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 $n, B\left(1^{n}\right)$ outputs a canonical n-bit prime $p_{n}$ 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].