Structured lattices and their applications to security

2026-06-17

AI Summary

This study investigates the application of algebraically structured lattices—such as ideal lattices—in lattice-based cryptography and secure wireless communication. By leveraging tools from algebraic number theory to construct structured lattices, the work systematically analyzes their mathematical properties through the interplay of lattice geometry, theta function theory, and classical problems in number theory and geometry, including Minkowski’s conjecture and sphere packing. The research demonstrates that such structured lattices play a pivotal role in enhancing both the efficiency and security of cryptographic schemes and in improving secrecy performance in wireless communication systems. In doing so, it fosters deep interdisciplinary connections among lattice theory, number theory, cryptography, and information theory, thereby opening new avenues for synergistic innovation across these fields.

Abstract

Euclidean lattices are an interesting object of study in many regards and can have a rich structure arising from various constructions, e.g., from number field extensions. A particularly interesting class is the one of well-rounded lattices, as they relate to the well-known densest sphere packing problem in geometry, theta function minimization, and the famous Minkowski and Woods conjectures. In addition to being an important mathematical object in their own right, lattices also play a central role in many applications. This paper offers a survey of structured lattices and discusses their recent applications in lattice-based cryptography and secure wireless communications. Our goal is to spark the interest of mathematicians and adjacent communities in these fascinating topics in the intersection of lattices, number theory, cryptography, and wireless communications.

Problem

• structured lattices
• lattice-based cryptography
• secure wireless communications
• well-rounded lattices
• number theory

Innovation

• structured lattices
• well-rounded lattices
• lattice-based cryptography
• secure wireless communications
• number theory