Abstract | In many cryptographic applications it is
necessary to generate elliptic curves (ECs) whose order
possesses certain properties. The method that is usually
employed for the generation of such ECs is the
so-called Complex Multiplication method. This method
requires the use of the roots of certain class field polynomials
defined on a specific parameter called the discriminant.
The most commonly used polynomials are
the Hilbert and Weber ones. The former can be used
to generate directly the EC, but they are characterized
by high computational demands. The latter have usually
much lower computational requirements, but they
do not directly construct the desired EC. This can be
achieved if transformations of their roots to the roots of the corresponding (generated by the same discriminant)
Hilbert polynomials are provided. In this paper we present
a variant of the Complex Multiplicationmethod that
generates ECs of cryptographically strong order. Our
variant is based on the computation of Weber polynomials.
We present in a simple and unifying manner a
complete set of transformations of the roots of aWeber
polynomial to the roots of its corresponding Hilbert
polynomial for all values of the discriminant. In addition,
we prove a theoretical estimate of the precision required
for the computation ofWeber polynomials for all values
of the discriminant.We present an extensive experimental
assessment of the computational efficiency of the
Hilbert and Weber polynomials along with their precision
requirements for various discriminant values and
we compare them with the theoretical estimates.We further
investigate the time efficiency of the new Complex
Multiplication variant under different implementations
of a crucial step of the variant. Our results can serve as
useful guidelines to potential implementers of EC cryptosystems
involving generation of ECs of a desirable
order on resource limited hardware devices or in systems
operating under strict timing response constraints |