Random Primes without Primality Testing

Abstract

Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of dynamic evaluation, applied to a randomly-chosen (composite) integer. The transformation we propose can apply to any algorithm in the algebraic RAM model, even allowing randomization. The resulting transformed algorithm avoids any primality tests and will, with constant positive probability, have the same result as the original computation modulo a randomly-chosen prime. As an application, we demonstrate how to compute the exact number of nonzero terms in an unknown integer polynomial in quasi-linear time. We also show how the same algorithmic transformation technique can be used for computing modulo random irreducible polynomials over a finite field.