Les fonctions élémentaires sont utilisées dans de nombreux codes de calcul haute performance. Bien que les bibliothèques mathématiques (libm) auxquelles font appel ces codes proposent en général plusieurs variétés d'une même fonction, celles-ci sont figées lors de leur implémentation. Cette caractéristique représente un frein à la performance des programmes qui les utilisent car elles sont conçues pour être polyvalentes au détriment d'optimisations spécifiques. De plus, la duplication de modèles partagés rend la maintenance de ces libms plus difficile et sujette à l'introduction de bugs. Un défi actuel est de proposer des "méta-outils" visant la génération automatique de code performant pour l'évaluation des fonctions élémentaires. Ces outils doivent permettre la réutilisation d'algorithmes efficaces et génériques pour différentes variétés de fonctions ou architectures matérielles. Il devient alors possible de générer des libms optimisées pour des besoins très spécifiques avec du code générateur factorisé, qui facilite sa maintenance. Dans un premier temps, nous proposons un algorithme original permettant de générer des tables sans erreur pour les fonctions trigonométriques et hyperboliques. Puis nous étudions les performances de schémas d'évaluation polynomiale vectorisés, premier pas vers la génération de fonctions vectorisées efficaces. Enfin, nous proposons une méta-implémentation d'un logarithme vectorisé, factorisant la génération de code pour différents formats et architectures. Ces contributions sont compétitives comparées à d'autres solutions, justifiant le développement de tels méta-codes. / Elementary mathematical functions are pervasive in many high performance computing programs. However, although the mathematical libraries (libms), on which these programs rely, generally provide several flavors of the same function, these are fixed at implementation time. Hence this monolithic characteristic of libms is an obstacle for the performance of programs relying on them, because they are designed to be versatile at the expense of specific optimizations. Moreover, the duplication of shared patterns in the source code makes maintaining such code bases more error prone and difficult. A current challenge is to propose "meta-tools" targeting automated high performance code generation for the evaluation of elementary functions. These tools must allow reuse of generic and efficient algorithms for different flavours of functions or hardware architectures. Then, it becomes possible to generate optimized tailored libms with factorized generative code, which eases its maintenance. First, we propose an novel algorithm that allows to generate lookup tables that remove rounding errors for trigonometric and hyperbolic functions. The, we study the performance of vectorized polynomial evaluation schemes, a first step towards the generation of efficient vectorized elementary functions. Finally, we develop a meta-implementation of a vectorized logarithm, which factors code generation for different formats and architectures. Our contributions are shown competitive compared to free or commercial solutions, which is a strong incentive to push for developing this new paradigm.

The number of opportunities for cooperative computation has exponentially been increasing with growing interaction via Internet technologies. These computations could occur between trusted partners, between partially trusted partners, or even between competitors. Most of the time, the communicating parties may not want to disclose their private data to the other principal while taking the advantage of collaboration, hence concentrating on the results rather than private and perhaps useless data values. For performing such computations, one party must know inputs from all the participants / however if none of the parties can be trusted enough to know all the inputs, privacy will become a primary concern. Hence the techniques for Secure Multiparty Computation (SMC) are quite relevant and practical to overcome such kind of privacy gaps. The subject of SMC has evolved from earlier solutions of combinational logic circuits to the recent proposals of anonymity-enabled computation. In this thesis, we put together the significant research that has been carried out on SMC. We demonstrate the concept by concentrating on a specific technique called Oblivious Polynomial Evaluation (OPE) together with concrete examples. We put critical issues, challenges and the level of adaptation achieved before the researchers. We also provide some future research opportunities based on the literature survey.

Understanding the structure and complexity of a polynomial family is a fundamental problem of arithmetic circuit complexity. There are various approaches like studying the lower bounds, which deals with nding the smallest circuit required to compute a polynomial, studying the orbit and stabilizer of a polynomial with respect to an invertible transformation etc to do this. We have a rich understanding of some of the well known polynomial families like determinant, permanent, IMM etc. In this thesis we study some of the structural properties of the polyno-mial family called the Nisan-Wigderson polynomial family. This polynomial family is inspired from a well known combinatorial design called Nisan-Wigderson design and is recently used to prove strong lower bounds on some restricted classes of arithmetic circuits ([KSS14],[KLSS14], [KST16]). But unlike determinant, permanent, IMM etc, our understanding of the Nisan-Wigderson polynomial family is inadequate. For example we do not know if this polynomial family is in VP or VNP complete or VNP-intermediate assuming VP 6= VNP, nor do we have an understanding of the complexity of its equivalence test. We hope that the knowledge of some of the inherent properties of Nisan-Wigderson polynomial like group of symmetries and Lie algebra would provide us some insights in this regard. A matrix A 2 GLn(F) is called a symmetry of an n-variate polynomial f if f(A x) = f(x): The set of symmetries of f forms a subgroup of GLn(F), which is also known as group of symmetries of f, denoted Gf . A vector space is attached to Gf to get the complete understanding of the symmetries of f. This vector space is known as the Lie algebra of group of symmetries of f (or Lie algebra of f), represented as gf . Lie algebra of f contributes some elements of Gf , known as continuous symmetries of f. Lie algebra has also been instrumental in designing e cient randomized equivalence tests for some polynomial families like determinant, permanent, IMM etc ([Kay12], [KNST17]). In this work we completely characterize the Lie algebra of the Nisan-Wigderson polynomial family. We show that gNW contains diagonal matrices of a speci c type. The knowledge of gNW not only helps us to completely gure out the continuous symmetries of the Nisan-Wigderson polynomial family, but also gives some crucial insights into the other symmetries of Nisan-Wigderson polynomial (i.e. the discrete symmetries). Thereafter using the Hessian matrix of the Nisan-Wigderson polynomial and the concept of evaluation dimension, we are able to almost completely identify the structure of GNW . In particular we prove that any A 2 GNW is a product of diagonal and permutation matrices of certain kind that we call block-permuted permutation matrix. Finally, we give explicit examples of nontrivial block-permuted permutation matrices using the automorphisms of nite eld that establishes the richness of the discrete symmetries of the Nisan-Wigderson polynomial family.

The aim of master's thesis is to get familiarized with the problems of measurement and analysis of seismic waves. Theoretical part deals with the description of seismic waves, especially their types, sources and properties. Attention was afterwards focused on the measurement systems of these waves, emphasis was placed on their principles and advantages. The practical part discusses methods of noise reduction and highlighting of significant events in measured data. At the end, individual methods are implemented into user-friendly graphical interface.