A note on conjugacy search and racks

TL;DR

This paper explores the relationship between conjugacy search problems in certain algebraic structures called racks and quasigroups, highlighting implications for cryptographic protocol security and implementation.

Contribution

It demonstrates that effective left conjugacy closed left quasigroups induce racks preserving conjugation, impacting cryptographic protocol security and implementation strategies.

Findings

Cryptographic security depends on infeasibility of conjugacy search in induced racks.

Protocols can be implemented directly using racks instead of quasigroups.

Exposition of the Anshel-Anshel-Goldfeld protocol in rack-based context.

Abstract

We show that for every effective left conjugacy closed left quasigroup, there is an induced rack that retains the conjugation structure of the left translations. This means that cryptographic protocols relying on conjugacy search can be secure only if conjugacy search of left translations is infeasible in the induced rack. We note that, in fact, protocols based on conjugacy search could be simply implemented using a rack. We give an exposition of the Anshel-Anshel-Goldfeld protocol in such a case.