Algorithmique de la réduction de réseaux et application à la recherche de pires cas pour l'arrondi de fonctions mathématiques
Damien Stehlé · Publications Et Travaux Academiques de Lorraine (Universite de Lorraine) · 2005
Euclidean lattices are a 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 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.