Cybernetics and Systems Analysis

, Volume 55, Issue 5, pp 731–741 | Cite as

Supersingular Twisted Edwards Curves over Prime Fields.* II. Supersingular Twisted Edwards Curves with the j-Invariant Equal to 663

  • A. V. BessalovEmail author
  • L. V. Kovalchuk


Theorems on the conditions for the existence of supersingular Edwards curves over a prime field with the j-invariant equal to 663 and with other values of j-invariants are formulated and proved. A generalization of the results obtained earlier is presented, which uses isomorphisms of curves in Legendre and Edwards forms.


supersingular curve complete Edwards curve twisted Edwards curve quadratic Edwards curve torsion pair order of point Legendre symbol quadratic residue quadratic nonresidue 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. V. Bessalov and L. V. Kovalchuk, “Supersingular twisted Edwards curves over prime fields. I. Supersingular twisted Edwards curves with j-invariants equal to zero and 123,” Cybernetics and Systems Analysis, Vol. 55, No. 3, 347–353 (2019).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. V. Bessalov, Elliptic Curves in Edwards Form and Cryptography [in Russian], Publ. House “Politekhnika” of the NTUU “Igor Sikorsky KPI,” Kyiv (2017).Google Scholar
  3. 3.
    D. J. Bernstein and T. Lange, “Faster addition and doubling on elliptic curves,” in: Advances in Cryptology — ASIACRYPT’2007 (Proc. 13th Int. Conf. on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia (December 2-6, 2007)); Lect. Notes Comp. Sci., Vol. 4833, 29–50, Springer, Berlin (2007).Google Scholar
  4. 4.
    A. J. Menezes, T. Okamoto, and S. A. Vanstone, “Reducing elliptic curve logarithms to logarithms in a finite field,” IEEE Transactions on Information Theory, Vol. 39, Iss. 5, 1639–1646 (1993).MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. C. Washington, Elliptic Curves: Number Theory and Cryptography, 2nd Edition, CRC Press, Boca Raton (2008).CrossRefGoogle Scholar
  6. 6.
    L. De Feo, D. Jao, and J. Plut, “Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies” J. Mathematical Cryptology, Vol. 8, No. 3, 209–247 (2014).MathSciNetzbMATHGoogle Scholar
  7. 7.
    D. Unruh, “Non-interactive zero-knowledge proofs in the quantum random oracle model,” in: Eurocrypt 2015, Vol. 9057, Springer, Berlin-Heidelberg (2015), pp. 755–784.Google Scholar
  8. 8.
    Y. Yoo, R. Azarderakhsh, A. Jalali, D. Jao, and V. Soukharev, “A post-quantum digital signature scheme based on supersingular isogenies,” Cryptology ePrint Archive, Report 2017/186, 2017. URL:
  9. 9.
    D. J. Bernstein, P. Birkner, M. Joye, T. Lange, and Ch. Peters, “Twisted Edwards curves,” in: IST Programme under Contract IST–2002–507932 ECRYPT and in Part by the National Science Foundation under Grant ITR–0716498 (2008), pp. 1–17.Google Scholar
  10. 10.
    A. V. Bessalov and O. V. Tsygankova, “Interrelation of families of points of high order on the Edwards curve over a prime field,” Problems of Information Transmission, Vol. 51, Iss. 4, 92–98 (2015).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. V. Bessalov and O. V. Tsygankova, “Classification of curves in the Edwards form over a prime field,” Applied Radio Electronics, Vol. 14, No. 4, 197–203 (2015).zbMATHGoogle Scholar
  12. 12.
    A. V. Bessalov and O. V. Tsygankova, “Number of curves in the generalized Edwards form with minimal even cofactor of the curve order,” Problems of Information Transmission, Vol. 53, Iss. 1, 101–111 (2017).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. V. Bessalov and A. B. Telizhenko, Cryptosystems Based on Elliptic Curves [in Russian], IVTs “Politekhnika,” Kyiv (2004).Google Scholar
  14. 14.
    F. Morain, Edwards curves and CM curves. ArXiv 0904/2243v1 [Math.NT] Apr. 15, 2009.Google Scholar
  15. 15.
    A. V. Bessalov and L. V. Kovalchuk, “Exact number of elliptic curves in the canonical form, which are isomorphic to Edwards curves over prime field,” Cybernetics and Systems Analysis, Vol. 51, No. 2, 165–172 (2015).MathSciNetCrossRefGoogle Scholar
  16. 16.
    O. Bespalov, “A generalization of the Gauss lemma about characters of pairs of elements of a finite prime field,” in: Proc. V. M. Glushkov Institute of Cybernetics of NASU and Kamianets-Podilskyi Ivan Ohiienko National University, Iss. 15 (2017), pp. 26–31.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Borys Grinchenko Kyiv University and Institute of Physics and Technology of the NTUU “Igor Sikorsky KPI”KyivUkraine
  2. 2.Institute of Physics and Technology of the NTUU “Igor Sikorsky KPI”KyivUkraine

Personalised recommendations