Conjugacy problem

Abstract: We give an easily handled algorithm for the word problem in each of Artin’s braid groups, Bn, based on Garside’s methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined in terms of positive braids, and apply this to compare braids with different powers ∆ of the fundamental half-twist braid ∆. This leads to an improvement of Garside’s conjugacy algorithm, using a much smaller finite subset of each conjugacy class, which we term the super summit set, to represent the class, in place of Garside’s summit set.

Abstract: We prove that for every finite rank Coxeter group there exists a polynomial (cubic) solution to the conjugacy problem.

Abstract: A simple heuristic approach to the conjugacy problem in braid groups is described. Although it does not provide a general solution to the latter problem, it demonstrates that various proposed key parameters for braid group based cryptographic primitives do not offer acceptable cryptographic security. We give experimental evidence that it is often feasible to reveal the secret data by means of a normal PC within a few minutes.

Abstract: 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 $\Gamma_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $\Gamma_x$ 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.