3-Message Zero Knowledge Against Human Ignorance

The notion of Zero Knowledge has driven the field of cryptography since its conception over thirty years ago. It is well established that two-message zero-knowledge protocols for NP do not exist, and that four-message zero-knowledge arguments exist under the minimal assumption of one-way functions. Resolving the precise round complexity of zero-knowledge has been an outstanding open problem for far too long. In this work, we present a three-message zero-knowledge argument system with soundness against uniform polynomial-time cheating provers. The main component in our construction is the recent delegation protocol for RAM computations (Kalai and Paneth, TCC 2016B and Brakerski, Holmgren and Kalai, ePrint 2016). Concretely, we rely on a threemessage variant of their protocol based on a key-less collision-resistant hash functions secure against uniform adversaries as well as other standard primitives. More generally, beyond uniform provers, our protocol provides a natural and meaningful security guarantee against real-world adversaries, which we formalize following Rogaway’s “human-ignorance” approach (VIETCRYPT 2006): in a nutshell, we give an explicit uniform reduction from any adversary breaking the soundness of our protocol to finding collisions in the underlying hash function. ∗Email: [email protected]. Research supported in part by DARPA Safeware Grant, NSF CAREER Award CNS-1350619, CNS-1413964 and by the NEC Corporation. †Email: [email protected]. Supported by the Israel Science Foundation (Grant No. 468/14), the Alon Young Faculty Fellowship, Binational Science Foundation (Grant No. 712307) and Google Faculty Research Award. ‡Email: [email protected]. §Email: [email protected]. ¶Email: [email protected]. Research supported in part by DARPA Grant number FA8750-11-2-0225, NSF CAREER Award CNS-1350619, NSF Grant CNS-1413964 (MACS: A Modular Approach to Computer Security), Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship, NEC Corporation and a Steven and Renee Finn Career Development Chair from MIT.