Counterexample to the Generalized Belfiore-Sole 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=pl is false, by proving that the 4-modular lattice C(4) = Z ⊕ p2Z ⊕ 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=pl for various 2-modular lattices and for C(4). We conjecture that this must hold true for all l-modular lattices."