New Distinguishers for Negation-Limited Weak Pseudorandom Functions

03/23/2022

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by   Zhihuai Chen, et al.

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We show how to distinguish circuits with log k negations (a.k.a k-monotone functions) from uniformly random functions in exp(Õ(n^1/3k^2/3)) time using random samples. The previous best distinguisher, due to the learning algorithm by Blais, Cannone, Oliveira, Servedio, and Tan (RANDOM'15), requires exp(Õ(n^1/2 k)) time. Our distinguishers are based on Fourier analysis on 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.