SaTC: CORE: Small: Number-theoretic aspects of lattice cryptology

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This award supports research in the mathematical underpinnings of cryptography. Since the late 1970s, widely-used cryptosystems have been developed based on the perceived difficulty of certain mathematical problems. New applications, as well as improvements in attacks on existing cryptosystems, call for a better understanding of these underlying mathematical problems. In particular, the potential to develop quantum computers (and new techniques they would be able to execute) puts many existing cryptosystems, such as the well known Rivest-Shamir-Adleman (RSA) system and Elliptic Curve Cryptography, at possible long-term risk. For this reason, it is important to study post-quantum alternatives, such as lattice-based cryptology (which is the focus of this project). Lattices also offer other potential benefits, including new types of functionalities such as the ability to perform some operations on encrypted data (which would strengthen cloud security).The project centers on the cryptographic strength of special types of lattices (such as ideal lattices or integer lattices) that are used in numerous recent proposed cryptographic schemes. The investigator and collaborators intend to study the distribution of these special lattices amongst larger natural families, to see whether or not they are distinguished by presently-unknown geometric features that might make attacks possible. They plan to use techniques from automorphic forms and algebraic number theory. Tail bounds (such as Banaszczyk's tail bound for Gaussian mass) will be studied for more general regions and functions appropriate to these lattices. From a different, computational perspective, the project will explore lattice-reduction algorithms tailored for these special types of lattices, and analyze how challenging bases of these lattices can be generated.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.