Perfect Zero-Knowledge PCPs for #P (2403.11941v2)

Abstract: We construct perfect zero-knowledge probabilistically checkable proofs (PZK-PCPs) for every language in #P. This is the first construction of a PZK-PCP for any language outside BPP. Furthermore, unlike previous constructions of (statistical) zero-knowledge PCPs, our construction simultaneously achieves non-adaptivity and zero knowledge against arbitrary (adaptive) polynomial-time malicious verifiers. Our construction consists of a novel masked sumcheck PCP, which uses the combinatorial nullstellensatz to obtain antisymmetric structure within the hypercube and randomness outside of it. To prove zero knowledge, we introduce the notion of locally simulatable encodings: randomised encodings in which every local view of the encoding can be efficiently sampled given a local view of the message. We show that the code arising from the sumcheck protocol (the Reed-Muller code augmented with subcube sums) admits a locally simulatable encoding. This reduces the algebraic problem of simulating our masked sumcheck to a combinatorial property of antisymmetric functions.

• The paper presents a construction of perfect zero-knowledge PCPs for every #P language using a masked sumcheck protocol based on the combinatorial nullstellensatz.
• It introduces locally simulatable encodings to efficiently simulate verifier views while maintaining non-adaptivity for honest verifiers.
• The results set a precedent for extending zero-knowledge proofs to complex computational problems, thereby enhancing cryptographic security.

Overview of Perfect Zero-Knowledge PCPs for #P

This paper addresses a significant question in theoretical computer science: Do perfect zero-knowledge probabilistic checkable proofs (PZK-PCPs) exist for non-trivial languages beyond those captured by bounded-error probabilistic polynomial time (BPP)? The authors provide an affirmative answer by constructing PZK-PCPs for every language in the class of function problems #P. This marks the first instance of such constructions for #P-complete problems outside the field of BPP.

The construction achieves both non-adaptivity for the honest verifier and zero-knowledge against adaptive polynomial-time malicious verifiers, overcoming limitations of previous zero-knowledge PCPs, which either required adaptation by the verifier or only offered statistical zero-knowledge guarantees. The significance of non-adaptivity lies in its alignment with the efficiency and robustness goals of cryptographic protocols, particularly when translating zero-knowledge proofs into real-world secure systems.

Technical Contributions

The paper's primary technical contribution is a novel form of the sumcheck protocol, termed the masked sumcheck PCP, which employs the combinatorial nullstellensatz. This adaptation introduces antisymmetric structures within the hypercube dataset being considered and builds randomness outside of it. The authors introduce the concept of locally simulatable encodings, which provide a means of efficiently and consistently simulating restricted views of encoded messages without needing to reference the entire message space.

To achieve zero knowledge, notably difficult algebraic structures arising from these antisymmetric constructions are tackled. The paper demonstrates that low-degree extensions from the sumcheck protocol can admit locally simulatable encodings, essential for maintaining zero-knowledge properties under verification by malicious actors.

Numerical Results and Implications

The construction is theoretically significant, offering a way to design zero-knowledge proofs for complex computational problems, providing evidence for the containment of #P in the landscape of complexity supported by perfect zero-knowledge protocols. While no explicit experimental or numerical results are given in terms of implementation scale or empirical validation, the exploration of these concepts in a theoretical framework demonstrates rigorous proofs and arguments supporting the feasibility of these constructions.

Projections for Future Research

The implications of this research extend significantly within theoretical computer science and cryptography. It offers a pathway for further extension in both refining the understanding of PZK-PCPs and expanding the scope of function problems they can address. Moreover, potential applications in cryptographic protocols could see security systems leverage these results to foster enhanced privacy constructs, particularly where information leakage is a critical risk.

In a broader context, this research prompts further exploration into other complexity classes and their compatibility with zero-knowledge and probabilistically checkable framework adaptations. Long-term, the application of these principles in quantum computing contexts also invites speculation, given the burgeoning field's need for robust and secure computational proofs.

This paper cogently contributes to closing significant gaps in theoretical understanding, presenting refined tools and methodologies that pave the way for future groundbreaking research in zero-knowledge proofs and their associated computational complexities.