Issue No. 11 - November (2008 vol. 57)

ISSN: 0018-9340

pp: 1469-1481

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2008.65

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

ABSTRACT

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.

INDEX TERMS

Elliptic curve cryptography, Field-programmable gate arrays, Koblitz curves, multiple-base expansions, parallel processing, sublinearity

CITATION

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.

doi:10.1109/TC.2008.65

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