Computing Hilbert class polynomials with the Chinese remainder theorem

We present a space-efficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P , based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|1/2+ log P ) space and has an expected running time of O(|D|1+ ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 1013 and h(D) up to 106. We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.