  Elliptic curves with a known number of points over a given prime field with n
elements are often needed for use in cryptography. In the context of primality
proving, Atkin and Morain suggested the use of the theory of complex
multiplication to construct such curves. One of the steps in this method is the
calculation of a root modulo n of the Hilbert class polynomial H(X) for a
fundamental discriminant D. The usual way is to compute H(X) over the integers
and then to find the root modulo n. We present a modified version of the
Chinese remainder theorem (CRT) to compute H(X) modulo n directly from the
knowledge of H(X) modulo enough small primes. Our complexity analysis suggests
that asymptotically our algorithm is an improvement over previously known
methods.
