**Scalar Multiplication on Koblitz Curves using $\tau^2-$NAF**

*Sujoy Sinha Roy and Chester Rebeiro and Debdeep Mukhopadhyay and Junko Takahashi and Toshinori Fukunaga*

**Abstract: **The paper proposes a $\tau^2-$NAF method for scalar multiplication on Koblitz curves,
which requires asymptotically $0.215m$ point additions in $GF(2^m)$.
For $\tau^2-$NAF method, point quading operation $(a\rightarrow a^4)$ is performed instead of point squarings.
The proposed method is faster than normal $\tau-$NAF method, which requires around $\frac{m}{3}$ point additions.
However, like width $w$ based $\tau-$NAF methods, there is an overhead of pre-computations in the $\tau^2-$NAF method.
For extended binary fields of small size, the $\tau^2-$NAF based scalar multiplication requires almost same number of point
additions as in width $4$ $\tau-$NAF method. Though, complexity wise, $\tau^2-$NAF based scalar multiplication and
width $4-\tau-$NAF based scalar multiplication are similar, but the techniques are different.

**Category / Keywords: **implementation / Koblitz curve, elliptic curve, scalar multiplication, tau^2 NAF

**Date: **received 15 Jun 2011

**Contact author: **sujoyetc at cse iitkgp ernet in, chester@cse iitkgp ernet in, debdeep@cse iitkgp ernet in, takahashi junko@lab ntt co jp, toshi fukunaga@hco ntt co jp

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**Version: **20110617:070953 (All versions of this report)

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