On the Maximum Flatness Factor over Unimodular Lattices

Abstract

The theta series of a lattice is a power series that characterizes the number of lattice vectors at certain norms. It is closely related to a critical quantity widely used in 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. This result advances the resolution of the open question, suggesting that any unimodular lattice, except those 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 theta series ratio. Consequently, all the findings concerning the theta series of a lattice extend to its flatness factor. Therefore, our results have applications to the Gaussian wiretap channel, the reverse Minkowski theorem, and lattice-based cryptography.