Alternative Digit Sets for Nonadjacent Representations
James A. Muir, Douglas R. Stinson · SIAM Journal on Discrete Mathematics · 2005
It is known that every positive integer n can be represented as a finite sum of the form $n={\textstyle\sum} a_i 2^{i}$, where $a_i\in \{0,1,-1\}$ for all i, and no two consecutive a i 's are nonzero. Such sums are called nonadjacent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. In this paper, we investigate if other digit sets of the form {0,1,x}, where x is an integer, provide each positive integer with a nonadjacent representation. If a digit set has this property, we call it a nonadjacent digit set (NADS). We present an algorithm to determine if {0,1,x} is an NADS and, if it is, we present an algorithm to efficiently determine the nonadjacent representation of any positive integer. We also present some necessary and sufficient conditions for {0,1,x} to be an NADS. These conditions are used to exhibit infinite families of integers x such that {0,1,x} is an NADS, as well as infinite families of x such that {0,1,x} is not an NADS.