We show that for every $\ell>1$, there is a counterexample to the $\ell$-modular secrecy function conjecture by Oggier, Solé, and Belfiore [ IEEE Trans. Inform. Theory, 62 (2016), pp. 5690--5708]. These counterexamples all satisfy the modified conjecture by Ernvall-Hytönen and Sethuraman [ IEEE Trans. Inform. Theory, 62 (2016), pp. 4514--4522]. Furthermore, we provide a method to prove or disprove the modified conjecture for any given $\ell$-modular lattice rationally equivalent to a suitable amount of copies of $\mathbb{Z}\oplus \sqrt{\ell}\,\mathbb{Z}$ with $\ell \in \{3,5,7,11,23\}$. We also provide a variant of the method for strongly $\ell$-modular lattices when $\ell\in \{6,14,15\}$.

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.

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 4-modular lattice C(4)= ℤ⊕√(2)ℤ ⊕ 2ℤ fails to satisfy this conjecture.

In the first part of this 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 at least 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 Solé, 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 by Belfiore and Solé. We will also get a conditional improvement on some Gaulter's results concerning the conjecture.

The point of this paper is to show that the secrecy function attains its maximum at$y=1$on all known extremal unimodular lattices and on some possibly existing even extremal unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Furthermore, we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice.

In this paper, the probability of Eve the Eavesdropper's correct decision is considered both in the Gaussian and Rayleigh fading wiretap channels when using lattice codes for the transmission. First, it is proved that the secrecy function determining Eve's performance attains its maximum at y = 1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Further, a very simple method to verify or disprove the conjecture on any given unimodular lattice is given. Second, preliminary analysis on the behavior of Eve's probability of correct decision in the fast fading wiretap channel is provided. More specifically, we compute the truncated inverse norm power sum factors in Eve's probability expression. The analysis reveals a performance-secrecy-complexity tradeoff: relaxing on the legitimate user's performance can significantly increase the security of transmission. The confusion experienced by the eavesdropper may be further increased by using skewed lattices, but at the cost of increased complexity.