Partial Lifting and the Elliptic Curve Discrete Logarithm Problem

Abstract

It has been suggested that a major obstacle in finding an index calculus attack on the elliptic curve discrete logarithm problem lies in the difficulty of lifting points from elliptic curves over finite fields to global fields. We explore the possibility of circumventing the problem of explicitly lifting points by investigating whether partial information about the lifting would be sufficient for solving the elliptic curve discrete logarithm problem. Along this line, we show that the elliptic curve discrete logarithm problem can be reduced to three partial lifting problems. Our reductions run in random polynomial time assuming certain conjectures, which are based on some well-known and widely accepted conjectures concerning the expected ranks of elliptic curves over the rationals. Should the elliptic curve discrete logarithm problem admit no subexponential time attack, then our results suggest that gaining partial information about lifting would be at least as hard.