On the Maximum Theta Series over Unimodular Lattices

2024-03-25

arXiv.org

AI Summary

This paper addresses the extremal problem of maximizing the theta series per unit volume over unimodular lattices in ℝⁿ, specifically determining whether the standard integer lattice ℤⁿ achieves the global maximum among all n-dimensional unimodular lattices. Introducing a novel analytic tool—the “secret ratio”—and combining lattice symmetry arguments, analytic number theory, and variational methods, the authors provide the first rigorous proof that ℤⁿ is the unique unimodular lattice attaining this maximum, and that its theta series strictly dominates those of all non-isometric unimodular lattices. The result fully resolves the Regev–Stephens-Davidowitz conjecture and establishes a fundamental connection to the Belfiore–Solé conjecture. Moreover, it furnishes the theoretical foundation for optimal bounds on the flatness factor and smoothing parameter in lattice-based cryptography.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 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.

Problem

• Determining maximum theta series over unimodular lattices
• Relating flatness factor to physical security and cryptography
• Proving Z^n achieves largest theta series among unimodulars

Innovation

• Unimodular lattice theta series analysis
• Theta series ratio comparison technique
• Integer lattice maximizes flatness factor