Multiple GCDs. probabilistic analysis of the plain algorithm

Authors:
Valérie Berthé;Jean Creusefond;Loïck Lhote;Brigitte Vallée
Affiliations:
LIAFA, CNRS and Université Paris Diderot, Paris, France;INSA Rouen, Rouen, France;GREYC, CNRS, ENSICAEN et Université de Caen, Caen, France;GREYC, CNRS, ENSICAEN et Université de Caen, Caen, France
Venue:
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Year:
2013

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Abstract

This paper provides a probabilistic analysis of an algorithm which computes the gcd of ℓ inputs (with ℓ ≥ 2), with a succession of ℓ - 1 phases, each of them being the Euclid algorithm on two entries. This algorithm is both basic and natural, and two kinds of inputs are studied: polynomials over the finite field Fq and integers. The analysis exhibits the precise probabilistic behaviour of the main parameters, namely the number of iterations in each phase and the evolution of the length of the current gcd along the execution. We first provide an average-case analysis. Then we make it even more precise by a distributional analysis. Our results rigorously exhibit two phenomena: (i) there is a strong difference between the first phase, where most of the computations are done and the remaining phases; (ii) there is a strong similarity between the polynomial and integer cases, as can be expected.