On sets with easy certificates and the existence of one-way permutations

Abstract

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. We give structural conditions that control the size of these classes. We also provide negative results showing that some of our positive claims are optimal. Our negative results are proven using a novel observation: The classic “wide spacing” oracle construction technique instantly yields non-bi-immunity results. Easy certificate classes are also a useful notion in the study of whether one-way functions exist. This is one of the most important open questions in cryptology. We extend the results of Grollmann and Selman [GS88] by obtaining a complete characterization regarding the existence of a certain type of one-way function—(partial) one-way permutations—in terms of easy certificate classes. By Grädel's recent results about one-way functions [Grä94], this also links statements about easy certificates of NP sets with statements in finite model theory. In addition, we give a condition necessary and sufficient for the existence of (total) one-way permutations.