Lifting and Elliptic Curve Discrete Logarithms

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Abstract

The difficulty of the elliptic curve discrete logarithm problem (ECDLP) underlies the attractiveness of elliptic curves for use in cryptography. The index calculus is a lifting algorithm that solves the classical finite field discrete logarithm problem in subexponential time, but no such algorithm is known in general for elliptic curves. It turns out that there are four distinct lifting scenarios that one can use in attempting to solve the ECDLP; the lifting field may be a local field or a global field, and the lifted points may be torsion points or nontorsion points. These choices lead to four quite different ways to try to solve the ECDLP via lifting. None of these approaches has led to a solution to the ECDLP, but each method has its own reasons for failing to work. In this article I survey the four ways of lifting the ECDLP, explain their similarities and their differences, and describe the distinct roadblocks that arise in each case.