A note on one-way functions and polynomial-time isomorphisms

Berman and Hartmanis [2] considered the question of whether all NP-complete sets are polynomial-time isomorphic. They essentially showed that all "natural" NPcomplete sets are in fact polynomial-time isomorphic and, based on this result, conjectured that all NP-complete sets are polynomial-time isomorphic. The conjecture remains open and is difficult since it implies that P ~ NP. More recently, Joseph and Young [5,8] have introduced new classes of NP-complete which they denote K~, where k is sets any natural number and f is any one-one, polynomially honest, and polynomial-time computable function. They have noted that k the sets Kf do not seem to be polynomialtime isomorphic to "natural" NP-complete sets unless the function f is also polynomial-time invertible. Thus, if f is a one-way function, then it seems that not all NP-complete sets are polynomialtime isomorphic. In fact, it is conjectured in [5,8] that if one-way functions exist, then not all NP-complete sets are polynomial-time isomorphic. Further, since it is believed that one-way functions exist, it is also conjectured in [5,8] that