Minimality and other properties of the width-w nonadjacent form

Let w \geq 2 be an integer and let D_w be the set of integers that includes zero and the odd integers with absolute value less than 2^{w-1}. Every integer n can be represented as a finite sum of the form n = \sum a_i 2^i, with a_i \in D_w, such that of any w consecutive a_i's at most one is nonzero. Such representations are called \emph{width-w nonadjacent forms} (w-NAFs). When w=2 these representations use the digits \{0,\pm1\} and coincide with the well-known \emph{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 (right-to-left) lexicographical ordering. We also generalize a result on w-NAFs and show that any base 2 representation of an integer, with digits in D_w, that has a minimal number of nonzero digits is at most one digit longer than its binary representation.