ICALP 2007: Automata, Languages and Programming pp 183-194

# On the Complexity of Hard-Core Set Constructions

• Chi-Jen Lu
• Shi-Chun Tsai
• Hsin-Lung Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

## 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.

## Keywords

Boolean Function IEEE Computer Society Majority Function Query Complexity Hard Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 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