Scalar multiplication on Koblitz curves using the frobenius endomorphism and its combination with point halving: Extensions and mathematical analysis

In this paper we prove the optimality and other properties of the τ-adic nonadjacent form: this expansion has been introduced in order to compute scalar multiplications on Koblitz curves efficiently. We also refine and extend results about double expansions of scalars introduced by Avanzi, Ciet and Sica in order to improve scalar multiplications further. Our double expansions are optimal and their properties are carefully analysed. In particular, we provide first- and second-order terms for the expected weight, determine the variance and prove a central limit theorem. Transducers for all the involved expansions are provided, as well as automata accepting all expansions of minimal weight.