The Isomorphism Conjecture Holds and One-Way Functions Exist Relative to an Oracle

Abstract

In this paper we demonstrate an oracle relative to which there are one-way functions but every paddable 1-li-degree collapses to an isomorphism type, thus yielding a relativized failure of the Joseph-Young Conjecture [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture is true but one-way functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK96]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the Isomorphism Conjecture and the existence of one-way functions. © 1997 Academic Press.