1. I. Finite Field Algebra

2. 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 “*”

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 Group G if for any a and b in G: a*b = b*a

4. 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

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. 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

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 . 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?

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 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

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. 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}

11. 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

12. Theorem (Read Only)
No two elements in two different Cosets of a subgroup H of 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. 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

17. 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

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)
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)

24. 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

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)