Algorithmique de la réduction de réseaux et
application à la recherche de pires cas pour l'arrondi de
fonctions mathématiques

Damien Stehlé 1
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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.
Document type :
Theses
Complete list of metadatas

https://tel.archives-ouvertes.fr/tel-00011150
Contributor : Damien Stehle <>
Submitted on : Monday, December 5, 2005 - 2:44:26 PM
Last modification on : Friday, May 10, 2019 - 12:23:09 PM
Long-term archiving on : Friday, April 2, 2010 - 10:37:01 PM

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  • HAL Id : tel-00011150, version 1

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Damien Stehlé. Algorithmique de la réduction de réseaux et
application à la recherche de pires cas pour l'arrondi de
fonctions mathématiques. Génie logiciel [cs.SE]. Université Henri Poincaré - Nancy I, 2005. Français. ⟨tel-00011150⟩

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