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 a 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.

翻译：格的Theta级数在文献中已被广泛研究，且与物理层安全及密码学领域广泛使用的关键量——平坦度因子（即格的平滑参数）密切相关。这两个领域提出了一个基础性问题：确定特定体积为一的格（即稳定格）集合上的（全局）最大Theta级数。本文证明了稳定格子类中的幺模格子类具有一个性质：整数格$\mathbb{Z}^{n}$在幺模格集合上实现了Theta级数的最大可能值。这一结果推进了Regev和Stephens-Davidowitz近期提出的猜想：任何与$\mathbb{Z}^{n}$不同构的幺模格，其Theta级数严格小于$\mathbb{Z}^{n}$的Theta级数。我们的方法主要基于研究幺模格Theta级数与$\mathbb{Z}^n$Theta级数之比（称为保密比）。我们将Regev-Stephens-Davidowitz猜想与文献中已知的另一个关于幺模格的猜想——Belfiore-Solé猜想——相关联。后者假设任意幺模格的保密比存在一个对称点，该点恰好是保密比全局最小值所在位置。