ISC 2007: Information Security pp 390-406 | Cite as

Efficient Quintuple Formulas for Elliptic Curves and Efficient Scalar Multiplication Using Multibase Number Representation

  • Pradeep Kumar Mishra
  • Vassil Dimitrov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4779)

Abstract

In the current work we propose two efficient formulas for computing the 5-fold (5P) of an elliptic curve point P. One formula is for curves over finite fields of even characteristic and the other is for curves over prime fields. Double base number systems (DBNS) have been gainfully exploited to compute scalar multiplication efficiently in ECC. Using the proposed point quintupling formulas one can use 2, 5 and 3, 5 (besides 2, 3) as bases of the double base number system. In the current work we propose a scalar multiplication algorithm, which uses a representation of the scalar using three bases 2, 3 and 5 and computes the scalar multiplication very efficiently. The proposed scheme is faster than all sequential scalar multiplication algorithms reported in literature.

Keywords

Elliptic Curve Cryptosystems Scalar Multiplication  Quintupling Efficient Curve Arithmetic 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avanzi, R.M., Sica, F.: Scalar Multiplication on Koblitz Curves using Double Bases. Tech Report. Available at http://eprint.iacr.org/2006/067
  2. 2.
    Avanzi, R.M., Dimitrov, V., Doche, C., Sica, F.: Extending Scalar Multiplication to Double Bases. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 130–144. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton (2005)Google Scholar
  4. 4.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  5. 5.
    Blake, I.F., Seroussi, G., Smart, N.P. (eds.): Advances in Elliptic Curves Cryptography. Cambridge University Press, Cambridge (2005)Google Scholar
  6. 6.
    Chevalier-Mames, B., Ciet, M., Joye, M.: Low-cost solutions for preventing simple side-channel analysis: Side-channel atomicity. IEEE Transactions on Computers 53(6), 760–768 (2004)CrossRefGoogle Scholar
  7. 7.
    Ciet, M., Lauter, K., Joye, M., Montgomery, P.L.: Trading inversions for multiplications in elliptic curve cryptography. Designs, Codes and Cryptography 39(2), 189–206 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ciet, M., Sica, F.: An Analysis of Double Base Number Systems and a Sublinear Scalar Multiplication Algorithm. In: Dawson, E., Vaudenay, S. (eds.) Mycrypt 2005. LNCS, vol. 3715, pp. 171–182. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptography. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Dahab, R., Lopez, J.: An Improvement of Guajardo-Paar Method for Multiplication on non-supersingular Elliptic Curves. In: SCCC 1998. Proceedings of the XVIII International Conference of the Chilean Computer Science Society, Antofagasta, Chile, November 12-14, 1998, pp. 91–95. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  12. 12.
    Dimitrov, V., Imbert, L., Mishra, P.K.: Efficient and Secure Elliptic Curve Point Multiplication Using Double Base Chain. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 59–79. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Dimitrov, V., Järvinen, K.U., Jacobson, M.J., Chan, W.F., Huang, Z.: FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 445–459. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Dimitrov, V.S., Jullien, G.A., Miller, W.C.: An algorithm for modular exponentiation. Information Processing Letters 66(3), 155–159 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dimitrov, V.S., Jullien, G.A., Miller, W.C.: Theory and applications of the double-base number system. IEEE Transactions on Computers 48(10), 1098–1106 (1999)CrossRefGoogle Scholar
  16. 16.
    Doche, C., Imbert, L.: Extended Double-Base Number System with applications to Elliptic Curve Cryptography. Tech Report, Conference version to appear in Indocrypt (2006), Available at http://eprint.iacr.org/2006/330
  17. 17.
    Doche, C., Icart, T., Kohel, D.: 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
  18. 18.
    Fong, K., Hankerson, D., Lòpez, J., Menezes, A.: Field inversion and point halving revisited. IEEE Transactions on Computers 53(8), 1047–1059 (2004)CrossRefGoogle Scholar
  19. 19.
    Guajardo, J., Paar, C.: Efficient Algorithms for Elliptic Curve Cryptosystems over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 342–356. Springer, Heidelberg (2000)Google Scholar
  20. 20.
    Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)MATHGoogle Scholar
  21. 21.
    Hankerson, D., Lòpez Hernandez, J., Menezes, A.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Itoh, K., Takenaka, M., Torii, N., Temma, S., Kurihara, Y.: Fast implementation of public-key cryptography on a DSP TMS320C6201. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 61–72. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  23. 23.
    Izu, T., Takagi, T.: Fast elliptic curve multiplications resistant against side channel attacks. IEICE Transactions Fundamentals E88-A(1), 161–171 (2005)CrossRefGoogle Scholar
  24. 24.
    Joye, M., Tymen, C.: Protections against differential analysis for elliptic curve cryptography – an algebraic approach. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 377–390. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  25. 25.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48(177), 203–209 (1987)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
  27. 27.
    Kocher, P., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)Google Scholar
  28. 28.
    Miller, V.S.: Uses of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–428. Springer, Heidelberg (1986)Google Scholar
  29. 29.
    Tijdeman, R.: On the maximal distance between integers composed of small primes. Compositio Mathematica 28, 159–162 (1974)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Pradeep Kumar Mishra
    • 1
  • Vassil Dimitrov
    • 1
  1. 1.Centre for Information Security and Cryptography University of Calgary, CalgaryCanada

Personalised recommendations