Polynomial-Time Pseudodeterministic Construction of Primes

• Published in: 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) pp. 1261 - 1270
• Main Authors: Chen, Lijie, Lu, Zhenjian, Oliveira, Igor C., Ren, Hanlin, Santhanam, Rahul
• Format: Conference Proceeding
• Language: English
• Published: IEEE 06.11.2023
• Subjects: Computer science; explicit construction; Generators; hardness vs. randomness; Optimization; pseudodeterministic construction; Search problems
• ISSN: 2575-8454
• DOI: 10.1109/FOCS57990.2023.00074

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].