2 research outputs found
A Tight Parallel Repetition Theorem for Partially Simulatable Interactive Arguments via Smooth KL-Divergence
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 \u2797]) and public-coin protocols (Hastad, Pass, Wikstrom, and Pietrzak [TCC \u2710], Chung and Lu [TCC \u2710] and Chung and Pass [TCC \u2715]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikstrom [TCC \u2707]).
The only known round-preserving approach that applies to all interactive arguments is Haitner\u27s random-terminating transformation [SICOMP \u2713], 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