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• W2783162249 abstract "We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $varepsilon$-PRG for the class of size-$M$ depth-$d$ $mathsf{AC}^0$ circuits with seed length $log(M)^{d+O(1)}cdot log(1/varepsilon)$, and an $varepsilon$-PRG for the class of $S$-sparse $mathbb{F}_2$ polynomials with seed length $2^{O(sqrt{log S})}cdot log(1/varepsilon)$. 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: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new emph{pseudorandom multi-switching lemma}. We derandomize recently-developed emph{multi}-switching lemmas, which are powerful generalizations of H{aa}stad's switching lemma that deal with emph{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 [IMP12] and H{aa}stad [H{aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $mathsf{AC}^0$ and sparse $mathbb{F}_2$ polynomials." @default.
• W2783162249 created "2018-01-26" @default.
• W2783162249 creator A5014866889 @default.
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• W2783162249 date "2018-01-10" @default.
• W2783162249 modified "2023-09-26" @default.
• W2783162249 title "Improved pseudorandom generators from pseudorandom multi-switching lemmas" @default.
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• W2783162249 doi "https://doi.org/10.48550/arxiv.1801.03590" @default.
• W2783162249 hasPublicationYear "2018" @default.
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