Non-Minimality of the Width-$w$ Non-adjacent Form in Conjunction with Trace One $\tau$-adic Digit Expansions and Koblitz Curves in Characteristic Two

TL;DR

This paper demonstrates that the width-$w$ non-adjacent form expansion is not optimal for certain algebraic integer bases in characteristic two, using advanced Diophantine analysis techniques.

Contribution

It proves the non-minimality of width-$w$ non-adjacent forms for trace-one $ au$-adic expansions on Koblitz curves in characteristic two, employing Diophantine tools.

Findings

Width-$w$ NAF is not weight-minimal for the specified bases.

Uses linear forms in logarithms and Baker--Davenport reduction.

Provides theoretical proof of non-optimality in this setting.

Abstract

This article deals with redundant digit expansions with an imaginary quadratic algebraic integer with trace $\pm 1$ as base and a minimal norm representatives digit set. For $w\geq 2$ it is shown that the width-$w$ non-adjacent form is not an optimal expansion, meaning that it does not minimize the (Hamming-)weight among all possible expansions with the same digit set. One main part of the proof uses tools from Diophantine analysis, namely the theory of linear forms in logarithms and the Baker--Davenport reduction method.