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