The root extraction problem in braid group-based cryptography

TL;DR

This paper examines the root extraction problem in braid groups, analyzes its use in cryptographic schemes, and demonstrates their vulnerabilities through specific attacks.

Contribution

It provides a comprehensive overview of braid group-based cryptosystems and proves their insecurity by constructing effective attacks.

Findings

Most braid group-based cryptosystems are insecure against known attacks.

The paper details specific vulnerabilities in authentication and signature schemes.

It highlights the need for more secure cryptographic primitives in braid groups.

Abstract

The root extraction problem in braid groups is the following: given a braid $\beta \in \mathcal{B}_n$ and a number $k\in \mathbb{N}$, find $\alpha\in \mathcal{B}_n$ such that $\alpha^k=\beta$. In the last decades, many cryptosystems such as authentication schemes and digital signatures based on the root extraction problem have been proposed. In this paper, we first describe these cryptosystems built around braid groups. Then we prove that, in general, these authentication schemes and digital signature are not secure by presenting for each of them a possible attack.