A note on window τ-NAF algorithm

The window τ-adic algorithm of Solinas [Efficient arithmetic on Koblitz curves, Designs, Codes and Cryptography 19 (2000) 195] is the most powerful method for computing point multiplication for Koblitz curves. In this note, the existence of a more general window τ-adic form for each element in Z[τ] is obtained. In particular, this provides a proof of the termination of Solinas algorithm.