Problems of Information Transmission

, Volume 53, Issue 1, pp 92–101 | Cite as

Number of curves in the generalized Edwards form with minimal even cofactor of the curve order

Information Protection
  • 23 Downloads

Abstract

We analyze properties of points of orders 2, 4, and 8 of a curve in the generalized Edwards form. Arithmetic for group operations with singular points of these curves is introduced. We propose a classification of curves in the Edwards form into three disjoint classes. Formulas for the number of curves of order 4n of different classes are obtained. Works of other authors are critically analyzed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Edwards, H.M., A Normal Form for Elliptic Curves, Bull. Amer. Math. Soc. (N.S.), 2007, vol. 44, no. 3, pp. 393–422.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bessalov, A.V. and Tsygankova, O.V., The Performance of Group Operations on Twisted Edwards Curve over a Prime Field, Radiotekhnika, vol. 181, Kharkiv: KhNURE, 2015, pp. 58–63.Google Scholar
  3. 3.
    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., and Peters, C., Twisted Edwards Curves, Progress in Cryptology—AFRICACRYPT’2008 (Proc. 1st Int. Conf. on Cryptology in Africa, Casablanca, Morocco, June 11–14, 2008), Vaudenay, S., Ed., Lect. Notes Comp. Sci., vol. 5023, Berlin: Springer, 2008, pp. 389–405.Google Scholar
  4. 4.
    Bernstein, D.J. and Lange, T., Faster Addition and Doubling on Elliptic Curves, Advances in Cryptology— ASIACRYPT’2007 (Proc. 13th Int. Conf. on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia, Dec. 2–6, 2007), Kurosawa, K., Ed., Lect. Notes Comp. Sci., vol. 4833, Berlin: Springer, 2007, pp. 29–50.Google Scholar
  5. 5.
    Bessalov, A.V. and Tsygankova, O.V., Interrelation of Families of Points of High Order on the Edwards Curve over a Prime Field, Probl. Peredachi Inf., 2015, vol. 51, no. 4, pp. 92–98 [Probl. Inf. Trans. (Engl. Transl.), 2015, vol. 51, no. 4, pp. 391-397].MathSciNetMATHGoogle Scholar
  6. 6.
    Bessalov, A.V., Kriptosistemy na ellipticheskikh krivykh (Cryptosystems on Elliptic Curves), Kyiv: Politekhnika, 2004.Google Scholar
  7. 7.
    Bessalov, A.V. and Kovalchuk, L.V., Exact Number of Elliptic Curves in the Canonical Form, Which Are Isomorphic to Edwards Curves over Prime Field, Kibernet. Sistem. Anal., 2015, vol. 51, no. 2, pp. 3–12 [Cybernet. Systems Anal. (Engl. Transl.), 2015, vol. 51, no. 2, pp. 165–172].MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.Borys Grinchenko Kyiv UniversityKyivUkraine
  2. 2.Institute of Physics and TechnologyNational Technical University of Ukraine “Kyiv Polytechnic Institute,”KyivUkraine

Personalised recommendations