norm residue symbols and conductors. J. Number Theory,86(2):196–209, 2001. [103, 118]
[1925] J. T. Sheats. The Riemann hypothesis for the Goss zeta function for Fq[T ]. J.Number Theory, 71(1):121–157, 1998. [456]
Miscellaneous applications 805
[1926] G. B. Sherwood, S. S. Martirosyan, and C. J. Colbourn. Covering arrays of higherstrength from permutation vectors. J. Combin. Des., 14(3):202–213, 2006.[549]
[1927] I. P. Shestakov and U. U. Umirbaev. The Nagata automorphism is wild. Proc. Natl.Acad. Sci. USA, 100(22):12561–12563 (electronic), 2003. [652]
[1928] G. Shimura and Y. Taniyama. Complex multiplication of abelian varieties and itsapplications to number theory, volume 6 of Publications of the MathematicalSociety of Japan. The Mathematical Society of Japan, Tokyo, 1961. [251, 255]
[1929] K. Shiratani and M. Yamada. On rationality of Jacobi sums. Colloq. Math.,73(2):251–260, 1997. [103, 118]
[1930] I. Shmulevich, E. R. Dougherty, S. Kim, and W. Zhang. Probabilistic booleannetworks: a rule-based uncertainty model for gene regulatory networks. Bioin-formatics, 18(2):261–274, February 2002. [685]
[1931] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete log-arithms on a quantum computer. SIAM J. Comput., 26(5):1484–1509, 1997.[633, 634]
[1932] V. Shoup. Removing Randomness From Computational Number Theory. PhD thesis,University of Wisconsin, Madison, 1989. [298, 299]
[1933] V. Shoup. New algorithms for finding irreducible polynomials over finite fields.Mathematics of Computation, 54(189):435–447, Jan. 1990. [297, 298, 299]
[1934] V. Shoup. On the deterministic complexity of factoring polynomials over finitefields. Inform. Process. Lett., 33(5):261–267, 1990. [146]
[1935] V. Shoup. Fast construction of irreducible polynomials over finite fields. Journal ofSymbolic Computation, 17(5):371–391, 1994. [297, 299]
[1936] V. Shoup. A computational introduction to number theory and algebra. CambridgeUniversity Press, Cambridge, second edition, 2009. [20]
[1937] I. Shparlinski. On the distribution of irreducible trinomials. to appear in Canad.Math. Bull.. [57]
[1938] I. Shparlinski. Finding irreducible and primitive polynomials. Applicable Algebra inEngineering, Communication and Computing, 4(4):263–268, Dec. 1993. [298,299]
[1939] I. Shparlinski. On the linear complexity of the power generator. Des. Codes Cryp-togr., 23(1):5–10, 2001. [278, 281]
[1940] I. Shparlinski. Cryptographic applications of analytic number theory, volume 22 ofProgress in Computer Science and Applied Logic. Birkhauser Verlag, Basel,2003. Complexity lower bounds and pseudorandomness. [19, 20, 141, 279, 281]
[1941] I. Shparlinski. On the exponential sum-product problem. Indag. Math. (N.S.),19(2):325–331, 2008. [130]
[1942] I. Shparlinski and A. Winterhof. Noisy interpolation of sparse polynomials in finitefields. Appl. Algebra Engrg. Comm. Comput., 16(5):307–317, 2005. [141]
[1943] I. E. Shparlinski. Computational and algorithmic problems in finite fields, volume 88of Mathematics and its Applications (Soviet Series). Kluwer Academic Pub-lishers Group, Dordrecht, 1992. [19, 20]
[1944] I. E. Shparlinski. A deterministic test for permutation polynomials. Comput. Com-plexity, 2(2):129–132, 1992. [173, 185]
[1945] I. E. Shparlinski. Finite fields: theory and computation, volume 477 of Mathematicsand its Applications. Kluwer Academic Publishers, Dordrecht, 1999. The meet-ing point of number theory, computer science, coding theory and cryptography.
806 Handbook of Finite Fields
[19, 20]
[1946] I. E. Shparlinski. Bounds of Gauss sums in finite fields. Proc. Amer. Math. Soc.,132(10):2817–2824 (electronic), 2004. [98, 118]
[1947] I. E. Shparlinski. On the number of zero trace elements in polynomial bases for F2n .Rev. Mat. Complut., 18(1):177–180, 2005. [77, 79]
[1948] I. E. Shparlinski. Playing ‘hide-and-seek’ with numbers: the hidden number problem,lattices and exponential sums. In Public-key cryptography, volume 62 of Proc.Sympos. Appl. Math., pages 153–177. Amer. Math. Soc., Providence, RI, 2005.[141]
[1949] I. E. Shparlinski. On some dynamical systems in finite fields and residue rings.Discrete Contin. Dyn. Syst., 17(4):901–917, 2007. [282, 289]
[1950] I. E. Shparlinski. On the distribution of angles of the Salie sums. Bull. Austral.Math. Soc., 75(2):221–227, 2007. [113, 118]
[1951] I. E. Shparlinski. On the distribution of Kloosterman sums. Proc. Amer. Math.Soc., 136(2):419–425 (electronic), 2008. [113, 118]
[1952] I. E. Shparlinski. On the distribution of arguments of Gauss sums. Kodai Math. J.,32(1):172–177, 2009. [97, 118]
[1953] I. E. Shparlinski˘i. On primitive polynomials. Problemy Peredachi Informatsii,23(3):100–103, 1987. [40, 66, 68]
[1954] I. Shparlinskiy. On some problems of theory of finite fields, June 1990. [298, 299]
[1955] F. Shuqin and H. Wenbao. Primitive polynomials over finite fields of characteristictwo. Appl. Algebra Engrg. Comm. Comput., 14(5):381–395, 2004. [62]
[1956] T. Siegenthaler. Correlation-immunity of nonlinear combining functions for crypto-graphic applications. IEEE Trans. Inform. Theory, 30(5):776–780, 1984. [201,204]
[1957] D. Silva, F. R. Kschischang, and R. Kotter. A rank-metric approach to error controlin random network coding. IEEE Trans. Inform. Theory, 54(9):3951–3967,2008. [701]
[1958] J. H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. [19, 20,334, 351]
[1959] J. H. Silverman. The arithmetic of dynamical systems, volume 241 of GraduateTexts in Mathematics. Springer, New York, 2007. [282, 283, 289]
[1960] J. H. Silverman. Variation of periods modulo p in arithmetic dynamics. New YorkJ. Math., 14:601–616, 2008. [287, 289]
[1961] J. H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts inMathematics. Springer-Verlag, New York, second edition, 2009. [19, 20, 334,335, 337, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 350, 351]
[1962] J. H. Silverman. A survey of local and global pairings on elliptic curves and abelianvarieties. In Pairing-Based Cryptography (PAIRING 2010), volume 6478 ofLecture Notes in Comput. Sci., pages 377–396. Springer, Berlin, 2010. [346,351]
[1963] J. H. Silverman and J. Tate. Rational points on elliptic curves. UndergraduateTexts in Mathematics. Springer-Verlag, New York, 1992. [19, 20, 334, 351]
[1964] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt. Spread SprctrumCommunications Handbook. McGraw-Hill, Inc., 2002. [264]
[1965] J. Singer. A theorem in finite projective geometry and some applications to number
[1966] M. Ska lba. Points on elliptic curves over finite fields. Acta arithmetica, 117(3):293–301, 2005. [678]
[1967] C. Small. Solution of Waring’s problem mod n. Amer. Math. Monthly, 84(5):356–359, 1977. [168, 170]
[1968] C. Small. Sums of powers in large finite fields. Proc. Amer. Math. Soc., 65(1):35–36,1977. [168, 170]
[1969] C. Small. Waring’s problem mod n. Amer. Math. Monthly, 84(1):12–25, 1977. [168,170]
[1970] C. Small. Diagonal equations over large finite fields. Canad. J. Math., 36(2):249–262, 1984. [165, 166, 170]
[1971] C. Small. Permutation binomials. Internat. J. Math. Math. Sci., 13(2):337–342,1990. [179, 185]
[1972] C. Small. Arithmetic of finite fields, volume 148 of Monographs and Textbooks inPure and Applied Mathematics. Marcel Dekker Inc., New York, 1991. [19, 20,164, 170, 179, 185]
[1973] N. P. Smart. The discrete logarithm problem on elliptic curves of trace one. J.Cryptology, 12(3):193–196, 1999. [351]
[1974] N. P. Smart. The exact security of ECIES in the generic group model. In B. Honary,editor, Cryptography and Coding, volume 2260 of Lecture Notes in ComputerScience, pages 73–84, Berlin, 2001. Springer-Verlag. [667]
[1975] N. Smart et al. ECRYPT II yearly report on algorithms and keysizes (2009-2010).Technical Report D.SPA.13, European Network of Excellence in CryptologyII, 2010. [666, 675]
[1976] B. Smeets. The linear complexity profile and experimental results on a random-ness test of sequences over the field Fq. presented at IEEE Int. Symp. onInformation Theory 1988, June 19–24. [275, 281]
[1977] B. Smeets and W. Chambers. Windmill generators: a generalization and an obser-vation of how many there are. In Advances in cryptology—EUROCRYPT’88,volume 330 of Lecture Notes in Comput. Sci., pages 325–330. Springer, Berlin,1988. [37, 38]
[1978] M. H. M. Smid. Duadic codes. IEEE Trans. Inform. Theory, 33(3):432–433, 1987.[581, 602]
[1979] S. L. Snover. The uniqueness of the Nordstrom-Robinson and the Golay binarycodes. ProQuest LLC, Ann Arbor, MI, 1973. Thesis (Ph.D.)–Michigan StateUniversity. [601, 602]
[1980] I. M. Sobol’. Distribution of points in a cube and approximate evaluation of integrals(Russian). Z. Vycisl. Mat. i Mat. Fiz., 7:784–802, 1967. [373, 379, 382, 383]
[1981] A. B. Sørensen. Projective Reed-Muller codes. IEEE Trans. Inform. Theory,37(6):1567–1576, 1991. [587, 602]
[1982] S. Sperber. On the p-adic theory of exponential sums. Amer. J. Math., 108(2):255–296, 1986. [126, 396, 402]
[1984] M. Stamp and C. F. Martin. An algorithm for the k-error linear complexity ofbinary sequences with period 2n. IEEE Trans. Inform. Theory, 39(4):1398–1401, 1993. [271, 274, 281]
808 Handbook of Finite Fields
[1985] H. M. Stark and A. A. Terras. Zeta functions of finite graphs and coverings. Adv.Math., 121(1):124–165, 1996. [545]
[1986] A. Steel. Conquering inseparability: primary decomposition and multivariate fac-torization over algebraic function fields of positive characteristic. J. SymbolicComput., 40(3):1053–1075, 2005. [306, 311]
[1987] L. J. Steggles, R. Banks, O. Shaw, and A. Wipat. Qualitatively modelling andanalysing genetic regulatory networks: a Petri net approach. Bioinformatics,23:336–343, 2007. [685]
[1988] A. Stein. Sharp upper bounds for arithmetic in hyperelliptic function fields. J.Ramanujan Math. Soc., 16(2):119–203, 2001. [357, 360]
[1989] A. Stein. Explicit infrastructure for real quadratic function fields and real hyperel-liptic curves. Glas. Mat. Ser. III, 44(64)(1):89–126, 2009. [357, 360]
[1990] S. A. Stepanov. On the number of polynomials of a given form that are irreducibleover a finite field. Mat. Zametki, 41(3):289–295, 456, 1987. [46, 49]
[1991] S. A. Stepanov. Arithmetic of algebraic curves. Monographs in Contemporary Math-ematics. Consultants Bureau, New York, 1994. Translated from the Russianby Irene Aleksanova. [19, 20, 142]
[1992] J. Stern, D. Pointcheval, J. Malone-Lee, and N. Smart. Flaws in applying proofmethodologies to signature schemes. In M. Yung, editor, Advances in Cryp-tology — CRYPTO 2002, volume 2442 of Lecture Notes in Computer Science,pages 93–110, Berlin, 2002. Springer-Verlag. [667]
[1993] H. Stichtenoth. Algebraic function fields and codes. Universitext. Springer-Verlag,Berlin, 1993. [19, 20]
[1994] H. Stichtenoth. Transitive and self-dual codes attaining the Tsfasman-Vladut-Zinkbound. IEEE Trans. Inform. Theory, 52(5):2218–2224, 2006. [372]
[1995] H. Stichtenoth. Algebraic function fields and codes, volume 254 of Graduate Textsin Mathematics. Springer-Verlag, Berlin, second edition, 2009. [19, 20, 166,170, 317, 319, 320, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 333, 361,362, 363, 364, 365, 367, 368, 369, 370, 371, 372, 605, 612]
[1996] H. Stichtenoth and C. P. Xing. Excellent nonlinear codes from algebraic functionfields. IEEE Trans. Inform. Theory, 51:4044–4046, 2005. [612]
[1997] L. Stickelberger. On a new property of the discriminants of algebraic number fields.(Ueber eine neue Eigenschaft der Discriminanten algebraischer Zahlkorper.).Verh. d. intern. Math.-Congr., 1:182–193, 1897. [35, 36, 38, 41]
[1998] B. Stigler. Polynomial dynamical systems in systems biology. In Modeling andsimulation of biological networks, volume 64 of Proc. Sympos. Appl. Math.,pages 53–84. Amer. Math. Soc., Providence, RI, 2007. [282, 289]
[1999] D. R. Stinson. On bit-serial multiplication and dual bases in GF(2m). IEEE Trans.Inform. Theory, 37(6):1733–1736, 1991. [77, 79]
[2000] D. R. Stinson. Cryptography. CRC Press Series on Discrete Mathematics and itsApplications. Chapman & Hall/CRC, Boca Raton, FL, second edition, 2002.Theory and practice. [19, 20]
[2001] D. R. Stinson. Combinatorial designs: Constructions and analysis. Springer-Verlag,New York, 2004. [20, 511, 558]
[2002] D. R. Stinson. Cryptography. Discrete Mathematics and its Applications (BocaRaton). Chapman & Hall/CRC, Boca Raton, FL, third edition, 2006. Theoryand practice. [19, 20, 634]
[2003] D. R. Stinson, R. Wei, and L. Zhu. New constructions for perfect hash families and
Miscellaneous applications 809
related structures using combinatorial designs and codes. J. Combin. Des.,8(3):189–200, 2000. [552]
[2004] K.-O. Stohr and J. F. Voloch. Weierstrass points and curves over finite fields. Proc.London Math. Soc. (3), 52(1):1–19, 1986. [366, 367]
[2005] T. Stoll. Complete decomposition of Dickson-type polynomials and related Dio-phantine equations. J. Number Theory, 128(5):1157–1181, 2008. [240, 242]
[2006] T. Storer. Cyclotomy and difference sets. Lectures in Advanced Mathematics, No.2. Markham Publishing Co., Chicago, Ill., 1967. [516, 519]
[2007] W. W. Stothers. On permutation polynomials whose difference is linear. GlasgowMath. J., 32(2):165–171, 1990. [184, 185]
[2008] D. R. Stoutemyer. Which polynomial representation is best? In Proceedings ofthe 1984 MACSYMA Users’ Conference: Schenectady, New York, July 23–25,1984, pages 221–243, 1984. [301, 311]
[2009] V. Strassen. Vermeidung von Divisionen. J. Reine Angew. Math., 264:182–202,1973. [309, 311]
[2010] S. J. Suchower. Subfield permutation polynomials and orthogonal subfield systemsin finite fields. Acta Arith., 54(4):307–315, 1990. [187, 188]
[2011] S. J. Suchower. Polynomial representations of complete sets of frequency hyperrect-angles with prime power dimensions. J. Combin. Theory Ser. A, 62(1):46–65,1993. [466, 467]
[2012] B. Sudakov, E. Szemeredi, and V. H. Vu. On a question of ErdHos and Moser. DukeMath. J., 129(1):129–155, 2005. [130]
[2013] M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. J.Complexity, 13(1):180–193, 1997. [597, 598, 602]
[2014] M. Sugita, M. Kawazoe, and H. Imai. Grobner basis based cryptanalysis of sha-1. Cryptology ePrint Archive, Report 2006/098, 2006. http://eprint.iacr.org/. [665]
[2015] Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa. A method for solvingkey equation for decoding Goppa codes. Information and Control, 27:87–99,1975. [594, 602]
[2016] J. Sun and O. Y. Takeshita. Interleavers for turbo codes using permutation poly-nomials over integer rings. IEEE Trans. Inform. Theory, 51(1):101–119, 2005.[184, 185]
[2017] Q. Sun. The number of solutions of certain diagonal equations over finite fields.Sichuan Daxue Xuebao, 34(4):395–398, 1997. [166, 170]
[2018] Q. Sun and D. Q. Wan. On the solvability of the equation∑ni=1 xi/di ≡ 0 (mod 1)
and its application. Proc. Amer. Math. Soc., 100(2):220–224, 1987. [165, 166,170]
[2019] Q. Sun and D. Q. Wan. On the Diophantine equation∑ni=1 xi/di ≡ 0 (mod 1).
Proc. Amer. Math. Soc., 112(1):25–29, 1991. [167, 170]
[2020] Z.-W. Sun. On value sets of polynomials over a field. Finite Fields Appl., 14(2):470–481, 2008. [167, 170, 192]
[2021] T. Sunada. L-functions in geometry and some applications. In Curvature and topol-ogy of Riemannian manifolds (Katata, 1985), volume 1201 of Lecture Notes inMath., pages 266–284. Springer, Berlin, 1986. [545]
[2022] T. Sunada. Fundamental groups and Laplacians. In Geometry and analysis onmanifolds (Katata/Kyoto, 1987), volume 1339 of Lecture Notes in Math., pages248–277. Springer, Berlin, 1988. [545]
810 Handbook of Finite Fields
[2023] A. V. Sutherland. Genus 1 point counting in essentially quartic time and quadraticspace, Sept. 2010. Slides, http://math.mit.edu/~drew/NYU0910.pdf. [670]
[2024] A. V. Sutherland. Genus 1 point-counting record modulo a 5000+digit prime, July 2010. Posting to the Number Theory List,http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1007&L=
nmbrthry&T=0&F=&S=&P=287. [670]
[2025] R. G. Swan. Factorization of polynomials over finite fields. Pacific J. Math.,12:1099–1106, 1962. [36, 38, 41, 66, 68]
[2026] P. Sziklai. On small blocking sets and their linearity. J. Combin. Theory Ser. A,115(7):1167–1182, 2008. [472, 475]
[2027] T. SzHonyi. On the number of directions determined by a set of points in an affineGalois plane. J. Combin. Theory Ser. A, 74(1):141–146, 1996. [469, 475]
[2028] T. SzHonyi. Blocking sets in Desarguesian affine and projective planes. Finite FieldsAppl., 3(3):187–202, 1997. [472, 475]
[2029] T. SzHonyi. Around Redei’s theorem. Discrete Math., 208/209:557–575, 1999.Combinatorics (Assisi, 1996). [474, 475]
[2030] P. Szusz. On a problem in the theory of uniform distribution. Comptes RendusPremier Congres Hongrois, pages 461–472, 1952. (in Hungarian). [285, 289]
[2031] L. Taelman. Special L-values of t-motives: a conjecture. Int. Math. Res. Not. IMRN,(16):2957–2977, 2009. [452, 453]
[2032] L. Taelman. A Dirichlet unit theorem for Drinfeld modules. Math. Ann., 348(4):899–907, 2010. [452, 453]
[2033] L. Taelman. The Carlitz shtuka. J. Number Theory, 131(3):410–418, 2011. [453]
[2034] L. Taelman. A herbrand-ribet theorem for function fields. Preprint, 2011. [455]
[2035] L. Taelman. Special l-values of drinfeld modules. To appear in Ann. of Math., 2011.[454]
[2036] Y. Taguchi. The Tate conjecture for t-motives. Proc. Amer. Math. Soc.,123(11):3285–3287, 1995. [452]
[2037] T. Takagi, T. Okamoto, E. Okamoto, and T. Okamoto, editors. Pairing-BasedCryptography — Pairing 2007, volume 4575 of Lecture Notes in ComputerScience, Berlin, 2007. Springer-Verlag. [670]
[2038] T. Takahashi. Good reduction of elliptic modules. J. Math. Soc. Japan, 34(3):475–487, 1982. [452]
[2039] O. Y. Takeshita. Permutation polynomial interleavers: an algebraic-geometric per-spective. IEEE Trans. Inform. Theory, 53(6):2116–2132, 2007. [184, 185]
[2040] A. Tamagawa. The Tate conjecture and the semisimplicity conjecture for t-modules.Surikaisekikenkyusho Kokyuroku, (925):89–94, 1995. Algebraic number theoryand arithmetic geometry (Japanese) (Kyoto, 1994). [452]
[2041] Y. Tan, A. Pott, and T. Feng. Strongly regular graphs associated with ternary bentfunctions. J. Combin. Theory Ser. A, 117(6):668–682, 2010. [219, 224]
[2042] T. Tao. Structure and randomness. American Mathematical Society, Providence,RI, 2008. Pages from year one of a mathematical blog. [262]
[2043] T. Tao and V. Vu. Additive combinatorics, volume 105 of Cambridge Studies inAdvanced Mathematics. Cambridge University Press, Cambridge, 2006. [20,130]
[2044] V. Tarokh, N. Seshadri, and A. R. Calderbank. Space-time codes for high data ratewireless communication: performance criterion and code construction. IEEE
[2045] J. Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134–144, 1966. [342, 344, 351, 363, 367]
[2046] J. T. Tate. The arithmetic of elliptic curves. Invent. Math., 23:179–206, 1974. [334,351]
[2047] D. E. Taylor. The geometry of the classical groups, volume 9 of Sigma Series inPure Mathematics. Heldermann Verlag, Berlin, 1992. [440, 442, 445, 446, 447]
[2048] A. Terras. Fourier analysis on finite groups and applications, volume 43 of LondonMathematical Society Student Texts. Cambridge University Press, Cambridge,1999. [258, 260, 262, 533, 545]
[2049] F. Thaine. On Gaussian periods that are rational integers. Michigan Math. J.,50(2):313–337, 2002. [98, 118]
[2050] D. Thakur. Multizeta in function field arithmetic. In Proceedings of Banff Workshop.European Mathematical Society (EMS), Zurich. [454, 456]
[2051] D. S. Thakur. Function field arithmetic. World Scientific Publishing Co. Inc., RiverEdge, NJ, 2004. [19, 20, 448]
[2052] J. A. Thas. Normal rational curves and k-arcs in Galois spaces. Rend. Mat. (6),1:331–334, 1968. [498]
[2053] J. A. Thas. The affine plane AG(2, q), q odd, has a unique one point extension.Invent. Math., 118(1):133–139, 1994. [501]
[2054] N. Theriault. Index calculus attack for hyperelliptic curves of small genus. InAdvances in cryptology—ASIACRYPT 2003, volume 2894 of Lecture Notes inComput. Sci., pages 75–92. Springer, Berlin, 2003. [360]
[2055] E. Thome. Fast computation of linear generators for matrix sequences and appli-cation to the block Wiedemann algorithm. In Proceedings of the 2001 Inter-national Symposium on Symbolic and Algebraic Computation, pages 323–331(electronic), New York, 2001. ACM. [436]
[2056] T. M. Thompson. From error-correcting codes through sphere packings to simplegroups, volume 21 of Carus Mathematical Monographs. Mathematical Associ-ation of America, Washington, DC, 1983. [590, 602]
[2057] T. Tian and W.-F. Qi. Typical primitive polynomials over integer residue rings.Finite Fields Appl., 15(6):796–807, 2009. [59]
[2058] A. Tietavainen. On diagonal forms over finite fields. Ann. Univ. Turku. Ser. A INo., 118:10, 1968. [164, 170]
[2059] A. Tietavainen. On the nonexistence of perfect codes over finite fields. SIAM J.Appl. Math., 24:88–96, 1973. [572, 583, 602]
[2060] A. Tietavainen. A short proof for the nonexistence of unknown perfect codes overGF(q), q > 2. Ann. Acad. Sci. Fenn. Ser. A I, (580):6, 1974. [572, 601, 602]
[2061] A. Topuzoglu and A. Winterhof. Pseudorandom sequences. In Topics in geometry,coding theory and cryptography, volume 6 of Algebr. Appl., pages 135–166.Springer, Dordrecht, 2007. [282, 283, 289]
[2062] A. Toth. On the evaluation of Salie sums. Proc. Amer. Math. Soc., 133(3):643–645(electronic), 2005. [117, 118]
[2063] J. Tromp, L. Zhang, and Y. Zhao. Small weight bases for Hamming codes. InComputing and combinatorics (Xi’an, 1995), volume 959 of Lecture Notes inComput. Sci., pages 235–243. Springer, Berlin, 1995. [58, 59]
[2064] T. T. Truong. Degree complexity of a family of birational maps. II. Exceptional
[2065] M. Tsfasman, S. Vladut, and D. Nogin. Algebraic geometric codes: basic notions,volume 139 of Mathematical Surveys and Monographs. American MathematicalSociety, Providence, RI, 2007. [19, 20]
[2066] M. A. Tsfasman and S. G. Vladut. Algebraic-geometric codes, volume 58 of Math-ematics and its Applications (Soviet Series). Kluwer Academic PublishersGroup, Dordrecht, 1991. Translated from the Russian by the authors. [19, 20,610, 612]
[2067] M. A. Tsfasman, S. G. Vladut, and T. Zink. Modular curves, Shimura curves, andGoppa codes, better than Varshamov-Gilbert bound. Math. Nachr., 109:21–28,1982. [367, 368, 372]
[2068] M. A. Tsfasman, S. G. Vladut, and T. Zink. Modular curves, Shimura curves, andGoppa codes, better than Varshamov-Gilbert bound. Math. Nachr., 109:21–28,1982. [611, 612]
[2069] S. Tsujii, A. Fujioka, and T. Itoh. Generalization of the public key cryptosystembased on the difficulty of solving a system of non-linear equations. In Proc.10th Symposium on Information Theory and Its applications, pages JA5–3,1987. [652]
[2070] S. Tsujii, K. Kurosawa, T. Itoh, A. Fujioka, and T. Matsumoto. A public keycryptosystem based on the difficulty of solving a system of nonlinear equations.ICICE Transactions (D) J69-D, 12:1963–1970, 1986. [652]
[2071] W. J. Turner. Black box linear algebra with the linbox library. PhD thesis, 2002.[432, 436]
[2072] G. Turnwald. Permutation polynomials of binomial type. In Contributions to generalalgebra, 6, pages 281–286. Holder-Pichler-Tempsky, Vienna, 1988. [174, 179,185]
[2073] G. Turnwald. A new criterion for permutation polynomials. Finite Fields Appl.,1(1):64–82, 1995. [173, 184, 185, 189, 192]
[2074] G. Turnwald. On Schur’s conjecture. J. Austral. Math. Soc. Ser. A, 58(3):312–357,1995. [183]
[2075] R. J. Turyn. The linear generation of Legendre sequence. J. Soc. Indust. Appl.Math., 12:115–116, 1964. [279, 281]
[2076] R. J. Turyn. Character sums and difference sets. Pacific J. Math., 15:319–346, 1965.[517, 518, 519]
[2077] R. J. Turyn. Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulsecompression, and surface wave encodings. J. Combinatorial Theory Ser. A,16:313–333, 1974. [696, 701]
[2078] S. Uchiyama. Sur le nombre des valeurs distinctes d’un polynome a coefficients dansun corps fini. Proc. Japan Acad., 30:930–933, 1954. [192]
[2079] S. Uchiyama. Sur les polynomes irreductibles dans un corps fini. II. Proc. JapanAcad., 31:267–269, 1955. [43, 49]
[2080] D. Ulmer. Jacobi sums, Fermat Jacobians, and ranks of abelian varietiesover towers of function fields. Math. Res. Lett., 14(3):453–467, 2007.http://people.math.gatech.edu/ ulmer/research/papers/2007c-correction.pdf.[103, 118]
[2081] A. Valette. Graphes de Ramanujan et applications. Asterisque, (245):Exp. No. 829,4, 247–276, 1997. Seminaire Bourbaki, Vol. 1996/97. [532, 545]
[2082] E. R. van Dam and D. Fon-Der-Flaass. Codes, graphs, and schemes from nonlinear
Miscellaneous applications 813
functions. European J. Combin., 24(1):85–98, 2003. [211, 213]
[2083] G. van der Geer and M. van der Vlugt. Reed-Muller codes and supersingular curves.I. Compositio Math., 84(3):333–367, 1992. [402]
[2084] G. van der Geer and M. van der Vlugt. On the existence of supersingular curves ofgiven genus. J. Reine Angew. Math., 458:53–61, 1995. [401, 402]
[2085] G. van der Geer and M. van der Vlugt. Quadratic forms, generalized Hammingweights of codes and curves with many points. J. Number Theory, 59(1):20–36, 1996. [163]
[2086] G. van der Geer and M. van der Vlugt. An asymptotically good tower of curves overthe field with eight elements. Bull. London Math. Soc., 34(3):291–300, 2002.[371, 372]
[2087] G. van der Geer and M. van der Vlugt. Tables of curves with many points.http://www.science.uva.nl/ geer/tables-mathcomp21.pdf, 2009. [364, 367]
[2088] M. van der Put. A note on p-adic uniformization. Nederl. Akad. Wetensch. Indag.Math., 49(3):313–318, 1987. [456]
[2089] J. H. van Lint. Introduction to coding theory, volume 86 of Graduate Texts inMathematics. Springer-Verlag, Berlin, third edition, 1999. [19, 20, 499, 561,563, 568, 570, 571, 572, 573, 574, 583, 584, 585, 586, 589, 602]
[2090] J. H. van Lint and R. M. Wilson. A course in combinatorics. Cambridge UniversityPress, Cambridge, 1992. [20, 548]
[2091] P. C. van Oorschot and M. J. Wiener. Parallel collision search with cryptanalyticapplications. J. Cryptology, 12(1):1–28, 1999. [629, 634]
[2092] T. van Trung and S. Martirosyan. New constructions for IPP codes. Des. CodesCryptogr., 35(2):227–239, 2005. [552]
[2093] P. van Wamelen. New explicit multiplicative relations between Gauss sums. Int. J.Number Theory, 3(2):275–292, 2007. [103, 118]
[2094] R. Varshamov. Estimate of the number of signals in error correcting codes. Dokl.Akad. Nauk. SSSR, 117:739–741, 1957. [571, 601, 602]
[2095] R. Varshamov. A general method of synthesizing irreducible polynomials over Galoisfields. Soviet Math. Dokl., 29(2):334–336, 1984. [297, 299]
[2096] R. R. Varshamov. A certain linear operator in a Galois field and its applications(Russian). Studia, Sci. Math. Hunger., 8:5–19, 1973. [32, 34]
[2097] R. R. Varshamov. Operator substitutions in a Galois field and their applications(Russian). Dokl. Akad. Nauk SSSR;, 211:768–771, 1973. [32, 34]
[2098] R. R. Varshamov. A general method of synthesis for irreducible polynomials overGalois fields. Dokl. Akad. Nauk SSSR, 275(5):1041–1044, 1984. [32, 33, 34]
[2099] R. R. Varshamov and G. Garakov. On the theory of self-dual polynomials over aGalois field (Russian). Bull. Math. Soc. Sci. Math. R. S. Roumania (N.S.),13:403–415, 1969. [31, 34]
[2100] R. C. Vaughan and T. D. Wooley. Waring’s problem: a survey. In Number theory forthe millennium, volume III, pages 301–340. A. K. Peters, Natick, MA, 2002.[413, 414]
[2101] A. Veliz-Cuba, A. S. Jarrah, and R. Laubenbacher. Polynomial algebra of discretemodels in systems biology. Bioinformatics, 26(13):1637–1643, July 2010. [684,685, 688, 692]
[2102] A. Venkateswarlu and H. Niederreiter. Improved results on periodic multisequenceswith large error linear complexity. Finite Fields Appl., 16(6):463–476, 2010.
814 Handbook of Finite Fields
[276, 281]
[2103] F. Vercauteren. Computing zeta functions of hyperelliptic curves over finite fieldsof characteristic 2. In Advances in cryptology—CRYPTO 2002, volume 2442of Lecture Notes in Comput. Sci., pages 369–384. Springer, Berlin, 2002. [358,360]
[2104] F. Vercauteren. Optimal pairings. IEEE Transactions on Information Theory,56(1):455–461, 2010. [673]
[2105] E. R. Verheul. Evidence that XTR is more secure than supersingular elliptic curvecryptosystems. Journal of Cryptology, 17(4):277–296, 2004. [672]
[2106] C.-M. Viallet. Algebraic dynamics and algebraic entropy. Int. J. Geom. MethodsMod. Phys., 5(8):1373–1391, 2008. [282, 283, 289]
[2107] C. M. Viallet. Integrable lattice maps: QV , a rational version of Q4. Glasg. Math.J., 51(A):157–163, 2009. [282, 283, 289]
[2108] G. D. Villa Salvador. Topics in the theory of algebraic function fields. Mathematics:Theory & Applications. Birkhauser Boston Inc., Boston, MA, 2006. [317, 333,367]
[2109] G. Villard. Further analysis of coppersmith’s block wiedemann algorithm for thesolution of sparse linear systems (extended abstract). In Proceedings of the1997 international symposium on Symbolic and algebraic computation, ISSAC’97, pages 32–39, New York, NY, USA, 1997. ACM. [436]
[2110] G. Villard. Computing the Frobenius normal form of a sparse matrix. In Computeralgebra in scientific computing (Samarkand, 2000), pages 395–407. Springer,Berlin, 2000. [432, 436]
[2111] G. Villard. Algorithmique en algebre lineaire exacte. Memoire d’habilitation, Uni-versite Claude Bernard Lyon 1, 2003. [432, 436]
[2112] L. A. Vinh. The szemeredi-trotter type theorem and the sum-product estimate infinite fields. Eur. J. Combinatorics, 32:1177–1181, 2011. [133]
[2113] I. M. Vinogradov. Representation of an odd number as a sum of three primes.Comptes Rendus (Doklady), 15:191–294, 1937. [411, 414]
[2114] U. Vishne. Factorization of trinomials over Galois fields of characteristic 2. FiniteFields Appl., 3(4):370–377, 1997. [37, 38, 41]
[2115] S. G. Vleduts and Y. I. Manin. Linear codes and modular curves. In Currentproblems in mathematics, Vol. 25, Itogi Nauki i Tekhniki, pages 209–257. Akad.Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984. [458]
[2116] S. G. Vladut and V. G. Drinfeld. The number of points of an algebraic curve.Funktsional. Anal. i Prilozhen., 17(1):68–69, 1983. [367]
[2117] J. F. Voloch. On the order of points on curves over finite fields. Integers, 7:A49, 4,2007. [69, 70]
[2118] J. F. Voloch. Symmetric cryptography and algebraic curves. In Algebraic geometryand its applications, volume 5 of Ser. Number Theory Appl., pages 135–141.World Sci. Publ., Hackensack, NJ, 2008. [207, 213]
[2119] J. F. Voloch. Elements of high order on finite fields from elliptic curves. Bull. Aust.Math. Soc., 81(3):425–429, 2010. [70]
[2120] J. von zur Gathen. Irreducible polynomials over finite fields. In Proc. 6th Conf.Foundations of Software Technology and Theoretical Computer Science, volume241 of Springer Lecture Notes in Computer Science, pages 252–262, Delhi,India, 1986. [298, 299]
[2121] J. von zur Gathen. Tests for permutation polynomials. SIAM J. Comput., 20(3):591–
Miscellaneous applications 815
602, 1991. [173, 185]
[2122] J. von zur Gathen. Values of polynomials over finite fields. Bull. Austral. Math.Soc., 43(1):141–146, 1991. [191, 192]
[2123] J. von zur Gathen. Irreducible trinomials over finite fields. Math. Comp.,72(244):1987–2000 (electronic), 2003. [38]
[2124] J. von zur Gathen. Counting decomposable multivariate polynomials. Appl. AlgebraEngrg. Comm. Comput., 22:165–185, 2011. [53, 54, 55]
[2125] J. von zur Gathen, J. L. Imana, and C. K. Koc, editors. Arithmetic of fi-nite fields, volume 5130 of Lecture Notes in Computer Science, Berlin,2008. Springer. Available electronically at http://www.springerlink.com/
content/978-3-540-69498-4. [20]
[2126] J. von zur Gathen, M. Karpinski, and I. Shparlinski. Counting curves and theirprojections. Comput. Complexity, 6(1):64–99, 1996/97. [403, 406]
[2127] J. von zur Gathen and M. Nocker. Polynomial and normal bases for finite fields. J.Cryptology, 18(4):337–355, 2005. [42]
[2128] J. von zur Gathen and V. Shoup. Computing Frobenius maps and factoring poly-nomials. Computational Complexity, 2(3):187–224, 1992. [295, 299]
[2129] J. von zur Gathen and I. Shparlinski. Orders of Gauss periods in finite fields. Appl.Algebra Engrg. Comm. Comput., 9(1):15–24, 1998. [69, 70]
[2130] J. von zur Gathen and I. Shparlinski. Constructing elements of large order infinite fields. In Applied algebra, algebraic algorithms and error-correcting codes(Honolulu, HI, 1999), volume 1719 of Lecture Notes in Comput. Sci., pages404–409. Springer, Berlin, 1999. [70]
[2131] J. von zur Gathen and I. Shparlinski. Gauß periods in finite fields. In Finite fieldsand applications (Augsburg, 1999), pages 162–177. Springer, Berlin, 2001. [69,70]
[2132] C. H. Waddington. Canalisation of development and the inheritance of acquiredcharacters. Nature, 150:563–564, 1942. [690]
[2133] L. I. Wade. Certain quantities transcendental over GF (pn, x). Duke Math. J.,8:701–720, 1941. [458]
[2134] A. Wagner. On finite affine line transitive planes. Math. Z., 87:1–11, 1965. [480,486]
[2135] R. J. Walker. Determination of division algebras with 32 elements. In Proc. Sympos.Appl. Math., Vol. XV, pages 83–85. Amer. Math. Soc., Providence, R.I., 1963.[227, 229]
[2136] D. Wan. On the Riemann hypothesis for the characteristic p zeta function. J.Number Theory, 58(1):196–212, 1996. [456]
[2137] D. Wan. Generators and irreducible polynomials over finite fields. Math. Comp.,66(219):1195–1212, 1997. [44, 49, 126, 127]
[2138] D. Wan. Computing zeta functions over finite fields. In Finite fields: theory, appli-cations, and algorithms (Waterloo, ON, 1997), volume 225 of Contemp. Math.,pages 131–141. Amer. Math. Soc., Providence, RI, 1999. [405, 406]
[2139] D. Wan. Dwork’s conjecture on unit root zeta functions. Ann. of Math. (2),150(3):867–927, 1999. [394]
[2140] D. Wan. Higher rank case of Dwork’s conjecture. J. Amer. Math. Soc., 13(4):807–852 (electronic), 2000. [394]
[2141] D. Wan. Rank one case of Dwork’s conjecture. J. Amer. Math. Soc., 13(4):853–908
816 Handbook of Finite Fields
(electronic), 2000. [394]
[2142] D. Wan. Rationality of partial zeta functions. Indag. Math. (N.S.), 14(2):285–292,2003. [156, 158]
[2143] D. Wan. Variation of p-adic Newton polygons for L-functions of exponential sums.Asian J. Math., 8(3):427–471, 2004. [398, 400, 402]
[2144] D. Wan. Mirror symmetry for zeta functions. In Mirror symmetry. V, volume 38of AMS/IP Stud. Adv. Math., pages 159–184. Amer. Math. Soc., Providence,RI, 2006. With an appendix by C. Douglas Haessig. [154, 158]
[2145] D. Wan. Algorithmic theory of zeta functions over finite fields. In Algorithmicnumber theory: lattices, number fields, curves and cryptography, volume 44 ofMath. Sci. Res. Inst. Publ., pages 551–578. Cambridge Univ. Press, Cambridge,2008. [406]
[2146] D. Wan. Lectures on zeta functions over finite fields. In Higher-dimensional ge-ometry over finite fields, volume 16 of NATO Sci. Peace Secur. Ser. D Inf.Commun. Secur., pages 244–268. IOS, Amsterdam, 2008. [151, 154, 158]
[2147] D. Wan. Modular counting of rational points over finite fields. Found. Comput.Math., 8(5):597–605, 2008. [404, 406]
[2148] D. Q. Wan. On a problem of Niederreiter and Robinson about finite fields. J.Austral. Math. Soc. Ser. A, 41(3):336–338, 1986. [184, 185]
[2149] D. Q. Wan. Permutation polynomials over finite fields. Acta Math. Sinica (N.S.),3(1):1–5, 1987. [174, 179, 185]
[2150] D. Q. Wan. Zeros of diagonal equations over finite fields. Proc. Amer. Math. Soc.,103(4):1049–1052, 1988. [166, 170]
[2151] D. Q. Wan. An elementary proof of a theorem of Katz. Amer. J. Math., 111(1):1–8,1989. [157, 158]
[2152] D. Q. Wan. Permutation polynomials and resolution of singularities over finite fields.Proc. Amer. Math. Soc., 110(2):303–309, 1990. [174]
[2153] D. Q. Wan. A generalization of the carlitz conjecture. In Finite fields, coding the-ory, and advances in communications and computing (Las Vegas, NV, 1991),volume 141 of Lecture Notes in Pure and Appl. Math., pages 431–432. Dekker,New York, 1993. [174, 185]
[2154] D. Q. Wan. Newton polygons of zeta functions and L functions. Ann. of Math. (2),137(2):249–293, 1993. [398, 402]
[2155] D. Q. Wan. A p-adic lifting lemma and its applications to permutation polynomials.In Finite fields, coding theory, and advances in communications and computing(Las Vegas, NV, 1991), volume 141 of Lecture Notes in Pure and Appl. Math.,pages 209–216. Dekker, New York, 1993. [173, 184, 185, 189, 192]
[2156] D. Q. Wan. A classification conjecture about certain permutation polynomials. InFinite fields: Theory, Applications and Algorithms, volume 168 of Contempo-rary Math., pages 401–402. 1994. [184, 185]
[2157] D. Q. Wan. Permutation binomials over finite fields. Acta Math. Sinica (N.S.),10(Special Issue):30–35, 1994. [174, 179, 185]
[2158] D. Q. Wan. A Chevalley-Warning approach to p-adic estimates of character sums.Proc. Amer. Math. Soc., 123(1):45–54, 1995. [157, 158]
[2159] D. Q. Wan. Minimal polynomials and distinctness of Kloosterman sums. FiniteFields Appl., 1(2):189–203, 1995. Special issue dedicated to Leonard Carlitz.[111, 118]
[2160] D. Q. Wan and R. Lidl. Permutation polynomials of the form xrf(x(q−1)/d) and
Miscellaneous applications 817
their group structure. Monatsh. Math., 112(2):149–163, 1991. [177, 185]
[2161] D. Q. Wan, G. L. Mullen, and P. J.-S. Shiue. Erratum: “The number of permutationpolynomials of the form f(x) + cx over a finite field”. Proc. Edinburgh Math.Soc. (2), 38(2):i, 1995. [467]
[2162] D. Q. Wan, G. L. Mullen, and P. J.-S. Shiue. The number of permutation polyno-mials of the form f(x) + cx over a finite field. Proc. Edinburgh Math. Soc. (2),38(1):133–149, 1995. [184, 185, 467]
[2163] D. Q. Wan, P. J.-S. Shiue, and C. S. Chen. Value sets of polynomials over finitefields. Proc. Amer. Math. Soc., 119(3):711–717, 1993. [173, 185, 190, 191, 192]
[2164] Z.-X. Wan. Geometry of Classical Groups over Finite Fields. Science Press, Beijing,second edition, 2002. [19, 20, 439, 441, 443, 446, 447]
[2165] Z.-X. Wan. Lectures on finite fields and Galois rings. World Scientific PublishingCo. Inc., River Edge, NJ, 2003. [3, 17, 18, 19, 20]
[2166] Z.-X. Wan. Finite fields and Galois rings. World Scientific Publishing Co. Inc.,Singapore, 2012. [19, 20]
[2167] L. Wang. On permutation polynomials. Finite Fields Appl., 8(3):311–322, 2002.[178, 185]
[2168] L. Wang and Y. Zhu. F [x]-lattice basis reduction algorithm and multisequencesynthesis. Sci. China Ser. F, 44(5):321–328, 2001. [275, 281]
[2169] L.-C. Wang and F.-H. Chang. Tractable rational map cryptosystem (version 2).http://eprint.iacr.org/2004/046, ver. 20040221:212731. [657]
[2170] L.-C. Wang and F.-H. Chang. Tractable rational map cryptosystem (version 4).http://eprint.iacr.org/2004/046, ver. 20060203:065450. [657]
[2171] L.-C. Wang, Y.-H. Hu, F. Lai, C.-Y. Chou, and B.-Y. Yang. Tractable rational mapsignature. In Public key cryptography—PKC 2005, volume 3386 of LectureNotes in Comput. Sci., pages 244–257. Springer, Berlin, 2005. [655]
[2172] L.-C. Wang, B.-Y. Yang, Y.-H. Hu, and F. Lai. A “medium-field” multivariatepublic-key encryption scheme. In Topics in cryptology—CT-RSA 2006, volume3860 of Lecture Notes in Comput. Sci., pages 132–149. Springer, Berlin, 2006.[657, 660]
[2173] L.-P. Wang and H. Niederreiter. Enumeration results on the joint linear complexityof multisequences. Finite Fields Appl., 12(4):613–637, 2006. [275, 276, 281]
[2174] L.-P. Wang, Y.-F. Zhu, and D.-Y. Pei. On the lattice basis reduction multisequencesynthesis algorithm. IEEE Trans. Inform. Theory, 50(11):2905–2910, 2004.[275, 281]
[2175] M. Wang. Linear complexity profiles and continued fractions. In Advances incryptology—EUROCRYPT ’89 (Houthalen, 1989), volume 434 of Lecture Notesin Comput. Sci., pages 571–585. Springer, Berlin, 1990. [274, 281]
[2176] M. Wang and I. F. Blake. Bit serial multiplication in finite fields. SIAM J. DiscreteMath., 3(1):140–148, 1990. [75, 79]
[2177] M. Z. Wang. Linear complexity profiles and jump complexity. Inform. Process.Lett., 61(3):165–168, 1997. [274, 281]
[2178] P. S. Wang. An improved multivariate polynomial factoring algorithm. Math.Comp., 32(144):1215–1231, 1978. [304, 311]
[2179] P. S. Wang and L. P. Rothschild. Factoring multivariate polynomials over theintegers. Math. Comp., 29:935–950, 1975. [304, 311]
[2180] Q. Wang. Cyclotomic mapping permutation polynomials over finite fields. In
818 Handbook of Finite Fields
Sequences, subsequences, and consequences, volume 4893 of Lecture Notes inComput. Sci., pages 119–128. Springer, Berlin, 2007. [176, 177, 178, 185]
[2182] Q. Wang. On generalized lucas sequences. Contemporary Math., 531:127–141, 2010.[178, 184, 185]
[2183] Q. Wang, K. Wang, and Z. Dai. Implementation of multi-continued fraction al-gorithm and application to multi-sequence linear synthesis. In Sequences andtheir applications—SETA 2006, volume 4086 of Lecture Notes in Comput. Sci.,pages 248–258. Springer, Berlin, 2006. [275, 281]
[2184] Q. Wang and J. L. Yucas. Dickson polynomials over finite fields. submitted. [236,239, 240, 241, 242]
[2185] Y. Wang. Linear complexity versus pseudorandomness: on Beth and Dai’s result. InAdvances in cryptology—ASIACRYPT’99 (Singapore), volume 1716 of LectureNotes in Comput. Sci., pages 288–298. Springer, Berlin, 1999. [280, 281]
[2186] K. L. Wantz. A new class of unitals in the Hughes plane. Geom. Dedicata, 70(2):125–138, 1998. [484, 486]
[2187] L. C. Washington. Introduction to cyclotomic fields, volume 83 of Graduate Textsin Mathematics. Springer-Verlag, New York, second edition, 1997. [100, 109,118]
[2188] L. C. Washington. Elliptic curves. Discrete Mathematics and its Applications (BocaRaton). Chapman & Hall/CRC, Boca Raton, FL, 2003. Number theory andcryptography. [19, 20]
[2189] L. C. Washington. Elliptic curves. Discrete Mathematics and Its Applications(Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2008.Number theory and cryptography. [19, 20, 334, 351]
[2190] W. C. Waterhouse. Abelian varieties over finite fields. Ann. Sci. Ecole Norm. Sup.(4), 2:521–560, 1969. [341, 342, 347, 351]
[2191] W. A. Webb. Waring’s problem in GF[q,x]. Acta Arith., 22:207–220, 1973. [413,414]
[2192] C. Wei and Q. Sun. The least integer represented by∑ni=1 xi/di and its application.
[2193] Q. Wei and Q. Zhang. On strong orthogonal systems and weak permutation polyno-mials over finite commutative rings. Finite Fields Appl., 13(1):113–120, 2007.[188]
[2194] S. Wei, G. Chen, and G. Xiao. A fast algorithm for determining the linear complexityof periodic sequences. In Information security and cryptology, volume 3822 ofLecture Notes in Comput. Sci., pages 202–209. Springer, Berlin, 2005. [274,281]
[2195] S. Wei, G. Xiao, and Z. Chen. A fast algorithm for determining the linear complexityof a binary sequence with period 2npm. Sci. China Ser. F, 44(6):453–460, 2001.[274, 281]
[2196] S. Wei, G. Xiao, and Z. Chen. A fast algorithm for determining the minimal polyno-mial of a sequence with period 2pn over GF(q). IEEE Trans. Inform. Theory,48(10):2754–2758, 2002. [274, 281]
[2197] A. Weil. On some exponential sums. Proc. Nat. Acad. Sci. U. S. A., 34:204–207,1948. [119, 127]
[2198] A. Weil. Sur les courbes algebriques et les varietes qui s’en deduisent. Actualites
[2199] L. Welch. Lower bounds on the maximum cross correlation of signals. IEEE Trans.Inform. Theory, 20(3):397–399, 1974. [266]
[2200] L. R. Welch and E. R. Berlekamp. Error correction for algebraic block codes. U. S.Patent 4,633,470 (1986). [594, 602]
[2201] E. J. Weldon, Jr. Euclidean geometry cyclic codes. In Combinatorial Mathematicsand its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C.,1967), pages 377–387. Univ. North Carolina Press, Chapel Hill, N.C., 1969.[588, 596, 602]
[2202] C. Wells. The degrees of permutation polynomials over finite fields. J. CombinatorialTheory, 7:49–55, 1969. [175, 176, 185]
[2203] G. Weng, W. Qiu, Z. Wang, and Q. Xiang. Pseudo-Paley graphs and skew Hadamarddifference sets from presemifields. Des. Codes Cryptogr., 44(1-3):49–62, 2007.[231, 234]
[2204] G. Weng and X. Zeng. Further results on planar do functions and commutativesemifields. submitted. [227, 229, 233, 234]
[2205] R. C. Whaley, A. Petitet, and J. J. Dongarra. Automated empirical optimizationsof software and the ATLAS project. Parallel Computing, 27(1–2):3–35, Jan.2001. http://www.netlib.org/utk/people/JackDongarra/PAPERS/atlas_
pub.pdf. [427, 436]
[2206] D. H. Wiedemann. Solving sparse linear equations over finite fields. IEEE Trans.Inform. Theory, 32(1):54–62, 1986. [432, 436]
[2207] D. Wiedermann. An iterated quadratic extension of GF(2). Fibonacci Quart.,26(4):290–295, 1988. [33, 34]
[2208] M. J. Wiener and R. J. Zuccherato. Faster attacks on elliptic curve cryptosystems.In S. Tavares and H. Meijer, editors, Selected Areas in Cryptography — SAC’98, volume 1556 of Lecture Notes in Computer Science, pages 190–100, Berlin,1999. Springer-Verlag. [668]
[2209] M. L. H. Willems and J. A. Thas. A note on the existence of special Laguerrei-structures and optimal codes. European J. Combin., 4(1):93–96, 1983. [499]
[2210] M. Willett. Matrix fields over GF(q). Duke Math. J., 40:701–704, 1973. [418, 424]
[2211] K. S. Williams. On general polynomials. Canad. Math. Bull., 10:579–583, 1967.[190, 192]
[2212] K. S. Williams. On exceptional polynomials. Canad. Math. Bull., 11:279–282, 1968.[189, 192]
[2213] R. M. Wilson. Cyclotomy and difference families in elementary abelian groups. J.Number Theory, 4:17–47, 1972. [506, 507]
[2214] S. Winograd. On multiplication of 2×2 matrices. Linear Algebra and Appl., 4:381–388, 1971. [428, 436]
[2215] A. Winterhof. On Waring’s problem in finite fields. Acta Arith., 87(2):171–177,1998. [169, 170]
[2216] A. Winterhof. A note on Waring’s problem in finite fields. Acta Arith., 96(4):365–368, 2001. [169, 170]
[2217] A. Winterhof. A note on the linear complexity profile of the discrete logarithm infinite fields. In Coding, cryptography and combinatorics, volume 23 of Progr.Comput. Sci. Appl. Logic, pages 359–367. Birkhauser, Basel, 2004. [279, 281]
820 Handbook of Finite Fields
[2218] A. Winterhof and C. van de Woestijne. Exact solutions to Waring’s problem forfinite fields. Acta Arith., 141(2):171–190, 2010. [169, 170]
[2219] E. Wirsing. Thin essential components. In Topics in number theory (Proc. Colloq.,Debrecen, 1974), pages 429–442. Colloq. Math. Soc. Janos Bolyai, Vol. 13.North-Holland, Amsterdam, 1976. [148]
[2220] E. Witt. Uber steinersche systeme. Abh. Math. Sem. Univ. Hamburg, 12:265–275,1938. [501]
[2221] C. Wolf, A. Braeken, and B. Preneel. Efficient cryptanalysis of RSE(2)PKC andRSSE(2)PKC. In 2004, volume 3352 of Lecture Notes in Computer Science,pages 294–309, Sept. 8–10 2004. Extended version: http://eprint.iacr.org/2004/237. [655]
[2222] J. K. Wolf. Adding two information symbols to certain nonbinary BCH codes andsome applications. Bell System Tech. J., 48:2405–2424, 1969. [581, 602]
[2223] J. Wolfmann. Formes quadratiques et codes a deux poids. C. R. Acad. Sci. ParisSer. A-B, 281(13):Aii, A533–A535, 1975. [163]
[2224] J. Wolfmann. The number of solutions of certain diagonal equations over finitefields. J. Number Theory, 42(3):247–257, 1992. [165, 170]
[2225] M. Wu, X. Yang, and C. Chan. A dynamic analysis of irs-pkr signaling in liver cells:A discrete modeling approach. PLoS ONE, 4(12):e8040, 12 2009. [683, 692]
[2226] P.-C. Wu. Random number generation with primitive pentanomials. ACM Trans.Modeling and Computer Simulation, 11(4):346–351, 2001. [67, 68]
[2227] G. Xiao and S. Wei. Fast algorithms for determining the linear complexity of periodsequences. In Progress in cryptology – INDOCRYPT 2002, number 2551, pages12–21, 2002. [274, 281]
[2228] G. Xiao, S. Wei, K. Y. Lam, and K. Imamura. A fast algorithm for determiningthe linear complexity of a sequence with period pn over GF(q). IEEE Trans.Inform. Theory, 46(6):2203–2206, 2000. [274, 281]
[2229] G. Z. Xiao and J. L. Massey. A spectral characterization of correlation-immunecombining functions. IEEE Trans. Inform. Theory, 34(3):569–571, 1988. [201,204]
[2230] C. P. Xing. Goppa geometric codes achieving the Gilbert-Varshamov bound. IEEETrans. Inform. Theory, 51:259–264, 2005. [611, 612]
[2231] C. P. Xing and H. Niederreiter. A construction of low-discrepancy sequences usingglobal function fields. Acta Arith., 73:87–102, 1995. [382, 383]
[2232] C. P. Xing, H. Niederreiter, and K. Y. Lam. A generalization of algebraic-geometrycodes. IEEE Trans. Inform. Theory, 45:2498–2501, 1999. [606, 612]
[2233] C. P. Xing and S. L. Yeo. New linear codes and algebraic function fields over finitefields. IEEE Trans. Inform. Theory, 53:4822–4825, 2007. [607, 612]
[2234] T. Yan. The geobucket data structure for polynomials. J. Symbolic Comput.,25(3):285–293, 1998. [301, 311]
[2235] B.-Y. Yang and J.-M. Chen. All in the XL family: theory and practice. In Infor-mation security and cryptology—ICISC 2004, volume 3506 of Lecture Notes inComput. Sci., pages 67–86. Springer, Berlin, 2005. [664, 665]
[2236] B.-Y. Yang and J.-M. Chen. Building secure tame-like multivariate public-key cryp-tosystems: The new TTS. In ACISP 2005, volume 3574 of Lecture Notes inComputer Science, pages 518–531. Springer, July 2005. [655, 656, 662]
[2237] B.-Y. Yang, J.-M. Chen, and Y.-H. Chen. TTS: High-speed signatures on a low-cost smart card. In CHES 2004, volume 3156 of Lecture Notes in Computer
Miscellaneous applications 821
Science, pages 371–385. Springer, 2004. [655]
[2238] J. Yang and Z. Dai. Linear complexity of periodically repeated random sequences.Acta Math. Sinica (N.S.), 11(Special Issue):1–7, 1995. A Chinese summaryappears in Acta Math. Sinica 39 (1996), no. 1, 140. [276, 281]
[2239] J. Yang, S. X. Luo, and K. Q. Feng. Gauss sum of index 4. II. Non-cyclic case. ActaMath. Sin. (Engl. Ser.), 22(3):833–844, 2006. [106, 118]
[2240] J. Yang and L. Xia. Complete solving of explicit evaluation of Gauss sums in theindex 2 case. Sci. China Math., 53(9):2525–2542, 2010. [106, 118]
[2241] R. Yang. Newton polygons of L-functions of polynomials of the form xd+λx. FiniteFields Appl., 9(1):59–88, 2003. [399, 402]
[2242] S. M. Yang and L. L. Qi. On improved asymptotic bounds for codes from globalfunction fields. Des. Codes Cryptogr., 53:33–43, 2009. [612]
[2243] M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. AlgebraicDiscrete Methods, 2(1):77–79, 1981. [434, 436]
[2244] Y. Ye. A hyper-Kloosterman sum identity. Sci. China Ser. A, 41(11):1158–1162,1998. [111, 118]
[2245] A. M. Youssef and G. Gong. Hyper-bent functions. In Advances in cryptology—EUROCRYPT 2001 (Innsbruck), volume 2045 of Lecture Notes in Comput.Sci., pages 406–419. Springer, Berlin, 2001. [221, 224]
[2246] J. Yu. Transcendence and Drinfel’d modules. Invent. Math., 83(3):507–517, 1986.[458]
[2247] J. Yu. On periods and quasi-periods of Drinfel’d modules. Compositio Math.,74(3):235–245, 1990. [458]
[2248] J.-D. Yu. Variation of the unit root along the Dwork family of Calabi-Yau varieties.Math. Ann., 343(1):53–78, 2009. [394, 402]
[2249] J. Yuan, C. Carlet, and C. Ding. The weight distribution of a class of linear codesfrom perfect nonlinear functions. IEEE Trans. Inform. Theory, 52(2):712–717,2006. [221, 224]
[2250] J. Yuan and C. Ding. Four classes of permutation polynomials of F2m . Finite FieldsAppl., 13(4):869–876, 2007. [182, 185]
[2251] J. Yuan, C. Ding, H. Wang, and J. Pieprzyk. Permutation polynomials of the form(xp − x+ δ)s + L(x). Finite Fields Appl., 14(2):482–493, 2008. [182, 185]
[2252] P. Yuan. More explicit classes of permutation polynomials of F33m . Finite FieldsAppl., 16(2):88–95, 2010. [182, 185]
[2253] P. Yuan and X. Zeng. A note on linear permutation polynomials. Finite FieldsAppl., in press. [172, 185]
[2254] J. L. Yucas. Irreducible polynomials over finite fields with prescribedtrace/prescribed constant term. Finite Fields Appl., 12(2):211–221, 2006. [25,30, 49]
[2255] J. L. Yucas and G. L. Mullen. Irreducible polynomials over GF(2) with prescribedcoefficients. Discrete Math., 274(1-3):265–279, 2004. [26, 27, 30, 48, 49]
[2256] H. Zassenhaus. On Hensel factorization I. J. Number Theory, 1(1):291–311, 1969.[304, 311]
[2257] H. Zassenhaus. Polynomial time factoring of integral polynomials. ACM SIGSAMBull., 15(2):6–7, 1981. [306, 311]
[2258] X. Zeng, X. Zhu, and L. Hu. Two new permutation polynomials with the form
[2259] Z. Zha, G. M. Kyureghyan, and X. Wang. Perfect nonlinear binomials and theirsemifields. Finite Fields Appl., 15(2):125–133, 2009. [233, 234]
[2260] Z. Zha and X. Wang. New families of perfect nonlinear polynomial functions. J.Algebra, 322(11):3912–3918, 2009. [233, 234]
[2261] Q. Zhang. Polynomial functions and permutation polynomials over some finitecommutative rings. J. Number Theory, 105(1):192–202, 2004. [185, 188]
[2262] Z. Zhao and X. Cao. A note on the reducibility of binary affine polynomials. Des.Codes Cryptogr., 57(1):83–90, 2010. [37, 38]
[2263] K. Zhou. A remark on linear permutation polynomials. Finite Fields Appl.,14(2):532–536, 2008. [172, 185]
[2264] H. J. Zhu. p-adic variation of L functions of one variable exponential sums. I. Amer.J. Math., 125(3):669–690, 2003. [399, 402]
[2265] H. J. Zhu. Asymptotic variation of L functions of one-variable exponential sums.J. Reine Angew. Math., 572:219–233, 2004. [399, 401, 402]
[2266] H. J. Zhu. L-functions of exponential sums over one-dimensional affinoids: Newtonover Hodge. Int. Math. Res. Not., (30):1529–1550, 2004. [397, 399, 402]
[2267] N. Zierler. Primitive trinomials whose degree is a Mersenne exponent. Informationand Control, 15:67–69, 1969. [66, 68]
[2268] M. E. Zieve. Some families of permutation polynomials over finite fields. Int. J.Number Theory, 4(5):851–857, 2008. [179, 185]
[2269] M. E. Zieve. On some permutation polynomials over Fq of the form xrh(x(q−1)/d).Proc. Amer. Math. Soc., 137(7):2209–2216, 2009. [176, 177, 179, 185]
[2270] M. E. Zieve. Classes of permutation polynomials based on cyclotomy and an additiveanalogue. In Additive Number Theory, pages 355–361. Springer, 2010. [177,180, 185]
[2271] T. Zink. Degeneration of Shimura surfaces and a problem in coding theory. InFundamentals of computation theory (Cottbus, 1985), volume 199 of LectureNotes in Comput. Sci., pages 503–511. Springer, Berlin, 1985. [367]
[2272] R. Zippel. Probabilistic algorithms for sparse polynomials. In EUROSAM ’79:Proceedings of the International Symposium on Symbolic and Algebraic Com-putation, number 72 in Lecture Notes in Comput. Sci., pages 216–226. Springer-Verlag, 1979. [310, 311]
[2273] R. Zippel. Newton’s iteration and the sparse Hensel algorithm (Extended Abstract).In SYMSAC ’81: Proceedings of the fourth ACM Symposium on Symbolic andAlgebraic Computation, pages 68–72, New York, 1981. ACM Press. [310, 311]
[2274] Z. Zlatev. Computational methods for general sparse matrices, volume 65 of Math-ematics and its Applications. Kluwer Academic Publishers Group, Dordrecht,1991. [434, 436]
[2275] D. Zywina. Explicit class field theory for global function fields. 2011. [450]
Dedekind eta function, 114Dedekind’s different theorem, 327degree
of an isogeny, 339degree zero divisor class group, 354degree zero divisor class number, 355Deligne’s theorem, 387, 390dense polynomial representation, 302density of primes, 408dependency graph, 684derivation, 330derivative, 206Desarguesian, 467design, 590
divisor (of a function field), 320canonical class, 322canonical divisor, 322, 331class group Cl0(F ), 362degree of a divisor, 320dimension of a divisor, `(A), 322divisor class [D], 321divisor class group Cl(F ), 321divisor group Div(F ), 320divisor of a differential, 331divisor of poles (x)∞, 321equivalent divisors, 321positive divisor, 320prime divisor, 320principal divisor, 321principal divisor div(x), 321zero divisor (x)0, 321
function field, 317constant field, 317elliptic function field, 317, 331Fermat function field, 365Giulietti–Korchmaros function field, 366Hermitian, 604Hermitian function field, 365hyperelliptic function field, 317, 331maximal function field, 365rational function field, 317, 319, 321, 323
Petri net, 688phase space, 684Picard–Fuchs differential operator, 349Picard group
of elliptic curve, 341place, 318
completely splitting place, 325degree of a place, 319extension of a place, 324
place at infinity, 319pole of x, 319prime element at a place, 318ramification index, 324ramified extension, 325rational place, 319, 361
number of rational places N(F ), 361relative degree, 324residue class field of a place, 319residue class map, 319unramified extension, 325zero of x, 319
normal, 60primitive, 43strong primitive normal, 64absolute value of, 408affine, 13all one, 40characteristic, 178Chebychev conjugate, 246complete mapping, 181Dembowski-Ostrom, 232Dickson, 191, 246Dickson polynomial of the first kind, 235Dickson polynomial of the second kind,
tame ramification, 327wild, 253wild ramification, 327
ramification locus (of a tower), 369Rank, 429, 432–435rank, 652rational, 652rational function
composition factor definition field, 252composition factors, 243cyclic conjugate, 250decomposable, 243exceptional over Fq, 244exceptional over a number field, 244permutation over Fq, 247Redei, 250separable, 247tame, 245
