Computing Hilbert class polynomials with the Chinese Remainder Theorem (2010)
by
Andrew V. Sutherland
| Citations: | 13 - 1 self |
BibTeX
@MISC{Sutherland10computinghilbert,
author = {Andrew V. Sutherland},
title = {Computing Hilbert class polynomials with the Chinese Remainder Theorem },
year = {2010}
}
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Abstract
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+ɛ log P) space and has an expected running time of O(|D | 1+ɛ). 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.
