Random subgroups of braid groups: an approach to cryptanalysis of a braid group based cryptographic protocol

Authors:
Alexei Myasnikov;Vladimir Shpilrain;Alexander Ushakov
Affiliations:
Department of Mathematics, McGill University, Quebec, Montreal;Department of Mathematics, The City College of New York, New York, NY;Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ
Venue:
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
Year:
2006

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Abstract

Motivated by cryptographic applications, we study subgroups of braid groups Bn generated by a small number of random elements of relatively small lengths compared to n. Our experiments show that “most” of these subgroups are equal to the whole Bn, and “almost all” of these subgroups are generated by positive braid words. We discuss the impact of these experimental results on the security of the Anshel-Anshel-Goldfeld key exchange protocol [2] with originally suggested parameters as well as with recently updated ones.