Resilient functions: Optimized, simplified, and generalized

Peter Ivanov; Emanuele Viola

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arXiv · 2406.19467

Resilient functions: Optimized, simplified, and generalized

Peter Ivanov·Emanuele Viola

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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.

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Peter Ivanov, Emanuele Viola. 2024-06-27. Resilient functions: Optimized, simplified, and generalized. https://arxiv.org/abs/2406.19467

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