541 research outputs found
Finding lower bounds on the complexity of secret sharing schemes by linear programming
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants.
By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing schemePeer ReviewedPostprint (author's final draft
Moduli spaces of Dirac operators for finite spectral triples
The structure theory of finite real spectral triples developed by Krajewski
and by Paschke and Sitarz is generalised to allow for arbitrary KO-dimension
and the failure of orientability and Poincare duality, and moduli spaces of
Dirac operators for such spectral triples are defined and studied. This theory
is then applied to recent work by Chamseddine and Connes towards deriving the
finite spectral triple of the noncommutative-geometric Standard Model.Comment: AMS-LaTeX, 60 pp. Revised version of qualifying year project
(Master's thesis equivalent), BIGS, University of Bonn. V2: Final version
with minor corrections, to appear in the Proceedings of the Workshop on
Quantum Groups and Noncommutative Geometry, M. Marcolli and D. Parashar
(eds.
Algebraic symmetries of generic dimensional periodic Costas arrays
In this work we present two generators for the group of symmetries of the
generic dimensional periodic Costas arrays over elementary abelian
groups: one that is defined by multiplication on
dimensions and the other by shear (addition) on dimensions. Through
exhaustive search we observe that these two generators characterize the group
of symmetries for the examples we were able to compute. Following the results,
we conjecture that these generators characterize the group of symmetries of the
generic dimensional periodic Costas arrays over elementary abelian
groups
Torsion pairs in a triangulated category generated by a spherical object
We extend Ng's characterisation of torsion pairs in the 2-Calabi-Yau
triangulated category generated by a 2-spherical object to the characterisation
of torsion pairs in the w-Calabi-Yau triangulated category, , generated by
a w-spherical object for any integer w. Inspired by the combinatorics of
for w < 0, we also characterise the torsion pairs in a certain w-Calabi-Yau
orbit category of the bounded derived category of the path algebra of Dynkin
type A.Comment: v2: 36 pages, 11 figures, added Section 4 which deals with extensions
whose outer terms are decomposable, minor changes in presentation, accepted
in J. Algebr
Polynomial knot and link invariants from the virtual biquandle
The Alexander biquandle of a virtual knot or link is a module over a
2-variable Laurent polynomial ring which is an invariant of virtual knots and
links. The elementary ideals of this module are then invariants of virtual
isotopy which determine both the generalized Alexander polynomial (also known
as the Sawollek polynomial) for virtual knots and the classical Alexander
polynomial for classical knots. For a fixed monomial ordering , the
Gr\"obner bases for these ideals are computable, comparable invariants which
fully determine the elementary ideals and which generalize and unify the
classical and generalized Alexander polynomials. We provide examples to
illustrate the usefulness of these invariants and propose questions for future
work.Comment: 12 pages; version 3 includes corrected figure
Finite dimensional quantum group covariant differential calculus on a complex matrix algebra
Using the fact that the algebra M(3,C) of 3 x 3 complex matrices can be taken
as a reduced quantum plane, we build a differential calculus Omega(S) on the
quantum space S defined by the algebra C^\infty(M) \otimes M(3,C), where M is a
space-time manifold. This calculus is covariant under the action and coaction
of finite dimensional dual quantum groups. We study the star structures on
these quantum groups and the compatible one in M(3,C). This leads to an
invariant scalar product on the later space. We analyse the differential
algebra Omega(M(3,C)) in terms of quantum group representations, and consider
in particular the space of one-forms on S since its elements can be considered
as generalized gauge fields.Comment: 11 pages, LaTeX, uses diagrams.st
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