Visible to the public EAGER: Number Theory and CryptograpghyConflict Detection Enabled

Project Details

Performance Period

Aug 01, 2016 - Jul 31, 2018


University of Colorado at Boulder

Award Number

This award supports the principal investigator's research in number theory and its cryptographic applications. Number theory serves as the basis for modern cryptography and internet security. The underlying mathematical theories, of elliptic curves and integer factorization, have been studied for centuries. This project includes components of basic research, concerning the relationship between geometry and number theory, as well as foundational research in cryptographic applications, with an eye toward the advent of quantum computers. In particular, the cryptographic community is currently searching for cryptosystems secure against quantum computation (a property not shared by the systems currently in use), and the PI will investigate one of the prime candidates from a number-theoretical perspective. The broader impacts of the project involve the mentoring of students and women in mathematics through early research collaboration. This will take place through summer research experiences for undergraduate and early graduate students, and a mentoring research collaboration conference for women. The PI will also reach out the general public through art and software.

The scientific component of the project has three parts. In the first, the PI will evaluate the security of the lattice-based cryptographic Ring Learning with Errors problem by developing and addressing questions concerning the lattice properties of number fields and their subfields and ideals. She will determine the extent of known number-theoretical attacks on the problem, as it relates to known parameters of provable security and suggested parameters for implementation. The second concerns Bianchi groups, as providing a geometric view of imaginary quadratic fields. The PI will develop connections from orbits of the Bianchi group to class groups, continued fractions, Apollonian circle packings, quadratic forms, thin groups, abelian sandpiles and elliptic curves. Finally, the PI will also build on her prior work defining elliptic nets. Elliptic nets provide a different computational framework for elliptic curves and have given rise to new algorithms for pairing-based cryptography. The PI will extend this work to the context of abelian varieties, and investigate applications.