On the Maximum Theta Series over Unimodular Lattices

The theta series of a lattice has been extensively studied in the literature and is closely related to a critical quantity widely used in the fields of physical layer security and cryptography, known as the flatness factor, or equivalently, the smoothing parameter of a lattice. Both fields raise the fundamental question of determining the (globally) maximum theta series over a particular set of volume-one lattices, namely, the stable lattices. In this work, we present a property of unimodular lattices, a subfamily of stable lattices, to verify that the integer lattice $\mathbb{Z}^{n}$ achieves the largest possible value of theta series over the set of unimodular lattices. Such a result moves towards proving the conjecture recently stated by Regev and Stephens-Davidowitz: any unimodular lattice, except for those lattices isomorphic to $\mathbb{Z}^{n}$, has a strictly smaller theta series than that of $\mathbb{Z}^{n}$. Our techniques are mainly based on studying the ratio of the theta series of a unimodular lattice to the theta series of $\mathbb{Z}^n$, called the secrecy ratio. We relate the Regev and Stephens-Davidowitz conjecture with another conjecture for unimodular lattices, known in the literature as the Belfiore-Sol{\'e} conjecture. The latter assumes that the secrecy ratio of any unimodular lattice has a symmetry point, which is exactly where the global minimum of the secrecy ratio is achieved.