Complete addition formulas on the level four theta model of elliptic curves

  • Emmanuel FouotsaEmail author
  • Oumar Diao


The addition formula provided for the recently discovered level four theta model of elliptic curve (Diao and Fouotsa in Afr Math 26(3):283–301, 2015) is not complete; meaning that it does not work for all inputs. In this work we provide three alternative addition formulas to solve this problem. The obtained formulas are unified in the sense that they can be used for both addition of points and doubling. These formulas are also valid both in odd and even characteristics. In particular our new formula in binary fields improves the one previously obtained on this curve (Diao and Fouotsa, 2015). We compute the cost of each formula. We also provide a sage and magma codes (Fouotsa and Diao in Sage and magma code for the verification of addition formulas, algorithm for cost computation and completeness of addition formulas., 2018) to ensure the correctness of all formulas and algorithms in this work.


Elliptic curves Theta model Addition formulas Complete formula 

Mathematics Subject Classification

14H52 1990S 



The authors are very grateful for the reviewers’s comments which enable to improve this work.


  1. 1.
    Bernstein, D.J., Lange, T.: Explicit-formulae database.
  2. 2.
    Bernstein, D.J., Lange, T., Farashahi, R.R.: Binary edwards curves. In: Cryptographic Hardware and Embedded Systems-CHES 2008. In: 10th International Workshop, Washington, D.C., USA, August 10–13, 2008. Proceedings, Lecture Notes in Computer Science, pp. 244–265. Springer (2008)Google Scholar
  3. 3.
    Boneh, D., Franklin, M.K.: Identity-based encryption from the Weil pairing. In: Advances in Cryptology-CRYPTO 2001, 21st Annual International Cryptology Conference, Santa Barbara, California, USA, August 19–23, 2001, Proceedings, pp. 213–229 (2001)Google Scholar
  4. 4.
    Devigne, J., Joye, M.: Binary Huff curves. In: Topics in Cryptology—CT-RSA 2011, LNCS Springer, vol. 6558 , pp. 340–355 (2011)CrossRefGoogle Scholar
  5. 5.
    Diao, O., Fouotsa, E.: Arithmetic of the level four theta model of elliptic curves. Afr. Math. 26(3), 283–301 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory 32, 644–654 (1976)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fouotsa, E.: Parallelizing pairings on hessian elliptic curves. Arab J. Math. Sci. 25(1), 29–42 (2019)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fouotsa, E., Diao, O.: Sage and magma code for the verification of addition formulas, algorithm for cost computation and completeness of addition formulas (2018).
  9. 9.
    Gamal, T.E.: A public key cryptosystem and signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory 31, 473–496 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Izu, T., Takagi, T.: Exceptional procedure attack on elliptic curve cryptosystems. In: PKC 2003, pp. 224–239. Springer (2003)Google Scholar
  11. 11.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203–209 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kohel, D.: Efficient arithmetic on elliptic curves in characteristic 2. In: INDOCRYPT 2012, LNCS Springer, vol. 7668, pp. 378–398 (2012)zbMATHGoogle Scholar
  13. 13.
    Libert, B., Quisquater, J.: Identity based undeniable signatures. In: Topics in Cryptology—CT-RSA 2004, The Cryptographers’ Track at the RSA Conference 2004, San Francisco, CA, USA, February 23–27, 2004, Proceedings, pp. 112–125 (2004)Google Scholar
  14. 14.
    Lubicz, D., Robert, D.: Efficient pairing computation with theta functions. In: Algorithmic Number Theory (2010), LNCS Springer, vol. 6197, pp. 251–269 (2010)Google Scholar
  15. 15.
    Miller V.S.: New directions in cryptography. In: Use of Elliptic Curves in Cryptography. Advances in Cryptology—CRYPTO’85, vol. 218, pp. 417–426 (1986)Google Scholar
  16. 16.
    Nafissatou, D., Emmanuel, F.: An encoding for the theta model of elliptic curves. Innov. Interdiscip. Solut. Underserved Areas 249, 224–235 (2019)Google Scholar
  17. 17.
    Nitaj, A., Fouotsa, E.: A new attack on rsa and demytko’s elliptic curve cryptosystem. J. Discrete Math. Sci. Cryptogr. 22(3), 391–409 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Higher Teacher Training CollegeThe University of BamendaBambiliCameroon
  2. 2.Department of SecuriyDrodiSecVigneux-Sur-SeineFrance

Personalised recommendations