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Low-Dimensional Lattice Basis Reduction Revisited

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Low-Dimensional Lattice Basis Reduction Revisited

Auteurs : Phong Q. Nguyen [France] ; Damien Stehlé [France]

Source :

Lecture Notes in Computer Science [ 0302-9743 ]

RBID : ISTEX:1072C83244519A5AB766FC3862FBA017026012B0

Descripteurs français

Pascal (Inist)
- Algorithme glouton, Algorithme optimal, Algorithme rapide, Arithmétique, Complexité algorithme, Cubique, Diagramme Voronoï, Géométrie algorithmique, Modèle 2 dimensions, Modèle 3 dimensions, Méthode polynomiale, Métrique Minkowski, Problème NP difficile, Processus Gauss, Théorie euclidienne, Treillis.

English descriptors

KwdEn :
- Algorithm complexity, Arithmetics, Computational geometry, Cubics, Euclidean theory, Fast algorithm, Gaussian process, Greedy algorithm, Lattice, Minkowski metric, NP hard problem, Optimal algorithm, Polynomial method, Three dimensional model, Two dimensional model, Voronoï diagram.

Abstract

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.

Url:

https://api.istex.fr/ark:/67375/HCB-30MF799R-1/fulltext.pdf

DOI: 10.1007/978-3-540-24847-7_26

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Low-Dimensional Lattice Basis Reduction Revisited

Low-Dimensional Lattice Basis Reduction Revisited

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