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Open AccessArticle

The Root Extraction Problem for Generic Braids

Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France
Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland EH14 4AS, UK
Departamento de Álgebra, Universidad de Sevilla, 41012 Sevilla, Spain
Departamento de Ciencias Integradas, Universidad de Huelva, 21007 Huelva, Spain
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1327;
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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