Semigroups and one-way functions

Abstract

We study the complexity classes P and NP through a semigroup fP ("polynomial-time functions"), consisting of all polynomially balanced polynomial-time computable partial functions. The semigroup fP is non-regular if and only if P ≠ NP. The one-way functions considered here are based on worst-case complexity (they are not cryptographic); they are exactly the non-regular elements of fP. We prove various properties of fP, e.g. that it is finitely generated. We define reductions with respect to which certain universal one-way functions are fP-complete.