Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2-dimensional generalization. This paper surveys the main properties of functional operators, -- transfer operators -- due to Ruelle and Mayer (also following Lévy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the "Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions and of the "digital tree" algorithm for completely sorting n real numbers by means of ...

A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a "generating" operator. Its dominant spectral objects are linked with important parameters of the source such as the entropy, and play a central role in all the results. 1 Introduction. In information theory contexts, data items are (infinite) words that are produced by a common mechanism, called a source. Realistic sources are often complex objects. We work here inside a quite general framework of sources related to dynamical systems theory which goes beyond the cases of memoryless and Markov sources. This model can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, they attain a high level of generality. A probabilistic dynamical source ...

We develop a general framework for the analysis of algorithms of a broad Euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. 1

We obtain new results regarding the precise average bit-complexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be 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 an entire class of gcd-like algorithms. Keywords: Average-case Analysis of algorithms, Bit-Complexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the average-case around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...

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.

This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the average--case analysis of algorithms. The series should comprise the following chapters; 1. Symbolic Enumeration and Ordinary Generating Functions; 2. Labelled Structures and Exponential Generating Functions; 3. Parameters and Multivariate Generating Functions; 4. Complex Asymptotic Methods; 5. Singularity Analysis of Generating Functions; 6. Saddle Point Asymptotics; 7. Mellin Transform Asymptotics; 8. Functional Equations and Generating Functions; 9. Multivariate Asymptotics and Combinatorial Distributions. Chapters 1--3 have been issued as INRIA Research Report 1888 ("The Average Case Analysis of Algorithms: Counting and Generating Functions", 116 pages, 1993). Chapters 4--5 as INRIA Research Report 2026 ("The Average Case Analysis of Algorithms: Complex Asymptotics and Generating Functions", 100 pages, 1993). The present report corresponds to Chapter 6 of the series.

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.

Abstract. We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron’s formula and quasi-powers theorems. Such dynamical analyses can be used to perform the average-case analysis of algorithms, but also (dynamical) analysis in distribution. 1. Introduction. Computing the Greatest Common Divisor [Gcd

We provide here a complete average--case analysis of the three algorithms for computing the Jacobi symbol, for positive odd integers less than N . We analyse the average number of steps used for each of the algorithms. The average values are shown to be asymptotic to A1 log N or A2 log N for two of them, whereas it is asymptotic to A3 log 2 N for the third algorithm. The three constants A i are related to the invariant measure of the Perron-Frobenius operator linked to the dynamical system. More precisely, they can be expressed with the entropy of the system. 1 Introduction. The Jacobi symbol, introduced in [24], is a very important tool in algebra, since it is related to quadratic characteristics of modular arithmetics. Interest in its efficient computation is now reawakened with its utilisation in primality tests [40] or more generally in cryptography. The Jacobi symbol intervenes in the definition of the Quadratic Residuality Problem, and many cryptographic primitives are based o...