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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computing Hilbert class polynomials with the Chinese remainder theorem
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by Andrew V. Sutherland PDF
Math. Comp. 80 (2011), 501-538


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 $O(|D|^{1/2+\epsilon }\log {P})$ space and has an expected running time of $O(|D|^{1+\epsilon })$. 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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Additional Information
  • Andrew V. Sutherland
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 852273
  • ORCID: 0000-0001-7739-2792
  • Email:
  • Received by editor(s): March 3, 2009
  • Received by editor(s) in revised form: September 10, 2009
  • Published electronically: May 17, 2010
  • © Copyright 2010 by the author
  • Journal: Math. Comp. 80 (2011), 501-538
  • MSC (2010): Primary 11Y16; Secondary 11G15, 11G20, 14H52
  • DOI:
  • MathSciNet review: 2728992