Close to Uniform Prime Number Generation With Fewer Random Bits

TL;DR

This paper analyzes prime number generation methods requiring fewer random bits, demonstrating their effectiveness under various assumptions and showing they produce nearly uniform prime distributions efficiently.

Contribution

It introduces variants of a prime generation algorithm that use fewer random bits and achieve near-uniform output, with proofs under multiple assumptions including unproven conjectures and unconditional results.

Findings

Requires fewer random bits than traditional methods

Produces output distribution close to uniform

Valid under multiple assumptions, including unproven conjectures

Abstract

In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a<q$ coprime to $q$, and choose $p$ of the form $a+t\cdot q$, where only $t$ is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H.L. Montgomery, made precise by Friedlander and Granville; then the Extended Riemann Hypothesis; and finally fully unconditionally using the Barban-Davenport-Halberstam theorem. We argue that this approach has a number of desirable properties compared to previous algorithms.