Counterexample to the Generalized Belfiore–Solé Secrecy Function Conjecture for $l$ -Modular Lattices

In this paper, we show that the secrecy function conjecture that states that the maximum of the secrecy function of an l-modular lattice occurs at 1/√l is false, by proving that the four-modular lattice C
⁽⁴⁾
= ℤ ⊕√2ℤ ⊕ 2ℤ fails to satisfy this conjecture. After this, we indicate how the secrecy function must be modified in the l-modular case to provide a more meaningful comparison for l-modular lattices, and show that this new secrecy function indeed has a maximum at 1/√l for various two-modular lattices and for C
⁽⁴⁾
. We conjecture that this must hold true for all l-modular lattices.