Paper 2007/441

Faster Group Operations on Elliptic Curves

Huseyin Hisil, Kenneth Koon-Ho Wong, Gary Carter, and Ed Dawson


This paper improves implementation techniques of Elliptic Curve Cryptography. We introduce new formulae and algorithms for the group law on Jacobi quartic, Jacobi intersection, Edwards, and Hessian curves. The proposed formulae and algorithms can save time in suitable point representations. To support our claims, a cost comparison is made with classic scalar multiplication algorithms using previous and current operation counts. Most notably, the best speedup is obtained in the case of Jacobi quartic curves which also lead to one of the most efficient scalar multiplications benefiting from the proposed 2M + 5S + 1D (i.e. 2 multiplications, 5 squarings, and 1 multiplication by a curve constant) point doubling and 7M + 3S + 1D point addition algorithms. Furthermore, the new addition algorithm provides an efficient way to protect against side channel attacks which are based on simple power analysis (SPA).

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Published elsewhere. Unknown where it was published
Efficient elliptic curve arithmeticunified additionside channel attack.
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h hisil @ isi qut edu au
2009-03-11: last of 4 revisions
2007-12-05: received
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      author = {Huseyin Hisil and Kenneth Koon-Ho Wong and Gary Carter and Ed Dawson},
      title = {Faster Group Operations on Elliptic Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2007/441},
      year = {2007},
      note = {\url{}},
      url = {}
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