**A Tight Parallel Repetition Theorem for Partially Simulatable Interactive Arguments via Smooth KL-Divergence**

*Itay Berman and Iftach Haitner and Eliad Tsfadia*

**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 (Hastad, Pass, Wikstrom, and Pietrzak [TCC '10], Chung and Lu [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 Wikstrom [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 $1-\varepsilon$, then the $n$-parallel repetition of its random-terminating variant has soundness error $(1-\varepsilon)^{\varepsilon n / m^4}$ (omitting constant factors). Hastad et al. have generalized this result to partially simulatable interactive arguments, showing that the $n$-fold repetition of an $m$-round $\delta$-simulatable argument of soundness error $1-\varepsilon$ has soundness error $(1-\varepsilon)^{\varepsilon \delta^2 n / m^2}$. When applied to random-terminating arguments, the Hastad 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 $(1-\varepsilon)^{n / m}$, only an $m$ factor from the optimal rate of $(1-\varepsilon)^n$ achievable in public-coin and three-message arguments. The result generalizes to $\delta$-simulatable arguments, for which we prove a bound of $(1-\varepsilon)^{\delta n / m}$. 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.

**Category / Keywords: **foundations / parallel repetition; interactive argument; partially simulatable; smooth KL-divergence

**Date: **received 14 Apr 2019, last revised 2 Jun 2020

**Contact author: **eliadtsfadia at gmail com, iftachh at gmail com

**Available format(s): **PDF | BibTeX Citation

**Note: **In this version we extended the result to partially simulatable interactive arguments, and rewrote most parts of previous version.

**Version: **20200602:102109 (All versions of this report)

**Short URL: **ia.cr/2019/393

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