On the complexity of hard-core set constructions

We study a fundamental result of Impagliazzo (FOCS'95) known as the hard-core set lemma. Consider any function f : {0,1}ⁿ -→ {0,1} which is "mildly-hard", in the sense that any circuit of size s must disagree with f on at least δ fraction of inputs. Then the hardcore set lemma says that f must have a hard-core set H of density 6 on which it is "extremely hard"", in the sense that any circuit of size s′ = 0(s/(1/ε² log(1/εδ))) must disagree with f on at least (1-ε)/2 fraction of inputs from H. There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low-level complexity class. We introduce two models of hard-core set constructions, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models. First, we show that in any strongly black-box construction, one can only prove the hardness of a hard-core set for smaller circuits of size at most s′ = O(s/(1/ε log j)). Next, we show that any weakly black-box construction must be inherently non-uniform -to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with Ω{1/ε log |G|) bits of advice. Finally, we show that weakly black-box constructions in general cannot be realized in a low-level complexity class such as AC⁰;[p] -the assumption that f is hard for AC⁰[p] is not sufficient to guarantee the existence of a hard-core set.