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 $m \times m$ nonsingular polynomial matrix of degree $d$ in expected $\widetilde{\mathcal{O}}(m^\omega d)$ field operations, where $\omega$ is the exponent of matrix multiplication and $\widetilde{\mathcal{O}}(\cdot)$ indicates that logarithmic factors are omitted. This is the first algorithm in $\widetilde{\mathcal{O}}(m^\omega d)$ for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case $d \le \lceil \sigma/m \rceil$ where $\sigma$ is the generic determinant bound, with $\sigma / m$ bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes $\widetilde{\mathcal{O}}(m^\omega \lceil \sigma/m \rceil)$, 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 $\widetilde{\mathcal{O}}(m^{\omega-1} \sigma)$ field operations, where $m$ is the number of unknowns and $\sigma$ 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

Computing a Lattice Basis Revisited

International audienc

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