TY - JOUR

T1 - Complexity of Hard-Core Set Proofs

AU - Lu, Chi Jen

AU - Tsai, Shi-Chun

AU - Wu, Hsin Lung

PY - 2011/1/1

Y1 - 2011/1/1

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 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 must disagree with f on at least (1 - )/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. 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 bits of advice. Finally, we show that weakly black-box proofs 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 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 must disagree with f on at least (1 - )/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. 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 bits of advice. Finally, we show that weakly black-box proofs 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.

KW - black-box proofs

KW - Hard-core set

KW - hardness amplification

UR - http://www.scopus.com/inward/record.url?scp=85027956216&partnerID=8YFLogxK

U2 - 10.1007/s00037-011-0003-7

DO - 10.1007/s00037-011-0003-7

M3 - Article

AN - SCOPUS:85027956216

VL - 20

SP - 145

EP - 171

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 1

ER -