Combinatorial aspects of braids with applications to cryptography

Abstract

This thesis is a collection of different results on braids, and draws connections between them. We first introduce braids by showcasing a number of equivalent ways of describing what a braid is, and how those representations are related. Then, while uncovering enumerative properties of the positive braid monoid, we consider algorithms to compute the lcm of a set of braids. This leads to more than one elegant solution to the word problem. We explore some efficient algorithms which solve the word problem for braids, and then also explore the conjugacy problem and the cryptosystems that rely on the hardness of it in their proofs of security.