Sharp Estimates for the Main Parameters of the Euclid Algorithm

Abstract

We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid algorithm, and we study in particular the distribution of the bit-complexity which involves two main parameters : digit–costs and length of continuants. We perform a “dynamical analysis” which heavily uses the dynamical system underlying the Euclidean algorithm. Baladi and Vallée [2] have recently designed a general framework for “distributional dynamical analysis”, where they have exhibited asymptotic gaussian laws for a large class of digit–costs. However, this family contains neither the bit–complexity cost nor the length of continuants. We first show here that an asymptotic gaussian law also holds for the length of continuants at a fraction of the execution. There exist two gcd algorithms, the standard one which only computes the gcd, and the extended one which also computes the Bezout pair, and is widely used for computing modular inverses. The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the extended algorithm. We prove that the bit–complexity of the extended Euclid algorithm asymptotically follows a gaussian law, and we exhibit the speed of convergence towards the normal law. We describe also conjectures [quite plausible], under which we can obtain an asymptotic gaussian law for the plain bit-complexity, or a sharper estimate of the speed of convergence towards the gaussian law.