The Root Extraction Problem in Braid Group-Based Cryptography

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