63 research outputs found
Computing the Rank Profile Matrix
The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm
Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations
We give a Las Vegas algorithm which computes the shifted Popov form of an nonsingular polynomial matrix of degree in expected
field operations, where is the
exponent of matrix multiplication and
indicates that logarithmic factors are omitted. This is the first algorithm in
for shifted row reduction with arbitrary
shifts.
Using partial linearization, we reduce the problem to the case where is the generic determinant bound, with bounded from above by both the average row degree and the average column
degree of the matrix. The cost above becomes , improving upon the cost of the fastest previously
known algorithm for row reduction, which is deterministic.
Our algorithm first builds a system of modular equations whose solution set
is the row space of the input matrix, and then finds the basis in shifted Popov
form of this set. We give a deterministic algorithm for this second step
supporting arbitrary moduli in
field operations, where is the number of unknowns and is the sum
of the degrees of the moduli. This extends previous results with the same cost
bound in the specific cases of order basis computation and M-Pad\'e
approximation, in which the moduli are products of known linear factors.Comment: 8 pages, sig-alternate class, 5 figures (problems and algorithms
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
Structure of the Afferent Terminals in Terminal Ganglion of a Cricket and Persistent Homology
We use topological data analysis to investigate the three dimensional spatial structure of the locus of afferent neuron terminals in crickets Acheta domesticus. Each afferent neuron innervates a filiform hair positioned on a cercus: a protruding appendage at the rear of the animal. The hairs transduce air motion to the neuron signal that is used by a cricket to respond to the environment. We stratify the hairs (and the corresponding afferent terminals) into classes depending on hair length, along with position. Our analysis uncovers significant structure in the relative position of these terminal classes and suggests the functional relevance of this structure. Our method is very robust to the presence of significant experimental and developmental noise. It can be used to analyze a wide range of other point cloud data sets
Chemical and biological evaluation of Amazonian medicinal plant Vouacapoua americana Aubl.
Vouacapoua americana (Fabaceae) is an economically important tree in the Amazon region and used for its highly resistant heartwood as well as for medicinal purposes. Despite its frequent use, phytochemical investigations have been limited and rather focused on ecological properties than on its pharmacological potential. In this study, we investigated the phytochemistry and bioactivity of V. americana stem bark extract and its constituents to identify eventual lead structures forfurther drug development. Applying hydrodistillation and subsequent GC-MS analysis, we investigated the composition of the essential oil and identified the 15 most abundant components. Moreover, the diterpenoids deacetylchagresnone (1), cassa-13(14),15-dien-oic acid (2), isoneocaesalpin H (3), (+)-vouacapenic acid (4), and (+)-methyl vouacapenate (5) were isolated from the stem bark, with compounds 2 and 4 showing pronounced effects on Methicillin-resistant Staphylococcus aureus and Enterococcus faecium, respectively. During the structure elucidation of deacetylchagresnone (1), which was isolated from a natural source for the first time, we detected inconsistencies regarding the configuration of the cyclopropane ring. Thus, the structure was revised for both deacetylchagresnone (1) and the previously isolated chagresnone. Following our works on Copaifera reticulata and Vatairea guianensis, the results of this study further contribute to the knowledge of Amazonian medicinal plants
Cryptanalysis of the New CLT Multilinear Map over the Integers
Multilinear maps serve as a basis for a wide range of cryptographic applications. The first candidate construction of multilinear maps was proposed by Garg, Gentry, and Halevi in 2013, and soon afterwards, another construction was suggested by Coron, Lepoint, and Tibouchi (CLT13), which works over the integers. However, both of these were found to be insecure in the face of so-called zeroizing attacks, by Hu and Jia, and by Cheon, Han, Lee, Ryu and Stehlé. To improve on CLT13, Coron, Lepoint, and Tibouchi proposed another candidate construction of multilinear maps over the integers at Crypto 2015 (CLT15).
This article presents two polynomial attacks on the CLT15 multilinear map, which share ideas similar to the cryptanalysis of CLT13. Our attacks allow recovery of all secret parameters in time polynomial in the security parameter, and lead to a full break of the CLT15 multilinear map for virtually all applications
Certified Dense Linear System Solving
The following problems related to linear systems are studied: finding a diophantine solution; finding a rational solution; proving no diophantine solution exists; proving no rational solution exists. These problems are reduced, via randomization, to that of computing an expected constant number of rational solutions of square nonsingular systems using adic lifting. The bit complexity of the latter problem is improved by incorporating fast arithmetic and fast matrix multiplication. The resulting randomized algorithm for certified dense linear system solving has substantially better asymptotic complexity than previous algorithms for either rational or diophantine linear system solving
On Lattice Reduction for Polynomial Matrices
A simple algorithm for transformation to weak Popov form -- essentially lattice reduction for polynomial matrices -- is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank profile and determinant computation; unimodular triangular factorization; transformation to Hermite and Popov canonical form; rational and diophantine linear system solving; short vector computation
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