On the Use of Weber Polynomials in Elliptic Curve Cryptography

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 the Weber 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.