Braid Group Cryptography

TL;DR

This paper surveys cryptosystems based on braid groups, detailing their mathematical foundations, existing algorithms, security attacks, and future research directions in non-commutative group cryptography.

Contribution

It provides a comprehensive overview of braid group cryptography, including algorithms, security analyses, and potential future developments in the field.

Findings

Overview of braid group properties and Garside normal form

Summary of known cryptosystems and their vulnerabilities

Discussion of future research directions in non-commutative cryptography

Abstract

In the last decade, a number of public key cryptosystems based on com- binatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of cryptosystems which are based on other non-commutative groups).