A simple unpredictable pseudo random number generator
SIAM Journal on Computing
On the worst case of three algorithms for computing the Jacobi symbol
Journal of Symbolic Computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Continued fraction algorithms, functional operators, and structure constants
Theoretical Computer Science
Analytic Analysis of Algorithms
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
A Unifying Framework for the Analysis of a Class of Euclidean Algorithms
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction
Combinatorics, Probability and Computing
Dynamical Analysis of the Parametrized Lehmer–Euclid Algorithm
Combinatorics, Probability and Computing
Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
Journal of Symbolic Computation
| Hi-index | 0.00 |
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.