Some Results on Average-Case Hardness Within the Polynomial Hierarchy

Abstract. We prove several results about the average-case complexity of problems in the Polynomial Hierarchy (PH). We give a connection among average-case, worst-case, and non-uniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worst-case then it is either hard on the average (in the sense of Levin) or it is non-uniformly hard (i.e. it does not have small circuits). Recently, Gutfreund, Shaltiel and Ta-Shma (IEEE Conference on Computational Complexity, 2005) showed an interesting worst-case to averagecase connection for languages in NP, under a notion of average-case hardness defined using uniform adversaries. We show that extending their connection to hardness against quasi-polynomial time would imply that NEXP doesn’t have polynomial-size circuits. Finally we prove an unconditional average-case hardness result. We show that for each k, there is an explicit language in P Σ2 which is hard on average for circuits of size n k.

Non-uniform hardness for np via black-box adversaries

We may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistiguishable from SAT to every polynomial-time bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to, and produces, for a given input length, a Boolean formula on which differs from SAT. A key step for obtaining this result is a new proof of the main result by Gutfreund, Shaltiel, and Ta-Shma reducing average-case hardness to worst-case hardness via uniform adversaries that know the algorithm they fool. The new adversary we construct has the feature of being black-box on the algorithm it fools, so it makes sense in the non-uniform setting as well. Our proof makes use of a refined analysis of the learning algorithm of Bshouty et al.