Special Issue On Worst-case Versus Average-case Complexity Editors' Foreword
Computational Complexity
Hardness amplification proofs require majority
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On the (Im)Possibility of Arthur-Merlin Witness Hiding Protocols
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
The complexity of zero knowledge
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
Relativized worlds without worst-case to average-case reductions for NP
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Hardness amplification within NP against deterministic algorithms
Journal of Computer and System Sciences
Hardness Amplification Proofs Require Majority
SIAM Journal on Computing
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
On efficient zero-knowledge PCPs
TCC'12 Proceedings of the 9th international conference on Theory of Cryptography
Relativized Worlds without Worst-Case to Average-Case Reductions for NP
ACM Transactions on Computation Theory (TOCT)
Black-box reductions and separations in cryptography
AFRICACRYPT'12 Proceedings of the 5th international conference on Cryptology in Africa
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
On the power of nonuniformity in proofs of security
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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We show that if an NP-complete problem has a nonadaptive self-corrector with respect to any samplable distribution, then coNP is contained in NP/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow [SIAM J. Comput., 22 (1993), pp. 994-1005] show the same conclusion under the stronger assumption that an NP-complete problem has a nonadaptive random self-reduction. A self-corrector for a language L with respect to a distribution $\cal D$ is a worst-case to average-case reduction that transforms any given algorithm that correctly decides $L$ on most inputs (with respect to $\cal D$) into an algorithm of comparable efficiency that decides L correctly on every input. A random self-reduction is a special case of a self-corrector, where the reduction, given an input $x$, is restricted to only making oracle queries that are distributed according to $\cal D$. The result of Feigenbaum and Fortnow depends essentially on the property that the distribution of each query in a random self-reduction is independent of the input of the reduction. Our result implies that the average-case hardness of a problem in NP or the security of a one-way function cannot be based on the worst-case complexity of an NP-complete problem via nonadaptive reductions (unless the polynomial hierarchy collapses).