Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

  • Reza R. Farashahi
  • Hongfeng Wu
  • Chang-An Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7707)


This paper presents new explicit formulae for the point doubling, tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinates require 3M + 2C, 8M + 1C + 1D and 4M + 4C + 1D respectively, where M, C and D is the cost of a field multiplication, a cubing and a multiplication by a constant. Finally, we present several examples of ordinary elliptic curves in characteristic three for high security levels.


Elliptic curve Hessian curve scalar multiplication cryptography 


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  1. 1.
    Avanzi, R., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press (2005)Google Scholar
  2. 2.
    Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. Cambridge University Press (2005)Google Scholar
  3. 3.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography, vol. 265. Cambridge University Press, New York (1999)zbMATHGoogle Scholar
  4. 4.
    Bernstein, D.J., Kohel, D., Lange, T.: Twisted Hessian Curves,
  5. 5.
    Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of Numbers Generated by Addition in Formal Groups and New Primality and Factorization Tests. Advances in Applied Mathematics 7(4), 385–434 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Farashahi, R.R., Joye, M.: Efficient Arithmetic on Hessian Curves. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 243–260. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Fouquet, M., Gaudry, P., Harley, R.: An Extension of Satoh’s Algorithm and its Implementation. J. Ramanujan Math. Soc. 15, 281–318 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hankerson, D., Menezes, A.J., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer (2004)Google Scholar
  9. 9.
    Hesse, O.: Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln. Journal für Die Reine und Angewandte Mathematik 10, 68–96 (1844)CrossRefGoogle Scholar
  10. 10.
    Hisil, H., Carter, G., Dawson, E.: New Formulae for Efficient Elliptic Curve Arithmetic. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 138–151. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Kim, K.H., Kim, S.I., Choe, J.S.: New Fast Algorithms for Arithmetic on Elliptic Curves over Fields of Characteristic Three. Cryptology ePrint Archive, Report 2007/179 (2007)Google Scholar
  13. 13.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48, 203–209 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Koblitz, N.: An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 327–337. Springer, Heidelberg (1998)Google Scholar
  15. 15.
    López, J., Dahab, R.: Improved Algorithms for Elliptic Curve Arithmetic in tex2html_wrap_inline116. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 201–212. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    Negre, C.: Scalar Multiplication on Elliptic Curves Defined over Fields of Small Odd Characteristic. In: Maitra, S., Veni Madhavan, C.E., Venkatesan, R. (eds.) INDOCRYPT 2005. LNCS, vol. 3797, pp. 389–402. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Satoh, T.: The canonical lift of an Ordinary Elliptic Curve over a Finite Field and its Point Counting. J. Ramanujan Math. Soc. 15, 247–270 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Smart, N.P.: The Hessian Form of an Elliptic Curve. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 118–125. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Smart, N.P., Westwood, E.J.: Point Multiplication on Ordinary Elliptic Curves over Fields of Characteristic Three. Appl. Algebra Eng. Commun. Comput. 13(6), 485–497 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Reza R. Farashahi
    • 1
    • 2
  • Hongfeng Wu
    • 3
  • Chang-An Zhao
    • 4
  1. 1.Dept. of ComputingMacquarie UniversitySydneyAustralia
  2. 2.Dept. of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  3. 3.College of SciencesNorth China University of TechnologyBeijingChina
  4. 4.School of Computer Science and Educational SoftwareGuangzhou UniversityGuangzhouChina

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