Hardness vs. randomness within alternating time

We study the complexity of building pseudorandom generators (PRGs) with logarithmic seed length from hard functions. We show that, starting from a function f:{0,1}^l→{0,1} that is mildly hard on average, i.e. every circuit of size 2^Ω(l) fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0,1}^O(logn)→{0,1}ⁿ computable in ATIME(O(1), logn)=alternating time O(logn) with O(1) alternations. Such a PRG implies BP·AC₀=AC₀ under DLOGTIME-uniformity. On the negative side, we prove a tight lower bound on black-box PRG constructions that are based on worst-case hard functions. We also prove a tight lower bound on black-box worst-case hardness amplification, which is the problem of producing an average-case hard function starting from a worst-case hard one. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and list-decodable codes.