Probabilistic analyses of the plain multiple gcd algorithm

Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ−1ℓ−1 phases (ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the average-case and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases.