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Title:
A Novel Pre-Computation Scheme of Window $\tau$NAF for Koblitz Curves
Authors: Wei Yu, Saud Al Musa, Guangwu Xu, Bao Li
Abstract:Let E_a: y^2+xy=x^3+ax^2+1/ \mathbb{F}_{2^m} be a Koblitz curve. The window \tau-adic nonadjacent-form (window $\tau$NAF) is currently the standard representation system to perform scalar multiplications on E_a by utilizing the Frobenius map \tau. Pre-computation is an important part for the window $\tau$NAF. In this paper, we first introduce \mu\bar{\tau}-operations in lambda coordinates (\mu=(-1)^{1-a} and \bar{\tau} is the complex conjugate of the complex representation of \tau). Efficient formulas of \mu\bar{\tau}-operations are then derived and used in a novel pre-computation scheme to improve the efficiency of scalar multiplications using window $\tau$NAF. Our pre-computation scheme costs $7$M$+5$S, $26$M$+16$S, and $66$M$+36$S for window $\tau$NAF with width 4, 5, and 6 respectively whereas the pre-computation with the state-of-the-art technique costs $11$M$+8$S, $43$M$+18$S, and $107$M$+36$S. Experimental results show that our pre-computation is about 60\% faster, compared to the best pre-computation in the literature. It also shows that we can save from 2.5\% to 4.9\% on the scalar multiplications using window $\tau$NAF with our pre-computation.
ePrint: https://eprint.iacr.org/2017/1020
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