Abstract | In many cryptographic applications it is necessary to generate
elliptic curves (ECs) with certain security properties. These curves
are commonly constructed using the Complex Multiplication method
which typically uses the roots of Hilbert or Weber polynomials. The former
generate the EC directly, but have high computational demands,
while the latter are faster to construct but they do not lead, directly, to
the desired EC. In this paper we present in a simple and unifying manner
a complete set of transformations of the roots of a Weber polynomial to
the roots of its corresponding Hilbert polynomial for all discriminant values
on which they are defined. Moreover, we prove a theoretical estimate
of the precision required for the computation of Weber polynomials. Finally,
we experimentally assess the computational efficiency of theWeber
polynomials along with their precision requirements for various discriminant
values and compare the results with the theoretical estimates. Our
experimental results may be used as a guide for the selection of the most
efficient curves in applications residing in resource limited devices such as
smart cards that support secure and efficient Public Key Infrastructure
(PKI) services. |