Cryptanalysis of the Algebraic Eraser and short expressions of permutations as products

On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the _Algebraic Eraser_ scheme for key agreement over an insecure channel. This scheme is based on semidirect products of algebraic structures, and uses a novel hybrid of infinite and finite noncommutative groups. They also introduced the_Colored Burau Key Agreement Protocol (CBKAP)_, a concrete realization of this scheme. We present an efficient method to extract the shared key out of the public information provided by CBKAP, assuming that the keys are chosen with standard distributions. Our methods come from probabilistic group theory, and seem to have not been used before in cryptanalysis. Of independent interest may be a simple heuristic algorithm we propose for finding short expressions of permutations as products of given random permutations. According to heuristic analysis supported by experiments, our algorithm gives expressions of length O(n^2log n) in running time O(n^4log n).