Non-Malleable Affine Extractors with Small Error and Complexity Lower Bounds

2026-10-01

AI Summary

This study addresses the difficulty of constructing non-malleable affine extractors at constant entropy rates and the absence of lower bounds for related computational models. It proposes explicit construction techniques for affine extractors and directional extractors, combined with derandomization algorithms, to achieve efficient constructions. For the first time, this work realizes explicit non-malleable affine extractors under constant entropy rates, attaining linear output length and exponentially small error. This significantly enhances tamper resilience and eliminates logarithmic losses in derandomization. Furthermore, it establishes a linear depth lower bound for oblivious decision trees and constructs CNF formulas requiring exponential-size Res(+) refutations, thereby providing new theoretical foundations for proof complexity.

Abstract

We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<η<1$ and $t$, we also obtain entropy threshold $C_{η,t}n/\log n$, output length $\lfloor n^{1-η}\rfloor$, and error $2^{-n^{1-η}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-Ω(n)}$ against weakly read-once linear branching programs of size $2^{Ω(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $Ω_δ((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-δ}$, where $0<δ<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-Ω(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<ξ<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-ξ)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=Ω(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).

Problem

• non-malleable affine extractors
• complexity lower bounds
• proof complexity
• branching programs
• derandomization

Innovation

• Non-Malleable Affine Extractors
• Complexity Lower Bounds
• Linear Branching Programs
• Proof Complexity
• Derandomization