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Clemens Heuberger; Daniel Krenn
Optimality of the Width-$w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
Journal de théorie des nombres de Bordeaux, 25 no. 2 (2013), p. 353-386, doi: 10.5802/jtnb.840
Article PDF | Reviews MR 3228312 | Zbl 1282.11005
Class. Math.: 11A63, 94A60
Keywords: $\tau$-adic expansions, width-$w$ non-adjacent forms, redundant digit sets, elliptic curve cryptography, Koblitz curves, Frobenius endomorphism, scalar multiplication, Hamming weight, optimality, imaginary quadratic bases

Résumé - Abstract

We consider digit expansions $\sum _{j=0}^{\ell -1} \Phi ^j(d_j)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF).

This result is then applied to imaginary quadratic bases, which are used for scalar multiplication in elliptic curve cryptography. Both an algorithmic criterion and generic answers for various cases are given. Imaginary quadratic integers of trace at least $3$ (in absolute value) have optimal $w$-NAFs for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm \sqrt{-3})/2$ (four cases) and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau =\pm 1\pm i$ (again four cases), optimality depends on the parity of $w$. Computational results for small trace are given.

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