Some results related to the conjecture by Belfiore and Sol\'e

TL;DR

This paper explores the relationship between kissing number and secrecy gain in unimodular lattices, providing new theoretical results, conditional calculations, and examples under the Belfiore and Solé conjecture.

Contribution

It establishes new links between lattice properties and secrecy gain, proves results for unimodular lattices, and offers conditional insights assuming the Belfiore and Solé conjecture.

Findings

Secrecy gain increases as the number of shortest vectors decreases in certain lattices.

Existence of lattices with same shortest vector length and kissing number but different secrecy gains.

Conditional results on lattice properties assuming the Belfiore and Solé conjecture.

Abstract

In the first part of the paper, we consider the relation between kissing number and the secrecy gain. We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. We will also consider the situations with shorter vectors. Furthermore, assuming the conjecture by Belfiore and Sol\'e, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. We will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains. Finally, we consider some cases of a question by Elkies by providing an answer for a special class of lattices assuming the conjecture of Belfiore and Sol\'e. We will also get a conditional improvement on some Gaulter's results concerning the conjecture.