An average density of τ-adic Naf (τ-NAF) representation: An alternative proof

In order to improve the efficiency of scalar multiplications on elliptic koblitz curves,expansions of the scalar to a complex base associated with the frobenius endomorphism are commonly used. one such expansion is the τ-adic non adjacent form (τ-naf),introduced by solinas (1997). some properties of this expansion,such as the average density,are well known. however in the literature there is no description on the same sequences occuring as length-l naf's and length-l τ-naf's to proof that the average density is approximately 1/3 . in this paper we provide an alternative proof of this fact.