Minimality and other properties of the width-w nonadjacent form

Let w ≥ 2 be an integer and let Dw be the set of integers which includes zero and the odd integers with absolute value less than 2. Every integer n can be represented as a finite sum of the form n = P ai2 , with ai ∈ Dw, such that of any w consecutive ai’s at most one is nonzero. Such representations are called width-w nonadjacent forms (w-NAFs). When w = 2 these representations use the digits {0,±1} and coincide with the well known nonadjacent forms. Width-w nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the w-NAF. We show that w-NAFs have a minimal number of nonzero digits and we also give a new characterization of the w-NAF in terms of a lexicographical ordering. We also generalize a result on w-NAF and show that any base 2 representation of an integer, with digits in Dw, that has a minimal number of nonzero digits is at most one digit longer than its binary representation.