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Polynomial-Time Pseudodeterministic Construction of Primes

Lijie Chen, Zhenjian Lu, Igor C. Oliveira, Hanlin Ren, Rahul Santhanam

TL;DR

The paper resolves the longstanding open question of pseudodeterministic polynomial-time prime construction by proving an infinitely-often result for primes and extending the framework to any dense, easy property. It introduces a novel bootstrapping approach that recursively applies uniform hardness–randomness tradeoffs via an improved Chen–Tell hitting-set generator, combined with a uniform-learning reconstruction of a Shaltiel–Umans generator. The core technical advance lies in achieving polynomial-time pseudodeterminism through a layered-polynomial arithmetization of uniform circuits and a non-black-box, nontrivial reconstruction process, yielding outputs canonical on infinitely many lengths and nearly canonical on all lengths. This work bridges complexity theory, randomness extraction, and number theory, providing a powerful method to derandomize search problems in a principled, scalable way with potential broad applicability beyond primes. It also demonstrates that output can be sensitive to the exact decision procedure used for Q, reflecting a nuanced non-black-box behavior intrinsic to the construction.

Abstract

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.

Polynomial-Time Pseudodeterministic Construction of Primes

TL;DR

The paper resolves the longstanding open question of pseudodeterministic polynomial-time prime construction by proving an infinitely-often result for primes and extending the framework to any dense, easy property. It introduces a novel bootstrapping approach that recursively applies uniform hardness–randomness tradeoffs via an improved Chen–Tell hitting-set generator, combined with a uniform-learning reconstruction of a Shaltiel–Umans generator. The core technical advance lies in achieving polynomial-time pseudodeterminism through a layered-polynomial arithmetization of uniform circuits and a non-black-box, nontrivial reconstruction process, yielding outputs canonical on infinitely many lengths and nearly canonical on all lengths. This work bridges complexity theory, randomness extraction, and number theory, providing a powerful method to derandomize search problems in a principled, scalable way with potential broad applicability beyond primes. It also demonstrates that output can be sensitive to the exact decision procedure used for Q, reflecting a nuanced non-black-box behavior intrinsic to the construction.

Abstract

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 such that, for infinitely many values of , outputs a canonical -bit prime with high probability. More generally, we prove that for every dense property of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying . 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.
Paper Structure (45 sections, 16 theorems, 77 equations)

This paper contains 45 sections, 16 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Proof Ideas
  4. Subexponential-time constructions of DBLP:conf/stoc/OliveiraS17.
  5. Win-win arguments.
  6. Recursive application of uniform hardness-randomness tradeoffs.
  7. Technical Overview
  8. The Chen--Tell ChenT21b targeted HSG ("ideal version").
  9. Bootstrapping the win-win argument.
  10. Infinitely-Often Pseudodeterministic Polynomial-Time Constructions
  11. Informal description of the algorithm and correctness.
  12. Improving the Chen--Tell Targeted Hitting Set Generator
  13. Non-black-box behavior.
  14. Modified Shaltiel--Umans Generator with Uniform Learning Reconstruction
  15. Further remarks about the proof.
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $Q \subseteq \{0,1\}^{*}$ be a language with the following properties: Then there exist a probabilistic polynomial-time algorithm $B$ and a sequence $\{x_n\}_{n \in \mathbb{N}_{\ge 1}}$ of $n$-bit strings in $Q$ such that the following conditions hold:

Theorems & Definitions (35)

  • Theorem 1.1: Infinitely-Often Polynomial-Time Pseudodeterministic Constructions
  • Corollary 1.2: Infinitely-Often Polynomial-Time Pseudodeterministic Construction of Primes
  • Corollary 1.3: Infinitely Many Primes with Efficient Succinct Descriptions
  • Definition 2.1: Logspace-Computable Functions
  • Definition 2.3: $\alpha$-Space-Uniform Circuits
  • Definition 2.5: Avoiding and Distinguishing
  • Theorem 3.1: Infinitely-Often Polynomial-Time Pseudodeterministic Constructions
  • Theorem 3.1: Improved Chen--Tell Hitting Set Generator
  • proof : Proof of \ref{['theo:intro_main']}
  • Corollary 3.2: Pseudodeterministic Polynomial-Time Construction with $1$ Bit of Advice that Succeeds Infinitely Often
  • ...and 25 more