Multiple GCD's. Probabilistic analysis of the plain algorithm
Résumé
The paper provides a probabilistic analysis of an algorithm
which computes the gcd of α inputs (with α >= 2), with a
succession of α - 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 rigor-
ously 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.
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