I. Finite Field Algebra

Binary Operation G is closed under “*” G is a set of elements “*” A binary operation on G is a rule that assigns to each pair of elements a and b a uniquely defined element c G is closed under “*”

Groups A set G on which a binary operation “*” is defined is called a Group if: The binary operation is associative G contains an identity element e (a *e = e *a = a) For any element a in G, there exists an inverse element a’ in G (a *a’ = a’ *a = e) Commutative Group G if for any a and b in G: a*b = b*a

Theorems The identity element in a group G is unique Proof If we have two identity elements e and e’ in G, Then, e’ =e’ * e =e  e, e’ are identical The inverse of any element in a group G is unique Proof If we have two inverse elements a’ and a’’ for a in G, Then, a’ =a’ *e =a’ *(a*a’’)  a’, a’’ are identical

Example: Modulo-2 Addition The set G={0,1} is a group of order 2 under modulo-2 addition Modulo-2 addition is associative The identity element is 0 The inverse of 0 is 0 in G The inverse of 1 is 1 in G Modulo-2 Addition

Example: Modulo-m Addition The set G={0,1,2,…,m-1} is a group of order m under modulo-m addition Modulo-m addition is associative The identity element is 0 The inverse of i is m-i in G Modulo-m Addition + i j =r i+j=qm+r, 0≤r<m-1

Example: Modulo-p Multiplication G={1,2,…,p-1}, p is a prime number, is a group of order p under modulo-p multiplication Modulo-p Multiplication . i j =r i.j=qp+r, 0≤r<p-1 Modulo-5 Multiplication Modulo-5 multiplication is associative The identity element is 1 The inverse of 1 is 1 in G The inverse of 2 is 3 in G The inverse of 3 is 2 in G The inverse of 4 is 4 in G . 1 2 3 4 . Proof?

SubGroups Define a set G as a group under a binary operation *, A subset H is called a subgroup if H is closed under the binary operation * For any element a in H, the inverse of a is also in H Example: Let G be the set of rational numbers constitute a group under real addition. Therefore, The set of integers H is a proper (i.e., H ≠G) subgroup under real addition

Cosets H is a subgroup of a group G under binary operation * If the group G is commutative, a *H =H *a is simply labeled as: a Coset of H

3 + H ={3,7,11,15}= 7 + H + H ={0,4,8,12} 1 + H ={1,5,9,13} 2 + H Example G={0,1,2,…,15} under modulo-16 addition H={0,4,8,12} is a subgroup of G why? The coset 3 + H ={3,7,11,15}= 7 + H Four Distinct and Disjoint Cosets of H + H ={0,4,8,12} 1 + H ={1,5,9,13} 2 + H ={2,6,10,14} 3 + H ={3,7,11,15}

Theorem (Read Only) Let H be a subgroup of a group G with binary operation *. No two elements in a Coset of H are identical

Theorem (Read Only) No two elements in two different Cosets of a subgroup H of a group G are identical

Properties of Cosets Every element in G appears in one and only one of distinct Cosets of H All the distinct Cosets of H are disjoint The union of all distinct Cosets of H forms the group G

Fields Let F be a set of elements on which two binary operations called addition “+” and multiplication “.” are defined. The set F and the two binary operations represent a field if: F is a commutative group under addition. The identity element with respect to addition is called the zero element (denoted by 0) The set of nonzero elements in F is a commutative group under multiplication. The identity element with respect to multiplication is called the unit element (denoted the 1 element) Multiplication is distributive over addition: a.(b+c) = a.b + a.c, a, b, c in F

Basic Properties of Fields a.0=0.a=0 If a,b≠0, a.b≠0 a.b=0 and a≠0 imply that b=0 -(a.b)=(-a).b=a.(-b) If a≠0, a.b=a.c imply that b=c

Galois Field of the order 2 Binary Field GF(2) Modulo-2 Addition Modulo-2 Multiplication + 1 . 1 F={0,1} is a Finite field of order 2 under modulo-2 addition and modulo-2 multiplication Galois Field of the order 2

Subtraction and Division (GF(7)) Modulo-7 Addition Modulo-7 Multiplication + 1 2 3 4 5 6 . 1 2 3 4 5 6 Ex: 3-6=3+(-6)=3+1=4 Ex: 3/2=3.2-1 =3.4=5

Characteristic of a Finite Field GF(q) (Read)

Theorem (Read Only) Proof

The order of a Field Element (Read)

Theorem (Read Only) Let a be a nonzero element of a finite field GF(q). Then aq-1=1 Proof

Theorem (Read Only) Let a be a nonzero element in a finite field GF(q). Let n be the order of a. Then n divides q-1 Proof

A Primitive Element of GF(q) A nonzero element a is said to be primitive if the order of a is q-1 Example: GF(7) 31=3 32=2 33=6 34=4 35=5 36=1 41=4 42=2 43=1 Order of element 4 is 3 which is a factor of 6 Element 4 is not a primitive element of GF(7) Order of element 3 is 6 Element 3 is a primitive element of GF(7)

Binary Field Arithmetic Polynomials of Degree 1 over GF(2) Polynomials of Degree 2 over GF(2) Polynomials of Degree n over GF(2) X X2 2n Polynomials over GF(2) with degree n 1+X 1+X2 X+X2 1+X+X2

Addition of Two Polynomials over GF(2) Example: g(X) = 1+X+X3+X5 f(X) = 1+X2+X3+X4+X7 g(X)+f(X) = X+X2+X4+X5+X7

Division of Two Polynomials over GF(2) (Quotient q(X)) (Remainder r(X))

Irreducible Polynomials A polynomial p(X) over GF(2) of degree m is said to be irreducible over GF(2) if p(X) is not divisible by any polynomial over GF(2) of degree less than m but greater than 0

Theorem Any irreducible polynomial over GF(2) divides Xn+1 where n=2m-1 and m is the degree of the polynomial

Primitive Polynomials An irreducible polynomial p(X) of degree m is said to be primitive if the smallest positive integer n for which p(X) divides Xn+1 is n=2m-1 Example p(X)=X4+X+1 divides X15+1 but does not divide any Xn+1 for 1≤n<15 (Primitive) p(X)= X4+X3+X2+X+1 divides X5+1 (Irreducible but Not Primitive)

Useful Property of Polynomials over GF(2)