Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication

  • Patrick Longa
  • Catherine Gebotys
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5443)

Abstract

Recently, the new Multibase Non-Adjacent Form (mbNAF) method was introduced and shown to speed up the execution of the scalar multiplication with an efficient use of multiple bases to represent the scalar. In this work, we first optimize the previous method using fractional windows, and then introduce further improvements to achieve additional cost reductions. Moreover, we present new improvements in the point operation formulae. Specifically, we reduce further the cost of composite operations such as quintupling and septupling of a point, which are relevant for the speed up of multibase methods in general. Remarkably, our tests show that, in the case of standard elliptic curves, the refined mbNAF method can be as efficient as Window-w NAF using an optimal fractional window size. Thus, this is the first published method that does not require precomputations to achieve comparable efficiency to the standard window-based NAF method using precomputations. On other highly efficient curves as Jacobi quartics and Edwards curves, our tests show that the refined mbNAF currently attains the highest performance for both scenarios using precomputations and those without precomputations.

Keywords

Elliptic curve cryptosystem scalar multiplication multibase non-adjacent form double base number system fractional window 

References

  1. 1.
    Bernstein, D., 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., Lange, T.: Analysis and Optimization of Elliptic-Curve Single-Scalar Multiplication. Cryptology ePrint Archive, Report 2007/455 (2007)Google Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    Billet, O., Joye, M.: The Jacobi Model of an Elliptic Curve and Side-Channel Analysis. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34–42. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Ciet, M., Joye, M., Lauter, K., Montgomery, P.L.: Trading Inversions for Multiplications in Elliptic Curve Cryptography. Designs, Codes and Cryptography 39(2), 189–206 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dimitrov, V., Jullien, G., Miller, W.: Theory and Applications for a Double-Base Number System. ARITH 1997, p. 44 (1997)Google Scholar
  8. 8.
    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
  9. 9.
    Dimitrov, V., Mishra, P.K.: 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
  10. 10.
    Doche, C., Habsieger, L.: A Tree-Base Approach for Computing Double-Base Chains. In: Mu, Y., Susilo, W., Seberry, J. (eds.) ACISP 2008. LNCS, vol. 5107, pp. 433–446. Springer, Heidelberg (2008)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.
    Edwards, H.: A Normal Form for Elliptic Curves. Bulletin of the American Mathematical Society 44, 393–422 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Elmegaard-Fessel, L.: Efficient Scalar Multiplication and Security against Power Analysis in Cryptosystems based on the NIST Elliptic Curves over Prime Fields. Master Thesis, University of Copenhagen (2006)Google Scholar
  14. 14.
    Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)MATHGoogle Scholar
  15. 15.
    Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster Group Operations on Elliptic Curves. Cryptology ePrint Archive, Report 2007/441 (2007)Google Scholar
  16. 16.
    Hisil, H., Wong, K., Carter, G., Dawson, E.: An Intersection Form for Jacobi-Quartic Curves. Personal communication (2008)Google Scholar
  17. 17.
    Longa, P.: Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields. Master Thesis, University of Ottawa (2007), http://patricklonga.bravehost.com/publications.html
  18. 18.
    Longa, P.: ECC Point Arithmetic Formulae (EPAF), http://patricklonga.bravehost.com/jacobian.html
  19. 19.
    Longa, P., Gebotys, C.: Setting Speed Records with the (Fractional) Multibase Non-Adjacent Form Method for Efficient Elliptic Curve Scalar Multiplication. CACR Technical Report, CACR 2008-06, University of Waterloo (2008)Google Scholar
  20. 20.
    Longa, P., Gebotys, C.: Novel Precomputation Schemes for Elliptic Curve Cryptosystems. Cryptology ePrint Archive, Report 2008/526 (2008)Google Scholar
  21. 21.
    Longa, P., Miri, A.: Fast and Flexible Elliptic Curve Point Arithmetic over Prime Fields. IEEE Trans. Comp. 57(3), 289–302 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    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
  24. 24.
    Möller, B.: Improved Techniques for Fast Exponentiation. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 298–312. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  25. 25.
    Möller, B.: Fractional windows revisited:Improved signed-digit representations for efficient exponentiation. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 137–153. Springer, Heidelberg (2005)CrossRefGoogle 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

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