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21
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 28 (12 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
A Binary Recursive Gcd Algorithm
 Proceedings of ANTS’04, Lecture Notes in Computer Science
, 2004
"... Abstract. The binary algorithm is a variant of the Euclidean algorithm that performs well in practice. We present a quasilinear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modified version of the binary algorithm. The structure of our ..."
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Abstract. The binary algorithm is a variant of the Euclidean algorithm that performs well in practice. We present a quasilinear time recursive algorithm that computes the greatest common divisor of two integers by simulating a slightly modified version of the binary algorithm. The structure of our algorithm is very close to the one of the wellknown KnuthSchönhage fast gcd algorithm; although it does not improve on its O(M(n) log n) complexity, the description and the proof of correctness are significantly simpler in our case. This leads to a simplification of the implementation and to better running times. 1
Dynamical analysis of αEuclidean algorithms
 J. Algorithms
"... Abstract We study a class of Euclidean algorithms related to divisions where the remainder is constrained to belong to [α −1, α], for some α ∈ [0, 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the s ..."
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Cited by 9 (3 self)
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Abstract We study a class of Euclidean algorithms related to divisions where the remainder is constrained to belong to [α −1, α], for some α ∈ [0, 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the socalled "dynamical analysis" method, where dynamical systems are made a deep use of. Here, the dynamical systems of interest have an infinite number of branches and they are not Markovian, so that the general framework of dynamical analysis is more complex to adapt to this case than previously. 2002 Elsevier Science (USA). All rights reserved.
Euclidean dynamics
 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
, 2006
"... 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 ope ..."
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Cited by 4 (2 self)
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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 quasipowers theorems. Such dynamical analyses can be used to perform the averagecase analysis of algorithms, but also (dynamical) analysis in distribution.
Analysis of fast versions of the Euclid Algorithm
 Proceedings of ANALCO’07, Janvier 2007
"... There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the wors ..."
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There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the worst–case complexity of these fast versions provides the bound O(n(log n) 2 log log n). However, this estimate is based on some 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 Θ(n(log n) 2 log log n), with a precise remainder term. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain three results about the (plain) Euclid Algorithm which may be of independent interest. We precisely describe the evolution of the distribution during the execution of the (plain) Euclid Algorithm; we obtain a sharp estimate for the probability that all the quotients produced by the (plain) Euclid Algorithm are small enough; we also exhibit a strong regularity phenomenon, which proves that these interrupted algorithms are locally “similar ” to the total algorithm. This finally 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. 1
Delange’s Tauberian theorem and asymptotic normality of random ordered factorizations of integers
, 2009
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MontesRodŕıguez PerronFrobenius operators and the KleinGordon equation
 J. Eur. Math. Soc. (JEMS
"... Abstract. For a smooth curve and a set in the plane R 2 , let AC( ; ) be the space of finite Borel measures in the plane supported on , absolutely continuous with respect to arc length and whose Fourier transform vanishes on . Following [12], we say that ( , ) is a Heisenberg uniqueness pair if AC( ..."
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Abstract. For a smooth curve and a set in the plane R 2 , let AC( ; ) be the space of finite Borel measures in the plane supported on , absolutely continuous with respect to arc length and whose Fourier transform vanishes on . Following [12], we say that ( , ) is a Heisenberg uniqueness pair if AC( ; ) = {0}. In the context of a hyperbola , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets of a collection of solutions to the KleinGordon equation. In this work, we mainly address the issue of finding the dimension of AC( ; ) when it is nonzero. We will fix the curve to be the hyperbola x 1 x 2 = 1, and the set = α,β to be the latticecross α,β = (αZ × {0}) ∪ ({0} × βZ) , where α, β are positive reals. We will also consider + , the branch of x 1 x 2 = 1 where x 1 > 0. In
FINE COSTS FOR THE EUCLID ALGORITHM ON POLYNOMIALS AND FAREY MAPS
"... Abstract. This paper studies digitcost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bitcomplexity is defined with respect to the degree of the quotients; we focus here on a no ..."
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Abstract. This paper studies digitcost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bitcomplexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine ’ complexity (and on associated costs) which relies on the number of their nonzero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law. We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers ‘step by step ’ each nonzero monomial of the quotient, so its number of steps is closely related to the number of nonzero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field.