Integer factoring and modular square roots

TL;DR

This paper extends the classification of integer factoring and related problems within complexity classes PPA and PPP, showing reductions and derandomization under certain hypotheses, and explores related problems like modular square roots.

Contribution

It generalizes previous results to all integers using bounded arithmetic and establishes reductions to PPA and PPP, including derandomization under the generalized Riemann hypothesis.

Findings

Integer factoring reduces to PPA and PPP problems in randomized polynomial time.

Derandomization of reductions possible under the generalized Riemann hypothesis.

PPA contains problems like modular square root computation and quadratic nonresidue finding.

Abstract

Buresh-Oppenheim proved that the NP search problem to find nontrivial factors of integers of a special form belongs to Papadimitriou's class PPA, and is probabilistically reducible to a problem in PPP. In this paper, we use ideas from bounded arithmetic to extend these results to arbitrary integers. We show that general integer factoring is reducible in randomized polynomial time to a PPA problem and to the problem WEAKPIGEON in PPP. Both reductions can be derandomized under the assumption of the generalized Riemann hypothesis. We also show (unconditionally) that PPA contains some related problems, such as square root computation modulo n, and finding quadratic nonresidues modulo n.