On the cycling operation in braid groups

TL;DR

This paper provides a polynomial-time solution to the cycling problem in braid groups, demonstrates the equivalence of left and right ultra summit sets for rigid braids, and conjectures this extends to all elements in spherical type Artin-Tits groups.

Contribution

It proves cycling is surjective and computable efficiently, and establishes an isomorphism between left and right ultra summit sets for rigid braids, suggesting a broader conjecture.

Findings

Cycling is surjective and can be computed quickly.

Left and right ultra summit sets of rigid braids are isomorphic graphs.

Conjecture that this isomorphism holds for all elements in braid groups.

Abstract

The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type. On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USS), using left normal forms of braids. But one can equally use right normal forms and compute right-USS's. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USS's. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type.