Abstract Algebra In Cryptography

7/31/2019 Abstract Algebra In Cryptography

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Abstract AlgebraIN

Cryptography


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Field

A set of elements with two binary operations,called addition and multiplication

Obeys:

Closure under addition and multiplication Associativity of addition and multiplication

Commutativity of addition and multiplication

Additive and Multiplicative Identity

Distributive laws No Zero divisors

Additive and Multiplicative Inverse


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Finite Fields

All encryption algorithm, both symmetric and publickey, involve arithmetic operations on integers.

If one of the operation is division, then we need towork in arithmetic defined over fields.

Number of elements in finite fields must be a power ofa prime number : pn

Also known as Galois Fields

Denoted by: GF(pn)

In particular often used: GF(pn) n=1

GF(pn) p=2 & n1


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Galois Field (p)

GF(p) is a set of integers {0,1,.p-1} with

arithmetic operations modulo prime p

Forms a finite field

Since multiplication inverse is defined

We can perform addition, subtraction,

multiplication and division without leaving the

field GF(p).


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Arithmetic Modulo 7

Set of elements {0,1,2,3,4,5,6}

Addition:

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

Additive Identity

Additive Inverse

Both Additive Identity and Inverse


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Contd..

Multiplication:

X 0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6

2 0 2 4 6 1 3 5

3 0 3 6 2 5 1 4

4 0 4 1 5 2 6 3

5 0 5 3 1 6 4 26 0 6 5 4 3 2 1

Both Multiplicative Identity and Inverse

Multiplicative Inverse

Multiplicative Identity


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Modulo 8 Addition:

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

Additive Identity

Additive Inverse

Both Additive Identity and Inverse


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Modulo 8 Multiplication

X 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 6 4 2 6

7 0 7 6 5 4 3 2 1

Both Multiplicative Identity and Inverse

Multiplicative Inverse

Multiplicative Identity


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Observations

Modulo 7-

Multiplication inverse is defined for each element ofthe set.

Modulo 8- Multiplication inverse is defined only for 1,3,5,7

Hence, Arithmetic modulo 7 is a Finite Fieldwhere as Arithmetic modulo 8 is not a Finite Field

Arithmetic modulo 7 is a Galois Field of typeGF(pn) and can be represented as GF(7)

where p is a prime number and n = 1 .


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Polynomial Arithmetic

can be computed using polynomials

f(x) = anxn+an-1x

n-1+..+a1x+a0= aixi

Classes of polynomial arithmetic Ordinary polynomial arithmetic

Polynomial arithmetic in which coefficients are in

GF(p)

Polynomial arithmetic in which coefficients are in

GF(p) & polynomials are defined modulo a polynomial

m(x) whose highest power is some integer n.


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Polynomial Arithmetic with modulo

coefficients

Modulo is considered when computing value ofeach coefficients.

could be modulo any prime but we are interested

in modulo 2 i.e. coefficients are 0 or 1

Modulo 2 arithmetic

Addition - XOR

Multiplication first multiply using ordinarypolynomial multiplication then add using additionmodulo 2


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E.g. let f(x) = x3+x2 and g(x) = x2+x+1

f(x) + g(x) = x3 + x + 1

f(x) + g(x) = x3 + x2

X x2 + x + 1

x3 + x2

x4 + x3

x5 + x4

x5 + x2


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Polynomial Division

can write any polynomial in the form:

f(x) = q(x) g(x) + r(x)

can interpret r(x) as being a remainder

r(x) = f(x) mod g(x)

if have no remainder say g(x) divides f(x)

ifg(x) has no divisors other than itself & 1 say

it is irreducible (or prime) polynomial arithmetic modulo an irreducible polynomial

forms a field


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Modular Polynomial Arithmetic

Forms a Finite field

Coefficients are in GF(p)

Polynomials modulo an irreducible polynomial m(x).

Uses the set of all polynomials of degree n-1 or less over

the field Zp Arithmetic follows the ordinary rules of polynomial

arithmetic using the basic rules of algebra, with thefollowing two refinements:

Arithmetic on the coefficients is performed modulo p

If multiplication results in a polynomial of degreegreater than n-1, then the polynomial is reducedmodulo some irreducible polynomial m(x) of degree n.


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Galois Field GF(2n)

To construct, need to choose a irreducible

polynomial m(x) of degree n

Polynomials, with coefficients modulo 2

whose degree less than n

Must be reduced modulo an irreducible

polynomial of degree n ( in multiplication)

Can be uniquely represented by n binary bits.


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Example: GF(23)


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Computational Example

in GF(23) have (x2+1) is 1012 & (x2+x+1) is 1112

so addition is (x2+1) + (x2+x+1) = x

101 XOR 111 = 0102

and multiplication is (x+1).(x2+1) = x.(x2+1) + 1.(x2+1)

= x3+x+x2+1 = x3+x2+x+1

011.101 = (101)


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Generator for GF(23)

Irreducible polynomial m(x) = x3 + x + 1

+ 000 001 010 100 100 101 110 111

0 1 g1 g2 g3 g4 g5 g6

000 0 0 1 G g2 g+1 g2+g g2+g+1 g2+1

001 1 1 0 g+1 g2+1 g g2+g+1 g2+g g2

010 g1 g g+1 0 g2+g 1 g2 g2+1 g2+g+1

100 g2 g2 g2+1 g2+g 0 g2+g+1 g g+1 1

011 g3 g+1 g 1 g2+g+1 0 g2+1 g2 g2+g

110 g4 g2+g g2+g+1 g2 g g2+1 0 1 g+1

111 g5 g2+g+1 g2+g g2+1 g+1 g2 1 0 g

101 g6 g2+1 g2 g2+g+1 1 g2+g g+1 g 0

Abstract Algebra In Cryptography

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