Resilient functions: Optimized, simplified, and generalized
<|reference_start|>Resilient functions: Optimized, simplified, and generalized: 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.<|reference_end|>
@article{ivanov2024resilient,
    title={Resilient functions: Optimized, simplified, and generalized},
    author={Peter Ivanov, Emanuele Viola},
    journal={arXiv preprint arXiv:2406.19467},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.19467},
    primaryClass={cs.CC cs.DS}
}