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 \cite{FGM}, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups \cite{AAG}. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.