Counterexample to the $l$-modular Belfiore-Sol\'e Conjecture

TL;DR

This paper disproves a conjecture about the maximum point of the secrecy function in $l$-modular lattices by providing a counterexample and suggests a modified function that aligns better with observed maxima.

Contribution

It provides a counterexample to the $l$-modular Belfiore-Solé Conjecture and proposes a modified secrecy function that fits the observed maxima in certain cases.

Findings

Counterexample to the original conjecture with a 4-modular lattice

Modified secrecy function aligns with maxima in odd 2-modular lattices

Original conjecture does not hold universally for $l$-modular lattices

Abstract

We show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$-modular lattice occurs at $1/\sqrt{l}$ is false, by proving that the 4-modular lattice $C^(4) = \mathbb{Z} \oplus \sqrt{2}\mathbb{Z} \oplus 2\mathbb{Z}$ fails to satisfy this conjecture. We also indicate how the secrecy function must be modified in the $l$-modular case to have a more reasonable chance for it to have a maximum at $1/\sqrt{l}$, and show that the conjecture, modified with this new secrecy function, is true for various odd 2-modular lattices.