Efficient scalar multiplication in Abelian groups (which is an important
operation in public key cryptography) can be performed using digital
expansions. Apart from rational integer bases (double-and-add algorithm),
imaginary quadratic integer bases are of interest for elliptic curve
cryptography, because the Frobenius endomorphism fulfils a quadratic equation.
One strategy for improving the efficiency is to increase the digit set (at the
prize of additional precomputations). A common choice is the width\nbd-w
non-adjacent form (\wNAF): each block of w consecutive digits contains at
most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number
of non-zero digits, which translates in few costly curve operations. This paper
investigates the following question: Is the \wNAF{}-expansion optimal, where
optimality means minimising the weight over all possible expansions with the
same digit set?
The main characterisation of optimality of \wNAF{}s can be formulated in the
following more general setting: We consider an Abelian group together with an
endomorphism (e.g., multiplication by a base element in a ring) and a finite
digit set. We show that each group element has an optimal \wNAF{}-expansion if
and only if this is the case for each sum of two expansions of weight 1. This
leads both to an algorithmic criterion and to generic answers for various
cases.
Imaginary quadratic integers of trace at least 3 (in absolute value) have
optimal \wNAF{}s for w≥4. The same holds for the special case of base
(±3±−3)/2 and w≥2, which corresponds to Koblitz curves in
characteristic three. In the case of τ=±1±i, optimality depends on
the parity of w. Computational results for small trace are given