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Short Bases of Lattices over Number Fields

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Algorithmic Number Theory (ANTS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6197))

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Abstract

Lattices over number fields arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of ℤ-lattices, the choice of a nice, “short” (pseudo)-basis is important in many applications. In this article, we provide the first algorithm that computes such a “short” (pseudo)-basis. We utilize the LLL algorithm for ℤ-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to find a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a field invariant. As it runs in polynomial time, this provides an effective variant of Minkowski’s second theorem for lattices over number fields.

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Author information

Authors and Affiliations

  1. Magma Computer Algebra Group, School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia

    Claus Fieker & Damien Stehlé

  2. CNRS and Macquarie University,  

    Damien Stehlé

Authors
  1. Claus Fieker
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  2. Damien Stehlé
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Editor information

Editors and Affiliations

  1. LIP/ENS-Lyon, 46, allée d’Italie, 69364, Lyon, Cedex 07, France

    Guillaume Hanrot

  2. Laboratoire d’Informatique (CNRS/UMR 7650), École polytechnique, 91128, Palaiseau Cedex, France

    François Morain

  3. INRIA Nancy, project CARAMEL, 615 rue du jardin botanique, 54602, Villers-lès-Nancy Cedex,, France

    Emmanuel Thomé

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Fieker, C., Stehlé, D. (2010). Short Bases of Lattices over Number Fields. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_15

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  • DOI: https://doi.org/10.1007/978-3-642-14518-6_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14517-9

  • Online ISBN: 978-3-642-14518-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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