We present a variant of the complex multiplication method that generates elliptic curves of cryptographically strong order. Our variant is based on the computation ofWeber polynomials that require significantly less time and space resources than their Hilbert counterparts. We investigate the time efficiency and precision requirements for generating off-line Weber polynomials and its comparison to another variant based on the off-line generation of Hilbert polynomials. We also investigate the efficiency of our variant when the computation of Weber polynomials should be made on-line due to limitations in resources (e.g., hardware devices of limited space).We present trade-offs that could be useful to potential implementors of elliptic curve cryptosystems on resource-limited hardware devices.