My mathematical interests focus on arithmetic algebraic geometry, on algorithmic mathematics and on foundations of mathematics. Especially, I am interested in the structure and arithmetic of abelian varieties, curves and their coverings, the discrete logarithm problem in elliptic curves over finite fields and more generally the discrete logarithm problem in class groups of curves over finite fields, as well as the problem to solve systems of systems of polynomial equations. My algorithmic research often has a bearing upon cryptography.
One research focus in the last years was the discrete logarithm problem in elliptic curves. Among other results, I could show that there exists a sequence of finite fields Fq of strictly increasing cardinality over which the elliptic curve discrete logarithm problem can be solved in an expected time of exp(O(log(q)2/3)). Previously, no such sequence of finite fields was known. The algorithm is based on the well known index calculus method, and the relation generation is based on the solution of systems of multivariate polynomial equations.
Also, I am part of a team which is "world record holder" for the classical discrete logarithm problem: We computed a discrete logarithm for a "cryptographically suitable" modulus of 768 bit. The computation relies on software by Thorsten Kleinjung, the other team members are Arjen Lenstra, Christine Priplata and Colin Stahlke.
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