Preventing SPA/DPA in ECC Systems Using the Jacobi Form

  • P. -Y. Liardet
  • N. P. Smart
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2162)


In this paper we show how using a representation of an ellip-tic curve as the intersection of two quadrics in ℙ3 can provide a defence against Simple and Differental Power Analysis (SPA/DPA) style attacks. We combine this with a ‘random window’ method of point multiplication and point blinding. The proposed method offers considerable advantages over standard algorithmic techniques of preventing SPA and DPA which usually require a significant increased computational cost, usually more than double. Our method requires roughly a seventy percent increase in computational cost of the basic cryptographic operation, although we give some indication as to how this can be reduced. In addition we show that the Jacobi form is also more efficient than the standard Weierstrass form for elliptic curves in the situation where SPA and DPA are not a concern.


Elliptic Curve Smart Card Elliptic Curf Point Doubling Jacobi Form 
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.


  1. 1.
    I.F. Blake, G. Seroussi and N.P. Smart. Elliptic curves in cryptography. Cambridge University Press, 1999.Google Scholar
  2. 2.
    J.W.S. Cassels. Lectures on Elliptic Curves. LMS Student Texts, Cambridge University Press, 1991.Google Scholar
  3. 3.
    J.W.S. Cassels and E.V. Flynn. Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. Cambridge University Press, 1996.Google Scholar
  4. 4.
    D.V. Chudnovsky and G.V. Chudnovsky. Sequences of numbers generated by addition in formal groups and new primality and factorisation tests. Adv. in Appl. Math., 7, 385–434, 1987.MathSciNetCrossRefGoogle Scholar
  5. 5.
    H. Cohen, A. Miyaji and T. Ono. Efficient elliptic curve exponentiation using mixed coordinates. In Advances in Cryptology, ASIACRYPT 98. Springer-Verlag, LNCS 1514, 51–65, 1998.CrossRefGoogle Scholar
  6. 6.
    N.A. Howgrave-Graham and N.P. Smart. Lattice attacks on digital signature schemes. To appear Designs, Codes and Cryptography.Google Scholar
  7. 7.
    P. Kocher, J. Jaffe and B. Jun. Differential power analysis. In Advances in Cryptology, CRYPTO’ 99, Springer LNCS 1666, pp 388–397, 1999.Google Scholar
  8. 8.
    J.R. Merriman, S. Siksek, and N.P. Smart. Explicit 4-descents on an elliptic curve. Acta. Arith., 77, 385–404, 1996.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • P. -Y. Liardet
    • 1
  • N. P. Smart
    • 2
  1. 1.Dept. System EngineeringSTMicroelectronicsRoussetFrance
  2. 2.Dept. Computer ScienceUniversity of BristolBristolUK

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