Braid Group Cryptography

In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. Our tutorial is aimed at presenting these cryptosystems and some known attacks on them. We start with some basic facts on braid groups and on the Garside normal form of its elements. We then present some known algorithms for solving the word problem in the braid group. After that, we present the major public-key cryptosystems based on the braid group. We then discuss some of the known attacks on these cryptosystems. We finish with a discussion of future directions.