摘要：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 epsilon-PRG for the class of size-M depth-d AC^0 circuits with seed length log(M)^{d+O(1)}* log(1/epsilon), and an epsilon-PRG for the class of S-sparse F_2 polynomials with seed length 2^{O(sqrt{log S})}* log(1/epsilon). 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 [Impagliazzo et al., 2012] and Håstad [Johan Håstad, 2014]. 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.
