Paper 2008/060

Fast Algorithms for Arithmetic on Elliptic Curves Over Prime Fields

Nicholas T. Sullivan


We present here a thorough discussion of the problem of fast arithmetic on elliptic curves over prime order finite fields. Since elliptic curves were independently pro- posed as a setting for cryptography by Koblitz [53] and Miller [67], the group of points on an elliptic curve has been widely used for discrete logarithm based cryptosystems. In this thesis, we survey, analyse and compare the fastest known serial and parallel algorithms for elliptic curve scalar multiplication, the primary operation in discrete logarithm based cryptosystems. We also introduce some new algorithms for the basic group operation and several new parallel scalar multiplication algorithms. We present a mathematical basis for comparing the various algorithms and make recommendations for the fastest algorithms to use in different circumstances.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Thesis, University of Calgary, 2007
Elliptic Curve Cryptography
Contact author(s)
nicholas sullivan @ gmail com
2008-02-11: received
Short URL
Creative Commons Attribution


      author = {Nicholas T.  Sullivan},
      title = {Fast Algorithms for Arithmetic on Elliptic Curves Over Prime Fields},
      howpublished = {Cryptology ePrint Archive, Paper 2008/060},
      year = {2008},
      note = {\url{}},
      url = {}
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