On the Optimal Pre-Computation of Window $\tau$ NAF for Koblitz Curves

Koblitz curves have been an important subject of consideration for both theoretical andpractical interests. The window
$\tau$
-adic algorithm of Solinas (window
$\tau$
NAF) is the most powerful method for computing point multiplication for Koblitz curves.Pre-computation plays an important role in improving the performance of point multiplication. In this paper, the concept of optimal pre-computation for window
$\tau$
NAF is formulated. In this setting, an optimal pre-computation has some mathematically natural and clean forms, and requires
$2^{w-2}-1$
point additions and two evaluations of the Frobenius map
$\tau$
, where
$w$
is the window width. One of the main results of this paper is to construct an optimal pre-computation scheme for each window width
$w$
from
$4$
to
$15$
(more than practical needs).These pre-computations can be easily incorporated into implementations of window
$\tau$
NAF. The ideas in the paper can also be used to construct other suitable pre-computations. This paper includes a discussion of coefficient sets for window
$\tau$
NAF and the divisibility by powers of
$\tau$
through different approaches. Some issues of implementation are also discussed.