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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.
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Submitted on : Friday, January 12, 2018 - 3:23:58 PM
Last modification on : Tuesday, November 3, 2020 - 3:19:41 PM


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