Probabilistic Solutions of Equations in the Braid Group
Group Theory
2010-08-02 v4 Cryptography and Security
Geometric Topology
Abstract
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: The conjugacy problem, the group membership problem, the shortest representation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group. We use a memory-based extension of the standard length-based approach, which in principle can be applied to any group admitting an efficient, reasonably behaving length function.
Cite
@article{arxiv.math/0404076,
title = {Probabilistic Solutions of Equations in the Braid Group},
author = {D. Garber and S. Kaplan and M. Teicher and B. Tsaban and U. Vishne},
journal= {arXiv preprint arXiv:math/0404076},
year = {2010}
}
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