If NP Languages are Hard on the Worst-Case Then It is Easy to Find Their Hard Instances

We prove that if NP ⊆ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution. This is the first worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean that there exists one fixed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way functions and cryptography (even if not a sufficient assumption). Nevertheless, we do show that there is a fixed distribution on instances of NPcomplete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an efficient (...

• Type: Conference
• Year: 2005
• Where: COCO
• Authors: Dan Gutfreund, Ronen Shaltiel, Amnon Ta-Shma
• Keywords: Average-case Reductions, COCO 2005, Probabilistic Polynomial-time Algorithms, Samplable Distribution