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International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2003: Advances in Cryptology — EUROCRYPT 2003 pp 388–400Cite as

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Improved Algorithms for Efficient Arithmetic on Elliptic Curves Using Fast Endomorphisms

Improved Algorithms for Efficient Arithmetic on Elliptic Curves Using Fast Endomorphisms

  • Mathieu Ciet5,
  • Tanja Lange6,
  • Francesco Sica5 &
  • …
  • Jean-Jacques Quisquater5 
  • Conference paper
  • First Online: 01 January 2003
  • 3673 Accesses

  • 19 Citations

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

Abstract

In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas’ approach presented at CRYPTO ’97. Furthermore, many elliptic curve cryptosystems require the computation of k 0 P + k 1 Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki’s. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

Keywords

  • Elliptic Curve
  • Characteristic Polynomial
  • Elliptic Curf
  • Improve Algorithm
  • Hyperelliptic Curve

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The work described in this paper has been supported [in part] by the Commission of the European Communities through the IST Programme under Contract IST-1999-12324, http://www.cryptonessie.org/. The information in this document is provided as is, and no guarantee or warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability. The views expressed are those of the authors and do not represent an official view/position of the NESSIE project (as a whole).

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Author information

Authors and Affiliations

  1. UCL Crypto Group, Place du Levant, 3, B-1348, Louvain-la-Neuve, Belgium

    Mathieu Ciet, Francesco Sica & Jean-Jacques Quisquater

  2. Institute for Information Security and Cryptology (ITSC), Ruhr-Universität Bochum, Universitätsstraße 150, D-44780, Bochum, Germany

    Tanja Lange

Authors
  1. Mathieu Ciet
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  2. Tanja Lange
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  3. Francesco Sica
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  4. Jean-Jacques Quisquater
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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

Ciet, M., Lange, T., Sica, F., Quisquater, JJ. (2003). Improved Algorithms for Efficient Arithmetic on Elliptic Curves Using Fast Endomorphisms. 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_24

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  • DOI: https://doi.org/10.1007/3-540-39200-9_24

  • 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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