Optimality of the width-\(w\) non-adjacent form: general characterisation and the case of imaginary quadratic bases. (English. French summary) Zbl 1282.11005

Summary: 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 \(w\geq 4\). The same holds for the special case of base \((\pm 3\pm \sqrt{-3})/2\) (four cases) and \(w\geq 2\), 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.


11A63 Radix representation; digital problems
94A60 Cryptography
11R04 Algebraic numbers; rings of algebraic integers
14G50 Applications to coding theory and cryptography of arithmetic geometry


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