205 research outputs found
Length-Based Attacks for Certain Group Based Encryption Rewriting Systems
In this note, we describe a probabilistic attack on public key cryptosystems
based on the word/conjugacy problems for finitely presented groups of the type
proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes
use of the property that in the given group the word problem has a polynomial
time solution, while the conjugacy problem has no known polynomial solution. An
example is the braid group from topology in which the word problem is solvable
in polynomial time while the only known solutions to the conjugacy problem are
exponential. The attack in this paper is based on having a canonical
representative of each string relative to which a length function may be
computed. Hence the term length attack. Such canonical representatives are
known to exist for the braid group
Knuth-Bendix algorithm and the conjugacy problems in monoids
We present an algorithmic approach to the conjugacy problems in monoids,
using rewriting systems. We extend the classical theory of rewriting developed
by Knuth and Bendix to a rewriting that takes into account the cyclic
conjugates.Comment: This is a new version of the paper 'The conjugacy problems in monoids
and semigroups'. This version will appear in the journal 'Semigroup forum
The infimum, supremum and geodesic length of a braid conjugacy class
Algorithmic solutions to the conjugacy problem in the braid groups B_n were
given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions
yield two conjugacy class invariants which are known as `inf' and `sup'. A
problem which was left unsolved in both papers was the number m of times one
must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or
to be sure that it is already maximal (resp. minimal) for the given conjugacy
class. Our main result is to prove that m is bounded above by n-2 in the
situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the
first. As a corollary, we show that the computation of inf and sup is
polynomial in both word length and braid index, in both algorithms. The
integers inf and sup determine (but are not determined by) the shortest
geodesic length for elements in a conjugacy class, as defined by Charney, and
so we also obtain a polynomial-time algorithm for computing this geodesic
length.Comment: 15 pages. Journa
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
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
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