Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms

Autores
Cesaratto, Eda; Clément, Julien; Daireaux, Benoit; Lhote, Loick; Maume, Veronique; Vallée, Brigitte
Año de publicación
2009
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
This paper is an extended complete version of  ´´Analysis of fast versions of Euclid Algorithm´´ presented in ANALCO´07. Among the differences here we deal with several Fast multiplication algorithms and we give precise estimates of the constants involved. 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 with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.nhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.
Fil: Cesaratto, Eda. Centre National de la Recherche Scientifique; Francia. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Clément, Julien. Centre National de la Recherche Scientifique; Francia
Fil: Daireaux, Benoit. No especifíca;
Fil: Lhote, Loick. Centre National de la Recherche Scientifique; Francia
Fil: Maume, Veronique. Universite Lyon 2; Francia
Fil: Vallée, Brigitte. Centre National de la Recherche Scientifique; Francia
Materia
EUCLID ALGORITHMS
DIVIDE AND CONQUER ALGORITHMS
FAST MULTIPLICATION
ANALYSIS OF ALGORITHMS
TRANSFER OPERATORS
PERRON FORMULA
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/242416

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oai_identifier_str oai:ri.conicet.gov.ar:11336/242416
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Regularity of the Euclid Algorithm; application to the analysis of fast GCD AlgorithmsCesaratto, EdaClément, JulienDaireaux, BenoitLhote, LoickMaume, VeroniqueVallée, BrigitteEUCLID ALGORITHMSDIVIDE AND CONQUER ALGORITHMSFAST MULTIPLICATIONANALYSIS OF ALGORITHMSTRANSFER OPERATORSPERRON FORMULAhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This paper is an extended complete version of  ´´Analysis of fast versions of Euclid Algorithm´´ presented in ANALCO´07. Among the differences here we deal with several Fast multiplication algorithms and we give precise estimates of the constants involved. 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 with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.nhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.Fil: Cesaratto, Eda. Centre National de la Recherche Scientifique; Francia. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Clément, Julien. Centre National de la Recherche Scientifique; FranciaFil: Daireaux, Benoit. No especifíca;Fil: Lhote, Loick. Centre National de la Recherche Scientifique; FranciaFil: Maume, Veronique. Universite Lyon 2; FranciaFil: Vallée, Brigitte. Centre National de la Recherche Scientifique; FranciaAcademic Press Ltd - Elsevier Science Ltd2009-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/242416Cesaratto, Eda; Clément, Julien; Daireaux, Benoit; Lhote, Loick; Maume, Veronique; et al.; Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms; Academic Press Ltd - Elsevier Science Ltd; Journal Of Symbolic Computation; 44; 7; 7-2009; 726-7670747-7171CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.jsc.2008.04.018info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0747717108001193info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:53:54Zoai:ri.conicet.gov.ar:11336/242416instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:53:54.926CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
title Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
spellingShingle Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
Cesaratto, Eda
EUCLID ALGORITHMS
DIVIDE AND CONQUER ALGORITHMS
FAST MULTIPLICATION
ANALYSIS OF ALGORITHMS
TRANSFER OPERATORS
PERRON FORMULA
title_short Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
title_full Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
title_fullStr Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
title_full_unstemmed Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
title_sort Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
dc.creator.none.fl_str_mv Cesaratto, Eda
Clément, Julien
Daireaux, Benoit
Lhote, Loick
Maume, Veronique
Vallée, Brigitte
author Cesaratto, Eda
author_facet Cesaratto, Eda
Clément, Julien
Daireaux, Benoit
Lhote, Loick
Maume, Veronique
Vallée, Brigitte
author_role author
author2 Clément, Julien
Daireaux, Benoit
Lhote, Loick
Maume, Veronique
Vallée, Brigitte
author2_role author
author
author
author
author
dc.subject.none.fl_str_mv EUCLID ALGORITHMS
DIVIDE AND CONQUER ALGORITHMS
FAST MULTIPLICATION
ANALYSIS OF ALGORITHMS
TRANSFER OPERATORS
PERRON FORMULA
topic EUCLID ALGORITHMS
DIVIDE AND CONQUER ALGORITHMS
FAST MULTIPLICATION
ANALYSIS OF ALGORITHMS
TRANSFER OPERATORS
PERRON FORMULA
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv This paper is an extended complete version of  ´´Analysis of fast versions of Euclid Algorithm´´ presented in ANALCO´07. Among the differences here we deal with several Fast multiplication algorithms and we give precise estimates of the constants involved. 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 with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.nhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.
Fil: Cesaratto, Eda. Centre National de la Recherche Scientifique; Francia. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Clément, Julien. Centre National de la Recherche Scientifique; Francia
Fil: Daireaux, Benoit. No especifíca;
Fil: Lhote, Loick. Centre National de la Recherche Scientifique; Francia
Fil: Maume, Veronique. Universite Lyon 2; Francia
Fil: Vallée, Brigitte. Centre National de la Recherche Scientifique; Francia
description This paper is an extended complete version of  ´´Analysis of fast versions of Euclid Algorithm´´ presented in ANALCO´07. Among the differences here we deal with several Fast multiplication algorithms and we give precise estimates of the constants involved. 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 with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.nhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity mu(n) and stop the recursion at a depth slightly smaller than log n. A rough estimate of the worst--case complexity of these fast versions provides the bound O ( mu(n)log n). 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 Theta (mu(n) log n , with a precise remainder term, and estimates of the constant in the Theta--term. Our analysis applies to any cases when the cost mu(n) is of order Omega(n log n), and is valid both for the FFT multiplication algorithm of Schönhage--Stassen, but also for the new algorithm introduced quite recently by Fürer . 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 psi which plays a central rôle since it always appear at the beginning of each recursive call. This strong regularity phenomenon proves that the 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.
publishDate 2009
dc.date.none.fl_str_mv 2009-07
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/242416
Cesaratto, Eda; Clément, Julien; Daireaux, Benoit; Lhote, Loick; Maume, Veronique; et al.; Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms; Academic Press Ltd - Elsevier Science Ltd; Journal Of Symbolic Computation; 44; 7; 7-2009; 726-767
0747-7171
CONICET Digital
CONICET
url http://hdl.handle.net/11336/242416
identifier_str_mv Cesaratto, Eda; Clément, Julien; Daireaux, Benoit; Lhote, Loick; Maume, Veronique; et al.; Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms; Academic Press Ltd - Elsevier Science Ltd; Journal Of Symbolic Computation; 44; 7; 7-2009; 726-767
0747-7171
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jsc.2008.04.018
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0747717108001193
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Ltd - Elsevier Science Ltd
publisher.none.fl_str_mv Academic Press Ltd - Elsevier Science Ltd
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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