A tight parallel repetition theorem for partially simulatable interactive arguments via smooth kl-divergence

Itay Berman, Iftach Haitner, Eliad Tsfadia

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


Hardness amplification is a central problem in the study of interactive protocols. While “natural” parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS ’97]) and public-coin protocols (Håstad, Pass, Wikström, and Pietrzak [TCC ’10], Chung and Liu [TCC ’10] and Chung and Pass [TCC ’15]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikström [TCC ’07]). The only known round-preserving approach that applies to all interactive arguments is Haitner’s random-terminating transformation [SICOMP ’13], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original m-round protocol has soundness error 1-ε, then the n-parallel repetition of its random-terminating variant has soundness error (1-ε)ε n/m4 (omitting constant factors). Håstad et al. have generalized this result to partially simulatable interactive arguments, showing that the n-fold repetition of an m-round δ-simulatable argument of soundness error 1-ε has soundness error (1-ε)ε δ2 n/m2 . When applied to random-terminating arguments, the Håstad et al. bound matches that of Haitner. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the n parallel repetition is (1-ε)n/m, only an m factor from the optimal rate of (1-ε)n achievable in public-coin and three-message arguments. The result generalizes to δ-simulatable arguments, for which we prove a bound of (1-ε)δ n/m. This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol.

Original languageEnglish
Title of host publicationAdvances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings
EditorsDaniele Micciancio, Thomas Ristenpart
Number of pages30
StatePublished - 2020
Event40th Annual International Cryptology Conference, CRYPTO 2020 - Santa Barbara, United States
Duration: 17 Aug 202021 Aug 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12172 LNCS


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


  • Interactive argument
  • Parallel repetition
  • Partially simulatable
  • Smooth KL-divergence

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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