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Exact Number of Elliptic Curves in the Canonical Form, Which are Isomorphic to Edwards Curves Over Prime Field

Cybernetics and Systems Analysis volume 51, pages 165–172 (2015)Cite this article

Abstract

The necessary and sufficient conditions for the parameters of the curve in the canonical form with two points of order 4 are found. Two lemmas about the properties of quadratic residues are proved, using the Gauss scheme for quadratic residues and non-residues. Based on this lemmas, the exact formulas are derived for the number of elliptic curves with non-zero parameters a and b and two points of order 4 that are isomorphic to Edwards curves over the prime field. It is proved that for large fields the share of such curves is close to 1/4.

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References

  1. H. M. Edwards, “A normal form for elliptic curves,” Bull. Amer. Math. Soc., 44, No. 3, 393–422 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  2. D. J. Bernstein and T. Lange, “Faster addition and doubling on elliptic curves,” IST Programme under Contract IST–2002–507932 ECRYPT (2007), pp. 1–20.

  3. A. V. Bessalov, “The number of isomorphisms and twist pairs of Edwards curves over a prime field,” Radiotekhnika, 167, 203–208 (2011).

    Google Scholar 

  4. A. V. Bessalov, A. I. Gur’yanov, and A. A. Dikhtenko, “Edwards curves of almost prime order over extensions of small prime fields,” Priklad. Radioelektronika, 11, No. 2, 225–227 (2012).

    Google Scholar 

  5. A. V. Bessalov and A. A. Dikhtenko, “Cryptosecure Edwards curves over prime fields,” Priklad. Radioelektronika, 12, No. 2, 285–291 (2013).

    Google Scholar 

  6. H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, Cambridge Univ. Press (2008).

  7. A. V. Bessalov and A. B. Telizhenko, Cryptosystems on Elliptic Curves: A Manual [in Russian], IVTs Politekhnika, Kyiv (2004).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Physics and Technology, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine

    A. V. Bessalov

  2. Institute of Special Communication and Information Security, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine

    L. V. Kovalchuk

Authors
  1. A. V. Bessalov
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  2. L. V. Kovalchuk
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Corresponding author

Correspondence to A. V. Bessalov.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2015, pp. 3–12.

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Bessalov, A.V., Kovalchuk, L.V. Exact Number of Elliptic Curves in the Canonical Form, Which are Isomorphic to Edwards Curves Over Prime Field. Cybern Syst Anal 51, 165–172 (2015). https://doi.org/10.1007/s10559-015-9709-x

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  • DOI: https://doi.org/10.1007/s10559-015-9709-x

Keywords

  • canonical elliptic curve
  • Edwards curve
  • twist curve
  • curve parameters
  • isomorphism
  • quadratic residue
  • quadratic non-residue