My research interests (as well as the cryptography) cover the following areas: smooth curves of genus 1 and 2, abelian varieties, Shimura curves, and integral binary and ternary quadratic forms.  

   In my dissertation, I focus on studying the nature of (smooth) genus 2 curves. To do this, my method focuses on a special associated integral quadratic form, which encodes many useful properties of genus 2 curves, and this form makes the nature of curves clear. This form is the key ingredient of my Ph.D.  thesis as it allows for the translation of geometric problems into arithmetic ones. It was introduced by Kani (1994) upon observing that every curve C comes equipped with a canonically defined positive definite quadratic form. This result can be used to algebraically define the (usual) Humbert invariant (1899) and Humbert surfaces, and so Kani called the refined Humbert invariant the form he had introduced.