Constructing Elliptic Curves over Ramanujan's Class Invariants

Abstract

The Complex Multiplication (CM) method is a widely used technique for constructing elliptic curves over finite fields. The key point in this method is parameter generation of the elliptic curve and root compution of a special type of class polynomials. However, there are several class polynomials which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we provide a method which can construct elliptic curves by Ramanujan's class invariants. We described the algorithm for the construction of elliptic curves (ECs) over imaginary quadratic field and given the transformation from their roots to the roots of the corresponding Hilbert polynomials. We compared the efficiency in the use of this method and other methods.