Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser 1 posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an unconditional polynomial-time randomized algorithm B such that, for infinitely many values of n, B (1^n) outputs a canonical n-bit prime p₍ with high probability. More generally, we prove that for every dense property Q of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying Q. This improves upon a subexponential-time construction of Oliveira and Santhanam 2. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell 3, using a variant of the Shaltiel-Umans generator 4.