The Cost of Statistical Security in Proofs for Repeated Squaring

Cody Freitag, Ilan Komargodski

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In recent years, the number of applications of the repeated squaring assumption has been growing rapidly. The assumption states that, given a group element x, an integer T, and an RSA modulus N, it is hard to compute x2T mod N - or even decide whether y =? x2T mod N - in parallel time less than the trivial approach of simply computing T squares. This rise has been driven by efficient proof systems for repeated squaring, opening the door to more efficient constructions of verifiable delay functions, various secure computation primitives, and proof systems for more general languages. In this work, we study the complexity of statistically sound proofs for the repeated squaring relation. Technically, we consider proofs where the prover sends at most k ≥ 0 elements and the (probabilistic) verifier performs generic group operations over the group Z*N. As our main contribution, we show that for any (one-round) proof with a randomized verifier (i.e., an MA proof) the verifier either runs in parallel time Ω(T/(k + 1)) with high probability, or is able to factor N given the proof provided by the prover. This shows that either the prover essentially sends p, q such that N = p · q (which is infeasible or undesirable in most applications), or a variant of Pietrzak's proof of repeated squaring (ITCS 2019) has optimal verifier complexity O(T/(k + 1)). In particular, it is impossible to obtain a statistically sound one-round proof of repeated squaring with efficiency on par with the computationally-sound protocol of Wesolowski (EUROCRYPT 2019), with a generic group verifier. We further extend our one-round lower bound to a natural class of recursive interactive proofs for repeated squaring. For r-round recursive proofs where the prover is allowed to send k group elements per round, we show that the verifier either runs in parallel time Ω(T/(k + 1)r) with high probability, or is able to factor N given the proof transcript.

Original languageAmerican English
Title of host publication4th Conference on Information-Theoretic Cryptography, ITC 2023
EditorsKai-Min Chung
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages4:1-4:23
Number of pages23
ISBN (Electronic)9783959772716
DOIs
StatePublished - Jul 2023
Event4th Conference on Information-Theoretic Cryptography, ITC 2023 - Aarhus, Denmark
Duration: 6 Jun 20238 Jun 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume267

Conference

Conference4th Conference on Information-Theoretic Cryptography, ITC 2023
Country/TerritoryDenmark
CityAarhus
Period6/06/238/06/23

Keywords

  • Cryptographic Proofs
  • Lower Bounds
  • Repeated Squaring

All Science Journal Classification (ASJC) codes

  • Software

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