COMPUTING HILBERT CLASS POLYNOMIALS WITH THE CHINESE REMAINDER THEOREM

We present a space-efficient algorithm to compute the Hilbert class polynomial H-D (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(vertical bar D vertical bar(1/2+epsilon) log P) space and has an expected running time of O(vertical bar D vertical bar(1+epsilon)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with vertical bar D vertical bar as large as 10(13) and h(D) up to 10(6). We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.