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Artículo
On the cycling operation in braid groups
Autor/es | González-Meneses López, Juan
Gebhardt, Volker |
Departamento | Universidad de Sevilla. Departamento de álgebra |
Fecha de publicación | 2008-09-06 |
Fecha de depósito | 2016-06-15 |
Publicado en |
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Resumen | 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 ... 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 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. |
Agencias financiadoras | Ministerio de Educación y Ciencia (MEC). España European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER) |
Identificador del proyecto | MTM2004-07203-C02-01 |
Cita | González-Meneses López, J. y Gebhardt, V. (2008). On the cycling operation in braid groups. Discrete Applied Mathematics, 156 (16), 3072-3090. |
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