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Lecture 10

Extension Fields:

Properties and Construction

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Finite field: GF(4)

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Finite field: GF(4)

One way to define GF(4): the (binary)polynomials over GF(2) modulo a prime

polynomial over GF(2) of degree 2. The only prime polynomial of degree 2over GF(2) is 2! ! ".

GF(4) #$%& "& & !"'&polynomials of

degree 2 modulo 2

! ! ".Operation tables for GF(4) using thispolynomial representation an be found

Obvious substitutions 2& ! " *

yields operation tables of last page.

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Finite field: GF(4)

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GF(4) (ontinued)

+rithmeti modulo 2! ! " ise,uivalent to replaing all ourrenesof 2! ! " by %. -n partiular2!

! " # % 2

# ! " (over GF(2)) Thus 2and all higher powers of

modulo 2! ! " an be replaed by apolynomial of degree ".

/very non0ero element of GF(4) is apower of :

GF(4) # $%& "& & 2# ! "'.

-n other wordsxis a primitive element,which we can denote b .

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Representations of GF(4)PolynomialNotation

BinaryNotation

Integer

Notation

EponentialNotation

! !! ! !

1 !1 1 !

1! 2 1

"1 11 3 2

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GF(4) (ontinued) /very element in GF(4) is a linear

ombination of the basis vetors "

and

GF(4) # $% " ! "'

Therefore multipliation in GF(4) is

determined by the produts of "and .

1roduts of the basis vetors

define multipliation

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GF(4) (onluded)

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Finite field arithmeti: GF()

To define GF() we need a primepolynomial f() over GF(2) of degree *.

3onstant oeffiient f%must be "otherwise f() is divisible by .

1arity of oeffiients must be odd&otherwise " is a fator.

Of the moni polynomials of degree *two satisfy the above re,uirements:

*! ! " & *! 2! ".

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GF() 5oth are prime beause they have no

fators of degree " (i.e. or ! ").

/ither an be used to define arithmetiin GF(). +rithmeti tables are slightlysimpler for *! ! ".

6hen GF() # binary polynomials

modulo*! ! " the 7ey e,uation is

*! ! " # % *# ! "&where

the element is the polynomial .

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1owers in GF()

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GF() /very non0ero element of GF() is

a power of . i.e. is a primitive

element. /very element of GF() is a binary

linear ombination of $"& & 2'.

8ene GF() # $ %" 2 9.'

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GF() operation tables

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GF(;)

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1owers in GF(;)

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Primitive Polynomials

An irreducible polynomialp(x)overGF(p)of degreemis primitiveif the

smallest positive integer nfor,whichp(x)dividesxn-1is n=pm-1

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& primiti'e polynomialis an irre()*i+le

polynomial p() of (egree m o'er GF(p),a'ing a primiti'e element of GF(pm)as a root-

.,is efinition means t,at/ if0

1- p() is irre*i+le o'er GF(p)/2- is primiti'e in GF(pm)/ an3- p() !/t,en/ p() is a primiti'e polynomialan

1 1

m

q

=