Paper 2019/393

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

Itay Berman, 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 $$-round protocol has soundness error $$, then the $$-parallel repetition of its random-terminating variant has soundness error $$ (omitting constant factors). Hastad et al. have generalized this result to partially simulatable interactive arguments, showing that the $$-fold repetition of an $$-round $$-simulatable argument of soundness error $$ has soundness error $$. When applied to random-terminating arguments, the Hastad et al. bound matches that of Haitner. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the $$ parallel repetition is $$, only an $$ factor from the optimal rate of $$ achievable in public-coin and three-message arguments. The result generalizes to $$-simulatable arguments, for which we prove a bound of $$. 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.

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