On the Optimal Pre-Computation of Window τNAF for Koblitz Curves

Koblitz curves have been a nice subject of consideration for both theoretical and practical interests. The window τ -adic algorithm of Solinas (window τNAF) is the most powerful method for computing point multiplication for Koblitz curves. Precomputation plays an important role in improving the performance of point multiplication. In this paper, the concept of optimal pre-computation for window τNAF is formulated. In this setting, an optimal pre-computation has some mathematically natural and clean forms, and requires 2w−2 − 1 point additions and two evaluations of the Frobenius map τ , 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 τNAF. The ideas in the paper can also be used to construct other suitable pre-computations. This paper also includes a discussion of coefficient sets for window τNAF and the divisibility by powers of τ through different approaches.