Integer factoring and modular square roots

https://doi.org/10.1016/j.jcss.2015.08.001Get rights and content
Under an Elsevier user license
open archive

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 WEAKPIGEONPPP. 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.

Keywords

Integer factoring
Search problem
PPA
Quadratic residue
Bounded arithmetic

Cited by (0)

1

Supported by grant IAA100190902 of GA AV ČR, Center of Excellence CE-ITI under the grant P202/12/G061 of GA ČR, and RVO: 67985840. Part of the research was done while visiting the Isaac Newton Institute in Cambridge.