Computation of Centralizers in Braid groups and Garside groups

Abstract

We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in [9], are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups [2]. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.

On the centralizer of generic braids

“…Following the ideas in [17], we see that in order to compute a generating set for Z(x), we just need to know Γ x . Suppose that a ∈ Z(x).…”

“…One generator is pseudo-Anosov, usually a root of x, and the other generator is periodic, and can be chosen to be a root of the Garside element ∆ 2 . But in order to compute the generators of Z(x), the procedure in [17] is usually applied.…”

“…end if 18: else 19: Compute Z(x) applying the algorithm in [17] (using SC(x)). 20: end if element is immediately obtained.…”

Braid Group Cryptography

“…The algorithm for computing centralizers presented in [45] is based on computing the Super Summit Set, which is hard in general (note that actually the Super Summit Set can be replaced by the Ultra Summit Set and the Sliding Circuits set in Franco and González-Meneses' algorithm [60]). Hence, Longrigg and Ushakov use some subgroup of the centralizer which is much easier to work with.…”

“…For computing the centralizers, Alice and Bob should use the algorithm from [45], but actually they have to compute only some elements from them and not the whole sets.…”