Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cryptography and Underlying Algebraic
Structures
Groups, Finite Fields and Cryptography
Joseph Spring
Cryptography and Underlying Algebraic Structures
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Outline
1
Introduction
2
The Communication Channel
3
Symmetric and Asymmetric Cipher Systems
4
The Underlying Mathematical Structure
The Integer Factorisation Problem
The Discrete Logarithm Problem
5
Groups
Cyclic Groups
Groups and Primitives
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Introduction
The RSA encryption scheme is a Asymmetric key encrytion
scheme based on the Integer Factorisation Problem (IFP), in
contrast to the Diffie-Hellman Asymmetric key agreement
scheme which is based on the Discrete Logarithm Problem
(DLP). For Symmetric key encryption schemes we consider the
Vernam (One time pad) and discuss in overview the Data
Encryption Scheme (DES)
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
The Integer Factorisation Problem
The Discrete Logarithm Problem
Outline
1
Introduction
2
The Communication Channel
3
Symmetric and Asymmetric Cipher Systems
4
The Underlying Mathematical Structure
The Integer Factorisation Problem
The Discrete Logarithm Problem
5
Groups
Cyclic Groups
Groups and Primitives
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
The Integer Factorisation Problem
The Discrete Logarithm Problem
Integer Factorisation Problem
IFP
Definition
The Integer Factorisation Problem may be expressed as
follows:
Let n ∈ N+ denote a positive integer. The integer factorisation
problem is to find prime numbers p 1 , p2 , . . . , pr and indices
e1 , e2 , . . . , er such that:
n = p1e1 p2e2 . . . prer
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
The Integer Factorisation Problem
The Discrete Logarithm Problem
Outline
1
Introduction
2
The Communication Channel
3
Symmetric and Asymmetric Cipher Systems
4
The Underlying Mathematical Structure
The Integer Factorisation Problem
The Discrete Logarithm Problem
5
Groups
Cyclic Groups
Groups and Primitives
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
The Integer Factorisation Problem
The Discrete Logarithm Problem
Discrete Logarithm Problem
DLP
Definition
The Discrete Logarithm Problem may be expressed as follows:
Given a multiplicative cyclic group (G, ∗), and an element
β ∈ G find r such that
β = αr mod p
r is said to be the logarithm of β and is sometimes expressed
as logα (β). p is taken to be a prime number.
Note: G is generated by a primative element α, i.e. G
Joseph Spring
Cryptography
=
<α>
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Outline
1
Introduction
2
The Communication Channel
3
Symmetric and Asymmetric Cipher Systems
4
The Underlying Mathematical Structure
The Integer Factorisation Problem
The Discrete Logarithm Problem
5
Groups
Cyclic Groups
Groups and Primitives
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Groups
Definition
Let (G, ∗)denote a set of elements G together with an operator
* denoting addition or multiplication. ∀ a, b, c ∈ G
Closure Law
(1)
Associative Law
(2)
∃ e ∈ G s.t. a ∗ e = e ∗ a = a
Identity Law
(3)
∃ a−1 s.t. a ∗ a−1 = a−1 ∗ a = e
Inverse Law
(4)
Commutative Law
(5)
a ∈ G, b ∈ G ⇒ a ∗ b ∈ G
a ∗ (b ∗ c) = (a ∗ b) ∗ c
a∗b =b∗a
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Any set G with operator * satisfying (1) - (4) above is said
to form a Group
Any group (G,*) which also satisfies (5) is said to be an
Abelian Group (named after Nils Abel)
Definition
A cyclic group is a group that can be generated by a single
member α of the group. α is referred to as a primtive for the
group
We recall that given a cyclic group (G,*), with G =< α > in
which α is a primitive for the group G, there exists a prime
number p such that αp−1 = 1 and hence higher powers of
α generate the α r with 0 < r ≤ p − 1
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Example
Let Z∗13 denote the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Then
Z∗13 =< 2 > under the binary operation of multiplication.
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Outline
1
Introduction
2
The Communication Channel
3
Symmetric and Asymmetric Cipher Systems
4
The Underlying Mathematical Structure
The Integer Factorisation Problem
The Discrete Logarithm Problem
5
Groups
Cyclic Groups
Groups and Primitives
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Groups
Groups and Primitives
Example
Find a primitive for the multiplicative group (Z ∗7 , ∗).
(Z∗7 = {1, 2, 3, 4, 5, 6} so we are looking for 1 ≤ a ≤ 6 such that
a generates the integers between 1 and 6 inclusive.
Clearly 1 is not the required primitive.
For 2 we have: 2 1 = 2, 22 = 4,23 = 8 = 1mod7, so
< 2 >= {1, 2, 4}. Hence 2 is not a primitive for (Z ∗7 , ∗).
For 3 we have: 3 1 = 3, 32 = 9 = 2mod7, 33 = 6mod7,
34 = 4mod7, 35 = 5mod7 and 3 6 = 1mod7, hence 3 is a
primitive for the multiplicative group (Z∗7 , ∗).
< 3 >= {3, 2, 6, 4, 5, 1} = {1, 2, 3, 4, 5, 6}
Joseph Spring
Cryptography
Introduction
The Communication Channel
Symmetric and Asymmetric Cipher Systems
The Underlying Mathematical Structure
Groups
Cyclic Groups
Groups and Primitives
Example
Can you find any other primitives for the multiplicative group
(Z∗7 , ∗)?
Joseph Spring
Cryptography