# Improved Upper Bounds on the Hermite and KZ Constants

## TL;DR

This paper introduces improved upper bounds on the Hermite and KZ constants, which are crucial for understanding lattice properties in cryptography and communications, offering sharper estimates than previous results.

## Contribution

The authors develop new linear upper bounds on the Hermite and KZ constants, improving upon recent bounds and enhancing lattice analysis tools.

## Key findings

- Sharper upper bounds on Hermite constant
- Enhanced bounds on KZ constant
- Applications demonstrated with examples

## Abstract

The Korkine-Zolotareff (KZ) reduction is a widely used lattice reduction strategy in communications and cryptography. The Hermite constant, which is a vital constant of lattice, has many applications, such as bounding the length of the shortest nonzero lattice vector and orthogonality defect of lattices. The KZ constant can be used in quantifying some useful properties of KZ reduced matrices. In this paper, we first develop a linear upper bound on the Hermite constant and then use the bound to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the first two authors. Some examples on the applications of the improved upper bounds are also presented.