Fiat Shamir via List-Recoverable Codes (or: Parallel Repetition of GMW is not Zero-Knowledge)

In a seminal work, Goldreich, Micali and Wigderson (CRYPTO '86) demonstrated the wide applicability of zero-knowledge proofs by constructing such a proof system for the NP-complete problem of graph 3-coloring. A long-standing open question has been whether parallel repetition of their protocol preserves zero knowledge. In this work, we answer this question in the negative, assuming a standard cryptographic assumption (i.e., the hardness of learning with errors (LWE)). Leveraging a connection observed by Dwork, Naor, Reingold, and Stockmeyer (FOCS '99), our negative result is obtained by making positive progress on a related fundamental problem in cryptography: securely instantiating the Fiat-Shamir heuristic for eliminating interaction in public-coin interactive protocols. A recent line of work has shown how to instantiate the heuristic securely, albeit only for a limited class of protocols. Our main result shows how to instantiate Fiat-Shamir for parallel repetitions of much more general interactive proofs. In particular, we construct hash functions that, assuming LWE, securely realize the Fiat-Shamir transform for the following rich classes of protocols: 1) The parallel repetition of any "commit-and-open"protocol (such as the GMW protocol mentioned above), when a specific (natural) commitment scheme is used. Commit-and-open protocols are a ubiquitous paradigm for constructing general purpose public-coin zero knowledge proofs. 2) The parallel repetition of any base protocol that (1) satisfies a stronger notion of soundness called round-by-round soundness, and (2) has an efficient procedure, using a suitable trapdoor, for recognizing "bad verifier randomness"that would allow the prover to cheat. Our results are obtained by establishing a new connection between the Fiat-Shamir transform and list-recoverable codes. In contrast to the usual focus in coding theory, we focus on a parameter regime in which the input lists are extremely large, but the rate can be small. We give a (probabilistic) construction based on Parvaresh-Vardy codes (FOCS '05) that suffices for our applications.