A Parallel Repetition Theorem for Any Interactive Argument

A fundamental question in the study of protocols is characterizing the effect parallel repetition has on the soundness error. While parallel repetition reduces the soundness error in interactive proofs and in special cases of interactive arguments (e.g., three-message protocols [M. Bellare, R. Impagliazzo, and M. Naor, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, IEEE, Washington, DC, 2007, p. 374], and public-coin protocols [J. Hästad et al., Proceedings of the 7th Theory of Cryptography Conference, Zurich, Switzerland, 2010, pp. 1--18]), Bellare, Impagliazzo, and Naor gave an example of an interactive argument for which parallel repetition does not reduce the soundness error at all. We show that by slightly modifying any interactive argument, in a way that preserves its completeness and only slightly deteriorates its soundness, we get a protocol for which parallel repetition does reduce the error (at a weakly exponential rate). In this modified version, the verifier flips at the beginning of each round an $(1 - \frac1{2m},\frac1{2m})$ biased coin (i.e., 1 is tossed with probability $1/{2m}$), where $m$ is the round complexity of the (original) protocol. If the outcome is one, the verifier halts the interaction and accepts. Otherwise, it sends the same message that the original verifier would. At the end of the protocol (if reached), the verifier accepts if and only if the original verifier does.