Implementing-cryptography-using-python

106 Chapter 4 ■ Cryptographic Math and Frequency Analysis
while not is_prime(p, 128):
p = generate_prime_candidate(length)
return p
print('Generate the value of n using two prime numbers p and q:')
print('\n')
print(generate_prime_number())
print('\n')
Figure 4.2: Generate_large_primes.py
Basic Group Theory
In abstract algebra and other fields of mathematics, group theory studies the
algebraic structures known as groups. The concept of a group is central to abstract
algebra: other familiar algebraic structures, such as vector spaces, rings, and fields,
can all perform as groups endowed with additional operations and axioms. Group
theory comes into play as you explore the Diffie-Hellman and RSA encryption
systems in Chapter 8. For now, consider the idea that elements in groups (e.g.,
integers) have orders (i.e., cycle lengths), and this lets you use modular arithmetic
to speed up significant computations. Group theory is exploited to build cryptographic
schemes and the associated cryptanalysis.
To start your exploration into the basics of group theory, we define a group
as a set, G, together with a binary operation * (called the group law of G) that
combines any two elements to form another element. The operation may be multiplication,
addition, rotation, composition, or any other action that consumes
two elements and returns one.
To be eligible as a group, the set and operation (G, *) requires four properties,
known as the group axioms.
■■
Closure: a, b ∈ G implies a * b ∈ G
■■
Identity: There exists e ∈ G such that a * e = e * a = a for all a ∈ G
■■
Inverse: For each a ∈ G, there exists a −1 ∈ G such that a * a −1 = a −1 * a = e
■■
Associativity: a, b, c ∈ G implies (a * b) * c = a* (b * c)