Abstract. Several key agreement protocols are based on the following Generalized Conjugacy Search Problem: Find, given elements b 1 , . . . , b n andin a nonabelian group G, the conjugator x. In the case of subgroups of the braid group B N , Hughes and Tannenbaum suggested a length-based approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated. We introduce several effective realizations of this approach. In particular, a length function is defined on B N which possesses significantly better properties than the natural length associated to the Garside normal form. We give experimental results concerning the success probability of this approach, which suggest that an unfeasible computational power is required for this method to successfully solve the Generalized Conjugacy Search Problem when its parameters are as in existing protocols.

Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: The conjugacy problem, the group membership problem, the shortest presentation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group.We use a memory-based extension of the standard length-based approach, which in principle can be applied to any group admitting an efficient, reasonably behaving length function.

One of the most interesting questions about a group is whether its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists, and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometric one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition in order to give a new approach for solving its word problem. Our algorithm, although not better in complexity, is faster in comparison with known algorithms for short braid words, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms. © 2002 Elsevier Science (USA)

Abstract. The braid group Bn, endowed with Artin's presentation, admits an antiautomorphism Bn → Bn, such that v → v is defined by reading braids in reverse order (from right to left instead of left to right). We prove that the map Bn → Bn, v → vv is injective. We also give some consequences arising due to this injectivity.

No abstract