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. Academic Press

New Public-Key Cryptosystem Using Braid Groups

“…In this section, we will briefly introduce the notion of braids and give evidence that the braid groups can also play important roles in cryptography. The general reference for braid theory is the Birman's book [5] and for the word problem and conjugacy problem, see [6,13,14,16]. This section is composed as follows: §2.1 is the definition of the braid groups.…”

“…In late sixties, Garsides solved the word problem after exploring the properties of the semigroup of positive words in [16] and his idea was improved by Thurston [14], Elrifai-Morton [13] and BirmanKo-Lee [6]. They showed that there is a fast algorithm to compute the canonical form, which is unique for their results briefly.…”

“…The adversary may try to use a mathematical solution to the conjugacy problem by Garside [16], Thurston [14], Elrifai-Morton [13] and Birman-Ko-Lee [6]. But the known algorithms find an element a ∈ B +r , not in LB .…”

An Efficient Implementation of Braid Groups

“…There are theoretically similar solutions to the word and conjugacy problems in B n for both presentations [1][2][3]. The band-generator presentation has a computational advantage over the Artin as far as the word problem is concerned.…”

“…In this paper we discuss implementation issues of the braid group given by either the Artin presentation [2] or the band-generator presentation [1]. Due to the analogy between the two presentations, our implementations on the two presentations are basically identical, except the low-level layer consisting of data structures and algorithms for canonical factors, which play the role of the building blocks for braids.…”