On the Complexity of Hard-Core Set Constructions

Abstract

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 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’= ${O} {(s/({{1}\over{\varepsilon^2}}\log(\frac{1}{\varepsilon\delta})))}$ 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 $s'=O(s/(\frac{1}{\varepsilon^2}log\frac{1}{\delta}))$. 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 $\Omega(\frac{1}{\varepsilon}log|G|)$ bits of advice. Finally, we show that weakly black-box constructions in general cannot be realized in a low-level complexity class such as AC ⁰[p] — the assumption that f is hard for AC ⁰[p] is not sufficient to guarantee the existence of a hard-core set.