Index calculus is the best known method for solving discrete logarithm problem(DLP) in general groups. In the elliptic curve case this method uses lifting and dependence relation among lifted rational points to solve DLP. In this paper, we propose an algorithm to find such dependence relation in rank one case.

This paper provides a polynomial time algorithm for finding integer number d such that T = dS in the group of Q-rational points on a given elliptic curve E defined over Q (ECDLP) where T, S ∈ E(Q). In this method, first we determine the biggest integer number n such that S ∈ 2 n E(Q) and then using halving and another algorithms, we solve this problem.