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Open AccessArticle

The Root Extraction Problem for Generic Braids

1
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France
2
Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland EH14 4AS, UK
3
Departamento de Álgebra, Universidad de Sevilla, 41012 Sevilla, Spain
4
Departamento de Ciencias Integradas, Universidad de Huelva, 21007 Huelva, Spain
*
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1327; https://doi.org/10.3390/sym11111327
Received: 24 September 2019 / Accepted: 17 October 2019 / Published: 23 October 2019
(This article belongs to the Special Issue Interactions between Group Theory, Symmetry and Cryptology)
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 ) n 3 log n ) . 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 . View Full-Text
Keywords: braid groups; algorithms in groups; group-based cryptography braid groups; algorithms in groups; group-based cryptography
MDPI and ACS Style

Cumplido, M.; González-Meneses, J.; Silvero, M. The Root Extraction Problem for Generic Braids. Symmetry 2019, 11, 1327.

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