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

Abstract

Motivated by cryptographic applications, we study subgroups of braid groups B n 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 B n, 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. © International Association for Cryptologic Research 2006.