A double-digit Lehmer-Euclid algorithm for finding the GCD of long integers
Journal of Symbolic Computation - Special issue on design and implementation of symbolic computation systems
Practical integer division with Karatsuba complexity
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
On convergence rates in the central limit theorems for combinatorial structures
European Journal of Combinatorics
Parallel Implementation of Schönhage's Integer GCD Algorithm
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Dynamical analysis of a class of Euclidean algorithms
Theoretical Computer Science - Latin American theoretical informatics
Dynamical Analysis of the Parametrized Lehmer–Euclid Algorithm
Combinatorics, Probability and Computing
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Analytic Combinatorics
Sharp estimates for the main parameters of the euclid algorithm
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Pseudorandomness of a random kronecker sequence
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
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There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schonhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity @m(n) and stop the recursion at a depth slightly smaller than lgn. A rough estimate of the worst-case complexity of these fast versions provides the bound O(@m(n)logn). Even the worst-case estimate is partly based on heuristics and is not actually proven. Here, we provide a precise probabilistic analysis of some of these fast variants, and we prove that their average bit-complexity on random inputs of size n is @Q(@m(n)logn), with a precise remainder term, and estimates of the constant in the @Q-term. Our analysis applies to any cases when the cost @m(n) is of order @W(nlogn), and is valid both for the FFT multiplication algorithm of Schonhage-Strassen, but also for the new algorithm introduced quite recently by Furer [Furer, M., 2007. Faster integer Multiplication. In: Proceedings of STOC'07. pp. 57-66]. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain two main results about the (plain) Euclid Algorithm, which are of independent interest. We precisely describe the evolution of the distribution of numbers during the execution of the (plain) Euclid Algorithm, and we exhibit an (unexpected) density @j which plays a central role since it always appears at the beginning of each recursive call. This strong regularity phenomenon proves that the interrupted algorithms are locally ''similar'' to the total algorithm. This ultimately leads to the precise evaluation of the average bit-complexity of these fast algorithms. This work uses various tools, and is based on a precise study of generalised transfer operators related to the dynamical system underlying the Euclid Algorithm.