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.