Research Interests Juliana v. Belding Number Theoretic Algorithms for Elliptic Curves with Cryptographic Applications

An elliptic curve over a field K is given by an equation of the form y2 = x3 + Ax+B. There is a natural way to add any two points on the curve to get a third point, and therefore the set of points of the curve with coordinates in K form a group, denoted E(K). Elliptic curves have long fascinated mathematicians, as they can be approached from many angles, including complex analysis, number theory and algebraic geometry. In the past twenty years, elliptic curves have gained even more attention: elliptic curves over Q play a key role in the proof of Fermat’s Last Theorem, while elliptic curves over finite fields come into play when we exchange private information securely over the internet. My work relates directly to this latter role, the use of elliptic curves in cryptography. Since first proposed in 1985, much research has been devoted to the problem of constructing “cryptographic” elliptic curves. For a finite field K of p elements, the group E(K) is finite, of order N roughly the same size as the prime p. The discrete logarithm problem (DLP) is: