Generating Prime Order Elliptic Curves: Difficulties and Efficiency Considerations

We consider the generation of prime order elliptic curves (ECs) over a prime field Fp 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 Fp. In this paper we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order sup...

• Type: Conference
• Year: 2004
• Where: ICISC
• Authors: Elisavet Konstantinou, Aristides Kontogeorgis, Yannis C. Stamatiou, Christos D. Zaroliagis
• Keywords: Corresponding Hilbert Polynomials, ICISC 2004, ICISC 2007, Prime Order, Weber Polynomials