New Distinguishers for Negation-Limited Weak Pseudorandom Functions

Zhihuai Chen; Siyao Guo; Qian Li; Chengyu Lin; Xiaoming Sun

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arXiv · 2203.12246

New Distinguishers for Negation-Limited Weak Pseudorandom Functions

Zhihuai Chen·Siyao Guo·Qian Li·Chengyu Lin·Xiaoming Sun

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Abstract

We show how to distinguish circuits with $\log k$ negations (a.k.a $k$-monotone functions) from uniformly random functions in $\exp\left(\tilde{O}\left(n^{1/3}k^{2/3}\right)\right)$ time using random samples. The previous best distinguisher, due to the learning algorithm by Blais, Cannone, Oliveira, Servedio, and Tan (RANDOM'15), requires $\exp\big(\tilde{O}(n^{1/2} k)\big)$ time. Our distinguishers are based on Fourier analysis on \emph{slices of the Boolean cube}. We show that some "middle" slices of negation-limited circuits have strong low-degree Fourier concentration and then we apply a variation of the classic Linial, Mansour, and Nisan "Low-Degree algorithm" (JACM'93) on slices. Our techniques also lead to a slightly improved weak learner for negation limited circuits under the uniform distribution.

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BibTeX RIS

Zhihuai Chen, Siyao Guo, Qian Li, Chengyu Lin, Xiaoming Sun. 2022-03-23. New Distinguishers for Negation-Limited Weak Pseudorandom Functions. https://arxiv.org/abs/2203.12246

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