# On the KZ Reduction

## TL;DR

This paper analyzes properties of KZ reduced matrices, proposes tighter bounds on their characteristics, and introduces a faster, more reliable KZ reduction algorithm especially effective for ill-conditioned matrices.

## Contribution

The paper provides new theoretical bounds on KZ reduced matrices and develops a novel, more efficient KZ reduction algorithm with improved numerical stability.

## Key findings

- New upper bounds on Hermit and KZ constants.
- Polynomially and exponentially smaller bounds on matrix properties.
- Faster, more reliable KZ reduction algorithm for ill-conditioned matrices.

## Abstract

The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around $\frac{7}{8}$ times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is {\em polynomially} smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even {\em polynomially and exponentially} smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang {\em et al.} Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang {\em et al.}, especially when the basis matrix is ill conditioned.