Computing Hilbert class polynomials with the Chinese Remainder Theorem
Number Theory
2013-11-25 v4
Abstract
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.
Cite
@article{arxiv.0903.2785,
title = {Computing Hilbert class polynomials with the Chinese Remainder Theorem},
author = {Andrew V. Sutherland},
journal= {arXiv preprint arXiv:0903.2785},
year = {2013}
}
Comments
37 pages, corrected a typo that misstated the heuristic complexity