Improved pseudorandom generators from pseudorandom multi-switching lemmas
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We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an ε-PRG for the class of size-M depth-d AC^0 circuits with seed length (M)^d+O(1)·(1/ε), and an ε-PRG for the class of S-sparse F_2 polynomials with seed length 2^O(√( S))·(1/ε). 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 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 [IMP12] and Håstad [Hås14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC^0 and sparse F_2 polynomials.
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