Implementing-cryptography-using-python

Chapter 4 ■ Cryptographic Math and Frequency Analysis 103
Figure 4.1 shows the use the Miller-Rabin primality test to display all prime
numbers between 1 and 1000.
Figure 4.1: Miller.py test
Now that you have a better understanding of primality tests, you can use
the Miller-Rabin primality test to verify if much larger numbers are prime. In
the next code listing, you can test to make sure that a specific prime satisfies
a specific set of bits such as 1,024 or 2,048; large primes such as these will be
used in stronger encryption that will be introduced in Chapter 8. In the next
section, you will see the is_prime() function used to ensure that the numbers
being generated are prime.
The code will use the prime number theorem to generate an approximation
of π(n) because when n tends to infinity, π(n)/(n/ln(n)) = 1. The probability that
a randomly chosen number is prime is 1/ln(n). This is due to the fact that there
are n positive integers ≤ n and approximately n/ln(n) primes, and (n/ln(n))/
n = (1/ln(n)). To get an understanding of what the theorem states, examine the
probability of finding a prime number of 1,024 bits. Using the prime number
theorem, you will see that the probability is 1/(ln(2 1,024 )) = (1/710). Prime numbers
have to be odd since 2 will divide into any even number; the single exception
is the number 2. When attempting to generate a large prime, you can increase
the probability by 2, so on average, to generate a 1,024-bit prime number, you
may need to test 355 randomly generated numbers.
Examine the following Python code to test large prime numbers:
def is_prime(n, k=128):
""" Test if a number is prime
Args:
n -- int -- the number to test
k -- int -- the number of tests to do
return True if n is prime
"""
# Test if n is not even.
# But take care, 2 is prime !
if n == 2 or n == 3:
return True