Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
2003
AI
The paper explores the word, conjugacy, and Markov problems within the context of braid groups, specifically Bn. The authors provide foundational definitions and historical context for these problems, with a focus on the significance of knot theory and the relationships among the problems. They present an algorithm for solving the conjugacy problem in Bn, along with its computational complexity, highlighting advancements and unresolved aspects in this area of research.
Advances in Mathematics, 2002
One of the most interesting questions about a group is if 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 geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, 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.
Advances in Mathematics, 1998
A new approach to the word and conjugacy problems in the braid groups Abstract A new presentation of the n-string braid group B n is studied. Using it, a new solution to the word problem in B n is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing. * Partially supported by NSF Grant 94-02988. This paper was completed during a visit by the first author to MSRI. She thanks MSRI for its hospitality and partial support, and thanks Barnard College for its support under a Senior Faculty Research Leave. † This work was initiated during the second author's sabbatical visit to Columbia University in 1995-6. He thanks Columbia University for its hospitality during that visit.
Braids, 2009
In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of cryptosystems which are based on other non-commutative groups).
University College London, Ph.D. Thesis, 2002
We introduce the braid groups in their connection to knot theory and investigate several of their properties. Based on term rewriting systems, which we review, we find new solutions to the word and conjugacy problems in the braid groups. A similar problem asks for the minimal length word for an equivalence class in a given braid group which we prove to be NP-complete (after a review of this concept) and present a new algorithm for it. As this algorithm takes an exponentially increasing amount of time, we construct an algebraic approximation algorithm which we find to work well. We consider several methods of approximating the minimal word via computer simulation of the braid strings moving under the influence of certain forces. Using the theory of tangles which we also review, we construct a new notation for knots which is usable by a computer. From this notation, we construct an efficient algorithm to find the braid or plat whose closure is ambient isotopic to any given knot. Finally, we apply the computer software developed for these problems to the solar coronal heating problem by simulating magnetic flux tubes. We also present a number of incidental results that were found along the way of researching these problems.
2004
The conjugacy problem in finitely presented groups is an important old problem that has gained prominence recently via group theoretical encryption methods. The theory of rewriting systems is extended to allow conjugacy problems in certain groups to be solved using rewriting methods. We construct explicit rewrite systems to solve the word and conjugacy problems in the braid groups. We show that the complexity of both algorithms is polynomial-time. This resolves an old problem in low-dimensional topology and shows that the braid group is not a suitable platform group for cryptography, against expectations in the literature.
Arxiv preprint cs/0507066, 2005
The idea of using the braid group as a platform for cryptosystems was introduced in [2]. In recent years have emerged several proposals for secure cryptographical schemes using noncommutative groups, in particular Artin's braid groups [1, 2, 4, 5, and 7]. Braid groups are more ...
Computing Research Repository, 2004
After some excitement generated by recently suggested public key exchange protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether choosing a different group, or a class of groups, can remedy the situation; (2) whether some other "hard" problem from combinatorial group theory can be used, instead of the conjugacy search problem, in a public key exchange protocol. Another question that we address here, although somewhat vague, is likely to become a focus of the future research in public key cryptography based on symbolic computation: (3) whether one can efficiently disguise an element of a given group (or a semigroup) by using defining relations.
Journal of Algebra, 2003
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 [3]. 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).
Artin's braid groups have been recently suggested as a new source for public-key cryptography. In this paper we propose the first group signature schemes based on the conjugacy problem, decomposition problem and root problem in the braid groups which are believed to be hard problems.
Arxiv preprint arXiv: …, 2008
We give a new infinite family of group homomorphisms from the braid group B k to the symmetric group S mk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B k → Sn, common to all homomorphisms in our family, which we term good, and of which there are two types. We prove that all good homomorphisms B k → S mk of type 1 are included in the infinite family of homomorphisms we gave. For m = 3, we prove that all good homomorphisms B k → S 3k of type 2 are also included in this family. Finally, we refute a conjecture made in [MaSu05] regarding permutation representations of braids and give an updated conjecture.
Pacific Journal of Mathematics, 2016
We show a simple and easily implementable solution to the word problem for virtual braid groups.
ArXiv, 2017
In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, $B_3$, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-complete for braids with only three strands. The membership problem is decidable in NP for $B_3$, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for $B_4$ remains open. Finally we show that the freeness problem for semigroups of braids from $B_3$ is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of severa...
Filomat, 2020
It is known that if there exists a Gr?bner-Shirshov basis for a group G, then we say that one of the decision problem, namely the word problem, is solvable for G as well. Therefore, as the main target of this paper, we will present a (non-commutative) Gr?bner-Shirshov basis for the braid group associated with the congruence classes of complex reflection group G12 which will give us normal forms of the elements of G12 and so will obtain a new algorithm to solve the word problem over it.
Designs, Codes and Cryptography, 2007
Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this paper, we first survey all the major attacks on the braid cryptosystem and conclude that the attacks were successful because the current ways of random key generation almost always result in weaker instances of the conjugacy problem. We then propose several alternate ways of generating hard instances of the conjugacy problem for use braid cryptography.
2016
We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
2002
Using presentations of the braid groups B 3 and B 4 as towers of HNN extensions of the free group of rank 2, we obtain normal forms, Gröbner bases and rewriting systems for these groups.
The Quarterly Journal of Mathematics, 1994
We give an easily handled algorithm for the word problem in each of Artin's braid groups, B n , 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 ∆ r 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.
International Journal of Computer Applications, 2011
Braid groups were first introduced by Emil Artin in 1925. First cryptosystem, using Braid groups as a platform was discovered by Anshel et al in 2001. After the publication of this paper several cryptosystems on Braid groups had been designed. In this paper we have proposed a tripartite authenticated key agreement protocol using conjugacy problem which works in a braid group. We have proved that our protocol meet the security attributes under the assumption that the Braid Decomposition Problem (BDP) and the Conjugacy Search Problem (CSP) are hard in braid group.
Advances in Applied Mathematics, 2005
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.
International Journal of Network Security, 2008
In this paper we propose an authenticated key agreement, which works in a braid group. We prove that our protocol meet the security attributes under the assumption that the Conjugacy Search Problem (CSP) is hard in braid group. Keywords: Authenticated key ...
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.