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] Franco, N. and Gonzalez-Meneses, J.: Conjugacy problem for braid groups and Garside groups, to appear in Journal of Algebra. Available at http://arxiv.org/math.GT/0112310, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups [2] Anshel, I., Anshel, M. and Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, 287–291. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.