28 research outputs found
Cylindrical Graph Construction (definition and basic properties)
In this article we introduce the {\it cylindrical construction} for graphs
and investigate its basic properties. We state a main result claiming a weak
tensor-like duality for this construction. Details of our motivations and
applications of the construction will appear elsewhere
On nodal domains of finite reversible Markov processes and spectral decomposition of cycles
24 pagesInternational audienceLet be a reversible Markovian generator on a finite set . Relations between the spectral decomposition of and subpartitions of the state space into a given number of components which are optimal with respect to min-max or max-min Dirichlet connectivity criteria are investigated. Links are made with higher order Cheeger inequalities and with a generical characterization of subpartitions given by the nodal domains of an eigenfunction. These considerations are applied to generators whose positive rates are supported by the edges of a discrete cycle , to obtain a full description of their spectra and of the shapes of their eigenfunctions, as well as an interpretation of the spectrum through a double covering construction. Also, we prove that for these generators, higher Cheeger inequalities hold, with a universal constant factor 48
On The Isoperimetric Spectrum of Graphs and Its Approximations
In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the th isoperimetric constant and the number obtained by taking the minimum
over all -partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the -completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi () we study the close
relationships to the well-known -means algorithm and normalized cuts method
Strong Continuous Non-malleable Encoding Schemes with Tamper-Detection
A non-malleable encoding scheme is a keyless encoding scheme which is resilient to tampering attacks. Such a scheme
is said to be continuously secure if the scheme is resilient to attacks containing more than one tampering procedure. Also, such a
scheme is said to have tamper-detection property if any kind of
tampering attack is detected. In [S. Faust, et al., Continuous nonmalleable codes, TCC Proc., LNCS Vol. 8349, 2014.] a general
continuous non-malleable encoding scheme based on NIZK is
introduced which is secure in a strong model for which the
adversary receives a no-tamper as a response to its tampering
query if the decoding of the tampered codeword is identical to
the original message.
In this article we introduce a new strongly secure continuous
non-malleable encoding scheme with tamper-detection property
whose security is based on the existence of secure MAC’s. Moreover, we introduce and justify the importance of an intermediate security model called semi-strong continuous non-malleability, while we provide a secure semi-strong continuous non-malleable encoding scheme whose security is based on the existence of CCA-secure public-key encryption.
Considering the area of applications of encoding schemes in
tamper-proof devices, it is instructive to note that our proposed
schemes can be used to implement an algorithmic tamperdetection level as well as maintaining the security conditions