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}/sup l//spl rarr/{0,1} that is mildly hard on average, i.e. every circuit of size 2/sup /spl Omega/(l)/ fails to compute f on at least a 1/poly(l) fraction of inputs, we can build a PRG: {0,1}/sup O(logn)//spl rarr/{0,1}/sup n/ computable in ATIME(O(1), logn)=alternating time O(logn) with O(1) alternations. Such a PRG implies BP/spl middot/AC/sub 0/=AC/sub 0/ 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.