Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

TL;DR

This paper develops an efficient algorithm for testing linear isomorphism of Boolean functions using spectral norms, significantly improving query complexity bounds and establishing new lower bounds through cryptographic reductions.

Contribution

It introduces a spectral norm-based tolerant tester with improved query complexity and provides a nearly matching lower bound, advancing the understanding of linear invariance testing.

Findings

Query complexity improved from rac{(m/\u03c9)^24}{} to rac{(m/)^4}{}

Established a lower bound of (m^2) queries for constant , surpassing previous (\u220a ) bounds

Developed a novel reduction from communication complexity using cryptographic functions

Abstract

Given Boolean functions \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \), we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all \( x \). We study this problem in the tolerant property testing framework under the known--unknown model, where \( g \) is given explicitly and \( f \) is accessible only via oracle queries, meaning the algorithm may adaptively request the value of \( f(x) \) for inputs \( x \in \mathbb{F}_2^n \) of its choice. Given parameters \( \epsilon \ge 0 \) and \( \omega>0 \), the goal is to distinguish whether there exists \( A \in \mathrm{GL}_n(\mathbb{F}_{2})\) such that the normalized Hamming distance between \( f \) and \( g(Ax) \) is at most \( \epsilon \), or whether for every \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) the distance is at least \( \epsilon+\omega \). Our main result is a tolerant tester making \( \widetilde{O} \left( \left( m/\omega \right)^4 \right) \) queries to \( f \), where \( m \) is an upper bound on the spectral norm of \( g \), improving the previous \( \widetilde{O} \left( \left( m/\omega \right)^{24} \right) \) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of \( \Omega(m^2) \) for constant \( \omega \) (for example, \( \omega=1/4 \)), improving the prior \( \Omega(\log m) \) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.