Abstract
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-Solé 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.
Our technical contributions are three-fold.
• We show strong connections between the Belfiore-Solé and the Regev and Stephens-Davidowitz conjectures. Through a closed-form expression of the secrecy ratio of a unimodular lattice and a notion called U-shaped property, we can conclude that if the secrecy ratio of any unimodular lattice is U-shaped, then both the Belfiore-Solé and the Regev and Stephens-Davidowitz conjectures are satisfied. Hence, it suffices to consider the U-shaped-ness for unimodular lattices.
• We provide sufficient conditions to verify the U-shaped-ness for the secrecy ratio of any \(n\)-dimensional unimodular lattices as well as of a subfamily of unimodular lattices, the Construction A lattices obtained from self-dual codes. As a consequence, it confirms the statement that \(\mathbb{Z}^{n}\) achieves the largest flatness factor other than that of those families of unimodular lattices (except for those equivalent to \(\mathbb{Z}^{n}\)). A necessary condition for Construction A unimodular lattices to satisfy the Regev and Stephens-Davidowitz conjecture is also established.
• We consider a binary self-dual code of length \(n\) chosen uniformly at random and we show that the secrecy ratio of the Construction A unimodular lattice obtained from such a random self-dual code is U-shaped, indicating that one can always expect to obtain a strictly smaller flatness factor of a unimodular lattice obtained from a random self-dual code than that of \(\mathbb{Z}^{n}\). Moreover, numerical results are presented to compare flatness factors of the families of unimodular lattices we consider and \(\mathbb{Z}^{n}\).