Analysis of the Brun Gcd Algorithm

Abstract

We introduce and study a multiple gcd algorithm that is a natural extension of the usual Euclid algorithm, and coincides with it for two entries; it performs Euclidean divisions, between the largest entry and the second largest entry, and then re-orderings. This is the discrete version of a multidimensional continued fraction algorithm due to Brun. We perform the average-case analysis of this algorithm, and prove that the mean number of steps is linear with respect to the size of the entry. The method relies on dynamical analysis, and is based on the study of the underlying Brun dynamical system. The dominant constant of the analysis is related to the entropy of the system. We also compare this algorithm to another extension of the Euclid algorithm, proposed by Knuth, and already analyzed by the authors.