Ramanujanʹs class invariants and their use in elliptic curve cryptography

The Complex Multiplication (CM) method is a method frequently used for the generation of elliptic curves (ECs) over a prime field Fp. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. However, there are several polynomials, called 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 propose the use of a new class of polynomials which are derived from Ramanujanʹs class invariants tn. We explicitly describe the algorithm for the construction of the new polynomials and give the necessary transformation of their roots to the roots of the corresponding Hilbert polynomials. We provide a theoretical asymptotic bound for the bit precision requirements of all class polynomials and, also using extensive experimental assessments, we compare the efficiency of using the new polynomials against the use of the other class polynomials. Our comparison shows that the new class of polynomials clearly surpass all of the previously used polynomials when they are used in the generation of prime order elliptic curves.