TY - GEN

T1 - On the complexity of hard-core set constructions

AU - Lu, Chi Jen

AU - Tsai, Shi-Chun

AU - Wu, Hsin Lung

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38149136502&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-73420-8_18

DO - 10.1007/978-3-540-73420-8_18

M3 - Conference contribution

AN - SCOPUS:38149136502

SN - 3540734198

SN - 9783540734192

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 183

EP - 194

BT - Automata, Languages and Programming - 34th International Colloquium, ICALP 2007, Proceedings

Y2 - 9 July 2007 through 13 July 2007

ER -