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Issue No. 11 - November (2008 vol. 57)
ISSN: 0018-9340
pp: 1469-1481
Vassil S. Dimitrov , University of Calgary, Calgary
Kimmo U. Järvinen , Helsinki University of Technology , Espoo
Micheal J. Jacobson Jr. , University of Calgary, Calgary
Wai Fong (Andy) Chan , University of Calgary, Calgary
Zhun Huang , University of Calgary, Calgary
We describe algorithms for point multiplication on Koblitz curves using multiple-base expansions of the form $k = \sum \pm \tau^a (\tau-1)^b$ and $k= \sum \pm \tau^a (\tau-1)^b (\tau^2 - \tau - 1)^c.$ 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 $\tau$-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. We also show that implementations with very low computation latency are possible with the proposed method because parallel processing can be exploited efficiently.
Elliptic curve cryptography, Field-programmable gate arrays, Koblitz curves, multiple-base expansions, parallel processing, sublinearity

W. F. Chan, Z. Huang, K. U. Järvinen, V. S. Dimitrov and M. J. Jacobson Jr., "Provably Sublinear Point Multiplication on Koblitz Curves and Its Hardware Implementation," in IEEE Transactions on Computers, vol. 57, no. , pp. 1469-1481, 2008.
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