ZK-PCPs from Leakage-Resilient Secret Sharing

Zero-Knowledge PCPs (ZK-PCPs; Kilian, Petrank, and Tardos, STOC `97) are PCPs with the additional zero-knowledge guarantee that the view of any (possibly malicious) veri er making a bounded number of queries to the proof can be e ciently simulated up to a small statistical distance. Similarly, ZK-PCPs of Proximity (ZK-PCPPs; Ishai and Weiss, TCC `14) are PCPPs in which the view of an adversarial veri er can be e ciently simulated with few queries to the input. Previous ZK-PCP constructions obtained an exponential gap between the query complexity q of the honest veri er, and the bound q∗ on the queries of a malicious veri er (i.e., q = poly log (q∗)), but required either exponential-time simulation, or adaptive honest veri cation. This should be contrasted with standard PCPs, that can be veri ed non-adaptively (i.e., with a single round of queries to the proof). The problem of constructing such ZK-PCPs, even when q∗ = q, has remained open since they were rst introduced more than 2 decades ago. This question is also open for ZK-PCPPs, for which no construction with non-adaptive honest veri cation is known (not even with exponential-time simulation). We resolve this question by constructing the rst ZK-PCPs and ZK-PCPPs which simultaneously achieve e cient zero-knowledge simulation and non-adaptive honest veri cation. Our schemes have a square-root query gap, namely q∗/q = O ( √ n) where n is the input length. Our constructions combine the MPC-in-the-head technique (Ishai et al., STOC `07) with leakage-resilient secret sharing. Speci cally, we use the MPC-in-the-head technique to construct a ZK-PCP variant over a large alphabet, then employ leakage-resilient secret sharing to design a new alphabet reduction for ZK-PCPs which preserves zero-knowledge.