Implementing-cryptography-using-python

100 Chapter 4 ■ Cryptographic Math and Frequency Analysis
Fermat’s Little Theorem
Fermat’s little theorem is used in number theory to compute the powers of integers
modulo prime numbers. The theorem is a special case of Euler’s theorem.
(We explore Euler’s theorem later in this chapter.) Fermat’s little theorem states
let p be a prime number, and a be any integer.
or
If n is a prime number, then for every a, 1 < a < p − 1,
a p − 1 ≡ 1 (mod p)
a p − 1 % p = 1
To ensure this makes sense, let’s look at an example:
p = prime integer number
a = integer which is not a multiple of p
According to Fermat’s little theorem,
2 (17 − 1) ≡ 1 mod (17)
65,536 % 17 ≡ 1
This means (65,536 – 1) is a multiple of 17. This is proven by multiplying 17 *
3,855, which equals (65,536 – 1) or 65,535. If you know the modulo m is prime,
then you can also use Fermat’s little theorem to find the inverse. We will cover
this in more detail later in this chapter.
Here is a quick and easy function that will return whether an integer is
prime or not:
>>> def CheckIfProbablyPrime(x):
... return pow(2, x-1, x) == 1
>>> CheckIfProbablyPrime(19)
True
>>> CheckIfProbablyPrime(31)
True
>>> CheckIfProbablyPrime(589)
False
Miller-Rabin Primality Test
Similar to the other tests featured in this section, the Miller-Rabin primality
test is a primality test; it was first discovered by M. M. Artjuhov in 1967. It was
later rediscovered in 1976 by Gary L. Miller. Miller’s version of the test is deterministic.
Michael Rabin further modified the algorithm in 1980 to obtain an
unconditional probabilistic algorithm.