Public Key Cryptography based on Semigroup Actions

TL;DR

This paper generalizes Diffie-Hellman key exchange to abelian semigroup actions on finite sets, exploring new cryptographic constructions using semirings with open questions on security and orbit size analysis.

Contribution

It introduces a novel framework for cryptography based on semigroup actions and constructs practical examples using finite semirings, extending traditional elliptic curve methods.

Findings

Proposes a generalized Diffie-Hellman protocol for semigroup actions.

Constructs a practical semigroup action using finite semirings.

Highlights open problems in orbit size computation and security analysis.

Abstract

A generalization of the original Diffie-Hellman key exchange in $(\Z/p\Z)^*$ found a new depth when Miller and Koblitz suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general. In Section 2 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.