783 research outputs found
Probabilistic Solutions of Equations in the Braid Group
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.Comment: Small update
On an authentication scheme based on the Root Problem in the braid group
Lal and Chaturvedi proposed two authentication schemes based on the
difficulty of the Root Problem in the braid group. We point out that the first
scheme is not really as secure as the Root Problem, and describe an efficient
way to crack it. The attack works for any group.Comment: This paper has been withdrawn by the author. One of the claims is
incorrect as written. We are working on correcting and generalizing it. This
will be published in another pape
Group theory in cryptography
This paper is a guide for the pure mathematician who would like to know more
about cryptography based on group theory. The paper gives a brief overview of
the subject, and provides pointers to good textbooks, key research papers and
recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor
typographical changes. To appear in Proceedings of Groups St Andrews 2009 in
Bath, U
A new key exchange protocol based on the decomposition problem
In this paper we present a new key establishment protocol based on the
decomposition problem in non-commutative groups which is: given two elements
of the platform group and two subgroups (not
necessarily distinct), find elements such that . Here we introduce two new ideas that improve the security of key
establishment protocols based on the decomposition problem. In particular, we
conceal (i.e., do not publish explicitly) one of the subgroups , thus
introducing an additional computationally hard problem for the adversary,
namely, finding the centralizer of a given finitely generated subgroup.Comment: 7 page
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