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

Ákos G. Horváth

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.

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

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.

Paper Structure

This paper contains 5 sections, 5 theorems, 8 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Minkowski-reduced basis
  3. The admissible centering of a lattice
  4. Characterization of the Minkowski-reduced forms by a finite system of inequalities.
  5. The theorem

Key Result

Theorem 1

Let $L$ be an $n$-dimensional lattice, where $n\leq 6$, $m=\sum\limits_{i=1}^nx_ie_i$ be an arbitrary minimal vector of $L$, where $\{e_1, \ldots,e_n\}$ is a Minkowski-reduced basis of the lattice. Then, for an arbitrary index $i$, the following inequalities are satisfied:

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 4 more