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/\sqrt {l} is false, by proving that the four-modular lattice C^{(4)}= \mathbb {Z}\oplus \sqrt {2} \mathbb {Z}\oplus 2 \mathbb {Z} 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 for various two-modular lattices and for C^(4) . We conjecture that this must hold true for all l -modular lattices. [ABSTRACT FROM PUBLISHER]