Polynomial-Time Pseudodeterministic Constructions

A randomised 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 (2011) 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 randomised algorithm B such that, for infinitely many values of n, B(1(n)) 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 pseudodeterministic construction of Oliveira and Santhanam (2017). This talk will cover the main ideas behind these constructions and discuss their implications, such as the existence of infinitely many primes with succinct and efficient representations.