Signed MSB-Set Comb Method for Elliptic Curve Point Multiplication

  • Min Feng
  • Bin B. Zhu
  • Cunlai Zhao
  • Shipeng Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3903)


Comb method is an efficient method to calculate point multiplication in elliptic curve cryptography, but vulnerable to power-analysis attacks. Various algorithms have been proposed recently to make the comb method secure to power-analysis attacks. In this paper, we present an efficient comb method and its Simple Power Analysis (SPA)-resistant counterpart. We first present a novel comb recoding algorithm which converts an integer to a sequence of signed, MSB-set comb bit-columns. Using this recoding algorithm, the signed MSB-set comb method and a modified, SPA-resistant version are then presented. Measures and precautions to make the proposed SPA-resistant comb method resist all power-analysis attacks are also discussed, along with performance comparison with other comb methods. We conclude that our comb methods are among the most efficient comb methods in terms of number of precomputed points and computational complexity.


Elliptic Curve Elliptic Curf Point Addition Evaluation Stage Elliptic Curve Cryptography 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  2. 2.
    Lim, C., Lee, P.: More Flexible Exponentiation with Precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)Google Scholar
  3. 3.
    Kocher, P.C.: Timing Attacks on Implementations of Diffe-Hellman, RSA, DSS and Other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Kocher, P.C., Jaffe, J., Jun, B.: Differential Power Analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)Google Scholar
  5. 5.
    Coron, J.S.: Resistance Against Differential Power Analysis for Elliptic Curve Cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Okeya, K., Sakurai, K.: A Second-Order DPA Attack Breaks a Window-Method Based Countermeasure against Side Channel Attacks. In: Chan, A.H., Gligor, V.D. (eds.) ISC 2002. LNCS, vol. 2433, pp. 389–401. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Goubin, L.: A Refined Power-Analysis Attack on Elliptic Curve Cryptosystems. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 199–211. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Akishita, T., Takagi, T.: Zero-Value Point Attacks on Elliptic Curve Cryptosystem. In: Boyd, C., Mao, W. (eds.) ISC 2003. LNCS, vol. 2851, pp. 218–233. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Liardet, P.Y., 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
  10. 10.
    Joye, M., Quisquater, J.J.: Hessian Elliptic Curves and Side-Channel Attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Okeya, K., Kurumatani, H., Sakurai, K.: Elliptic Curves with the Montgomery- Form and Their Cryptographic Applications. In: Imai, H., Zheng, Y. (eds.) PKC 2000. LNCS, vol. 1751, pp. 238–257. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Fischer, W., Giraud, C., Knudsen, E.W., Seifert, J.-P.: Parallel Scalar Multiplication on General Elliptic Curve over Fp Hedged against Non-Differential Side- Channel Attacks. In: IACR, Cryptography ePrint Archieve 2002/007 (2002),
  13. 13.
    Izu, T., Takagi, T.: A Fast Parallel Elliptic Curve Multiplication Resistant against Side Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 280–296. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Brier, E., Joye, M.: Weierstrass Elliptic Curves and Side-Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Möller, B.: Securing Elliptic Curve Point Multiplication against Side-Channel Attacks, Addendum: Efficiency Improvement (2001),
  16. 16.
    Okeya, K., Takagi, T.: A More Flexible Countermeasure against Side Channel Attacks Using Window Method. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 397–410. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Hedabou, M., Pinel, P., Bébéteau, L.: A Comb Method to Render ECC Resistant against Side Channel Attacks (2004),
  18. 18.
    Hedabou, M., Pinel, P., Bébéteau, L.: Countermeasures for Preventing Comb Method Against SCA Attacks. In: Deng, R.H., et al. (eds.) ISPEC 2005. LNCS, vol. 3439, pp. 85–96. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Feng, M., Zhu, B., Xu, M., Li, S.: Efficient Comb Methods for Elliptic Curve Point Multiplication Resistant to Power Analysis,
  20. 20.
    Chevallier-Mames, B., Ciet, M., Joye, M.: Low-Cost Solutions for Preventing Simple Side-Channel Analysis: Side-Channel Atomicity. IEEE Transaction on Computers 53(6), 760–768 (2004)CrossRefGoogle Scholar
  21. 21.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Min Feng
    • 1
  • Bin B. Zhu
    • 2
  • Cunlai Zhao
    • 1
  • Shipeng Li
    • 2
  1. 1.School of Mathematical SciencesPeking Univ.BeijingChina
  2. 2.Microsoft Research AsiaBeijingChina

Personalised recommendations