Algorithmique de la réduction de réseaux et
application à la recherche de pires cas pour l'arrondi de
fonctions mathématiques
Identifieur interne : 005A83 ( Main/Merge ); précédent : 005A82; suivant : 005A84
Algorithmique de la réduction de réseaux et
application à la recherche de pires cas pour l'arrondi de
fonctions mathématiques
Auteurs : Damien Stehlé [France]Source :
Descripteurs français
- mix :
Abstract
Euclidean lattices are a particularly powerful tool for several
algorithmic topics, among which are cryptography and algorithmic
number theory. The contributions of this thesis are twofold:
we improve lattice basis reduction algorithms, and we introduce a new application of lattice reduction, in computer arithmetic. Concerning lattices, we consider both small dimensions (in dimension one, where the problem degenerates to a gcd
calculation, and in dimensions 2 to 4), and arbitrary dimensions, for
which we improve the classical LLL algorithm. Concerning the application, we make use of Coppersmith's method for computing
the small roots of multivariate modular polynomials, in order to find the worst cases for the rounding of mathematical functions, when the function, the rounding mode and the precision are fixed. We also generalise our technique to find input numbers that are simultaneously bad for two functions. These two
methods are expensive pre-computations, but once performed, they help speeding up the implementations of elementary mathematical functions in fixed precision, for example in double precision.
Most of the algorithms described in this thesis have been validated
experimentally. These implementations are available at the url
http://www.loria.fr/~stehle.
algorithmic topics, among which are cryptography and algorithmic
number theory. The contributions of this thesis are twofold:
we improve lattice basis reduction algorithms, and we introduce a new application of lattice reduction, in computer arithmetic. Concerning lattices, we consider both small dimensions (in dimension one, where the problem degenerates to a gcd
calculation, and in dimensions 2 to 4), and arbitrary dimensions, for
which we improve the classical LLL algorithm. Concerning the application, we make use of Coppersmith's method for computing
the small roots of multivariate modular polynomials, in order to find the worst cases for the rounding of mathematical functions, when the function, the rounding mode and the precision are fixed. We also generalise our technique to find input numbers that are simultaneously bad for two functions. These two
methods are expensive pre-computations, but once performed, they help speeding up the implementations of elementary mathematical functions in fixed precision, for example in double precision.
Most of the algorithms described in this thesis have been validated
experimentally. These implementations are available at the url
http://www.loria.fr/~stehle.
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arithmétique flottante</term><term>
méthode de Coppersmith</term><term>Réseaux euclidiens</term><term>algorithme LLL</term><term>analyse de complexité</term><term>analyse de complexité.</term><term>calcul de pgcd</term><term>dilemme du fabricant de tables</term><term>implantation de fonctions mathématiques
élémentaires</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Euclidean lattices are a particularly powerful tool for several
algorithmic topics, among which are cryptography and algorithmic
number theory. The contributions of this thesis are twofold:
we improve lattice basis reduction algorithms, and we introduce a new application of lattice reduction, in computer arithmetic. Concerning lattices, we consider both small dimensions (in dimension one, where the problem degenerates to a gcd
calculation, and in dimensions 2 to 4), and arbitrary dimensions, for
which we improve the classical LLL algorithm. Concerning the application, we make use of Coppersmith's method for computing
the small roots of multivariate modular polynomials, in order to find the worst cases for the rounding of mathematical functions, when the function, the rounding mode and the precision are fixed. We also generalise our technique to find input numbers that are simultaneously bad for two functions. These two
methods are expensive pre-computations, but once performed, they help speeding up the implementations of elementary mathematical functions in fixed precision, for example in double precision.
Most of the algorithms described in this thesis have been validated
experimentally. These implementations are available at the url
http://www.loria.fr/~stehle.</div></front></TEI></record>Pour manipuler ce document sous Unix (Dilib)
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