Some Inequalities Related to the Seysen Measure of a Lattice

TL;DR

This paper explores inequalities related to the Seysen measure of a lattice, providing explicit formulas and new bounds that enhance understanding of lattice basis orthogonality and reduction.

Contribution

It introduces new inequalities and explicit expressions for the Seysen measure, advancing theoretical understanding of lattice basis properties.

Findings

Derived explicit formulas for the Seysen measure.

Established new inequalities related to lattice orthogonality.

Enhanced theoretical framework for lattice reduction algorithms.

Abstract

Given a lattice $L$, a basis $B$ of $L$ together with its dual $B^*$, the orthogonality measure $S(B)=\sum_i ||b_i||^2 ||b_i^*||^2$ of $B$ was introduced by M. Seysen in 1993. This measure is at the heart of the Seysen lattice reduction algorithm and is linked with different geometrical properties of the basis. In this paper, we explicit different expressions for this measure as well as new inequalities.