On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the Algebraic Eraser scheme for key agreement over an insecure channel, using 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 general, efficient heuristic algorithms, which extract the shared key 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 are chosen with standard distributions. Our methods come from probabilistic group theory (permutation group actions and expander graphs). In particular, we provide a simple algorithm for finding short expressions of permutations in Sn, as products of given random permutations. Heuristically, our algorithm gives expressions of length O( n2logn), in time and space O( n3). Moreover, this is provable from the Minimal Cycle Conjecture, a simply stated hypothesis concerning the uniform distribution on Sn. Experiments show that the constants in these estimations are small. This is the first practical algorithm for this problem for n≥256. Algebraic Eraser is a trademark of SecureRF. The variant of CBKAP actually implemented by SecureRF uses proprietary distributions, and thus our results do not imply its vulnerability.
Bibliographical noteFunding Information:
This research was partially supported by the Oswald Veblen Fund, the Emmy Noether Research Institute for Mathematics, and the Minerva Foundation of Germany. We thank John Dixon for his proof of Theorem 8. We thank László Babai, Dorian Goldfeld, Stephen Miller, Ákos Seress, and Adi Shamir, for fruitful discussions. A special thanks is owed to Martin Kassabov, for comprehensive suggestions which improved the presentation of this paper, and for pointing out to us the present version of Step 2 of our algorithm. This version seems to be folklore, but we have initially used a less efficient variant. We also thank Alexander Hulpke and Stefan Kohl for useful information about GAP. Our full attack on the Algebraic Eraser was implemented using MAGMA .
- Algebraic Eraser
- Colored Burau Key Agreement Protocol (CBKAP)
- Expressions of permutations
- Minimal Cycle Conjecture
- Symmetric group