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Low-Dimensional Lattice Basis Reduction Revisited

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Algorithmic Number Theory (ANTS 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3076))

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Abstract

Most of the interesting algorithmic problems in the geometry of numbers are NP-hard as the lattice dimension increases. This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which is arguably the most natural lattice basis reduction algorithm, because it is a straightforward generalization of the well-known two-dimensional Gaussian algorithm. Our results are two-fold. From a mathematical point of view, we show that up to dimension four, the output of the greedy algorithm is optimal: the output basis reaches all the successive minima of the lattice. However, as soon as the lattice dimension is strictly higher than four, the output basis may not even reach the first minimum. More importantly, from a computational point of view, we show that up to dimension four, the bit-complexity of the greedy algorithm is quadratic without fast integer arithmetic: this allows to compute various lattice problems (e.g. computing a Minkowski-reduced basis and a closest vector) in quadratic time, without fast integer arithmetic, up to dimension four, while all other algorithms known for such problems have a bit-complexity which is at least cubic. This was already proved by Semaev up to dimension three using rather technical means, but it was previously unknown whether or not the algorithm was still polynomial in dimension four. Our analysis, based on geometric properties of low-dimensional lattices and in particular Voronoï cells, arguably simplifies Semaev’s analysis in dimensions two and three, unifies the cases of dimensions two, three and four, but breaks down in dimension five.

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Author information

Authors and Affiliations

  1. Département d’informatique, CNRS/École normale supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Phong Q. Nguyen

  2. LORIA/INRIA Lorraine, 615 rue du J. botanique, 54602, Villers-lès-Nancy, France

    Damien Stehlé

Authors
  1. Phong Q. Nguyen
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  2. Damien Stehlé
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, University of South Carolina , SC 29208, Columbia, USA

    Duncan Buell

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Nguyen, P.Q., Stehlé, D. (2004). Low-Dimensional Lattice Basis Reduction Revisited. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_26

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  • DOI: https://doi.org/10.1007/978-3-540-24847-7_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22156-2

  • Online ISBN: 978-3-540-24847-7

  • eBook Packages: Springer Book Archive

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