An Overview of Braid Group Cryptography

The past several years have seen an explosion of interest in the cryptographic applications of non-commutative groups. Braid groups in particular are especially desirable, as they provide difficult computational problems and can be implemented quite efficiently. Several different groups of researchers have proposed numerous cryptographic protocols that make use of braid groups, but unfortunately, flaws have been found in nearly every one. This expository paper discusses the specifications, attacks, and responses of both the Anshel, Anshel, and Goldfeld Commutator [2, 1] and the Cho et al. Diffie-Hellman Conjugacy [10] key exchange protocols.