The root extraction problem for generic braids
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-c...
| Authors: | , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2019 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/91652 |
| Online Access: | https://hdl.handle.net/11441/91652 https://doi.org/10.3390/sym11111327 |
| Access Level: | Open access |
| Keyword: | Braid groups Aalgorithms in groups Group-based cryptography |
| Summary: | 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)n3 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 ∆), consisting of either a single orbit conjugated to itself by ∆ or two orbits conjugated to each other by ∆. |
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