On the complexity of hard-core set constructions

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations

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 hardcore set lemma says that f must have a hard-core set H of density 6 on which it is "extremely hard"", in the sense that any circuit of size s′ = 0(s/(1/ε2 log(1/εδ))) 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/(1/ε log j)). 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 Ω{1/ε 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.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 34th International Colloquium, ICALP 2007, Proceedings
Pages183-194
Number of pages12
DOIs
StatePublished - 1 Dec 2007
Event34th International Colloquium on Automata, Languages and Programming, ICALP 2007 - Wroclaw, Poland
Duration: 9 Jul 200713 Jul 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4596 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference34th International Colloquium on Automata, Languages and Programming, ICALP 2007
CountryPoland
CityWroclaw
Period9/07/0713/07/07

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