## Abstract

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 language | English |
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Title of host publication | Advances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings |

Editors | Daniele Micciancio, Thomas Ristenpart |

Publisher | Springer |

Pages | 544-573 |

Number of pages | 30 |

ISBN (Print) | 9783030568764 |

DOIs | |

State | Published - 2020 |

Event | 40th Annual International Cryptology Conference, CRYPTO 2020 - Santa Barbara, United States Duration: 17 Aug 2020 → 21 Aug 2020 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12172 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 40th Annual International Cryptology Conference, CRYPTO 2020 |
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Country/Territory | United States |

City | Santa Barbara |

Period | 17/08/20 → 21/08/20 |

### Funding

Funders | Funder number |
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National Science Foundation | CNS-1413920, CNS-1350619 |

Army Research Office | W911NF-15-C-0236, W911NF-15-C-0226 |

Defense Advanced Research Projects Agency | |

European Research Council | 638121 |

Israel Science Foundation | 666/19 |

Check Point Institute for Information Security, Tel Aviv University |

## Keywords

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