Resilient functions: Optimized, simplified, and generalized

TL;DR

This paper presents new explicit constructions of resilient Boolean functions with improved parameters, including simplicity, generalization to biased distributions, and resilience to larger coalitions, advancing the design of robust cryptographic functions.

Contribution

It introduces simpler, generalized constructions of resilient functions with larger coalition resistance and smaller bias, extending previous work with novel analytical techniques.

Findings

Constructed depth-3 circuits resilient to coalitions of size $cn/ ext{log}^2 n$

Achieved resilience with bias $n^{-c}$, improving over constant bias

Utilized tail bounds for expander walks and Janson's inequality in proofs

Abstract

An $n$-bit boolean function is resilient to coalitions of size $q$ if any fixed set of $q$ bits is unlikely to influence the function when the other $n-q$ bits are chosen uniformly. We give explicit constructions of depth-$3$ circuits that are resilient to coalitions of size $cn/\log^{2}n$ with bias $n^{-c}$. Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.