The Root Extraction Problem for Generic Braids
Abstract
We show that, generically, finding the k-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid x on n strands and canonical length l, and an integer k>1, computes a k-th root of x, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is O(l(l+n)n3logn). The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee. This algorithm uses the fact that the ultra summit set of a braid is, generically, very small and symmetric (through conjugation by the Garside element Δ), consisting of either a single orbit conjugated to itself by Δ or two orbits conjugated to each other by Δ.
- Publication:
-
Symmetry
- Pub Date:
- October 2019
- DOI:
- 10.3390/sym11111327
- arXiv:
- arXiv:1909.10962
- Bibcode:
- 2019Symm...11.1327C
- Keywords:
-
- braid groups;
- algorithms in groups;
- group-based cryptography;
- Mathematics - Group Theory
- E-Print:
- 15 pages