Abstract | 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 supported by a thorough experimental study. In particular,
we show that such Weber polynomials have roots in F
p3 and present a
set of transformations for mapping roots of Weber polynomials in F
p3
to roots of their corresponding Hilbert polynomials in Fp. We also show
how a new class of polynomials, with degree equal to their corresponding
Hilbert counterparts (and hence having roots in Fp), 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.
Keywords: Elliptic Curve Cryptosystems, |