Global Methods for Discrete Logarithm Problems I: A Unified Approach for the Multiplicative Group and for Elliptic Curves over a Finite Field

Let A be a finite abelian group and x an element of A. Let y be in the subgroup generated by x, so that y = nx for some positive integer n. Recall that the discrete logarithm problem is to determine n in a computationally efficient way. The computational complexity of solving this problem when the bit size of the inputs is large is the basis of many public-key encryption schemes used today. Two of the most important examples of finite abelian groups that are used in public-key cryptography are the multiplicative group of a finite field and the group of points on an elliptic curve over a finite field (see [K] and [Mill] for the original papers and [KMV] for a survey of work as of 2000).