Advances in Alternative Non-adjacent Form Representations

From several decades, non-adjacent form (NAF) representations for integers have been extensively studied as an alternative to the usual binary number system where digits are in {0,1}. In cryptography, the non-adjacent digit set (NADS) {–1,0,1} is used for optimization of arithmetic operations in elliptic curves. At SAC 2003, Muir and Stinson published new results on alternative digit sets: they proposed infinite families of integers x such that {0,1,x} is a NADS as well as infinite families of integers x such that {0,1,x} is not a NADS, so called a NON-NADS. Muir and Stinson also provided an algorithm that determines whether x leads to a NADS by checking if every integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n \epsilon [0, \lfloor \frac{-x}{3} \rfloor]$\end{document} has a {0,1,x}-NAF. In this paper, we extend these results by providing generators of NON-NADS infinite families. Furthermore, we reduce the search bound from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor \frac{-x}{3} \rfloor$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor \frac{-x}{12} \rfloor$\end{document}. We introduce the notion of worst NON-NADS and give the complete characterization of such sets. Beyond the theoretical results, our contribution also aims at exploring some algorithmic aspects. We supply a much more efficient algorithm than those proposed by Muir and Stinson, which takes only 343 seconds to compute all x’s from 0 to –107 such that {0,1,x} is a NADS.