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A Zero-Knowledge PCP Theorem

Tom Gur, Jack O'Connor, Nicholas Spooner

Abstract

We show that for every polynomial q* there exist polynomial-size, constant-query, non-adaptive PCPs for NP which are perfect zero knowledge against (adaptive) adversaries making at most q* queries to the proof. In addition, we construct exponential-size constant-query PCPs for NEXP with perfect zero knowledge against any polynomial-time adversary. This improves upon both a recent construction of perfect zero-knowledge PCPs for #P (STOC 2024) and the seminal work of Kilian, Petrank and Tardos (STOC 1997).

A Zero-Knowledge PCP Theorem

Abstract

We show that for every polynomial q* there exist polynomial-size, constant-query, non-adaptive PCPs for NP which are perfect zero knowledge against (adaptive) adversaries making at most q* queries to the proof. In addition, we construct exponential-size constant-query PCPs for NEXP with perfect zero knowledge against any polynomial-time adversary. This improves upon both a recent construction of perfect zero-knowledge PCPs for #P (STOC 2024) and the seminal work of Kilian, Petrank and Tardos (STOC 1997).

Paper Structure

This paper contains 17 sections, 17 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Techniques
  3. Open problems
  4. Related work
  5. Preliminaries
  6. Coding theory
  7. PCPs
  8. Zero-knowledge PCPs
  9. Low-degree testing
  10. Zero-knowledge proof composition
  11. Locally computable proofs
  12. Alphabet reduction of PZK-PCPs
  13. Proof composition
  14. Robust PCPPs for polynomial summation
  15. Robust PZK-PCPPs for polynomial summation
  16. ...and 2 more sections

Key Result

Theorem 1

There exist PCPs for $\mathsf{NEXP}$ of exponential length with a non-adaptive verifier that reads $O(1)$ bits of the proof, which are perfect zero-knowledge against any efficient adversary.

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2: "Zero-knowledge PCP theorem"
  • Definition 1: Oracle 3-SAT
  • Definition 2: Locally computable algorithms (informal; see \ref{['def:loc-proofs']})
  • Definition 2.1: PCP
  • Definition 2.2: PCPP
  • Definition 2.3: Non-adaptive PCP verifiers
  • Definition 2.4: Accepting view of a PCP verifier
  • Definition 2.5: Accepting view of a PCP of proximity verifier
  • Definition 2.6: Robust soundness
  • ...and 40 more