FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers

  • V. S. Dimitrov
  • K. U. Järvinen
  • M. J. JacobsonJr.
  • W. F. Chan
  • Z. Huang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4249)


We describe algorithms for point multiplication on Koblitz curves using multiple-base expansions of the form k = ∑±τ a (τ–1) b and k= ∑±τ a (τ–1) b (τ 2τ– 1) c . We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first provably sublinear point multiplication algorithm on Koblitz curves. For the first type, we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of τ-adic non-adjacent form expansions. We present details of an innovative FPGA implementation of our algorithm and performance data demonstrating the efficiency of our method.


Point Multiplication Greedy Algorithm Elliptic Curve Clock Cycle Elliptic Curf 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • V. S. Dimitrov
    • 1
  • K. U. Järvinen
    • 2
  • M. J. JacobsonJr.
    • 3
  • W. F. Chan
    • 3
  • Z. Huang
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of CalgaryCalgary, AlbertaCanada
  2. 2.Signal Processing LaboratoryHelsinki University of TechnologyEspooFinland
  3. 3.Department of Computer ScienceUniversity of CalgaryCalgary, AlbertaCanada

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