Minimality and Other Properties of the Width-w Nonadjacent Form

Let$w \geq 2$be an integer and let Dwbe 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 width-w nonadjacent forms (w-NAFs). When w = 2 these representations use the digits$\{0, \pm 1\}$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 (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 Dw, that has a minimal number of nonzero digits is at most one digit longer than its binary representation.