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The LLL Algorithm pp 19–69Cite as

Hermite’s Constant and Lattice Algorithms

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Part of the book series: Information Security and Cryptography ((ISC))

Abstract

We introduce lattices and survey the main provable algorithms for solving the shortest vector problem (SVP), either exactly or approximately. In doing so, we emphasize a surprising connection between lattice algorithms and the historical problem of bounding a well-known constant introduced by Hermite in 1850, which is related to sphere packings. For instance, we present Lenstra–Lenstra–Lovász (LLL) as an (efficient) algorithmic version of Hermite’s inequality on Hermite’s constant. Similarly, we present blockwise generalizations of LLL as (more or less tight) algorithmic versions of Mordell’s inequality.

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Author information

Authors and Affiliations

  1. Département d’informatique, INRIA, Ecole normale supérieure, 45 rue d’Ulm, Paris, 75005, France

    Phong Q. Nguyen

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  1. Phong Q. Nguyen
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Editors and Affiliations

  1. Dépt. Mathématiques et d'Informatique, Ecole Normale Supérieure, rue d'Ulm 45, Paris CX 05, 75230, France

    Phong Q. Nguyen

  2. Dept. Informatique, Université de Caen, Caen CX, 14032, France

    Brigitte Vallée

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© 2009 Springer-Verlag Berlin Heidelberg

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Nguyen, P.Q. (2009). Hermite’s Constant and Lattice Algorithms. In: Nguyen, P., Vallée, B. (eds) The LLL Algorithm. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02295-1_2

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  • DOI: https://doi.org/10.1007/978-3-642-02295-1_2

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