Fiat-shamir for repeated squaring with applications to ppad-hardness and vdfs

Alex Lombardi, Vinod Vaikuntanathan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations


The Fiat-Shamir transform is a methodology for compiling a (public-coin) interactive proof system for a language L into a non-interactive argument system for L. Proving security of the Fiat-Shamir transform in the standard model, especially in the context of succinct arguments, is largely an unsolved problem. The work of Canetti et al. (STOC 2019) proved the security of the Fiat-Shamir transform applied to the Goldwasser-Kalai-Rothblum (STOC 2008) succinct interactive proof system under a very strong “optimal learning with errors” assumption. Achieving a similar result under standard assumptions remains an important open question. In this work, we consider the problem of compiling a different succinct interactive proof system: Pietrzak’s proof system (ITCS 2019) for the iterated squaring problem. We construct a hash function family (with evaluation time roughly (Formula Presented)) that guarantees the soundness of Fiat-Shamir for this protocol assuming the sub-exponential (Formula Presented)-hardness of the n-dimensional learning with errors problem. (The latter follows from the worst-case (Formula Presented) hardness of lattice problems.) More generally, we extend the “bad-challenge function” methodology of Canetti et al. for proving the soundness of Fiat-Shamir to a class of protocols whose bad-challenge functions are not efficiently computable. As a corollary (following Choudhuri et al., ePrint 2019 and Ephraim et al., EUROCRYPT 2020), we construct hard-on-average problems in the complexity class (Formula Presented) under the (Formula Presented)-hardness of the repeated squaring problem and the (Formula Presented)-hardness of the learning with errors problem. Under the additional assumption that the repeated squaring problem is “inherently sequential”, we also obtain a Verifiable Delay Function (Boneh et al., EUROCRYPT 2018) in the standard model. Finally, we give additional PPAD-hardness and VDF instantiations demonstrating a broader tradeoff between the strength of the repeated squaring assumption and the strength of the lattice assumption.

Original languageEnglish (US)
Title of host publicationAdvances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings
EditorsDaniele Micciancio, Thomas Ristenpart
Number of pages20
ISBN (Print)9783030568764
StatePublished - 2020
Externally publishedYes
Event40th Annual International Cryptology Conference, CRYPTO 2020 - Santa Barbara, United States
Duration: Aug 17 2020Aug 21 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12172 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference40th Annual International Cryptology Conference, CRYPTO 2020
Country/TerritoryUnited States
CitySanta Barbara

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science


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