Cryptographic Primitives in Artin Groups of Finite Type

Abstract

Braid groups can be viewed as Coxeter group "A_n" without relations of type "s_i^2=1$" Substitution of another Coxeter group results in a different Artin group with distinct relations. Due to the rising importance of non-commutative cryptography, we explore these generalized braid groups for potential cryptographic viability. This paper explores potential cryptographic applications, building up from the fundamentals in Coxeter and braid theory that are prerequisite. We implement cryptographic primitives on generalized braid groups associated with finite irreducible and affine Coxeter systems and quantify results. More specifically, we perform length analysis of keys in input versus output, quantify reducibility based on Coxeter graph and word length, and analyze disparities in results of different Artin groups. Then we discuss potential attacks and further findings. Lastly, we outline possible directions for future research.