Complexity of Hard-Core Set Proofs

Chi Jen Lu, Shi-Chun Tsai, Hsin Lung Wu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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 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 must disagree with f on at least (1 - )/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. 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 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.

Original languageEnglish
Pages (from-to)145-171
Number of pages27
JournalComputational Complexity
Issue number1
StatePublished - 1 Jan 2011


  • black-box proofs
  • Hard-core set
  • hardness amplification

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