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arXiv · 0903.2785

Computing Hilbert class polynomials with the Chinese Remainder Theorem

Andrew V. Sutherland
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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.

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Andrew V. Sutherland. 2009-03-16. Computing Hilbert class polynomials with the Chinese Remainder Theorem. https://doi.org/10.1090/s0025-5718-2010-02373-7

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