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.

following an idea of Benedict H.Gross [2] who presented an elliptic curve test for Mresenne numbers we propose a similar test.

In this paper, we propose a new certificateless identification scheme based on isogenies between elliptic curves that is a candidate for quantum-resistant problems. The proposed scheme has the batch verification property which allows verifying more than one identity by executing only a single challenge-response protocol.