Cryptosystems Using Automorphisms of Finitely Generated Free Groups

TL;DR

This paper presents new cryptosystems based on automorphisms of free groups, including a one-time pad-like system and an ElGamal-inspired system, utilizing Nielsen transformations for encryption.

Contribution

It introduces novel cryptosystems leveraging automorphisms of free groups and Nielsen transformations, expanding cryptographic methods beyond traditional approaches.

Findings

Proposes a one-time pad-like cryptosystem using automorphisms.

Develops an ElGamal-inspired public key cryptosystem.

Utilizes Nielsen transformations to generate automorphisms.

Abstract

This paper introduces a newly developed private key cryptosystem and a public key cryptosystem. In the first one, each letter is encrypted with a different key. Therefore, it is a kind of a one-time pad. The second one is inspired by the ElGamal cryptosystem. Both presented cryptosystems are based on automorphisms of free groups. Given a free group $F$ of finite rank, the automorphism group $Aut(F)$ can be generated by Nielsen transformations, which are the basis of a linear technique to study free groups and general infinite groups. Therefore Nielsen transformations are introduced.