Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The Continued Logarithm Algorithm –CL for short– introduced by Gosper in 1978 computes the gcd of two integers; it is seemingly very efficient, as it only performs shifts and subtractions. Shallit has studied its worst-case complexity in 2016 and showed it to be linear. In this talk I present our average-case analysis of the algorithm: we study its main parameters (number of iterations, total number of shifts) and obtain precise asymptotics for their mean values. Our 'dynamical' analysis involves the dynamical system underlying the algorithm, that produces continued fraction expansions whose quotients are powers of 2. Even though this CL system, which is a variation of the Euclidean Dynamical System, has already been studied by Chan (around 2005), the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying the CL system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the CL algorithm, with explicit constants.