# Low-Dimensional Lattice Basis Reduction Revisited

• Download: Full Version in ACM Transactions on Algorithms (2009). Extended Abstract published at ANTS ( .gz version)
• Authors: Phong Q. Nguyen and Damien Stehlé
• Algorithmic Number Theory -- Proceedings of ANTS-VI (June 13--18, 2004, Burlington, U.S.A.), D. Buell (Ed.), Lecture Notes in Computer Science, Springer-Verlag.
• 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 VoronoÃ¯ 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.

• Bibtex: @article{NgSt09, title={Low-Dimensional Lattice Basis Reduction Revisited}, author={P. Q. Nguyen and D. Stehl\'e}, journal={ACM Transactions on Algorithms}, year=2009, note={To appear} } and @inproceedings{NgSt04, AUTHOR = {Phong Q. Nguyen and Damien Stehl{\'e}}, TITLE = {Low-dimensional lattice basis reduction revisited}, PAGES = {338--357}, booktitle = "Proceedings of the 6th International Algorithmic Number Theory Symposium, (ANTS-VI)", publisher = "Springer", series = "LNCS", volume = {3076}, YEAR = {2004} }