Complexity of Hard-Core Set Proofs

We study a fundamental result of Impagliazzo (FOCS’95) known as the hard-core set lemma. Consider any function $${f:\{0,1\}^n\to\{0,1\}}$$ which is “mildly hard”, in the sense that any circuit of size s must disagree with f on at least a δ fraction of inputs. Then, the hard-core set lemma says that f must have a hard-core set H of density δ on which it is “extremely hard”, in the sense that any circuit of size $${s'=O(s/(\frac{1}{\epsilon^2}\log(\frac{1}{\epsilon\delta})))}$$ must disagree with f on at least $${(1-\epsilon)/2}$$ 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 proofs, 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 using any strongly black-box proof, one can only prove the hardness of a hard-core set for smaller circuits of size at most $${s'=O(s/(\frac{1}{\epsilon^2}\log\frac{1}{\delta}))}$$ . Next, we show that any weakly black-box proof 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 $${\Omega(\frac{1}{\epsilon}\log|G|)}$$ bits of advice. Finally, we show that weakly black-box proofs 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.