Shortly after the introduction of zero-knowledge proofs, Goldreich, Micali and Wigderson (CRYPTO '86) demonstrated their wide applicability by constructing zero-knowledge proofs 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 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 works 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:
- 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.
- 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.
