From Unit Groups to Lattice Packings: New Directions in Number Theory

This project will advance mathematics and cryptography by solving fundamental problems in number theory and developing new encryption methods for secure online communication. As quantum computers grow more powerful, they threaten to break many common cryptosystems that are currently used to protect information on the internet. To achieve security, modern cryptography relies on hard mathematical problems derived from objects called lattices. The new encryption methods in this project will employ the recently introduced class of simultaneous approximation lattices, which offer improved efficiency relative to generic lattices. Beyond cybersecurity, the research generated by this project will improve the performance of certain number theoretic algorithms used across the mathematical sciences and contribute to theoretical models in materials science and information theory. The project will also help train the next generation of STEM professionals by engaging undergraduate and doctoral students in theoretical research and programming. In addition to developing a new lattice-based cryptosystem alongside undergraduate and graduate students, this project will resolve several specific problems in algebraic and algorithmic number theory. First, the research will compute explicit full-rank unit subgroups in Galois monogenic number fields. Such subgroups allow for faster computation of ideal class groups and permit the "log-unit" approach to finding short vectors in ideal lattices. Second, the project will analyze the dynamics of Markoff triples over finite fields, aiming to resolve the McCullough-Wanderley and Baragar conjectures. Third, the project will prove new upper bounds on the p-torsion rank of ideal class groups in p-extensions of the rational number field and potentially beyond. Finally, the research will uncover the geometric relationship between the periodic packing density of a compact symmetric solid and the lattice packing density of its higher-dimensional cylindrical lifts. 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.