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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
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