Computing Hilbert class polynomials with the Chinese Remainder Theorem

16 Mar 2009  ·  Andrew V. Sutherland ·

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(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| 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.

PDF Abstract

Categories


Number Theory 11Y16 (Primary), 11G15, 11G20, 14H52 (Secondary)