Efficient computation of addition chains

Bergeron, F. ; Berstel, J. ; Brlek, S.

Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 21-38

Résumé

The aim of this paper is to present a unifying approach to the computation of short addition chains. Our method is based upon continued fraction expansions. Most of the popular methods for the generation of addition chains, such as the binary method, the factor method, etc..., fit in our framework. However, we present new and better algorithms. We give a general upper bound for the complexity of continued fraction methods, as a function of a chosen strategy, thus the total number of operations required for the generation of an addition chain for all integers up to $n$ is shown to be $O\left(n{log}^{2}n{\gamma }_n\right)$, where ${\gamma }_n$ is the complexity of computing the set of choices corresponding to the strategy. We also prove an analog of the Scholz-Brauer conjecture.