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Parameters for secure elliptic curve cryptosystem -improvements on Schoof s algorithm

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Public Key Cryptography (PKC 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1431))

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Abstract

The security of elliptic curve cryptosystem depends on the choice of an elliptic curve on which cryptographic operations are performed. Schoof's algorithm is used to define a secure elliptic curve, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies's method, isogeny cycles method, and baby-step-giant-step algorithm, we can count the number of rational points on an elliptic curve over GF(p) in a reasonable time, where p is a prime whose size is around 240-bit.

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References

  1. Atkin, A.O., The number of points on an elliptic curve modulo a prime, preprint, 1988.

    Google Scholar 

  2. Couveignes, J.-M., Dewaghe, L., Morain, F., Isogeny cycles and the Schoof-Elkies-Atkin algorithm, LIX/RR/96/03, 1996.

    Google Scholar 

  3. Couveignes, J,-M., Morain, F., Schoof's algorithm and isogeny cycles, In ANT-I,L.Adleman and M.-D.Huang, Eds., Lecture Notes in Computer Science, 877, pp.43–58, 1994.

    Google Scholar 

  4. Elkies, N.D., Explicit isogenies, preprint, 1991.

    Google Scholar 

  5. Izu, T., Kogure, J., Noro, M., Yokoyama, K., Secure Elliptic Curve Cryptosystem: Improvements on Schoof's Algorithm by Intelligent Choice System, in preparation.

    Google Scholar 

  6. Lercier, R., Algorithmique des courbes elliptiques dans les corps finis, Doctoral Thesis, L'école Polytechnique, 1997.

    Google Scholar 

  7. Lercier, R., Finding good random elliptic curves for cryptosystems definded over 257-01, In EURO-CRYPTO '97, W.Fumy, Ed., Lecture Notes in Computer Science, 1233, pp.379–392, 1997.

    Google Scholar 

  8. Lercier, R., Morain, F., Counting the number of points on elliptic curves over finite fields: strategy and performances, In EURO-CRYPTO '95, L.C.Guillou and J.-J.Quisquater, Eds., Lecture Notes in Computer Science, 921, pp.79–94, 1995.

    Google Scholar 

  9. Morain, F., Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques, J. Théor. Nombres Bordeaux 7 (1995) 255–282.

    MATH  MathSciNet  Google Scholar 

  10. Noro, M., Takeshima, T., ftp://endeavor.fujitsu.co.jp/pub/isis/asir

    Google Scholar 

  11. Schoof, R., Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985) 483–494.

    Article  MATH  MathSciNet  Google Scholar 

  12. Schoof, R., Counting points on elliptic curves over finite fields, J. Théor. Nombres Bordeaux 7 (1995) 219–254.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Fujitsu Laboratories Ltd., 4-1-1 Kamikodanaka, 211-8588, Nakahara-ku Kawasaki, Japan

    Tetsuya Izu, Masayuki Noro & Kazuhiro Yokoyama

  2. Fujitsu Ltd., Nikko Fudousan Bldg. 2-15-16 Shinyokohama, 222-0033, Kouhoku-ku Yokohama, Japan

    Jun Kogure

Authors
  1. Tetsuya Izu
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  2. Jun Kogure
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  3. Masayuki Noro
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  4. Kazuhiro Yokoyama
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Hideki Imai Yuliang Zheng

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© 1998 Springer-Verlag Berlin Heidelberg

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Izu, T., Kogure, J., Noro, M., Yokoyama, K. (1998). Parameters for secure elliptic curve cryptosystem -improvements on Schoof s algorithm. In: Imai, H., Zheng, Y. (eds) Public Key Cryptography. PKC 1998. Lecture Notes in Computer Science, vol 1431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054030

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  • DOI: https://doi.org/10.1007/BFb0054030

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64693-8

  • Online ISBN: 978-3-540-69105-1

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