On the coordinates of minimal vectors in a Minkowski-reduced basis

Finding the shortest non-zero vectors in a lattice is a computationally hard problem (NP-hard in general dimensions), making results in low dimensions particularly important in lattice reduction theory. This paper focuses on the coordinates of minimal lattice vectors when expressed in a Minkowski-re...

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Bibliographic Details
Published in:Extracta mathematicae Vol. 40; no. 1; pp. 27 - 41
Main Author: Horváth, Ákos G.
Format: Journal Article
Language:English
Published: University of Extremadura 2025
Subjects:
admissible centering
Dirichlet
Dirichlet-Voronoi cell
minimum vector of a lattice
Minkowski
Minkowski-reduction
positive quadratic forms
reduction
Voronoi cell
ISSN:0213-8743, 2605-5686
Online Access:Get full text
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Summary:Finding the shortest non-zero vectors in a lattice is a computationally hard problem (NP-hard in general dimensions), making results in low dimensions particularly important in lattice reduction theory. This paper focuses on the coordinates of minimal lattice vectors when expressed in a Minkowski-reduced basis. By applying Ryskov’s findings on admissible centerings and Tammela’s work characterizing Minkowski-reduced forms via a finite set of inequalities (up to dimension 6), we demonstrate sharp bounds on the absolute values of these coordinates. Specifically, we show that for dimensions n ≤ 6, the absolute values of the coordinates of any minimal vector with respect to a Minkowski-reduced basis are bounded by 1 (for n = 2, 3), 2 (for n = 4, 5), and 3 (for n = 6). This refines bounds implicitly available from Tammela’s results by combining geometric arguments from lattice theory, admissible centering theory, and reduction theory. Finding the shortest non-zero vectors in a lattice is a computationally hard problem (NP-hard in general dimensions), making results in low dimensions particularly important in lattice reduction theory. This paper focuses on the coordinates of minimal lattice vectors when expressed in a Minkowski-reduced basis. By applying Ryskov’s findings on admissible centerings and Tammela’s work characterizing Minkowski-reduced forms via a finite set of inequalities (up to dimension 6), we demonstrate sharp bounds on the absolute values of these coordinates. Specifically, we show that for dimensions n ≤ 6, the absolute values of the coordinates of any minimal vector with respect to a Minkowski-reduced basis are bounded by 1 (for n = 2, 3), 2 (for n = 4, 5), and 3 (for n = 6). This refines bounds implicitly available from Tammela’s results by combining geometric arguments from lattice theory, admissible centering theory, and reduction theory.
ISSN:0213-8743
2605-5686
DOI:10.17398/2605-5686.40.1.27