Previously, no general-purpose algorithm was known for the elliptic curve logarithm problem that ran in better than exponential time. In this paper we demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying hit e field. For the class of supersingular elliptic curves, the reduction takes probabilistic polynomial time, thus providing a probabilistic subexponential time algorithm for the former problem. The implications of our results to public key cryptography are discussed.