1. Monoids, Groups, Rings, Fields
Algebraic Structures
Monoids, Groups, Rings, Fields

2. Monoid
For a set G and an operator  : G × G → G, a pair (G, ·) is a monoid iff the following properties are satisfied:
Identity
There is e ∈ G such that for all a ∈ G, a · e = a.
Associativity
For all a, b, c ∈ G, a · (b · c)=(a · b) · c.
Algebraic Structures

3. Monoid
Closure
Associativity
Identity
Algebraic Structures

4. Example Let N be the set of non-negative integers.
(N, +) is a monoid because:
For any a and b in N, a + b is in N.
For any a, b and c in N, (a + b) + c = a + (b + c).
There is 0 such that for any a in N, a + 0 = a.
(N, ) is a monoid because:
For any a and b in N, a  b is in N.
For any a, b and c in N, (a  b)  c = a  (b  c).
There is 1 such that for any a in N, a  1 = a.
Algebraic Structures

5. Example Let N be the set of of non-negative integers.
(N, -) is not a monoid because:
There are a and b in N such that a - b is in not N.
There are a, b and c in N such that (a - b) - c  a - (b - c).
(N, ) is not a monoid because:
There are a and b in N, such that a  b is in not N.
There are a, b and c in N such that (a  b)  c  a  (b  c).
Algebraic Structures

6. Group
A monoid (G, ·) is a group iff for all a ∈ G, there exists an element b ∈ G such that a · b = e.
Let I be the set of integers.
(I, +) is a group because:
For any a and b in I, a + b is in I.
For any a, b and c in I, (a + b) + c = a + (b + c).
There is 0 such that for any a in I, a + 0 = a.
For any a in I, there is a-1 = -a such that a + a-1 = 0.
Algebraic Structures

7. Group
A monoid (G, ·) is a group iff for all a ∈ G, there exists an element b ∈ G such that a · b = e.
(I, ) is not a group because:
For any a and b in I, a  b is in I.
For any a, b and c in I, (a  b)  c = a  (b  c).
There is 1 such that for any a in I, a  1 = a.
For some a in I, there is no a-1 such that a  a-1 = 1.
Algebraic Structures

8. Group closure associativity identity inverse 2301233
Algebraic Structures

9. Commutative Group
A group (G, ·) is commutative or Abelian iff for all a, b ∈ G, a · b = b · a.
Let I be the set of integers.
(I, +) is a commutative group because:
it is a group.
For any a and b in I, a + b = b + a.
(I, ) is not a commutative group because:
it is not a group.
For any a and b in I, a  b = b  a.
Algebraic Structures

10. Commutative Group closure identity associative inverse commutative
Algebraic Structures

11. Relationship Monoid group Commutative group 2301233
Algebraic Structures

12. Ring
For a set R and binary operators · and + over R, the triple (R, +, ·) is a ring iff the following properties are satisfied: Commutative addition (R, +) is an Abelian group with identity element 0. Multiplication (R, ·) is a monoid with identity element 1. Distributivity For all a, b, c ∈ R, a · (b + c) = a · b + a · c.
Algebraic Structures

13. Field
A non-empty set F with two binary operation + (addition) and  (multiplication) is called a field if
(F, +) is a commutative (additive) group, and
(F – {0}, ) is a commutative (multiplicative) group.
Algebraic Structures

14. Cryptography and Finite Fields
Cryptography focuses on finite fields.
For any prime integer p and any integer n greater than or equal to 1, there is a unique field, called Galios field, with pn elements in it, denoted by GF(pn).
“Unique” means that any two fields with the same number of elements must be essentially the same, except perhaps for giving the elements of the field different names.
Algebraic Structures

15. Galois Fields in Cryptography
GF(p1) : ({0,1,2,…,p-1}, +, *) for integers modulo p. Example Let p = 7. Z7 = {0,1,2,3,4,5,6}. GF(7) = (Z7 , +, *). (Z7, +) is a commutative group with identity 0, and the inverse of a is 7-a. (Z7, *) is a commutative group with identity 1, and the inverse of a is x such that ax 1 mod 7.
Algebraic Structures

16. Galois Fields in AES GF(28) : (Z256, +, *) where Z256 = {0,1,…,255}.
Each element b=b7 b6 b5 b4 b3 b2 b1 b0in Z256
is a polynomial
b7 x7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0.
Algebraic Structures

17. AES Specifications
Input & output block length: 128 bits. State: 128 bits, arranged in a 4-by-4 matrix of bytes. Each byte is viewed as an element in a field.
A0,0
A0,1
A0,2
A0,3
A1,0
A1,1
A1,2
A1,3
A2,0
A2,1
A2,2
A2,3
A3,0
A3,1
A3,2
A3,3
Algebraic Structures

18. Addition in GF(28) a7 x7 + a6x6 +…+ a1x+ a0 b7 x7 + b6x6 +…+ b1x+ b0
a7 a6 a5 a4 a3 a2 a1 a0
b7 b6 b5 b4 b3 b2 b1 b0
a7 x a6x6 +… a1x a0
b7 x b6x6 +… b1x b0
(a7+b7)x7+ (a6+b6)x6+ …+ (a1+b1)x+ (a0+b0)
All additions of polynomial coefficient are modulo 2.
1 + 1 =0 1 – 1 = 0 1  1 = 0
1 + 0 = 1 1 – 0 = 1 1  0 = 0
0 + 1 = 1 0 – 1 = 1 0  1 = 0
0 + 0 = 0 0 – 0 = 0 0  0 = 0
Algebraic Structures

19. Multiplication in GF(28)
a7 x a6x6 +… a1x a0
b7 x b6x6 +… b1x b0
(a7 b0) x7 + (a6b0) x6+ …+ (a1b0) x+ (a0b0)
(a7 b1) x8 + (a6b1) x7 + (a5b1) x6+ …+ (a0b1)x
(a7 b2)x9 +(a6b2) x8 +(a5b2) x7+ (a4b2)x6 +… …
  (ai bj) xi+j .
i=0,…,7 j=0,…,7
Algebraic Structures

20. Multiplication in GF(28)
The result can be a degree k polynomial, where k  14. Divide the result by a degree 8 polynomial . AES uses x8 + x4 + x3 + x +1.
Algebraic Structures

21. Example x7 + x5 + x4 + x2 + x => (75421)
( ) * ( )
( ) * (6) = ( )
( ) * (4) = ( )
( ) * (1) = ( )
( ) * (0) = )
( )
Algebraic Structures

22. Example
(x13 + x10 + x9 + x8+ x5 + x4 + x3 + x )/ (x8 + x4 + x3 + x +1)
=> ( )/( )
( )
( ) * (5) = ( )
( )
( ) * (2) = ( )
the remainder ( )
Algebraic Structures