Characterizations of the Existence of Partial and Total One-Way Permutations

In this note, we study the easy certificate classes introduced by Hemaspaandra, Rothe, and Wechsung [HRW], with regard to the question of whether or not surjective oneway functions exist. This is an important open question in cryptology. We show that the existence of partial one-way permutations can be characterized by separating P from the class of UP sets that, for all unambiguous polynomial-time Turing machines accepting them, always have easy (i.e., polynomial-time computable) certificates. This extends results of Grollmann and Selman [GS88]. 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. Similarly, there exist surjective poly-one oneway functions if and only if there is a set L in P such that not all FewP machines accepting L always have easy certificates. We also establish a condition necessary and sufficient for the existence of (total) one-way permutations.