  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 conugacy problem given by the authors in a previous paper, are
two main steps for solving conjugacy systems, thus breaking recently discovered
cryptosystems based in braid groups. We also present the result of our
computations, where we notice that our algorithm yields surprisingly small
generating sets for the centralizers.
