M. Khalily Dermany Islamic Azad University. finite number of element important in number theory, algebraic geometry, Galois theory, cryptography,

Finite Field or Galois Field

M. Khalily DermanyIslamic Azad University

finite number of element important in number theory, algebraic geometry,

Galois theory, cryptography, coding theory and Quantum error correction

applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type

Finite fields are an active area of research, including recent results on the Kakeya conjecture and open problems on the size of the smallest primitive root.

Introduction

Semi group <S,+>◦ Associative: (x+y)+z=x+(y+z)

Monoid <S,+,e>◦ A semi group with identity: a + e = a

Group <S,+>◦ A Monoid with inverses : a + (−a) = e◦ The order of a group is the number of elements in the group.

Abelian group <S,+>◦ Commutativity: a + b = b + a

Ring <S,+,.>◦ <S,+> is Abelian and <S,.> is group◦ Distributivity: a · (b + c) = (a · b) + (a · c).

Field <S,+,.>◦ <S,+> and <S,.> is Abelian◦ Distributivity a · (b + c) = (a · b) + (a · c).

History

(Z,+), e = 0 and the inverse of i is −i. (Q − {0}, ·). e = 1 and the inverse of a/b is b/a. ({0, 1},⊕), where ⊕ is exclusive-OR operation. Additive group: ({0, 1, 2, . . . ,m − 1},), where

m ∈ Z+, and

◦ e=0◦ ∀0 < i < m, m − i is the inverse of i

Multiplicative group: ({1, 2, 3, . . . , p − 1},), where p is a prime and ij ≡ i . j mod p.

Example of Abelian Groups

Closure of F under addition and multiplication ◦ For all a, b in F, both a + b and a · b are in F (or

more formally, + and · are binary operations on F). Associativity of addition and multiplication

◦ For all a, b, and c in F, a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.

Commutativity of addition and multiplication ◦ For all a and b in F : a + b = b + a and a · b = b · a.

Field

Existence of additive and multiplicative identity elements ◦ There exists an element of F, called the additive

identity element and denoted by 0, such that for all a in F a + 0 = a

◦ Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F a · 1 = a

the additive identity and the multiplicative identity are required to be distinct.

Field

Existence of additive inverses and multiplicative inverses or subtraction and division operations exist.◦ For every a in F, there exists an element −a in F,

such that a + (−a) = 0◦ Similarly, for any a in F other than 0, there exists

an element a−1 in F, such that a · a−1 = 1. ◦ The elements a + (−b) and a · b−1 are also denoted

a − b and a/b, respectively Distributivity of multiplication over addition

◦ For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c)

Field

<Z ,+,.>◦ <Z,+> Abelian group?◦ <Z-{0},.> Abelian group?◦ Distributivity?

<Q ,+,.> <{0,1} ,,. > binary field GF(2) <{0, 1, 2, . . . ,m − 1},> prime field (GF(p)), where p

is a prime. <R ,+,.> field Q() consisting of numbers of the form

with a, b ∈ Q, where is a primitive third root of unity

Field

∀a, b ∈ F and a, b 0. Then a · b 0. a · b = 0 and a 0 imply that b = 0. Cancellation law: a 0 and a · b = a · c imply

that b = c.

Properties of Fields

The order of a field is the number of elements of the field.

A field with finite order is a finite field. =field with p elements, p a prime number. this field finds applications in

◦ computer science, especially in cryptography and coding theory.

It is possible to extend the prime field GF(p) to a field of pm elements, GF(pm), which is called an extension field of GF(p).

Finite Field or Galois field

example F4 is a field with four elements Inverse Identity

Some small finite fields

all finite fields must have prime power order ◦ there is no finite field with 6 elements.

In any field F with m elements, the equation xm=x is satisfied by all elements x of F .

In any prime size field, it can be proved that there is always at least one element whose powers constitute all the nonzero elements of the field. This element is said to be primitive.

Properties of Finite Fields

For example, in the field GF(7), the number 3 is primitive as

6 x 2=33 x 32= 35= 5

Properties of Finite Fields(cont.)

In GF(7) ◦ multiplicative inverse of any element as 3i as 3-i =

36-I

◦ multiplicative inverse of 4 () is 2 (32)◦ multiplicative inverse of 5 (35) is 3 (31).

primitive

GF(7)

GF(7)

subtract 6 from 3 , ◦ first use the addition table to find the additive

inverse of 6, which is 1. ◦ Then we add 1 to 3 to obtain the result ◦ 3-6=3+(-6)=3+1=4

divide 3 by 2. ◦ first find the multiplicative inverse of 2, which is

4,◦ multiply 3 by 4 to obtain the result◦ 3÷2=3.(2-1)=3.4=5.

GF(7)

polynomials whose coefficients are from the binary field GF(2)

The degree of a polynomial is the largest power of X with a nonzero coefficient.

Computations with Polynomials

There are two polynomials over GF(2) with degree 1

◦ X and 1+X. There are four polynomials over GF(2) with degree 2

◦ X2 , 1 + X2 , X + X2 , and 1 + X + X2

In general, there are 2n polynomials over GF(2) with degree n.

degree of a polynomial

Adding polynomials

Dividing polynomials

Dividing polynomials

A polynomial p(X) over GF(2) of degree m is said to be irreducible over GF(2) if p(X) is not devisable by any polynomial over GF(2) of degree less than m but greater than zero.

the four polynomials of degree 2 ◦ X2, X2 + 1 and X2 + X are not irreducible◦ X2 + X + 1??

Irreducible

For any m ≥ 1, there exists an irreducible polynomial of degree m which divides

Irreducible (cont.)

Irreducible (cont.)

An irreducible polynomial p(X) of degree m is said to be primitive if the smallest positive integer n for which p(X) divides is

p(X) = X4 + X + 1 divides X15 + 1 ◦ does not divide any Xn + 1 for 1≤ n<15.

Hence, X4 + X + 1 is a primitive polynomial. The polynomial X4+X3+X2+X+1 is

irreducible ◦ it is not primitive, since it divides X5+ 1

Primitive

Primitive

m > 1 a new symbol

GF(2m)

GF(2m) (cont.)

GF(2m) (cont.)

GF(2m) (cont.)

Let p(X) be a primitive polynomial of degree m over GF(2). We assume that p()=0. Since p(X) divides

If we replace X by in above equation, we obtain

GF(2m) (cont.)

Therefore, under the condition that p()=0, the set F becomes finite and contains the following elements:

GF(2m) (cont.)

Therefore, the set F* is a Galois field of 2m elements. Also GF(2) is a subfield of GF(2m).

GF(2m) (cont.)

primitive polynomial p(X) = 1 + X + X4 over GF(2). ◦ Set p() = Then .

The identity is used repeatedly to form the polynomial representations for the elements of GF(24).

Example GF(24)

Example GF(24)(cont.)

Example GF(24)(cont.)

Fact:

Example GF(24)(cont.)

primitive polynomial p(X) = 1 + X + X2 over GF(2).

Set p() = Then . The identity is used repeatedly to form the

polynomial representations for the elements of GF(22).

Example GF(22)

Example GF(22)(cont.)

X2 + 6X + 25 does not have roots◦ —3 + 4j ◦ —3 — 4j

This is also true for polynomials with coefficients from GF(2)

Properties of GF(2m)

For example, X4 + X3 + 1 is irreducible over GF(2) and therefore it does not have roots from GF(2).

it has four roots from the field GF(24). If we substitute the elements of GF(24)

given by Table into X4 + X3 + 1, we find that , , , are the roots

Properties of GF(2m) (cont.)

Properties of GF(2m) (cont.)

there must exist two positive integers m and n such that m < n and

There must exist a smallest positive integer λ such that λ

This integer λ is called the characteristic of the field GF(q).

λ is a prime.

GF(q)

If q λ, then q is a power of λ.

for any k,m < λ and k m.

λ distinct elements in GF(q)

GF(λ) is called a subfield of GF(q)

GF(q)

Any two finite fields with the same number of elements are isomorphic. That is, under some renaming of the elements of one of these, both its addition and multiplication tables become identical to the corresponding tables of the other one.

isomorphic