Signature Calculus and the Discrete Logarithm Problem for Elliptic Curves (Preliminary Version)

This is the third in a series of papers in which we develop a unified method for treating the discrete logarithm problem (DLP) in various contexts. In [HR1], we described a formalism using global duality for a unified approach to the DLP for the multiplicative group and for elliptic curves over finite fields. The main tool to be employed is what we call signature calculus. In [HR2], we used signature calculus to study the DLP for the group F∗p of invertible elements of the finite prime field, Fp. In this paper, we use the method to study the DLP for the group Ẽ(Fp) of rational points of an elliptic curve Ẽ defined over Fp. Recall that in this context, the DLP is formulated as follows: let #Ẽ(Fp) = ` be prime and Q a point in Ẽ(Fp) of order `. Suppose we are given another element R. Then the DLP is to determine n so that R = nQ in a computationally efficient way. The expected computational complexity of this problem is the basis of elliptic curve cryptography.