computational complexity

, Volume 20, Issue 1, pp 145–171 | Cite as

Complexity of Hard-Core Set Proofs

  • Chi-Jen Lu
  • Shi-Chun Tsai
  • Hsin-Lung Wu


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.


Hard-core set hardness amplification black-box proofs 

Subject classification

68Q05 68Q10 68Q17 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of Information ScienceAcademia SinicaTaipeiTaiwan
  2. 2.Department of Computer ScienceNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of Computer Science and Information EngineeringNational Taipei UniversityTaipeiTaiwan

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