Semigroups and one-way functions

TL;DR

This paper explores the algebraic structure of polynomial-time functions through semigroup theory, linking it to P vs NP and characterizing one-way functions as non-regular elements within this framework.

Contribution

It introduces a semigroup-based approach to analyze P and NP, establishing conditions for one-way functions and proving properties like finite generation of the function semigroup.

Findings

fP is finitely generated

P ≠ NP iff fP is non-regular

Universal one-way functions are fP-complete under certain reductions

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. Then P is not equal to NP iff fP is a non-regular semigroup. The one-way functions considered here are based on worst-case complexity (they are not cryptographic); they are 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.