Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

  • Nicolas Méloni
  • M. Anwar Hasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5747)


In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as \(\sum^n_{i=1}2^{b_i}3^{t_i}\) where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we propose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups.


Double-base number system Zeckendorf representation elliptic curve point scalar multiplication Yao’s algorithm 


  1. 1.
    Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: Optimizing Double-Base Elliptic-Curve Single-Scalar Multiplication. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 167–182. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Lange, T.: Explicit-formulas database, http://hyperelliptic.org/EFD
  3. 3.
    Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: Boztaş, S., Lu, H.-F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Bernstein, D.J., Lange, T.: Analysis and optimization of elliptic-curve single-scalar multiplication. In: Finite fields and applications: proceedings of Fq8, pp. 1–19 (2008)Google Scholar
  6. 6.
    Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Adv. Appl. Math. 7(4), 385–434 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Cryptography. Chapman and Hall, Boca Raton (2006)MATHGoogle Scholar
  8. 8.
    Dimitrov, V., Cooklev, T.: Two algorithms for modular exponentiation using nonstandard arithmetics. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 78(1), 82–87 (1995)Google Scholar
  9. 9.
    Dimitrov, V., Imbert, L., Mishra, P.K.: Efficient and secure elliptic curve point multiplication using double-base chains. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 59–78. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Doche, C., Icart, T., Kohel, D.R.: Efficient scalar multiplication by isogeny decompositions. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 191–206. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Doche, C., Imbert, L.: Extended Double-Base Number System with Applications to Elliptic Curve Cryptography. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 335–348. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Duquesne, S.: Improving the arithmetic of elliptic curves in the Jacobi model. Inf. Process. Lett. 104(3), 101–105 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Edwards, H.M.: A normal norm for elliptic curves. Bulletin of the American Mathematical Society 44, 393–422 (2007)CrossRefMATHGoogle Scholar
  14. 14.
    Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)MATHGoogle Scholar
  15. 15.
    Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 138–151. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Hisil, H., Koon-Ho Wong, K., Carter, G., Dawson, E.: An intersection form for jacobi-quartic curves. Personal communication (2008)Google Scholar
  17. 17.
    Liardet, P., Smart, N.P.: Preventing SPA/DPA in ECC systems using the jacobi form. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 391–401. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Longa, P., Gebotys, C.: Setting speed records with the (fractional) multibase non-adjacent form method for efficient elliptic curve scalar multiplication. Technical report, Department of Electrical and Computer Engineering University of Waterloo, Canada (2009)Google Scholar
  19. 19.
    Longa, P., Miri, A.: New composite operations and precomputation scheme for elliptic curve cryptosystems over prime fields. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 229–247. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Meloni, N.: New point addition formulae for ECC applications. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 189–201. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Mishra, P.K., Dimitrov, V.S.: Efficient quintuple formulas for elliptic curves and efficient scalar multiplication using multibase number representation. In: Garay, J.A., Lenstra, A.K., Mambo, M., Peralta, R. (eds.) ISC 2007. LNCS, vol. 4779, pp. 390–406. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Yao, A.C.: On the evaluation of powers. SIAM Journal on Computing 5(1), 100–103 (1976)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zeckendorf, E.: Représentations des nombres naturels par une somme de nombre de Fibonacci ou de nombres de Lucas. Bulletin de la Soci. Royale des Sciences de Liège, pp. 179–182 (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Nicolas Méloni
    • 1
  • M. Anwar Hasan
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of WaterlooCanada

Personalised recommendations