Analysis of the width-w non-adjacent form in conjunction with hyperelliptic curve cryptography and with lattices

Abstract

In this work the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball) is analysed. As bases, expanding endomorphisms with eigenvalues of the same absolute value are allowed. Applications of the main result are on numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and the result is needed for analysing the running time of such algorithms.The counting result itself is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange’s method.