We obtain new results regarding the precise average-- case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters ---digits and continuants--- that intervene in an entire class of gcd-like algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.

We study a class of Euclidean algorithms related to divisions where the remainder belongs to [α - 1, α], for some α 2 [0; 1]. The paper is devoted to the average-case analysis of these algorithms, in terms of number of steps or bit-complexity. This is a new instance of the so-called "dynamical analysis" method, where it is made a deep use of dynamical systems. Here, the dynamical systems of interest have an infinite of branches and they are not markovian, so that the general framework of dynamical analysis is more complex to adapt to this case.