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Efficient scalar multiplication in Abelian groups (which is an importantoperation in public key cryptography) can be performed using digitalexpansions. Apart from rational integer bases (double-and-add algorithm),imaginary quadratic integer bases are of interest for elliptic curvecryptography, because the Frobenius endomorphism fulfils a quadratic equation.One strategy for improving the efficiency is to increase the digit set (at theprize of additional precomputations). A common choice is the width\nbd-$w$non-adjacent form (\wNAF): each block of $w$ consecutive digits contains atmost one non-zero digit. Heuristically, this ensures a low weight, i.e.\ numberof non-zero digits, which translates in few costly curve operations. This paperinvestigates the following question: Is the \wNAF{}-expansion optimal, whereoptimality means minimising the weight over all possible expansions with thesame digit set?  The main characterisation of optimality of \wNAF{}s can be formulated in thefollowing more general setting: We consider an Abelian group together with anendomorphism (e.g., multiplication by a base element in a ring) and a finitedigit set. We show that each group element has an optimal \wNAF{}-expansion ifand only if this is the case for each sum of two expansions of weight 1. Thisleads both to an algorithmic criterion and to generic answers for variouscases.  Imaginary quadratic integers of trace at least 3 (in absolute value) haveoptimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base$(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves incharacteristic three. In the case of $\tau=\pm1\pm i$, optimality depends onthe parity of $w$. Computational results for small trace are given.

Optimality of the Width-$w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases