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.

. 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 algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of four algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods 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 Introduction Euclid's algorithm, discovered as early as 300BC, was analysed first in the worst case in 1733 by de Lagny, then in the average-case around 1969 independently by Heilbronn [8] and Dixon [5], and finally in distribut...

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...