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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/242416
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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 |
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CONICET Digital (CONICET) |
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CONICET Digital (CONICET) |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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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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13.070432 |