TWC: Small: Automorphic Forms and Harmonic Analysis Methods in Lattice Cryptology

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Cryptography is a fundamental part of cybersecurity, both in designing secure applications as well as understanding how truly secure they really are. Traditionally, the mathematical underpinnings of cryptosystems were based on difficult problems involving whole numbers (most famously, the apparent difficulty of factoring a product of two unknown primes back into those prime factors). More recently, several completely new types of cryptography have been proposed using the mathematical properties of lattices. For example, homomorphic encryption offers the possibility of manipulating encrypted data without revealing its contents. The research funded by the project seeks to apply new mathematical techniques to understand difficult properties of lattices and hence of these proposed systems -- properties that are seen as crucial for making them practically fast, in particular. The projects in the proposal involve using tools from automorphic forms (such as Borel-Harish-Chandra reduction theory, Eisenstein series, and change of base field) to examine concrete questions about short vectors in certain families of lattices. It also includes a study of cryptosystems based on nonabelian generalizations of lattices, and the geometry of BKZ-reduced bases in Coppersmith's method (including the Boneh-Durfee attack on small exponent RSA).