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Mathematics of Computation

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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“Chinese & Match”, an alternative to Atkin’s “Match and Sort” method used in the SEA algorithm
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by Antoine Joux and Reynald Lercier PDF
Math. Comp. 70 (2001), 827-836 Request permission

Abstract:

A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so-called “Match and Sort” method due to Atkin. This method is a “baby step/giant step” way to find the number of points among $C$ candidates with $O(\sqrt {C})$ elliptic curve additions. Observing that the partial information modulo Atkin’s primes is redundant, we propose to take advantage of this redundancy to eliminate the usual elliptic curve algebra in this phase of the SEA computation. This yields an algorithm of similar complexity, but the space needed is smaller than what Atkin’s method requires. In practice, our technique amounts to an acceleration of Atkin’s method, allowing us to count the number of points of an elliptic curve defined over $\mathbb {F}_{2^{1663}}$. As far as we know, this is the largest point-counting computation to date. Furthermore, the algorithm is easily parallelized.
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Additional Information
  • Antoine Joux
  • Affiliation: SCSSI, 18 rue du Dr. Zamenhoff, F-92131 Issy-les-Moulineaux, France
  • MR Author ID: 316495
  • Email: Antoine.Joux@ens.fr
  • Reynald Lercier
  • Affiliation: CELAR, Route de Laillé, F-35998 Rennes Armées, France
  • MR Author ID: 602270
  • ORCID: 0000-0002-0531-8945
  • Email: lercier@celar.fr
  • Received by editor(s): January 22, 1999
  • Received by editor(s) in revised form: April 20, 1999
  • Published electronically: March 2, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 827-836
  • MSC (2000): Primary 11Y16; Secondary 68Q25
  • DOI: https://doi.org/10.1090/S0025-5718-00-01200-X
  • MathSciNet review: 1680891