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 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 $p_n$ 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. 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, using a variant of the Shaltiel--Umans generator.

翻译：针对搜索问题的随机算法若以高概率产生该问题的固定规范解，则称为*伪确定性*算法。在该领域的开创性工作中，Gat与Goldwasser将"素数是否可在多项式时间内通过伪确定性方式构造"作为其主要开放问题。我们在无穷频繁情形下为该问题给出了肯定解答。具体而言，我们给出一个*无条件*的多项式时间随机算法$B$，使得对无穷多个$n$值，$B(1^n)$以高概率输出规范的$n$位素数$p_n$。更一般地，我们证明：对任何可在多项式时间内判定的字符串稠密性质$Q$，存在满足$Q$的字符串的无穷频繁伪确定性多项式时间构造方法。这改进了Oliveira与Santhanam的次指数时间构造。我们的构造采用了若干新思路，包括一种用于伪确定性构造的新型自举技术，以及利用Shaltiel-Umans生成器变体对Chen-Tell均匀硬度-随机性框架进行的定量优化。