Quasi-Linear Size Zero Knowledge from Linear-Algebraic PCPs.

The seminal result that every language having an interactive proof also has a zero-knowledge interactive proof assumes the existence of one-way functions. Ostrovsky and Wigderson [33] proved that this assumption is necessary: if one-way functions do not exist, then only languages in BPP have zero-knowledge interactive proofs. Ben-Or et al. [9] proved that, nevertheless, every language having a multi-prover interactive proof also has a zero-knowledge multi-prover interactive proof, unconditionally. Their work led to, among many other things, a line of work studying zero knowledge without intractability assumptions. In this line of work, Kilian, Petrank, and Tardos [28] defined and constructed zero-knowledge probabilistically checkable proofs (PCPs). While PCPs with quasilinear-size proof length, but without zero knowledge, are known, no such result is known for zero knowledge PCPs. In this work, we show how to construct "2-round" PCPs that are zero knowledge and of length (O) over tilde (K) where K is the number of queries made by a malicious polynomial time verifier. Previous solutions required PCPs of length at least K-6 to maintain zero knowledge. In this model, which we call duplex PCP (DPCP), the verifier first receives an oracle string from the prover, then replies with a message, and then receives another oracle string from the prover; a malicious verifier can make up to K queries in total to both oracles. Deviating from previous works, our constructions do not invoke the PCP Theorem as a blackbox but instead rely on certain algebraic properties of a specific family of PCPs. We show that if the PCP has a certain linear algebraic structure - which many central constructions can be shown to possess, including [2,4,15] - we can add the zero knowledge property at virtually no cost (up to additive lower order terms) while introducing only minor modifications in the algorithms of the prover and verifier. We believe that our linear-algebraic characterization of PCPs may be of independent interest, as it gives a simplified way to view previous well-studied PCP constructions.