A Zero Knowledge Sumcheck and its Applications

Many seminal results in Interactive Proofs (IPs) use algebraic techniques based on low-degree polynomials, the study of which is pervasive in theoretical computer science. Unfortunately, known methods for endowing such proofs with zero knowledge guarantees do not retain this rich algebraic structure. In this work, we develop algebraic techniques for obtaining zero knowledge variants of proof protocols in a way that leverages and preserves their algebraic structure. Our constructions achieve unconditional (perfect) zero knowledge in the Interactive Probabilistically Checkable Proof (IPCP) model of Kalai and Raz [KR08] (the prover first sends a PCP oracle, then the prover and verifier engage in an Interactive Proof in which the verifier may query the PCP). Our main result is a zero knowledge variant of the sumcheck protocol [LFKN92] in the IPCP model. The sumcheck protocol is a key building block in many IPs, including the protocol for polynomial-space computation due to Shamir [Sha92], and the protocol for parallel computation due to Goldwasser, Kalai, and Rothblum [GKR15]. A core component of our result is an algebraic commitment scheme, whose hiding property is guaranteed by algebraic query complexity lower bounds [AW09; JKRS09]. This commitment scheme can then be used to considerably strengthen our previous work [BCFGRS16] that gives a sumcheck protocol with much weaker zero knowledge guarantees, itself using algebraic techniques based on algorithms for polynomial identity testing [RS05; BW04]. We demonstrate the applicability of our techniques by deriving zero knowledge variants of well-known protocols based on algebraic techniques. First, we construct zero knowledge IPCPs for NEXP starting with the Multi-prover Interactive Proofs of Babai, Fortnow, and Lund [BFL91]. This result is a direct application of our zero knowledge sumcheck and our algebraic commitment scheme, augmented with the use of ‘randomized’ low-degree extensions. We also construct protocols in a more restricted model where the prover and verifier engage in a standard Interactive Proof with oracle access to a uniformly random low-degree polynomial (soundness holds with respect to any oracle). In this setting we achieve zero knowledge variants of the protocols of Shamir and of Goldwasser, Kalai, and Rothblum.