Generating Prime Order Elliptic Curves: Difficulties and Efficiency Considerations

Abstract

We consider the generation of prime order elliptic curves (ECs) over a prime field $\mathbb{F}_p$ using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones, uniquely determined by the CM discriminant D. In attempting to construct prime order ECs using Weber polynomials two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D ≡ 3 (mod 8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in $\mathbb{F}_p$. In this paper we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order supported by a thorough experimental study. In particular, we show that such Weber polynomials have roots in $\mathbb{F}_{p^3}$ and present a set of transformations for mapping roots of Weber polynomials in $\mathbb{F}_{p^3}$ to roots of their corresponding Hilbert polynomials in $\mathbb{F}_{p}$. We also show how a new class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in $\mathbb{F}_{p}$), can be used in the CM method to generate prime order ECs. Finally, we compare experimentally the efficiency of using this new class against the use of the aforementioned Weber polynomials.

This work was partially supported by the Action IRAKLITOS (Fellowships for Research in the University of Patras) with matching funds from EC and the Greek Ministry of Education.