On the Secrecy Gain of $\ell$-Modular Lattices

Abstract

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 $\ell\in \{6,14,15\}$.

Locations

• SIAM Journal on Discrete Mathematics - View
• arXiv (Cornell University) - View - PDF