Minimality and other properties of the width-𝑤 nonadjacent form

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

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