Actions of $E-$dense semigroups and an application to the discrete log problem

TL;DR

This paper explores the structure of $E$-dense acts over $E$-dense semigroups, extending inverse semigroup theory, and applies these concepts to analyze the discrete logarithm problem in cryptography.

Contribution

It introduces a new structural framework for $E$-dense acts and applies it to cryptographic problems, extending the theory of inverse semigroups to a broader class.

Findings

Structural description of $E$-dense acts similar to inverse semigroup acts

Application of semigroup theory to the discrete log problem

Use of completely regular semigroups in cryptographic analysis

Abstract

We describe the structure of $E-$dense acts over $E-$dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We consider an application to the discrete log problem in cryptography as well as an application to the same problem using completely regular semigroups.