Novel Precomputation Schemes for Elliptic Curve Cryptosystems

  • Patrick Longa
  • Catherine Gebotys
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5536)


We present an innovative technique to add elliptic curve points with the form P ±Q, and discuss its application to the generation of precomputed tables for the scalar multiplication. Our analysis shows that the proposed schemes offer, to the best of our knowledge, the lowest costs for precomputing points on both single and multiple scalar multiplication and for various elliptic curve forms, including the highly efficient Jacobi quartics and Edwards curves.


Elliptic curve cryptosystem scalar multiplication multiple scalar multiplication precomputation scheme conjugate addition 


  1. 1.
    Bernstein, D., 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
  2. 2.
    Bernstein, D., 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
  3. 3.
    Brown, M., Hankerson, D., Lopez, J., Menezes, A.: Software Implementation of the NIST Elliptic Curves over Prime Fields. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 250–265. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    FIPS PUB 186-2: Digital Signature Standard (DSS). National Institute of Standards and Technology (NIST) (2000)Google Scholar
  5. 5.
    Higuchi, A., Takagi, N.: A Fast Addition Algorithm for Elliptic Curve Arithmetic in GF(2n) using Projective Coordinates. Information Processing Letters 76(3), 101–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster Group Operations on Elliptic Curves. Cryptology ePrint Archive, Report 2007/441 (2007)Google Scholar
  7. 7.
    Hisil, H., Wong, K., Carter, G., Dawson, E.: An Intersection Form for Jacobi-Quartic Curves. Personal communication (2008)Google Scholar
  8. 8.
    Koblitz, N.: Elliptic Curve Cryptosystems. Mathematics of Computation, vol. 48, pp. 203–209 (1987)Google Scholar
  9. 9.
    Kuang, B., Zhu, Y., Zhang, Y.: An Improved Algorithm for uP+vQ using JSF3. In: Jakobsson, M., Yung, M., Zhou, J. (eds.) ACNS 2004. LNCS, vol. 3089, pp. 467–478. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Lim, C.H., Hwang, H.S.: Fast Implementation of Elliptic Curve Arithmetic in GF(p n). In: Imai, H., Zheng, Y. (eds.) PKC 2000. LNCS, vol. 1751, pp. 405–421. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Longa, P.: ECC Point Arithmetic Formulae (EPAF),
  12. 12.
    Longa, P., Miri, A.: Fast and Flexible Elliptic Curve Point Arithmetic over Prime Fields. IEEE Trans. Comp. 57(3), 289–302 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    López, J., Dahab, R.: Improved Algorithms for Elliptic Curve Arithmetic in GF(2n). Technical Report IC-98-39, Relatorio Técnico (1998)Google Scholar
  15. 15.
    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
  16. 16.
    Miller, V.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  17. 17.
    Möller, B.: Algorithms for Multi-exponentiation. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 165–180. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Okeya, K., Takagi, T., Vuillaume, C.: Efficient Representations on Koblitz Curves with Resistance to Side Channel Attacks. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 218–229. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Proos, J.: Joint Sparse Forms and Generating Zero Columns when Combing. Technical Report CORR 2003-23, University of Waterloo (2003)Google Scholar
  20. 20.
    Solinas, J.: Low-Weight Binary Representations for Pairs of Integers. Technical Report CORR 2001-41, University of Waterloo (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Patrick Longa
    • 1
  • Catherine Gebotys
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of WaterlooCanada

Personalised recommendations