437 research outputs found
Improved Agreeing-Gluing Algorithm
In this paper we study the asymptotical complexity of solving a system of sparse algebraic equations over finite fields. An equation is called sparse if it depends
on a bounded number of variables. Finding
efficiently solutions to the system of such equations is an underlying hard problem in
the cryptanalysis of modern ciphers. New deterministic
Improved Agreeing-Gluing Algorithm is introduced.
The expected
running time of the Algorithm on uniformly random instances of the problem is rigorously estimated. The estimate is at present the best theoretical
bound on the complexity of solving average instances of the
problem. In particular, this is a significant improvement over those in our earlier papers [20,21].
In sparse Boolean equations a gap between the present worst case and the average time complexity of the problem has significantly increased. Also we formulate Average Time Complexity Conjecture. If proved that will have far-reaching consequences in the field of cryptanalysis and in computing in general
A generalisation of the deformation variety
Given an ideal triangulation of a connected 3-manifold with non-empty
boundary consisting of a disjoint union of tori, a point of the deformation
variety is an assignment of complex numbers to the dihedral angles of the
tetrahedra subject to Thurston's gluing equations. From this, one can recover a
representation of the fundamental group of the manifold into the isometries of
3-dimensional hyperbolic space. However, the deformation variety depends
crucially on the triangulation: there may be entire components of the
representation variety which can be obtained from the deformation variety with
one triangulation but not another. We introduce a generalisation of the
deformation variety, which again consists of assignments of complex variables
to certain dihedral angles subject to polynomial equations, but together with
some extra combinatorial data concerning degenerate tetrahedra. This "extended
deformation variety" deals with many situations that the deformation variety
cannot. In particular we show that for any ideal triangulation of a small
orientable 3-manifold with a single torus boundary component, we can recover
all of the irreducible non-dihedral representations from the associated
extended deformation variety. More generally, we give an algorithm to produce a
triangulation of a given orientable 3-manifold with torus boundary components
for which the same result holds. As an application, we show that this extended
deformation variety detects all factors of the PSL(2,C) A-polynomial associated
to the components consisting of the representations it recovers.Comment: 47 pages, 26 figures. Rewrote introduction and added motivation
section based on referee's comments. Rewrote the section on retriangulation,
and added new result on small manifolds with a single cus
08431 Abstracts Collection -- Moderately Exponential Time Algorithms
From to , the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Joint morphological-lexical language modeling for processing morphologically rich languages with application to dialectal Arabic
Language modeling for an inflected language
such as Arabic poses new challenges for speech recognition and
machine translation due to its rich morphology. Rich morphology
results in large increases in out-of-vocabulary (OOV) rate and
poor language model parameter estimation in the absence of large
quantities of data. In this study, we present a joint
morphological-lexical language model (JMLLM) that takes
advantage of Arabic morphology. JMLLM combines
morphological segments with the underlying lexical items and
additional available information sources with regards to
morphological segments and lexical items in a single joint model.
Joint representation and modeling of morphological and lexical
items reduces the OOV rate and provides smooth probability
estimates while keeping the predictive power of whole words.
Speech recognition and machine translation experiments in
dialectal-Arabic show improvements over word and morpheme
based trigram language models. We also show that as the
tightness of integration between different information sources
increases, both speech recognition and machine translation
performances improve
Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles
R(W5 , K5) = 27
The two-color Ramsey number R(G , H) is defined to be the smallest integer n such that any graph F on n vertices contains either a subgraph isomorphic to G or the complement of F contains a subgraph isomorphic to H. Ramsey numbers serve to quantify many of the existing theorems of Ramsey theory, which looks at large combinatorial objects for certain given smaller combinatorial objects that must be present. In 1989 George R. T. Hendry presented a table of two-color Ramsey numbers R(G , H) for all pairs of graphs G and H having at most five vertices. This table left seven unsolved cases, of which three have since been solved. This thesis eliminates one of the remaining four cases, R(W5 , K5), where a K5 is the complete graph on five vertices and a W5 is a wheel of order 5, which can be pictured as a wheel having four spokes or as a cycle of length 4 having all four vertices adjacent to a central vertex. In this thesis we show R(W5, K5) to be equal to 27, utilizing a combinatorial approach with significant computations. Specifically we use a technique developed by McKay and Radziszowski to effectively glue together smaller graphs in an effort to prove exhaustively that no graph having 27 vertices exists that does not contain an independent set on five vertices or a subgraph isomorphic to W5. The previous best bounds for this case were 27 \u3c= R( W_5 , K_5 ) \u3c= 29
Using Local Reduction for the Experimental Evaluation of the Cipher Security
Evaluating the strength of block ciphers against algebraic attacks can be difficult. The attack methods often use different metrics, and experiments do not scale well in practice. We propose a methodology that splits the algebraic attack into a polynomial part (local reduction), and an exponential part (guessing), respectively. The evaluator uses instances with known solutions to estimate the complexity of the attacks, and the response to changing parameters of the problem (e.g. the number of rounds). Although the methodology does not provide a positive answer ("the cipher is secure"), it can be used to construct a negative test (reject weak ciphers), or as a tool of qualitative comparison of cipher designs. Potential applications in other areas of computer science are discussed in the concluding parts of the article
- …