Méthodes algorithmiques pour les réseaux algébriques

Abstract : This thesis deals with lattices, which are fundamental objects in many fields, such as number theory and cryptography.As a first step, we propose a generalization and an implantation of the Lenstra, Lenstra and Lov'asz algorithm (LLL algorithm) in the simple algebraic setting of lattices over quadratic imaginary and euclidean ring of integers.Then, we present the notions of algebraic lattices and Humbert forms, which are extensions of euclidean lattices and quadratic forms in a large algebraic setting. Introducing these objects leads us to develop and implant modifications of the Plesken and Souvignier algorithm. This algorithm efficiently solves the isometric lattices problem and the automorphism group computation problem for algebraic lattices.Eventually, we analyze in depth the complexity of this two algorithmic problems. We show that they are intimately related to similar problems on graphs. This reduction leads us to express unprecedented complexity bounds.
Document type :
Theses
Complete list of metadatas

Cited literature [68 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01563081
Contributor : Abes Star <>
Submitted on : Friday, January 12, 2018 - 3:23:58 PM
Last modification on : Wednesday, April 3, 2019 - 1:55:43 AM

File

CAMUS_2017_diffusion.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01563081, version 2

Collections

Citation

Thomas Camus. Méthodes algorithmiques pour les réseaux algébriques. Géométrie algébrique [math.AG]. Université Grenoble Alpes, 2017. Français. ⟨NNT : 2017GREAM033⟩. ⟨tel-01563081v2⟩

Share

Metrics

Record views

473

Files downloads

290