A new approach to the conjugacy problem in Garside groups

The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.