[Paper Review] Algorithmic Problems in the Braid Group

Elie Feder|ArXiv.org|May 14, 2003

Geometric and Algebraic Topology46 references3 citations

TL;DR

This PhD thesis investigates algorithmic problems in braid groups, presenting efficient solutions to the generalized word problem in cyclic subgroups using a multitape Turing machine model. It extends these methods to solve the word and conjugacy problems in cyclic amalgamations of braid groups, offering new theoretical results and applications in braid group cryptography, with complexity analyses provided throughout.

ABSTRACT

We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution and its complexity using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups. Again using a multitape Turing machine, we solve the word problem for the cyclic amalgamation of two braid groups. We analyze its complexity as well. We then turn to a more general study of the conjugacy problem in cyclic amalgamations. We revise and prove some theorems of Lipschutz[L1] and show their application to cyclic amalgamations of braid groups. We generalize this application to prove a new theorem regarding the conjugacy problem in cyclic amalgamations. We then discuss some application of braid groups, culminating in a section devoted to the discussion of braid group cryptography. We conclude with a discussion of some open questions that we would like to pursue in future research.

Motivation & Objective

• To address fundamental decision problems in braid groups, particularly the generalized word problem in cyclic subgroups.
• To extend algorithmic solutions to cyclic amalgamations of braid groups, solving the word problem and analyzing its complexity.
• To generalize conjugacy problem results in cyclic amalgamations using revised theorems from Lipschutz and proving a new theorem.
• To explore applications of braid group algorithms in cryptography, especially in the context of public-key cryptosystems.
• To identify and outline open problems for future research in braid group algorithmics.

Proposed method

• A multitape Turing machine model is employed to implement and analyze the time complexity of solving the generalized word problem in cyclic subgroups of braid groups.
• The word problem in cyclic amalgamations of two braid groups is solved using structural group-theoretic techniques and automata-based reasoning.
• Techniques from combinatorial group theory are applied to analyze the conjugacy problem in cyclic amalgamations, building on and revising existing theorems by Lipschutz.
• The paper uses normal forms and braid group presentations to derive decision procedures for word and conjugacy problems.
• Complexity bounds are derived by modeling computation steps on a multitape Turing machine, providing precise time complexity estimates.
• Applications to braid group cryptography are discussed, particularly in the context of key exchange protocols based on conjugacy search problems.

Experimental results

Research questions

• RQ1 How can the generalized word problem be solved efficiently in cyclic subgroups of braid groups, and what is its computational complexity?
• RQ2 What is the complexity of solving the word problem in cyclic amalgamations of two braid groups?
• RQ3 Can existing theorems on the conjugacy problem in amalgamated free products be extended and applied to braid groups?
• RQ4 What new theoretical results can be derived for the conjugacy problem in cyclic amalgamations of braid groups?
• RQ5 What are the implications of these algorithmic results for braid group-based cryptography?

Key findings

• An efficient algorithm is presented for solving the generalized word problem in cyclic subgroups of braid groups, with a complexity analysis provided via a multitape Turing machine model.
• The word problem in the cyclic amalgamation of two braid groups is decidable, and its time complexity is explicitly bounded using the same computational model.
• A new theorem is proven regarding the conjugacy problem in cyclic amalgamations of braid groups, generalizing prior results from Lipschutz.
• The paper revises and proves key theorems from Lipschutz's work, establishing their validity and applicability in the context of braid group amalgamations.
• Theoretical results are applied to braid group cryptography, suggesting potential for secure key exchange protocols based on conjugacy search problems.
• Several open problems in braid group algorithmics are identified, including extensions to more general amalgamations and improved complexity bounds.