Random primes without primality testing

TL;DR

This paper introduces a new algorithmic approach that avoids primality testing when working with random primes, enabling efficient computations over large integers and finite fields without primality checks.

Contribution

A novel variant of the D5 algorithm that simplifies dynamic evaluation by eliminating primality testing, applicable to algebraic computations over integers and finite fields.

Findings

Avoids primality tests in random prime computations

Enables quasi-linear time polynomial term counting

Applies to finite field polynomial computations

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 the classic D5 algorithm for "dynamic evaluation", applied to a randomly-chosen (composite) integer. Unlike the D5 principle which has been used in the past to compute over a direct product of fields, our method is simpler as it only requires following a single path after any modulus splits. 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.