One-Way Functions and Zero Knowledge

The fundamental theorem of Goldreich, Micali, and Wigderson (J. ACM 1991) shows that the existence of a one-way function is sufficient for constructing computational zero knowledge (CZK) proofs for all languages in NP. We prove its converse, thereby establishing characterizations of one-way functions based on the worst-case complexities of zero knowledge. Specifically, we prove that the following are equivalent: center dot A one-way function exists. center dot NP. CZK and NP is hard in the worst case. center dot CZK is hard in the worst case and the problem GapMCSP of approximating circuit complexity is in CZK. The characterization above also holds for statistical and computational zero-knowledge argument systems. We further extend this characterization to a proof system with knowledge complexity O(log n). In particular, we show that the existence of a one-way function is characterized by the worst-case hardness of CZK if GapMCSP has a proof system with knowledge complexity O( log n). We complement this result by showing that NP admits an interactive proof system with knowledge complexity omega( log n) under the existence of an exponentially hard auxiliary-input one-way function (which is a weaker primitive than an exponentially hard one-way function).We also characterize the existence of a robustly-often nonuniformly computable one-way function by the non-deterministic hardness of CZK under the weak assumption that PSPACE not subset of AM. We present two applications of our results. First, we simplify the proof of the recent characterization of a one-way function by NP-hardness of a meta-computational problem and the worst-case hardness of NP given by Hirahara (STOC'23). Second, we show that if NP has a laconic zero-knowledge argument system, then there exists a public-key encryption scheme whose security can be based on the worst-case hardness of NP. This improves previous results which assume the existence of an indistinguishable obfuscation.