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

TL;DR

This paper establishes that in six-dimensional Minkowski-reduced bases, the coordinates of minimal vectors are bounded by three, advancing understanding of lattice reduction and minimal vector properties.

Contribution

The paper proves a new bound on the coordinates of minimal vectors in six-dimensional Minkowski-reduced bases, refining previous results with combined geometric and theoretical methods.

Findings

Coordinates of minimal vectors are ≤ 3 in six-dimensional Minkowski-reduced bases

Improved bounds enhance lattice reduction techniques

Combines geometric and theoretical approaches for sharper results

Abstract

Finding the shortest vectors in a lattice is an NP-hard problem, so low-dimensional results also play an essential role in lattice reduction theory. Using Ryskov's result for the admissible centerings and Tammela's result for determining the Minkowski-reduced form, we prove that the absolute values of the coordinates of a minimal vector on a six-dimensional Minkowski-reduced basis are less than or equal to three. To sharpen P. Tammela's work, we combine some lattice geometry arguments with the aforementioned theoretical results.