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

TL;DR

This paper establishes a near-optimal exponential decay rate for the soundness error in parallel repetition of partially simulatable interactive arguments, using a novel analysis based on smooth KL-divergence.

Contribution

It proves a tight bound on the soundness error decay rate for parallel repetition of random-terminating and $ ext{delta}$-simulatable arguments, improving previous results significantly.

Findings

Parallel repetition reduces soundness error at rate $(1-p)^{n/m}$ for random-terminating arguments.

The bound extends to $ ext{delta}$-simulatable arguments with rate $(1-p)^{ ext{delta} n/m}$.

The analysis introduces a tight bound on a relaxed KL-divergence measure, applicable beyond parallel repetition.

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 and public-coin protocols, it fails to do so in the general case. 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 $1-p$, then the $n$-parallel repetition of its random-terminating variant has soundness error $(1-p)^{p n / m^4}$ (omitting constant factors). Hastad et al. [TCC '10] 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 $1-p$ has soundness error $(1-p)^{p \delta^2 n / m^2}$. 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 $(1-p)^{n / m}$, only an $m$ factor from the optimal rate of $(1-p)^n$ achievable in public-coin and three-message arguments. The result generalizes to $\delta$-simulatable arguments, for which we prove a bound of $(1-p)^{\delta n / m}$. 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.