Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
International Workshop on Approximation Algorithms for Combinatorial Optimization
We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F2 polynomials. Our main results are an ε-PRG for the class of size-M depth-d AC 0 circuits with seed length log(M ) d+O(1) • log(1/ε), and an ε-PRG for the class of S-sparse F2 polynomials with seed length 2 O( √ log S) • log(1/ε). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the
... 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 multiswitching lemma -a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family -achieves the parameters obtained by the (full randomness) multiswitching lemmas of Impagliazzo, Matthews, and Paturi  and Håstad . This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC 0 and sparse F2 polynomials.