Efficient Arithmetic on Hessian Curves

  • Reza R. Farashahi
  • Marc Joye
Conference paper

DOI: 10.1007/978-3-642-13013-7_15

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6056)
Cite this paper as:
Farashahi R.R., Joye M. (2010) Efficient Arithmetic on Hessian Curves. In: Nguyen P.Q., Pointcheval D. (eds) Public Key Cryptography – PKC 2010. PKC 2010. Lecture Notes in Computer Science, vol 6056. Springer, Berlin, Heidelberg


This paper considers a generalized form for Hessian curves. The family of generalized Hessian curves covers more isomorphism classes of elliptic curves. Over a finite field \(\mathbb{F}_q\), it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to ℤ/3ℤ.

This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters.

This paper also presents extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require 9M + 3S, where M is the cost of a field multiplication and S is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only 5M + 4S when the curve is chosen with small curve parameters.


Elliptic curves Hessian curves cryptography 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Reza R. Farashahi
    • 1
    • 2
  • Marc Joye
    • 3
  1. 1.Department of ComputingMacquarie UniversitySydneyAustralia
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  3. 3.Technicolor, Security Competence CenterCesson-Sévigné CedexFrance

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