18 research outputs found
Iterated LD-Problem in non-associative key establishment
We construct new non-associative key establishment protocols for all left
self-distributive (LD), multi-LD-, and mutual LD-systems. The hardness of these
protocols relies on variations of the (simultaneous) iterated LD-problem and
its generalizations. We discuss instantiations of these protocols using
generalized shifted conjugacy in braid groups and their quotients, LD-conjugacy
and -symmetric conjugacy in groups. We suggest parameter choices for
instantiations in braid groups, symmetric groups and several matrix groups.Comment: 30 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1305.440
Logspace computations for Garside groups of spindle type
M. Picantin introduced the notion of Garside groups of spindle type,
generalizing the 3-strand braid group. We show that, for linear Garside groups
of spindle type, a normal form and a solution to the conjugacy problem are
logspace computable. For linear Garside groups of spindle type with homogenous
presentation we compute a geodesic normal form in logspace.Comment: 22 pages; short version as v1. Terminolgy and title changed. In
particular, in previous versions we called Garside groups of spindle type
"rigid Garside groups
Complexity of relations in the braid group
We show that for any given n, there exists a sequence of words a_k in the
generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the
identity element of B_n, such that the number of braid relations of the form
sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass
from a_k to the empty word is quadratic with respect to the length of a_k
Double coset problem for parabolic subgroups of braid groups
We provide the first solution to the double coset problem (DCP) for a large
class of natural subgroups of braid groups, namely for all parabolic subgroups
which have a connected associated Coxeter graph. Update: We succeeded to solve
the DCP for all parabolic subgroups of braid groups.Comment: 8 pages. Update remark adde