A Novel Pre-Computation Scheme of Window τNAF for Koblitz Curves

Let Ea : y 2 + xy = x + ax + 1/F2m be a Koblitz curve. The window τ -adic nonadjacent-form (window τNAF) is currently the standard representation system to perform scalar multiplications on Ea by utilizing the Frobenius map τ . Pre-computation is an important part for the window τNAF. In this paper, we first introduce μτ̄ -operations in lambda coordinates (μ = (−1)1−a and τ̄ is the complex conjugate of the complex representation of τ). Efficient formulas of μτ̄ -operations are then derived and used in a novel pre-computation scheme to improve the efficiency of scalar multiplications using window τNAF. Our pre-computation scheme costs 7M+5S, 26M+16S, and 66M+36S for window τNAF with width 4, 5, and 6 respectively whereas the precomputation with the state-of-the-art technique costs 11M+8S, 43M+18S, and 107M+36S. 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 τNAF with our pre-computation.