Abstract:
The secrecy function conjecture of Belfiore and Solé for unimodular lattices, which arose from wireless communications using lattice coset coding on gaussian wiretap channels, claims that for an $n$-dimensional unimodular lattice $\Gamma$, the quotient $\vartheta_\Gamma/\vartheta_{\mathbb Z^n}$ has, on the positive imaginary axis, a unique global minimum at the symmetry point $i$. Similarly, for integers $\ell\geqslant2$, a natural extension of this conjecture would be to claim that for an $\ell$-modular lattice, the quotient $\vartheta_\Gamma/\vartheta_{(\ell^{1/4}\mathbb Z)^n}$ has, on the positive imaginary axis, a unique global minimum at the symmetry point $i/\sqrt\ell$. Alas, Ernvall-Hytönen and Sethuraman showed that this is not the case as the behaviour of this quotient for the 4-modular lattice $\mathbb Z\oplus\sqrt2\,\mathbb Z\oplus2\mathbb Z$ is the opposite to the conjectured behaviour: there is a unique global maximum at the point $y=1/2$.
In this talk, we discuss the secrecy function conjecture. In particular, we describe an infinite family of $\ell$-modular counterexamples with $\ell\geqslant2$, constructed from direct products of dilates of $\mathbb Z$, for which we show that the behaviour is again opposite to what the extended secrecy function conjecture requires. One of the key ideas is using convexity properties of classical $\vartheta$-functions. The results described are joint work with A.-M. Ernvall-Hytönen.
