In this paper we propose an authenticated key agreement, which works in a braid group. We prove that our protocol meet the security attributes under the assumption that the Conjugacy Search Problem (CSP) is hard in braid group.

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n > 3 strands, with respect to Garside’s generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic...

Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.

We prove that, for n ≥ 3, the minimal dimension of a model of the classifying space of the braid group Bn, and of the pure braid group Pn, with respect to the family of virtually cyclic groups is n.