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
An analysis of Lehmer's Euclidean GCD algorithm
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Practical integer division with Karatsuba complexity
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Dynamical analysis of α-Euclidean algorithms
Journal of Algorithms - Analysis of algorithms
Analytic Analysis of Algorithms
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
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
Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
Journal of Symbolic Computation
Sharp estimates for the main parameters of the euclid algorithm
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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The Lehmer–Euclid Algorithm is an improvement of the Euclid Algorithm when applied to large integers. The original Lehmer–Euclid Algorithm replaces divisions on multi-precision integers by divisions on single-precision integers. Here we study a slightly different algorithm that replaces computations on $n$-bit integers by computations on $\mu n$-bit integers. This algorithm depends on the truncation degree $\mu\in ]0, 1[$ and is denoted as the ${\mathcal{LE}}_\mu$ algorithm. The original Lehmer–Euclid Algorithm can be viewed as the limit of the ${\mathcal{LE}}_\mu$ algorithms for $\mu \to 0$. We provide here a precise analysis of the ${\mathcal{LE}}_\mu$ algorithm. For this purpose, we are led to study what we call the Interrupted Euclid Algorithm. This algorithm depends on some parameter $\alpha \in [0, 1]$ and is denoted by ${\mathcal E}_{\alpha}$. When running with an input $(a, b)$, it performs the same steps as the usual Euclid Algorithm, but it stops as soon as the current integer is smaller than $a^\alpha$, so that ${\mathcal E}_{0}$ is the classical Euclid Algorithm. We obtain a very precise analysis of the algorithm ${\mathcal E}_{\alpha}$, and describe the behaviour of main parameters (number of iterations, bit complexity) as a function of parameter $\alpha$. Since the Lehmer–Euclid Algorithm ${\mathcal {LE}}_\mu$ when running on $n$-bit integers can be viewed as a sequence of executions of the Interrupted Euclid Algorithm ${\mathcal E}_{1/2}$ on $\mu n $-bit integers, we then come back to the analysis of the ${\mathcal {LE}}_\mu$ algorithm and obtain our results.