Efficient algorithm to calculate Reed-Muller expansions over GF(4)

S. Rahardja and B.J. Falkowski

Abstract: A new algorithm to generate the full polarity matrix of fixed polarity Reed-Muller expansions over Galois fields of order 4, GF(4), has been developed. By using directly the truth vector of the original function, a recursive formula is developed to generate the whole polarity matrix. The algorithm uses the properties of the fured polarity matrix to speed up the calculation and reduce the number of necessary multipliers and adders. The computational complexity of the algorithm is compared with other works. It is shown that, for practical hardware implementations of quaternary functions, the new algorithm is better than all other existing algorithms. The fast flow diagrams for computation of the whole or partial matrix are also presented.

1 Introduction

Any binary and multiple-valued functions can be repre- sented by the corresponding Reed-Muller (RM) expan- sions and more general exclusive-OR sum of products [1-17]. This way of representation has recently drawn much interest among researchers, because these expansions have excellent properties for testability [4, 12, 141 when they are realised in circuits. In practical applications, many func- tions can be best realised by their corresponding RM expansions. There are two types of RM expansions for the binary case: the fmed and mixed polarity RM expansions. Expressing a binary function in its mixed polarity RM expansion may possibly yield a simpler final circuit design than the fmed counterparts. On the other hand, a fmed RM circuit needs only half of the number of inputs as compared to the mixed counterpart. The RM universal logic modules have been proposed for implementation of futed RM expansions [15]. In addition, the fmed polarity binary RM expansion has the properties of an easily testable network and, for an n-variable function, 3n + 4 tests are sufficient to fully test the circuit [14]. Similar analysis of testability for multiple-valued RM circuits has also been conducted in

The concepts of fEed and mixed polarity RM expan- sions have been extended generally to a finite field GF(q) [&I 11, which is known as the Galois field of order q = p”, where p is a prime number and m is a positive integer. The fast synthesis of ternary RM expansions has been analysed [6] and its computational costs have been compared with Green’s algorithms [9]. Fast generation of generalised RM expansions for prime fields that are called Tamari expan- sions [18] have been shown in [19]. Another computation- ally efficient method to calculate the polynomial Reed- Muller-Fourier expansions for Galois fields of prime

[4, 121.

~~~

0 IEE, 2001 ZEE Procee&gs online no. 20010650 DO1 10.1049/ipcds:20010650 Paper f i t received 15th August 2000 and in revised form 24th July 2001 The authors are with the School of Electrical & Electronic Engineerin& Nan- yang Technological University, 50 Nanyang Avenue, Singapore 639798

cardinality has been reported recently in [20]. The minimi- sation of RM forms based on extensions of Galois field GF(2”) has been presented in [17]. The algorithm shown in [ 171 directly converted the polynomial coefficients from one polarity to the adjacent one without permuting the truth table and computing RM transformations for each new polarity. The average number of field multiplications for each polarity conversion was m x 2m-2 [17]. The same approach of generation of all sets of polarities rearranged according to Gray code ordering was used in [lo] for RM expansions for the smallest composite field, the quaternary field GF(4). The detailed number of operations of addition and multiplication in GF(4) is derived in [lo]. Sirmlarly, fast computation of RM expansions over GF(4) have been recently investigated in [5] and compared with Green’s algorithms [lo].

Th~s paper presents another method of computing the polarity matrix of an RM expansion over GF(4) for n-vaii- able quaternary functions. To find optimal RM expansion, it is necessary to generate the whole polarity matrix and choose the optimal polarity vector. The computational cost of the new algorithm is more efficient than the previously introduced algorithm in [5] for n I 6. In [5], the computa- tion of the polarity matrix commences after the zeroth polarity of the corresponding RM expansion has been calculated. The new algorithm utilises the functional data directly. Considering this fact, the computational costs to compute the zeroth polarity of the RM expansion should be included in the complexity of the algorithm [5]. As such, the new algorithm is computationally more efficient than any other existing algorithms among all methods that generate the entire matrix for n 5 6. It should be noticed that the development of an efficient algorithm for small n is important from a practical point of view as there is good implementation of RM expansions for smaller numbers of variables [l, 15, 211.

2 Basic definitions

Binary and ternary RM expansions can be easily extended to the general case, q-valued switching circuits, provided q is a prime number. When q is nonprime, but such that q = pm, where p is a prime and m any positive integer, there

289 IEE Proc-Circuits Devices Syst., Vol. 148. No. 6, December 2001

exists another form of possible representations which uti- lises p" polynomials of degree m - 1 or less, with coeffi- cients from GF(p) as basic elements and the operations of polynomial addition and multiplication performed ;modulo a primitive polynomial of degree m with coefficients from GF(p) [7, 8, lo]. In other words, GFW) is merely an exten- sion field of GF(p) with all its p" elements identified as the roots of irreducible polynomials over GFb). The smallest nonprime field is GF(4). Its elements can be taken as 0, 1, A and B, where A and B represent the elements x and x + 1, respectively, and x is a root of an irreducible polynomial of degree 1 over GF(2). Definition I : The polarity number, denoted by kR, is defined as the decimal equivalence of a quartemary n-tuple formed by writing a 0, 1, A or B for each literal np, E), 2: or 2:. Definition 2: The general canonical quaternary RM expan- sion for an n-variable quaternary function in polarity (kR) takes the form

4"-1 r n i

where 2 is the literal of a variable x, 2" = {?I, X2, ..., n,}, A, B}, ml E (0, 1, 2, 3}, r = 0, 1, 2, ..., 4" - 1, and (ml m2 ... m,) denotes the respective quaternary expression of the decimal number r, i.e. (r)lo = (m, m2 ... mJ4. The addition and multiplication in eqn. 1 follow the table shown in Fig. 1.

np = 1, 2) = E/, n: = 21 x n/, n,3 = 2: x 21, U , E (0 , 1,

+ 0

1

A

B

Fig. 1

O I A B

O l A B

I O B A

A B 0 I

B A 1 0

Addition and multiplications tables for GF(4)

Definition 3: The coefficient vector of the functionA2,) for (kR) polarity is denoted as dk) where

and k R = 0, 1, 2, ..., 4" - 1. Definition 4: The polarity matrix of a quaternary function

is a 4" x 4" matrix mz,)] , where every row corre- sponds to a polarity vector d k R ) in a dlferent polarity ( k R ) . Mathematically,

where T represents the matrix transpose. Exumple I : Let a l-variable quaternary truth vector be given as F = [MOA]? The polarity matrix of the quater- nary function

r A 1 B A i

LA A A AA From the polarity matrix, it is obvious that the function is best represented by the RM expansion corresponding to

290

the polarity whch has the maximum number of zero elements. Property 1: Every element pq (row i, column 1) of the polar- ity matrix describes the coefficient uJ with polarity (kR) = (9, i.e. up). Definition 5: The optimal polarity ofA2,) is defined as the polarity whose coefficient vector d k R ) has the minimal number of nonzero coefficients pV Definition 6: The column coeficient vector of the quater- nary function Ai?,) for polarity (kc) is denoted as d k C )

where

and kc = 0, 1, 2, ..., 4" - 1. Property 2: When (kc) = (0), the resulting vector in eqn. 4 is simply the truth vector of the quaternary functionA2,). Property 3: Every element pg (row i, column j ) of the polar- ity matrix describes the coefficient bi with polarity (kc) = Q, i.e. b,O) uf).

Theorem I : The polarity matrix of a quaternary function A2,) is a 4" x 4" matrix m2,)] given by

Prooj The proof follows from properties 1 and 3 and defi- nitions 3, 4 and 6. 0

3

The new algorithm uses special properties of the quater- nary RM polarity matrix. When such a matrix is defined in a recursive way, there is a possibility of reducing the number of GF(4) addition and multiplication operations by eliminating repetitive operations through introduction of special subvectors. Hence, the hardware advantage of the new algorithm is obtained at the expense of its computa- tional time. The following definitions are necessary to present the new algorithm.

Let the polarity matrix of an n-variable quaternary func- tionAzn) be defined by eqns. 3 and 5. Definition 7: Let the column coefficient vector of the qua- ternary function be expressed with 4 subvectors as follows:

Recursive definition of the new algorithm

The column coeffcient vector thkc) contains 4" elements. Hence the subvector B$fLl1 contains 4"-' elements where the subscript ml takes the value of {O, 1, 2, 3). Property 4: Every subvector can be expressed by another 4 subvectors and the total number of elements in each of the new subvectors will be 4 times smaller than that of the orig- inal subvector. Mathematically,

and p 2 0. Property 5: Generally, there are @ elements in B& . When /3 = n, the subvector B&kl = B#$] = b@).

To generate a full polarity matrix efficiently, a recursive definition of a matrix is necessary. Ths Qp matrix is defined as follows in the form of Qpl matrices. Such an

IEE Proc.-Circuits Devices Syst., Vol. 148, No. 6, December 2001

approach reduces the number of GF(4) operations required in generating the whole polarity matrix at each stage of recursion. Definition 8: A Qp matrix is a recursive square matrix which is defined by eqn. 8. In ths equation, Qpl is a square matrix which is one order lower than the Qp matrix, and the subscript b takes the value of (0, 1, 2, 3 }, 1 I p I n or 0 2 (p- 1) 5 n - 1, i.e.

- -

QP-1 (B;;?: 01) QO-I(% 1)+qz3)

&a-1 (B:;?!,lj) Q P - ~ ( & P , I I + ~ I S )

Qa-1 (B:i?l',21) Qa-l(&a 31+443)

- Q P - ~ (B:;?i,3]) Q P - ~ ( R [ P . ~ I + Q ~ ~ )

4 A complete example

Let n = 2, so that A2-J = &.!!o a,co)[rr&, Ey^/] and let the quaternary truth vector be given as I; = [OB 1 1 1 BABAABAl O0AlT, therefore, the column coefficient vector of the function under zeroth polarity is simply b(O) = [OB1llBABAABAIOOA]T = F. By definition 7, b(O) is expressed as the subvectors, b(O) = [(OBll), (lBAB), (AABA), (1 OOA)]'.

29 1 IEE Proc.-Circuits Devices Syst., Vol. 148, No. 6, December 2001

Fig.2 Flow diagram for one remsive step of the new algorithm

I - 1 1 A I

Fig.3 Flow diirgvuamfor the calculation qf column 2

A

Fig.4 Flow diagramfor the calculatwn of colwnn 3

bAo) bA3)

Fig.5 How ctragrmnfor the cdculation of c o l m 4

By definition 9, when k , = 0, R[,,,, = (lOBA)T, R,,,,] = (BBB0IT, R [ I , ~ ] = (BABO)T and R[l,q = (OBIA)T. By defini- tion 8, property 6, and eqn. 9,

P [ f ( S a ) l = Q z (a"))

Qi(clB1l)T Q I ( A A O ~ ) ~ Q 1 ( 1 0 B l ) ~ Qi(AAOA)T

Q I ( ~ B A B ) ~ Q 1 ( O O O B ) T Q I ( B A B B ) ~ Qi(AAOA)T

&I (AABAIT &I (BBOO)T &i(ABBA)= &I (AAOA)T

QI (lOOA)T Q 1 ( 1 1 0 A ) ~ Q I ( O ~ B O ) ~ Q I ( A A O A ) ~

OAAB A011 lOAB A l B A

B l l R A101 OB15 A B l A

IOBR OB51 B A B B OOOA

lBOR l A A l 1!OB AAAA

l B A B OlAB 5111 A l B A

BOlR OAlB AOOl A B l A

A l B B OBBB B A 5 1 OOOA

BAOB B O O B B B A l AAAA

A B A l BBBO AABO A l B A

AA51 BBBO BABO A B l A

BOO1 O B B O BABO OOOA

Al l1 OB50 AABO AAAA

1BlF lAOA OBOA A l B A

OOAB lOAA l l A A A B l A

OlOR OBBA B A B A OOOA

AABB A l l A OOlA AAAA

polarity is do) = [B0010BBOBABOOOOA] which has only eight nonzero elements. The corresponding quaternary RM expansion is

292 IEE Proc.-Circuits Devices Syst., Vol. 148, No. 6. December 2001

5

Similar to the recently introduced fast algorithm [5], the computational cost of the new algorithm can be calculated in terms of the number of GF(4) additions and multiplica- tions. It must be noted that the previous algorithm [5] requires computation of zeroth polarity of the function prior to applying the algorithm. The computational cost of the new algorithm involves the generation of polarity matrix Pcf(ll-,)] from the recursive matrix Q,(b(O)). Theorem 3: The number of GF(4) additions required to calculate the polarity matrix for an n-variable quatemary function is (1 3/9)( 13" - 4"). Proof: For an n-variable quatemary function, the polarity matrix has an order of 4". The new algorithm utilises eqn. 9 to generate the matrix. The total number of GF(4) addi- tions comprises the number of GF(4) additions required to compute the intermediate vectors R,pc1 Vc and the number of GF(4) additions to compute the remaining coefficients of the matrix.

The number of GF(4) additions A , required to compute the intermediate vectors,

A1 = 4 x 4"-' + 13 x 4 x 4"-2

Computational cost of the new algorithm

+ 132 x 4 x 4n-3 + . . . + 13"-l x 4 x 4"-"

The number of GF(4) additions A2 required to compute the remaining coefficients of the matrix,

Hence, the total number of GF(4) additions required is (13/

Theorem 4: The number of GF(4) multiplications required to calculate the polarity matrix of an n-variable quaternary function is (2/9)(13" - 4"). Proof: When eqn. 9 is used in recursive matrix generation, the number of GF(4) multiplication required to generate the Qb matrix is equal to the number of GF(4) multiplica- tions required to generate all the intermediate vectors Rp,cl, for 0 I fl I (n - 1). The total number of GF(4) multiphca- tions M is given by

9)(13" - 4"). 0

M = 2 x 4"-l + 13 x 2 x 4"-2

+ 132 x 2 x 4n-3 + . . . + 13"-l x 2 x 4"-" n

T = l

Hence, the total number of GF(4) multiplications required is (2/9)( 13" - 4").

In the literature, there are a few algorithms for comput- ing the fuied polarity RM expansions over GF(4), namely the:direct, fast and Gray-code algorithms [ lo ] and the fast computation using row polarity matrix method proposed in [5]. It should be noted that the new algorithm discussed in this paper starts from the operational domain, i.e. the truth vector of a Boolean function. All the remaining four algorithms start always from the zeroth polarity to obtain the remaining 4" - 1 polarities of an n-variable quatemary function. In general, the objective of these algorithms is to obtain the full polarity matrix and, more importantly, the optimum polarity of the function. As such, the computa-

IEE Proc.-Circuits Devices Syst., Vol. 148, No. 6, December 2001

tional complexity of calculating the zeroth polarity matrix needs to be included into the computational costs of the algorithms. The best algorithm to generate the zeroth polarity is the fast quatemary RM transform [ lo ] with the computational complexity tabulated in Table 1. This com- plexity is then added to corresponding complexity of the algorithms for a fair comparison.

Table 1: Computational cost of fast quatemary Reed- Muller transform

Number of Number of additions multiplications n

1 7 4

2 56 32

3 336 192

4 1 792 1 024

5 8 960 5120

6 43 008 24 576

7 200 704 114688

The most efficient algorithm of Green, being the Gray- code algorithm, has a computational cost of 4"(4" - 1) additions and (4"/3)(4" - 1) multiplications. The computa- tional complexity for the row polarity matrix method [5] is (4/3)(13" - 4") additions and (3/8)(3" - 1)4n multiplications. The number of operations from Table 1 are added to the complexity of direct, fast, Gray-code order and the row polarity matrix algorithms, for fair comparison with the new algorithm that always starts from the truth vector. The final results are shown in Tables 2 and 3. From these two Tables, it is clear that the new algorithm requires the least number of GF(4) additions and multiplications for smaller n and the row polarity method [5] is ultimately the most computationally efficient for n 27.

6 Conclusion

A new algorithm to calculate the polarity matrix of quater- nary switchmg functions in RM expansions over GF(4) has been presented. The polarity coefficient matrices are gener- ated by means of recursive square matrices. By using recur- sive properties of quatemary polarity matrices expanded by columns, the new algorithm minimises necessary hardware resources for n I 6 at the expense of computational time. The computational complexity of the new algorithm has been evaluated in terms of the number of required GF(4) additions and multiplications. The new algorithm is espe- cially advantageous in comparison to all other methods that generate the entire matrix [S, lo], for problems when there is a small number of variables of quaternary func- tions for which polarity coefficient matrices of RM expan- sions over GF(4) are being generated. The number of GF(4) additions is slightly more than the row polarity matrix counterpart, but the respective number of GF(4) multiplications is however superior. It should be noted that the hardware implementation of GF(4) multiplication is more expensive than the GF(4) addition [21]. Hence, the new algorithm is very well suited for hardware in a manner similar to the implementation of the ASIC device generat- ing all fixed polarities of RM expansions for an arbitrary 4- variable Boolean function [l]. The new algorithm is useful when a full polarity matrix needs to be generated, the case when the best polarity of quatemary RM expansion is needed. As the new algorithm is the most efficient for n I 6, then the quaternary circuits with input size of up to 46 can

293

Table 2 Comparison of computational complexity for additions

Number of additions

n Direct Fast

1 22 19

2 492 344

3 9 291 5 520

4 1 63 98 7 87 808

5 2778155 1 400 576

6 45932163 22 382 592

7 74% 726 939 357978112

Gray code order

19

296

4 368

67 072

1056512

16816 128

268 61 9 776

Row polarity matrix

19

260

3 180

39 532

502 652

6 473 292

83 843 548

New algorithm

13

221

3 081

40 885

534 833

6966141

90613081 _ _ _ _ ~

Table 3: Comparison of computational complexity for multiplications

Number of additions

n Direct Fast Gray code order Row polarity matrix New algorithm

1 8 8 8 7 2

2 21 6 160 112 80 34

3 4 468 2 688 1 536 81 6 474

4 82 184 43 520 22 784 U 704 6 290

5 1420644 698 368 354 304 98 048 82 282

6 23732440 11 182080 5615616 1 142 784 1071 714

7 388996244 178 946 048 89 587 71 2 13 545 472 13 940 474

be built in the most optimal way with its help. It should be noticed that this quaternary hardware realisations corre- sponding to RM expansions over GF(4) are also very well testable [4, 121 and can simply represent arbitrary binary function with 10g~(4~) input variables after an appropriate encoding of binary to quaternary variables. When a certain coefficient vector needs to be evaluated, the row polarity matrix generation method is more suitable [5] and will eventually require a smaller number of GF(4) operations.

There exists another method to find RM expansions over GF(4) based on GF(4) decision trees and information theo- retic measures [16]. This method does not require the gener- ation of a whole polarity matrix and, for some functions, it is able to provide good results. One of the advantages of the method presented in this paper is that, in definition 9, the recursive formula allows efficient implementation in the form of fast parallel computation on four-valued graph based decision diagrams, in a manner similar to the proces- sor-farm paradigm which allows the calculation to be per- formed by a few processors in parallel [22]. Multiple-place decision diagrams (MDDs) have been shown to be very effective [23] representations of multiple-valued functions that allow the manipulation of functions having a large number of variables and efficiently calculate their different respective spectra [24]. Other graph-based representations of discrete functions which are well suited for GF(q) are functional decision diagrams (FDDs) [25]. FDDs have already been generalised to multiple-valued functions, based on both Galois field and Reed-Muller-Fourier expansions [24]. By using the graph-based decision diagram representation of RM expansions over GF(4), the new algorithm can be implemented in a similar way as the methods for the binary case described in [26].

7 Acknowledgments

The authors wish to thank the referees for their numerous helpful comments.

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