On the discrete logarithm problem in elliptic curves

It is well known that the classical discrete logarithm problem (the problem to compute indices modulo prime numbers) can be solved in subexponential expected time.\\In contrast, it is not known whether the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields (the elliptic curve discrete logarithm problem) can be solved in subexponential expected time. Indeed, it was the lack of an obvious algorithm for this computational problem which was faster than "generic" algorithms which lead Miller and Koblitz to suggest the use of the problem for cryptographic applications.\\In 2004 Gaudry gave a randomized algorithm with which one can - under some heuristic assumptions - solve the elliptic curve discrete logarithm problem over all finite fields with a fixed extension degree at least 3 faster than with generic algorithms. By using a similar algorithm, I have shown that there exists a sequence of finite fields (of strictly increasing cardinality) over which the elliptic curve discrete logarithm problem can be solved in subexponential time.\\Recently, I have been able to extend the result in such a way that now for more families of finite fields the elliptic curve disctrete logarithm problem can be solved in subexponential expexted time. A corresponding algorithm and its analysis will be outlined in the talk.