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.

In this article, we give a digital signature by using Lindner–Peikert cryptosystem. The security of this digital signature is based on the assumptions about hardness of Ring-LWE and Ring-SIS problems, along with providing public key and signature of compact (1–1.5 kilobytes) size. We prove the security of our signature scheme in the Quantum Random Oracle Model. Our cryptanalysis has been done based on methods of Aggarwal et al. and Chen et al.

The independence of Heegner points associated to distinct imaginary quadratic fields has been shown by Rosen and Silverman. In this paper we show the independence of Heegner points associated to orders with the same conductor in distinct imaginary quadratic fields.