I. Finite Field Algebra

1. I. Finite Field Algebra

2. Binary Operation G is a set of elements “*” A binary operation on G is a rulethat assigns to each pair of elements a and ba uniquely defined element c G is closed under “*”

3. 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 GroupGif for any a and b in G: a*b = b*a

4. Theorems The identity element in a group G is unique If we have two identity elements e and e’ in G, Then, e’ =e’ * e =ee, e’ are identical Proof 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

5. 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

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

7. 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 . j =r i 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 .

8. 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 ain 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

9. 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

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

11. Theorem (Read Only) Let Hbe a subgroup of a group G with binary operation *. No two elements in a Coset of Hare identical

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

13. 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

14. 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

15. 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

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

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

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

19. Theorem (Read Only) Proof

20. The order of a Field Element (Read)

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

22. 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

23. 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) 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)

24. Binary Field Arithmetic

25. 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

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

27. 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

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

29. 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)

30. Useful Property of Polynomials over GF(2)