## Abstract

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 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.

## Keywords

Hard-core set hardness amplification black-box proofs## Subject classification

68Q05 68Q10 68Q17## Preview

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