Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices

Les grandes illusions. Les ambivalences de l’expérience combattante de Jacques Bardoux, intellectuel et engagé volontaire (août 1914-février 1915)

En août 1914, bien que réformé, l’intellectuel Jacques Bardoux décide de s’engager dans le 99e Régiment d’Infanterie territoriale. Nommé sous-lieutenant en novembre, il prend part à la guerre des tranchées d’octobre 1914 à février 1915. Rejoignant les questionnements récents sur les comportements dans les tranchées et l’adhésion des individus au conflit, ce présent article vise à retracer l’expérience originale de cet intellectuel, en confrontant les diverses manifestations de son consentement patriotique à la réalité difficile de la guerre des tranchées, afin de mettre en lumière la stratégie d’embusquage mise en œuvre par cet engagé volontaire.In august 1914, although he was a discharged soldier, the intellectual Jacques Bardoux decided to engage in the 99th “Régiment d’Infanterie Territoriale”. Appointed “sous-lieutenant”, he fought in the trenches from october 1914 to february 1915. Linked to the recent questionning about consent and behaviours of soldiers in the trenches, this article tries to tell the experience of this intellectual, by comparing the manifestation of the patriotic consent of Jacques Bardoux and the reality of trench warfare, to finally focus on the strategy that he organised to shirk

Elimination-based certificates for triangular equivalence and rank profiles

International audienceIn this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrix-vector multiplications, a linear space, and a linear number of extra field operations, with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

International audienceWe present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures.Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization.Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebraover a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies

Dynamic proofs of retrievability with low server storage

Proofs of Retrievability (PoRs) are protocols which allow a client to store data remotely and to efficiently ensure, via audits, that the entirety of that data is still intact. A dynamic PoR system also supports efficient retrieval and update of any small portion of the data. We propose new, simple protocols for dynamic PoR that are designed for practical efficiency, trading decreased persistent storage for increased server computation, and show in fact that this tradeoff is inherent via a lower bound proof of time-space for any PoR scheme. Notably, ours is the first dynamic PoR which does not require any special encoding of the data stored on the server, meaning it can be trivially composed with any database service or with existing techniques for encryption or redundancy. Our implementation and deployment on Google Cloud Platform demonstrates our solution is scalable: for example, auditing a 1TB file takes just less than 5 minutes and costs less than $0.08 USD. We also present several further enhancements, reducing the amount of client storage, or the communication bandwidth, or allowing public verifiability, wherein any untrusted third party may conduct an audit

Représentations de la marginalité, marginalités en représentation : Marges et pouvoir (xviie-xxe siècles)

« En marge (de la société) : sans s’intégrer au groupe social et sans se soumettre à ses normes », nous indique le dictionnaire Larousse. Cette définition suggère que la « marge », en tant que milieu, implique pour celles et ceux qui s’y trouvent une mise à distance, un écart volontaire ou non par rapport à un centre. La marginalité est en ce sens une notion relative : elle gagne à être étudiée non pas seulement pour elle-même, mais aussi dans sa relation contrariée avec le groupe ou l’instit..

The mitochondrial DNA content of cumulus granulosa cells is linked to embryo quality

STUDY QUESTION: Could the mitochondrial DNA (mtDNA) content of cumulus granulosa cells (CGCs) be related to oocyte competence? SUMMARY ANSWER: The quality of embryos obtained during IVF procedures appears to be linked to mtDNA copy numbers in the CGCs. WHAT IS KNOWN ALREADY: Oocyte quality is linked to oocyte mtDNA content in the human and other species, and the mtDNA copy number of the oocyte is related to that of the corresponding CGCs. Moreover, the quantification of CGC mtDNA has recently been proposed as a biomarker of embryo viability. STUDY DESIGN SIZE, DURATION: An observational study was performed on 452 oocyte-cumulus complexes retrieved from 62 patients undergoing ICSI at the ART Center of the University Hospital of Angers, France, from January to May 2015. PARTICIPANTS/MATERIALS, SETTING, METHODS: The average mtDNA content of CGCs was assessed by using a quantitative real-time PCR technique. The relationship between CGC mtDNA content and oocyte maturity and fertilizability, on one hand, and embryo quality, on the other, was investigated using univariate and multivariate generalized models with fixed and mixed effects. MAIN RESULTS AND THE ROLE OF CHANCE: No relationship was found between CGC mtDNA content and oocyte maturity or fertilizability. In contrast, there was a significant link between the content of mtDNA in CGCs surrounding an oocyte and the embryo quality, with significantly higher mtDNA copy numbers being associated with good quality embryos compared with fair or poor quality embryos [interquartile range, respectively, 738 (250-1228) and 342 (159-818); P = 0.006]. However, the indication provided by the quantification of CGC mtDNA concerning the eventuality of good embryo quality was seriously subject to patient effect (AUC = 0.806, 95%CI = 0.719-0.869). The quantity of CGC mtDNA was influenced by BMI and smoking. LARGE SCALE DATA: N/A. LIMITATIONS REASONS FOR CAUTION: The quantification of CGC mtDNA may indicate embryo quality. However, since it is affected by patient specificity, it should be used with caution. It remains to be seen whether this marker could directly predict the implantation capacity of the embryo, which is the main objective in IVF practice. WIDER IMPLICATIONS OF THE FINDINGS: Our study suggests that the quantification of CGC mtDNA may be a novel biomarker of embryo viability. However, patient specificity makes it impossible to establish a general threshold value, valid for all patients. Nevertheless, further studies are needed to determine whether the quantification of CGC mtDNA may, in combination with the morpho-kinetic method, offer an additional criterion for selecting the best embryo for transfer from a given cohort. STUDY FUNDING/COMPETING INTEREST(S): This work was supported by the University Hospital of Angers, the University of Angers, France, and the French national research centres INSERM and the CNRS. There were no competing interests

Development and evaluation of a new methodology for Soft Tissue Artifact compensation in the lower limb

Skin Marker (SM) based motion capture is the most widespread technique used for motion analysis. Yet, the accuracy is often hindered by Soft Tissue Artifact (STA). This is a major issue in clinical gait analysis where kinematic results are used for decision-making. It also has a considerable influence on the results of rigid body and Finite Element (FE) musculoskeletal models that rely on SM-based kinematics to estimate muscle, contact and ligament forces. Current techniques devised to compensate for STA, in particular multi-body optimization methods, often consider simplified joint models. Although joint personalization with anatomical constraints has improved kinematic estimation, these models yet don’t represent a fully reliable solution to the STA problem, thus allowing us to envisage an alternative approach. In this perspective, we propose to develop a conceptual FE-based model of the lower limb for STA compensation and evaluate it for 66 healthy subjects under level walking motor task. Both hip and knee joint kinematics were analyzed, considering both rotational and translational joint motion. Results showed that STA caused underestimation of the hip joint kinematics (up to 2.2°) for all rotational DoF, and overestimation of knee joint kinematics (up to 12°) except in flexion/extension. Joint kinematics, in particular the knee joint, appeared to be sensitive to soft tissue stiffness parameters (rotational and translational mean difference up to 1.5° and 3.4 mm). Analysis of the results using alternative joint representations highlighted the versatility of the proposed modeling approach. This work paves the way for using personalized models to compensate for STA in healthy subjects and different activities