LEAPS-MPS: A study on the Representation Theory of Hermitian Lattices

The representation theory of lattices, often studied through their associated quadratic and Hermitian forms, underpins significant advances in both mathematics and computer science. Building on the classical works by Fermat, Legendre, Lagrange, Gauss, and other seminal figures, early research focused on solving quadratic equations and expressing natural numbers as sums of squares. Over centuries, this theory has expanded to more abstract settings such as number fields and function fields, and to examine whether there are transformations that map one lattice to another while preserving essential algebraic and geometric structures. In recent years, problems involving lattice representations, especially the Lattice Isomorphism Problem (LIP), have gained new significance due to their critical role in securing information against future quantum computing threats. For example, the HAWK digital signature scheme depends on the computational hardness of the LIP for certain structured lattices, underscoring the need for a deep, rigorous mathematical foundation. This project will investigate the representation theory of lattices from both theoretical and practical perspectives, bridging classical number theory with contemporary cryptographic challenges. Complementing its research component, the educational component includes a structured mentoring program designed to bring more students into the mathematical sciences by engaging students in computational number theory and lattice-based cryptography. Together, the research and mentoring activities will support emerging cybersecurity initiatives in San Antonio, Texas, a growing hub for cyber defense, where state and institutional efforts contribute to regional and national information security. At the technical level, this project will deepen theoretical understanding of Hermitian lattices and strengthen the cryptanalytic foundations of lattice-based post-quantum cryptographic systems. It will advance the theory of Hermitian lattices over number fields through two interconnected objectives: (1) computing the exact values of the g-invariant for unary and binary Hermitian lattices over imaginary quadratic fields, thereby generalizing the classical Pythagoras number to higher-dimensional structured lattices; and (2) cryptanalyzing the Module Lattice Isomorphism Problem (module-LIP) over CM fields from both arithmetic and geometric perspectives, employing genus theory and the s-hull of Hermitian lattices, respectively. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.