Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and

Abstract. In this paper we prove the optimality and other properties of the -adic nonadjacent form: this expansion has been introduced in order to efficiently compute scalar multiplications on Koblitz curves. We also refine and extend results about double expansions of scalars introduced by Avanzi, Ciet and Sica in order to further improve scalar multiplications. 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.

• Type: Journal
• Year: 2006
• Where: ALGORITHMICA
• Authors: Roberto Maria Avanzi, Clemens Heuberger, Helmut Prodinger
• Keywords: -adic Nonadjacent Form, ALGORITHMICA 2006, Double Expansions, Scalar Multiplications