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LLL: A Tool for Effective Diophantine Approximation

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The LLL Algorithm

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

Abstract

The purpose of this paper is to survey in a unified setting some of the results in diophantine approximation that the LLL algorithm can make effective in an efficient way. We mostly study the problems of finding good rational approximations to vectors of real and p-adic numbers, and of finding approximate linear relations between vectors of real numbers. We also discuss classical applications of those effective versions, among which Mertens’ conjecture and the effective solution of diophantine equations.

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Acknowledgements

Many thanks to Damien Stehlé for numerous discussions over the last few years about lattices, especially about fine behavior and advanced usage of LLL, and for many corrections, suggestions, and improvements to this survey; thanks also to Nicolas Brisebarre for many useful discussions and for his careful rereading of this survey.

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Authors and Affiliations

  1. INRIA/LORIA, Projet CACAO – Bâtiment A, 615 rue du jardin botanique, Villers-lès-Nancy Cedex, F-54602, France

    Guillaume Hanrot

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  1. Guillaume Hanrot
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Correspondence to Guillaume Hanrot .

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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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Hanrot, G. (2009). LLL: A Tool for Effective Diophantine Approximation. 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_6

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  • DOI: https://doi.org/10.1007/978-3-642-02295-1_6

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