SPA-Resistant Scalar Multiplication on Hyperelliptic Curve Cryptosystems Combining Divisor Decomposition Technique and Joint Regular Form

  • Toru Akishita
  • Masanobu Katagi
  • Izuru Kitamura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4249)

Abstract

Hyperelliptic Curve Cryptosystems (HECC) are competitive to elliptic curve cryptosystems in performance and security. Recently efficient scalar multiplication techniques using a theta divisor have been proposed. Their application, however, is limited to the case when a theta divisor is used for the base point. In this paper we propose efficient and secure scalar multiplication of a general divisor for genus 2 HECC over \(\mathbb{F}_{2^m}\). The proposed method is based on two novel techniques. One is divisor decomposition technique in which a general divisor is decomposed into two theta divisors. The other is joint regular form for a pair of integers that enables efficient and secure simultaneous scalar multiplication of two theta divisors. The marriage of the above two techniques achieves both about 19% improvement of efficiency compared to the standard method and resistance against simple power analysis without any dummy operation.

Keywords

hyperelliptic curve cryptosystems scalar multiplication theta divisor signed binary representation simple power analysis 

References

  1. 1.
    Cantor, D.G.: Computing in the Jacobian of a Hyperelliptic Curve. Mathematics of Computation 48(177), 95–101 (1987)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic Curve and Hyperelliptic Curve Cryptography. Chapman & Hall, Boca Raton (2005)CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Duquesne, S.: Montgomery Scalar Multiplication for Genus 2 Curves. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 153–168. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Fong, K., Hankerson, D., López, J., Menezes, A.: Field inversion and point halving revised. Technical Report CORR 2003-81 (2003), http://www.cacr.math.uwaterloo.ca/techreports/2003/corr2003-18.pdf
  6. 6.
    Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Harley, R.: Adding.txt, Doubling.c (2000), http://cristal.inria.fr/~harley/hyper/
  8. 8.
    Katagi, M., Akishita, T., Kitamura, I., Takagi, T.: Efficient Hyperelliptic Curve Cryptosystems Using Theta Divisors. IEICE Trans. Fundamentals E89-A(1), 151–160 (2006)CrossRefGoogle Scholar
  9. 9.
    Katagi, M., Kitamura, I., Akishita, T., Takagi, T.: Novel Efficient Implementations of Hyperelliptic Curve Cryptosystems Using Degenerate Divisors. In: Lim, C.H., Yung, M. (eds.) WISA 2004. LNCS, vol. 3325, pp. 345–359. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Kitamura, I., Katagi, M., Takagi, T.: A Complete Divisor Class Halving Algorithm for Hyperelliptic Curve Cryptosystems of Genus Two. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 146–157. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Koblitz, N.: Hyperelliptic Cryptosystems. Journal of Cryptology 1, 139–150 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lang, S.: Abelian Varieties. Springer, Heidelberg (1983)MATHCrossRefGoogle Scholar
  13. 13.
    Lange, T.: Formulae for Arithmetic on Genus 2 Hyperelliptic Curves. In: Applicable Algebra in Engineering, Communication and Computing, vol. 15, pp. 295–328. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Lim, C.H., Lee, P.J.: More Flexible Exponentiation with Precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Mamiya, H., Miyaji, A., Morimoto, H.: Efficient Countermeasure against RPA, DPA, and SPA. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 343–356. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Mumford, D.: Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser (1984)Google Scholar
  17. 17.
    Solinas, J.A.: Low-Weight Binary Representations for Pairs of Integers., Technical Report CORR 2001-41 (2001), http://www.cacr.math.uwaterloo.ca/techreports/2001/corr2001-41.ps

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Toru Akishita
    • 1
  • Masanobu Katagi
    • 1
  • Izuru Kitamura
    • 1
  1. 1.Information Technologies LaboratoriesSony CorporationTokyoJapan

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