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On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the\emph{Algebraic Eraser} scheme for key agreement over an insecure channel,using a novel hybrid of infinite and finite noncommutative groups. They alsointroduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concreterealization of this scheme.  We present general, efficient heuristic algorithms, which extract the sharedkey out of the public information provided by CBKAP. These algorithms are,according to heuristic reasoning and according to massive experiments,successful for all sizes of the security parameters, assuming that the keys arechosen with standard distributions.  Our methods come from probabilistic group theory (permutation group actionsand expander graphs). In particular, we provide a simple algorithm for findingshort expressions of permutations in $S_n$, as products of given randompermutations. Heuristically, our algorithm gives expressions of length$O(n^2\log n)$, in time and space $O(n^3)$. Moreover, this is provable from\emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning theuniform distribution on $S_n$. Experiments show that the constants in theseestimations are small. This is the first practical algorithm for this problemfor $n\ge 256$.  Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant ofCBKAP actually implemented by SecureRF uses proprietary distributions, and thusour results do not imply its vulnerability. See also arXiv:abs/12020598

Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser