11848 ch04

Cryptography and
Network Security
Chapter 4

Introduction
 will now introduce Algebraic structures
 of increasing importance in cryptography



1.
2.
3.

AES, Elliptic Curve, IDEA, Public Key
Common algebraic structures are:
Groups
Rings
Fields

Group
 a set of elements or “numbers”


may be finite or infinite

 with some operation whose result is also

in the set (closure)
 obeys:




associative law: (a.b).c = a.(b.c)
has identity e:
e.a = a.e = a
has inverses a-1: a.a-1 = e

 if commutative


a.b = b.a

then forms an abelian group

 The set of residue integers with the

addition operator, G = <Zn, +>, is a
commutative group. We can perform
addition and subtraction on the elements
of this set without moving out of the set.
 Checking the properties:
1. Closure is satisfied. 3+5 = 8
2. Associativity is satisfied. (3+5)+ 4=
3+(5+4)= 12
3. Commutativity is satisfied. 3+5= 5+3
4. Identity element exists. 3+0=0+3=3
5. Inverse exists for 3 its -3

 Finite group: A group is called a finite

group if the set has a finite number of
elements; other wise it is an infinte group.
 Order of a group: NO. of elements present
in the group.
 Subgroup: A subset H of a group G is a
subgroup of G if H itself is a group , with
respect to the operations on G.
 Is the group H= <Z10, +> a subgroup of
the group G= <Z12,+>????

Cyclic Group
 define exponentiation

as repeated

application of operator


example:

a3 = a.a.a

 and let identity be:

e=a0
 a group is cyclic if every element is a
power of some fixed element


ie b = ak

for some a and every b in group

 a is said to be a generator of the group

Cyclic Sub group
 If a subgroup can be generated using the

power of an element, the subgroup is
called the cyclic subgroup.




example: an = a.a.a.a.......a(n times)
The set made from this process is
referred to as <a>.
a0 = e.

 Four cyclic subgroups can be made from

group G= <Z6, +>
 They are H1=<{0},+>
 H2=<{0,2,4},+>
 H3=<{0,3},+>
 H4=G

 Suppose a group has only 4 elements

{1,3,7,9} and is denoted by Z10*.
 Find the elements of these subgroups.

 Suppose a group has only 4 elements

{1,3,7,9} and is denoted by Z10*.
 Find the elements of these subgroups.
H1=1
H2=1,9
H3=1,3,9,7

Ring





a set of “numbers”
with two operations (addition and multiplication)
which form:
an abelian group with addition operation
and multiplication:







has closure
is associative
distributive over addition:

a(b+c) = ab + ac

if multiplication operation is commutative, it
forms a commutative ring
if multiplication operation has an identity(a1 =
1a= a) and no zero divisors(ab=0 either a or
b=0), it forms an integral domain

Field
 a set of numbers
 with two operations(addition,

multiplication) which follows all the rules of
groups and rings and one more condition:
 MI= For each a in F, except zero, there is
an element a.a(^-1)= a(^-1).a =1
 have hierarchy with more axioms/laws


group -> ring -> field

Finite (Galois) Fields
 finite fields play a key role in cryptography
 can show number of elements in a finite

field must be a power of a prime pn
 known as Galois fields
 denoted GF(pn)
 in particular often use the fields:



GF(p)
GF(2n)

Galois Fields GF(p)
 GF(p) is the set of integers {0,1, … , p-1}

with arithmetic operations modulo prime p
 these form a finite field



since have multiplicative inverses
find inverse with Extended Euclidean algorithm

 hence arithmetic is “well-behaved” and can

do addition, subtraction, multiplication, and
division without leaving the field GF(p)

GF(7) Multiplication Example
× 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 2
6 0 6 5 4 3 2 1

Polynomial Arithmetic
 can compute using polynomials

f(x) = anxn + an-1xn-1 + … + a1x + a0 = ∑ aixi
• nb. not interested in any specific value of x
• which is known as the indeterminate

 several alternatives available




ordinary polynomial arithmetic
poly arithmetic with coords mod p
poly arithmetic with coords mod p and
polynomials mod m(x)

Ordinary Polynomial Arithmetic
 add or subtract corresponding coefficients
 multiply all terms by each other
 eg

let f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1
f(x) + g(x) = x3 + 2x2 – x + 3
f(x) – g(x) = x3 + x + 1
f(x) x g(x) = x5 + 3x2 – 2x + 2

Polynomial Arithmetic with
Modulo Coefficients
 when computing value of each coefficient

do calculation modulo some value


forms a polynomial ring

 could be modulo any prime
 but we are most interested in mod 2



ie all coefficients are 0 or 1
eg. let f(x) = x3 + x2 and g(x) = x2 + x + 1
f(x) + g(x) = x3 + x + 1
f(x) x g(x) = x5 + x2

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)
 if g(x) has no divisors other than itself & 1
say it is irreducible (or prime) polynomial
 arithmetic modulo an irreducible
polynomial forms a field

Polynomial GCD


can find greatest common divisor for polys




c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest
degree which divides both a(x), b(x)

can adapt Euclid’s Algorithm to find it:
Euclid(a(x), b(x))
if (b(x)=0) then return a(x);
else return
Euclid(b(x), a(x) mod b(x));



all foundation for polynomial fields as see next

Modular Polynomial
Arithmetic
 can compute in field GF(2 n)




polynomials with coefficients modulo 2
whose degree is less than n
hence must reduce modulo an irreducible poly
of degree n (for multiplication only)

 form

a finite field
 can always find an inverse


can extend Euclid’s Inverse algorithm to find

Computational
Considerations
 since coefficients are 0 or 1, can represent

any such polynomial as a bit string
 addition becomes XOR of these bit strings
 multiplication is shift & XOR


cf long-hand multiplication

 modulo reduction done by repeatedly

substituting highest power with remainder
of irreducible poly (also shift & XOR)

Computational Example



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



and multiplication is






(x2+1) + (x2+x+1) = x
101 XOR 111 = 0102
(x+1).(x2+1) = x.(x2+1) + 1.(x2+1)
= x3+x+x2+1 = x3+x2+x+1
011.101 = (101)<<1 XOR (101)<<0 =
1010 XOR 101 = 11112

polynomial modulo reduction (get q(x) & r(x)) is



(x3+x2+x+1 ) mod (x3+x+1) = 1.(x3+x+1) + (x2) = x2
1111 mod 1011 = 1111 XOR 1011 = 01002

11848 ch04

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11848 ch04

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