On worst-case to average-case reductions for NP problems

A. Bogdanov, L. Trevisan
44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.  
We show that if an NP-complete problem has a non-adaptive 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 (SICOMP 22:994-1005(SICOMP 22:994- , 1993 show the same conclusion under the stronger assumption that an NP-complete problem has a non-adaptive random self-reduction. A self-corrector for a language L with respect to a distribution D is a worst-case to averagecase
more » ... uction that transforms any given algorithm that correctly decides L on most inputs (with respect to 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 make oracle queries that are distributed according to 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 non-adaptive reductions (unless the polynomial hierarchy collapses).
doi:10.1109/sfcs.2003.1238205 dblp:conf/focs/BogdanovT03 fatcat:uc7vlkohbndm3pe4gqey425gvu