Verification of the Miller–Rabin probabilistic primality test

Abstract

Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the Miller–Rabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfies the stronger property of guaranteed termination. Completing the proof of correctness requires a significant body of group theory and computational number theory to be formalized in the theorem prover. Once verified, the primality test can either be executed in the logic (using rewriting) and used to prove the compositeness of numbers, or manually extracted to standard ML and used to find highly probable primes.