We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee . This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if… (More)
This paper is the second in a series (the others are  and ) in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set U SS(X) of an element X in a Garside group G is a finite set of elements in G, introduced in , which is a complete invariant of the conjugacy… (More)
We prove a conjecture due to Makanin: if α and β are elements of the Artin braid group B n such that α k = β k for some nonzero integer k , then α and β are conjugate. The proof involves the Nielsen-Thurston classification of braids.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in , are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups . We also present… (More)
We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in… (More)
An element in Artin's braid group B n is said to be periodic if some power of it lies in the center of B n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known… (More)
This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. Introduction A natural, but… (More)
There are recent cryptographic protocols that are based on Multiple Simultaneous Conjugacy Problems in braid groups. We improve an algorithm , due to Sang Jin Lee and Eonkyung Lee, to solve these problems, by applying a method developed by the author and Nuno Franco, originally intended to solve the Conjugacy Search Problem in braid groups.
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.