Addition chains using continued fractions

doi:10.1016/0196-6774(89)90036-9

Journal of Algorithms

Volume 10, Issue 3, September 1989, Pages 403-412

https://doi.org/10.1016/0196-6774(89)90036-9 Get rights and content

Abstract

This paper introduces a new algorithm for the evaluation of monomials in two variables x^ay^b based upon the continued fraction expansion of $a b$ . A method for fast explicit generation of addition chains of small length for a positive integer n is deduced from this Algorithm. As an illustration of the properties of the method, a Scholz-Brauer-like inequality p(N) ≤ nb + k + p(n + 1), is shown to be true whenever N is an integer of the form 2^k(1 + 2^b + … + 2^nb). Computer experimentation has shown that the length of the chains constructed are of optimal length for all integers up to 1000, with 29 exceptions for which the length is equal to the optimal length plus one.

References (8)

E.G Thurber
Addition chains and solutions of l(2n) = l(n) and l(2ⁿ − 1) = n + l(n) − 1
Discrete Math.
(1976)
A Brauer
On addition chains
Bull. Amer. Math. Soc.
(1939)
J. Berstel and S. Brlek, On the length of word chains, Inform. Process Lett. 26, September 87,...
D Dobkin et al.
Addition chain methods for the evaluation of specific polynomials
SIAM J. Comput.
(1980)

There are more references available in the full text version of this article.

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^∗: With the support of the NSERC of Canada Grant A9041 and of the FCAR of Québec Grant EQ1608.

^†: With the support of the PRC de Mathématiques et Informatique, France.

View full text

Addition chains using continued fractions

Abstract

Discrete Math.

On addition chains

Bull. Amer. Math. Soc.

Addition chain methods for the evaluation of specific polynomials

SIAM J. Comput.