## Generating Random Large Primes

### March 31, 2017

Sometimes you need to have a large prime, for testing, cryptography, or some other purpose. I’m talking about primes of several hundred to a few thousand digits that are certified — proven to be prime — rather than probable primes according to a Miller-Rabin or other probabilistic test. Henry Pocklington’s Criterion, which dates to 1914, gives us a way to find such primes quickly. Paulo Ribenboim explains it thus:

Let p be an odd prime, and let k be a positive integer, such that p does not divide k and 1 < k < 2(p + 1). Let N = 2kp + 1. Then the following conditions are equivalent:

1. N is a prime.
2. There exists an integer a, 1 < a < N, such that a(N−1)/2 ≡ −1 (mod N) and gcd(ak + 1, N) = 1.

This gives us an algorithm for generating large certified primes. Choose p a certified prime. Choose 1 ≤ k < 2 × p at random; we ignore the last two possibilities for k, so we don’t have to worry about k being a multiple of p. Compute N. For each a ∈ {2, 3}, test the conditions for primality. If you don’t find a prime, go back and choose a different random k. Once you have a prime N, “ratchet up” and restart the process with the new certified prime N as the p of the next step. Continue until N is big enough.

Your task is to write a program that generates random large primes. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.