One-Way Functions and the Berman-Hartmanis Conjecture

The Berman-Hartmanis conjecture states that all NP-complete sets are P-isomorphic each other. On this conjecture, we first improve the previous result of Agrawal and show that all NP-complete sets are P/poly-time computable 1,li-reducible to each other based on the assumption that there exist regular one-way functions that cannot be inverted by randomized polynomial-time algorithms. Secondly, we show that, besides the above assumption, if all one-way functions have some easy part to invert, then all NP-complete sets are P/poly-isomorphic to each other.