A New Approach to the Conjugacy Problem in Garside Groups

The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph ${\mathrm{\Gamma }}_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of ${\mathrm{\Gamma }}_x$ can be used instead of $S_x$ 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 are likely to have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.