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International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2003: Advances in Cryptology — EUROCRYPT 2003 pp 360–373Cite as

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Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

  • Reynald Lercier5 &
  • David Lubicz5 
  • Conference paper
  • First Online: 01 January 2003
  • 3552 Accesses

  • 3 Citations

Part of the Lecture Notes in Computer Science book series (LNCS,volume 2656)

Abstract

Let p be a small prime and q = p n. Let E be an elliptic curve over \( \mathbb{F}_q \) . We propose an algorithm which computes without any preprocessing the j-invariant of the canonical lift of E with the cost of O(log n) times the cost needed to compute a power of the lift of the Frobenius. Let μ be a constant so that the product of two n-bit length integers can be carried out in O(n μ) bit operations, this yields an algorithm to compute the number of points on elliptic curves which reaches, at the expense of a O(n 5/2) space complexity, a theoretical time complexity bound equal to O(n max(1.19,μ)+μ+1/2 log n). When the field has got a Gaussian Normal Basis of small type, we obtain furthermore an algorithm with O(log(n)n 2μ) time and O(n 2) space complexities. From a practical viewpoint, the corresponding algorithm is particularly well suited for implementations. We outline this by a 100002-bit computation.

Keywords

  • Elliptic curves
  • canonical lifts
  • AGM

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Author information

Authors and Affiliations

  1. CELAR, Route de Laillé, F-35570, Bruz, France

    Reynald Lercier & David Lubicz

Authors
  1. Reynald Lercier
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  2. David Lubicz
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Editor information

Editors and Affiliations

  1. Computer Science Department, Technion — Israel Institute of Technology, Haifa, 32000, Israel

    Eli Biham

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© 2003 International Association for Cryptologic Research

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Cite this paper

Lercier, R., Lubicz, D. (2003). Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time. In: Biham, E. (eds) Advances in Cryptology — EUROCRYPT 2003. EUROCRYPT 2003. Lecture Notes in Computer Science, vol 2656. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-39200-9_22

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  • DOI: https://doi.org/10.1007/3-540-39200-9_22

  • Published: 13 May 2003

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-14039-9

  • Online ISBN: 978-3-540-39200-2

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