The complexity of constructing pseudorandom generators from hard functions

Abstract

We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function $$f:\{ 0,1\} ^l \to \{ 0,1\} $$ computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant $$\epsilon > 0$$ such that every circuit of size $$2^{\epsilon l} $$ fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a $${\text{PRG}}:\{ 0,1\} ^{O(\log n)} \to \{ 0,1\} ^n $$ computable by DLOGTIME-uniform constant-depth circuits of size polynomial in n. Such a PRG implies $$BP \cdot AC^0 = AC^0 $$ under DLOGTIME-uniformity. On the negative side, we prove that starting from a worst-case hard function $$f:\{ 0,1\} ^l \to \{ 0,1\} $$ (i.e. there is a constant $$\epsilon > 0$$ such that every circuit of size $$2^{\epsilon l} $$ fails to compute f on some input) for every positive constant $$\delta < 1$$ there is no black-box construction of a $${\text{PRG}}:\{ 0,1\} ^{\delta n} \to \{ 0,1\} ^n $$ computable by constant-depth circuits of size polynomial in n. We also study worst-case hardness amplification, which is the related problem of producing an average-case hard function starting from a worst-case hard one. In particular, we deduce that there is no blackbox worst-case hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constant-depth circuits cannot compute good extractors and listdecodable codes.