Revised: December 30, 2020
Published: March 2, 2022
Abstract: [Plain Text Version]
We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse $\F_2$ polynomials. Our main results are an $\eps$-PRG for the class of size-$M$ depth-$d$ $\acz$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\eps)$, and an $\eps$-PRG for the class of $S$-sparse $\F_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\eps)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: substantially improving on the seed lengths of either PRG would require a breakthrough on longstanding and notorious circuit lower bound problems.
The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently developed multi-switching lemmas, which are powerful generalizations of Håstad's switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma—a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family—achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi (SODA'12) and Håstad (SICOMP 2014). This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\acz$ and sparse $\F_2$ polynomials.
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A preliminary version of this paper appeared in the Proceedings of the 23rd International Conference on Randomization and Computation (RANDOM 2019).
Licensed under a Creative Commons Attribution License (CC-BY)