Zero Knowledge for Free up to Linear Query Bounds: Quasilinear-Size Constant-Query Zero-Knowledge PCPs

Abstract

We show that for all polynomials $T$ and $b$, every relation in NTIME($T$) has a binary PCP with $O(1)$ non-adaptive queries, perfect completeness, soundness error 1/2 and length $(T+b)\cdot(\log(T+b))^{O(1)}$, which is perfect zero knowledge against every adaptive, possibly malicious verifier making at most $b$ queries. This improves the length $\tilde O(T+b^2)$ of Zeilberger, O'Connor, Gur and Spooner (2026), who observed that Dinur's gap amplification preserves zero knowledge. The simulator runs in expected polynomial time (or in strict polynomial time, with a failure symbol or with negligible statistical error). For $b\le T$ the length is $T\cdot\mathrm{polylog}\,T$, the form of the shortest known constant-query PCPs without zero knowledge. Unless NP $\subseteq$ BPP, the dependence on $b$ is optimal up to the polylogarithmic factor for polynomial-time simulators: zero knowledge is free up to linear query bounds. Zero-knowledge interactive oracle proofs achieved length $\tilde O(T+b)$ in 2016-17; we obtain it for PCPs. Gur, O'Connor and Spooner (STOC 2025) asked whether NP has such PCPs of quasilinear length (their Open Problem 1). This paper answers it for every polynomial query bound, with the simulators above in place of a strict polynomial-time perfect simulator. The proofs were developed with extensive AI assistance and have been checked by the authors. We mask the Reed-Solomon proof of proximity of Ben-Sasson and Sudan at every node of its recursion. By a locator lemma, at every node, whatever $k$ of its proof symbols reveal about its input is determined by $O(k)$ input values. We prove it with the coprime-order argument of Bafna and Vyas (2026); our treatment of domain points is new. Dinur's gap amplification gives constant query complexity. The query count and the polylogarithmic exponent are absolute but very large.