Efficient solutions to the braid isotopy problem

We describe the most efficient solutions to the word problem of Artin’s braid group known so far, i.e., in other words, the most efficient solutions to the braid isotopy problem, including the Dynnikov method, which could be especially suitable for cryptographical applications. Most results appear in literature; however, some results about the greedy normal form and the symmetric normal form and their connection with grid diagrams may have never been stated explicitly. Because they are both not too simple and not too complicated, Artin’s braid groups Bn have been and remain one of the most natural and promising platform groups for non-commutative group-based cryptography [2, 22, 13]. More precisely, braid groups are not too simple in that they lead to problems with presumably difficult instances, typically the conjugacy problem and the related conjugacy and multiple conjugacy search problems, and they are not too complicated in that there exist efficient solutions to the word problem, a preliminary requirement when one aims at practically computing in a group. It turns out that, since the founding paper [3] appeared in 1947, the word problem of braid groups—which is also the braid isotopy problem—has received a number of solutions: braid groups might even be the groups for which the number of known solutions to the word problem is currently the highest one. In this paper, we review some of these solutions, namely those that, at the moment, appear as the most efficient ones for practical implementation, and, therefore, the most promising ones for cryptographical applications. What makes the subject specially interesting is that these solutions relie on deep underlying structures that explain their efficiency. Five solutions are described, and they come in two families, namely those based on a normal form, and those that use no normal form. In the first family, we consider the so-called greedy normal form, both in its nonsymmetric and symmetric versions. In the second family, we consider the so-called subword reversing method, which, like the greedy normal forms, has a quadratic complexity, the handle reduction method, whose complexity remains unknown but which is very efficient in practice, and Dynnikov’s coordinization method, which relies on an entirely different, geometric approach, and might turn to be very efficient. In this description, we deliberately discard lots of alternative solutions which are intrinsically exponential in complexity. The paper is organized as follows. In Section 1, after setting the background, we describe the greedy normal form and the symmetric normal form, without providing explicit rules to compute them. All results in this section are standard. In Section 2, we introduce grid diagrams and explain—and prove—how to use such diagrams to 1991 Mathematics Subject Classification. 20F36, 94A60.