Paper 2026/2357

Rare-Event Decorrelation for Linear Tests of Low-Complexity Pseudorandom Functions

Abstract

Linear tests are a basic obstacle to pseudorandom functions built from low-depth circuits and secret linear or affine maps. Ball, Ducros, Erabelli, Kohl, Resch, and Scholl left open whether their bounded-query strong-PRF candidate in $\mathrm{AC}^0[2]$ resists general non-adaptive linear tests with polynomially many chosen queries. Boyle, Couteau, Gilboa, Ishai, Kohl, and Scholl left linear-attack resistance of their Sipser-based weak-PRF candidate open. We address both questions through a common correlation argument based on rare local events. Repeated Cauchy--Schwarz reduces the original correlation to a parity correlation of indicators of substantially rarer events while preserving the independence supplied by the underlying linear forms. Pairwise independence suffices to bound the resulting parity correlation by a constant below $1$, while higher-order independence gives stronger cancellation. For the candidate of Ball et al., this resolves the open problem for arbitrary polynomially many non-adaptive queries with unrestricted linear or affine dependencies. More generally, for every fixed $a\in(0,1)$ and $q\le2^{a(\log_2\lambda)^2}$, every such linear test has bias $\exp(-\lambda^{1-a-o(1)})$. For the Sipser-based family of Boyle et al., for every fixed $0<\varepsilon<2$, with overwhelming probability over $Q\le2^{(2-\varepsilon)(\log_2\lambda)^2}$ random samples, every linear attack has bias $\exp(-\lambda^{\varepsilon-o(1)})$. This establishes linear-attack resistance throughout this quasipolynomial-sample regime. Extending the guarantee to the conjectured subexponential weak-PRF regime remains open.