On the Complexity of Hard-Core Set Constructions

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

DOI: 10.1007/978-3-540-73420-8_18

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)
Cite this paper as:
Lu CJ., Tsai SC., Wu HL. (2007) On the Complexity of Hard-Core Set Constructions. In: Arge L., Cachin C., Jurdziński T., Tarlecki A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg

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 AC0[p] — the assumption that f is hard for AC0[p] is not sufficient to guarantee the existence of a hard-core set.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Chi-Jen Lu
    • 1
  • Shi-Chun Tsai
    • 2
  • Hsin-Lung Wu
    • 2
  1. 1.Institute of Information Science, Academia Sinica, TaipeiTaiwan
  2. 2.Department of Computer Science, National Chiao-Tung University, HsinchuTaiwan

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