New Distinguishers for Negation-Limited Weak Pseudorandom Functions
This work addresses a theoretical problem in computational complexity for researchers studying pseudorandom functions, representing an incremental improvement over prior results.
The paper tackles the problem of distinguishing circuits with limited negations from random functions, achieving a distinguisher with time complexity exp(Õ(n^{1/3}k^{2/3})), improving over the previous best of exp(Õ(n^{1/2}k)).
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.