Sublinear Scalar Multiplication on Hyperelliptic Koblitz Curves

  • Hugo Labrande
  • Michael J. JacobsonJr.
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7118)


Recently, Dimitrov et. al. [5] proposed a novel algorithm for scalar multiplication of points on elliptic Koblitz curves that requires a provably sublinear number of point additions in the size of the scalar. Following some ideas used by this article, most notably double-base expansions for integers, we generalize their methods to hyperelliptic Koblitz curves of arbitrary genus over any finite field, obtaining a scalar multiplication algorithm requiring a sublinear number of divisor additions.


Hyperelliptic Koblitz curves scalar multiplication double-base expansions 


  1. 1.
    Avanzi, R., Dimitrov, V., Doche, C., Sica, F.: Extending Scalar Multiplication Using Double Bases. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 130–144. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2006); MR2162716 (2007f:14020)Google Scholar
  3. 3.
    Dimitrov, V.S., Jullien, G.A., Miller, W.C.: An algorithm for modular exponentiation. Inform. Process. Lett. 66(3), 155–159 (1998); MR 1627991 (99d:94023)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dimitrov, V.S., Imbert, L., Zakaluzny, A.: Multiplication by a constant is sublinear. In: IEEE Symposium on Computer Arithmetic 2007, pp. 261–268 (2007)Google Scholar
  5. 5.
    Dimitrov, V.S., Järvinen, K.U., Jacobson Jr., M.J., Chan, W.F., Huang, Z.: Provably sublinear point multiplication on Koblitz curves and its hardware implementation. IEEE Trans. Comput. 57(11), 1469–1481 (2008); MR2464687 (2009j:68053) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Doche, C., Kohel, D.R., Sica, F.: Double-Base Number System for Multi-scalar Multiplications. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 502–517. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Enge, A.: Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time. Math. Comp. 71(238), 729–742 (2002); (electronic). MR 1885624 (2003b:68083)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Günther, C., Lange, T., Stein, A.: Speeding up the Arithmetic on Koblitz Curves of Genus Two. In: Stinson, D.R., Tavares, S. (eds.) SAC 2000. LNCS, vol. 2012, pp. 106–117. Springer, Heidelberg (2001); MR 1895585 (2003c:94024)CrossRefGoogle Scholar
  9. 9.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48(177), 203–209 (1987); MR 866109 (88b:94017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Koblitz, N.: CM-Curves with Good Cryptographic Properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)Google Scholar
  11. 11.
    Lange, T.: Efficient arithmetic on hyperelliptic curves, Ph.D. thesis, Universität-Gesamthochschule Essen, Essen, Germany (2001)Google Scholar
  12. 12.
    Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  13. 13.
    Solinas, J.A.: Efficient arithmetic on Koblitz curves. Des. Codes Cryptogr. 19(2-3), 195–249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tijdeman, R.: On the maximal distance between integers composed of small primes. Compositio. Math. 28, 159–162 (1974); MR 0345917 (49 #10646)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Vercauteren, F.: Computing Zeta Functions of Hyperelliptic Curves Over Finite Fields of Characteristic 2. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 369–384. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hugo Labrande
    • 1
  • Michael J. JacobsonJr.
    • 2
  1. 1.ENS LyonLyon Cedex 07France
  2. 2.Department of Computer ScienceUniversity of CalgaryCalgaryCanada

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