New Formulae for Efficient Elliptic Curve Arithmetic

  • Huseyin Hisil
  • Gary Carter
  • Ed Dawson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4859)


This paper is on efficient implementation techniques of Elliptic Curve Cryptography. In particular, we improve timings for Jacobi-quartic (3M+4S) and Hessian (7M+1S or 3M+6S) doubling operations. We provide a faster mixed-addition (7M+3S+1d) on modified Jacobi-quartic coordinates. We introduce tripling formulae for Jacobi-quartic (4M+11S+2d), Jacobi-intersection (4M+10S+5d or 7M+7S+3d), Edwards (9M+4S) and Hessian (8M+6S+1d) forms. We show that Hessian tripling costs 6M+4C+1d for Hessian curves defined over a field of characteristic 3. We discuss an alternative way of choosing the base point in successive squaring based scalar multiplication algorithms. Using this technique, we improve the latest mixed-addition formulae for Jacobi-intersection (10M+2S+1d), Hessian (5M+6S) and Edwards (9M+1S+ 1d+4a) forms. We discuss the significance of these optimizations for elliptic curve cryptography.


Elliptic curve efficient point multiplication doubling tripling DBNS 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Huseyin Hisil
    • 1
  • Gary Carter
    • 1
  • Ed Dawson
    • 1
  1. 1.Information Security Institute, Queensland University of Technology 

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