Garside groups were first introduced by Patrick Dehornoy and Luis Paris in 1999. Garside groups are a generalization of braid groups and were named after F.A. Garside to recognize his ground breaking work on solving the word and conjugacy problems in braid groups. Since their inception different authors have sought for better solutions to these same two problems. While several solutions have come forward proving that they are theoretically solvable, none have yet been proposed that can efficiently solve these problems in practice. Aside from the intrinsic value of solving these problems to better understand the group, there have been several proposed braid based cryptosystems. A practical solution for these problems in Garside groups would also apply to braid groups and would show a viable deterministic attack on these cryptosystems. While there is little to no expectation of a secure braid based cryptosystem, it is still an open question and worth investigation. In this paper we will examine an efficient solution to the word problem in Garside groups, and we will examine the progress made thus far in solving the word problem, the conjugacy decision problem, and the conjugacy search problem.
