efficient and practical algorithms for sequential modular ... · hal id: inria-00072855 submitted...

26
ISSN 0249-6399 ISRN INRIA/RR--3804--FR+ENG apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Efficient and Practical Algorithms for Sequential Modular Decomposition Elias Dahlhaus, Jens Gustedt, Ross M. McConnell N˚3804 Novembre 1999 THÈME 1

Upload: others

Post on 23-May-2020

5 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--38

04--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Efficient and Practical Algorithms forSequential Modular Decomposition

Elias Dahlhaus, Jens Gustedt, Ross M. McConnell

N˚3804

Novembre 1999

THÈME 1

Page 2: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and
Page 3: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

������������ �������� ��������� �������������� ��!�"$#�%&�'�( �)+*������� ��-,.�/�0*��� '�213��4�5"768��#9�:������

;'<>=>?A@CBD?AE�<FEG?AH�@JILK�MANLO�@�PQH�@SRTNLU�R&V&KXWZYA@J@0[]\�[_^a`+YbO�O�Nc<><ed

fhgjiTkmlon�prq5s&t�lvuLwbxylTz5t:{bt:z�iTkml&t|~}��L� lTz5q�s&t�s&�juat

q�uL�����a}:z5��l�}�l&��gjlT}��Sgjl3�0�������L��pr������lTkZ��}�l8nv�����0p����D�9uL��l&t

�o�4�v�v�c�������  ¢¡Q£�¤a¥�¦¨§ �L©ªuL�«w��j�b¬­}�l&�Tz�l&����}SuL��g¯®±°¢²´³~µ�¶�·5¬¨t�u¸t�lTz�¹º�L©»��lT}:z:¬¨�&l&t+z:g9uLz�g9uv��l�z:gjl0tSuLkmlt�lTz��L©~�jlT¬­��g¼���a}�t�¬­�½³¿¾'¹ÁÀ�fhgjl ¡m£v¤a¥A¦ÃÂaÄm¤¼§�Å�£a¡�Æb£LÇJÈ>ÉeÈÊ£aË ¬¨t�umw���¬¨Ì�wjlD��l&�&�akQ����t:¬­z:¬¨�a���L©~z:gjl0��lT}:z:¬¨�&l&t�¬­�¼z���jl&t:z�l&��km���bw�ͨl&tvÀ~νl���¬­��lZuQt:¬­kQ��ͨlT}�uL����}���ua�Sg-z��Qt�l&Ì�wjlT�¼z:¬ÏuLÍ4Í­¬­�jlvuL} дz:¬­kmlCkm�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a�XÀÑyÒ¼Ó�Ô�Õ0Ö �c×���� km�A�bw�ÍÏuL}5��l&�&�akQ����t�¬­z:¬¨�a�XØjÍ­¬­�jlvuL}�z:¬­kmlZuLÍ­���a}:¬­z:g�kmt

ÙÃÚXÛÝÜßÞvà'ÛháXâ­Ü´ãÊä&å

æ�çéè ê�ë:ì�íïî�ð�íJñ'êcòcë´èôó4õvö ÷øè ùLöôèúïùLû+çéè ê�ë:ìcíSî9ü�úSë´ýcè ñ�úSë´÷øö þ ÿ���ùc÷���ì�íïîjð�í��øí��ïùLèJÿ¼ð�í��øí��JùcèJÿ��èÝó´ñ�úïù�&ì�� �������� úïùLû ���������� íïóÊóßúS÷øùLèJÿ����������! �úïùaû#"ò��&ó´è���$ôþ � úïùcö%&ÿ'&cóßúïùcö èJÿ�(�)�*�+-,�.�/�+�0�)�120�3�425�6#7�89,�:�6Lì; çéè ê¼ì�íïî�ð�íJñ'êcòcë´èôó4õvö ÷øè ùLöôèúïùLû=<9ù'�J÷øùLèôèôó´÷øù��cÿ���ùc÷���ì�íïîjð�í��øíïóßúïûcí5úSë�çéè ù>�TèÝó:ÿvçéè ù>�Tèôó:ÿLð �@? ��A>BDCFE�G�G����H�éõ � ì

Unité de recherche INRIA LorraineTechnopôle de Nancy-Brabois, Campus scientifique,

615 rue de Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY (France)Téléphone : 03 83 59 30 30 - International : +33 3 3 83 59 30 30

Télécopie : 03 83 27 83 19 - International : +33 3 83 27 83 19Antenne de Metz, technopôle de Metz 2000, 4 rue Marconi, 55070 METZ

Téléphone : 03 87 20 35 00 - Internationala: +33 3 87 20 35 00Télécopie : 03 87 76 39 77 - International : +33 3 87 76 39 77

Page 4: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

�����5�'���:��!�"$�#260�� �������)�*��#2���2�é���� ��4�# 68��*0��� ����4��"76o��#�� ������ "$�/�0*��� ������¢#��)�*�������������

��� ���� � � �_km�A�bw�ͨl8��� w��Á��}SuL��gjl8�j�a�bÐß�a}:¬¨lT��z�s-® ° ²´³~µ�¶�·0l&t:z�w���lT�jt�lTkZ��ͨl8¹ ��l8t��akQkmlTz�tD̼w�¬�9uL}:zSuL��l¸Í¨lDk� TkmlmlT�jt�lTkZ��ͨl¸��lQ���a¬¨t:¬­�jtClT�_³�¾�¹ÁÀ���u ��s&�&�akQ����t:¬­z:¬¨�a�¯km�A�bw�ÍÏuL¬­}�lml&t:z/w��jlm��s&�&�akQ����t:¬­z:¬¨�a�w���¬¨Ì�wjl0��l&t�t��akQkmlTz�t�lT�-km�A�bw�ͨl&t�lTk0�����¨z�s&t�À����awjt+���a���j�a�jt�w��jl0uL����}��A��gjlDt:¬­kQ��Í­¬��9s&l��QÍÏum��s&�&�akQ����t:¬­z:¬¨�a�km�A�bw�ÍÏuL¬­}�lZlT�yz�lTkQ�jt5Í­¬­�jsvuL¬­}�l�À� Ö �v� Ô ��� � � ��s&�&�akQ����t:¬­z:¬¨�a�-km�A�bw�ÍÏuL¬­}�l�Ø9uLÍ­���a}:¬­z:g�kml&t�Í­¬­�jsvuL¬­}�l&t

Page 5: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË �

� � ��9���/��*0��������

�«���bw�ÍÏuL}m��l&�&�akQ����t:¬­z:¬¨�a� ¬¨tou«z:}�l&lJдͭ¬���l���l&�&�akQ����t�¬­z:¬¨�a� �L©�u«��}SuL��g z:g9uLz�� l���l&t��T}:¬­��l ��lTͨ���CÀÁÎ]gjlT� z:gjl}�l&t:w�Í­z:¬­���-z:}�l&lD¬¨t/�j�a�¼z:}:¬­�A¬ÏuLÍ´ØX¬­z/��}�����¬¨��l&t�u�� u&{ z���uL����Í­{�u �b¬­��¬¨��lJÐÝuL�j�AÐß�&�a�jÌ�wjlT}0t�z:}SuLz�lT��{�z���u8��w�kZ��lT}Z�L©�+|éÐß�&�akQ��ͨlTz�l«��}��a��ͨlTkmtvÀ¿fhg¼wjt�Ø�u�ÍÏuL}:��l���w�kZ��lT}y�L©/�+|éÐß�&�akQ��ͨlTz�l���}��a��ͨlTkmt �vuL� ��l�t��aÍ­��l&� }�l&�Tw�}�t�¬­��lTÍ­{�a� z:gjl�z:}�l&l�¬­� z:¬­kml���}��a���a}:z:¬¨�a�9uLÍ/z��Áz:gjl��¼w�k0��lT}��L©Z��lT}:z:¬¨�&l&t z:¬­kml&t����! #"%$SØ&�'gjlT}�l(' ¬¨toz:gjl�k¸ucxb¬­kZw�k��w�kZ��lT}+�L©~��g�¬­Í¨�b}�lT� �L©»uQ�j�A��l/¬­�-z:g�¬¨t5z:}�l&l�Ø)�'g�¬¨�Sg���¬­��l&t5}:¬¨t�lCz��Q���aÍ­{A�j�akQ¬ÏuLÍ�uLÍ­���a}:¬­z:g�kmt�©e�a}5��}SuL��gjt*�'gj��t�l��l&�&�akQ����t:¬­z:¬¨�a�-z:}�l&l&t�uL}�lC�L©����aw��j��l&����lT��}�l&l�À

��lTz�+ ��lT�j�az�l z:gjl ��w�kZ��lT}¸�L©���lT}:z:¬¨�&l&t�تuL�j��ͨlTz-, ��lT�j�az�l z:gjl �¼w�kZ��lT}¸�L©�l&�b��l&tvÀ¯fhgjlT}�l g9uv��ly��l&lT�u��¼w�k0��lT}Q�L© .¸²+�/�·JØ0.m²+�1v·JØ2.m²+�,�·JØ~uL�j�3.¸²+�4&·0uLÍ­���a}:¬­z:g�kmtD©Ê�a} �j�j�b¬­��� km�A�bw�ÍÏuL}Q��l&�&�akQ����t:¬­z:¬¨�a�XØ»t:wj��guat65 wjlT}�uL�j�7�98ag�}:¬­���9: nv�����<;JØ>=~g�}�lT�b©elTwj�Sg¼z/lTz�uLÍ´À?: nv���L�@;JØBA��aÍ­w�kZ��¬¨�C: nv�<D�D�;JØ>E�uL��¬­��uL�j�7� uLw�}�lT}�: nv�<Da�<;JØ�«��F �a���jlTÍ­ÍG: nv����H<;JØI�-w�ͭͨlT}'uL�j�CJ���¬­��}Sua�(: nv�����<;JØ)J�z�lT¬­�jlT}K: nv�����<;JØbt��akml��L©4z:gjlTk3©Ê�a} t���l&�T¬ÏuLÍ��vuat�l&t �a}ª��lT�jlT} ÐuLÍ­¬�L�uLz:¬¨�a�jt��L©4z:gjl���}��a��ͨlTk�À»fhgjl+�&�az:}�l&l/��l&�&�akQ����t:¬­z:¬¨�a�y�L©��&�a��}SuL��gjt�uL�j�8z:gjl�t�lT}:¬¨l&t д�9uL}SuLͭͨlTÍX��l&�&�akQ����t:¬­z:¬¨�a��L©4tSlT}:¬¨l&t д�9uL}SuLͭͨlTÍG�9uL}:z:¬ÏuLÍG�a}���lT}�thuL}�l�t:��l&�T¬ÏuLÍ��vuat�l&th�a�o��}SuL��gjt�uL�j�8�b¬­��}SuL��gjtvØb}�l&t:��l&�Tz:¬­��lTÍ­{�ØA©e�a}0�'g�¬¨��gyÍ­¬­�jlvuL} Ðz:¬­kml/t��aÍ­w�z:¬¨�a�jt�g9u&��l/��l&lT� ��¬­��lT�XØjt�l&lKF �a}:�jlT¬­Í�lTz�uLÍ´À�: nv����H<;JØIM»uLͨ��l&t�lTz�uLÍ´À�: nv�����<;JÀ0.¸²+ONP,RQ�S�T2+4·JØb���aw��j��t©Ê�a}~uL}:��¬­z:}SuL}:{Zw��j�b¬­}�l&�Tz�l&�D��}SuL��gjtU�ªlT}�lhz:gjlT�D��¬­��lT�D¬­�VF �aw�}:��¬¨lT}~uL�j��E�uL��¬­�7: nv�����<;JؼuL�j�QuL�V.m²+WN�,YXª²,�µ�+4·:·���aw��j��� uat'��¬­��lT� ¬­�YJ���¬­��}Sua�Z: nv�����<;JÀfhgjl �j}�t:z Í­¬­�jlvuL} дz:¬­kmlZuLÍ­���a}:¬­z:g�k[� uath��¬­��lT� ¬­������F �a���jlTÍ­Í�uL�j�7J���¬­�bÐ}Sua�\: nv���L�@;JÀ   � uLÍ­z��a��lTz:gjlT}m�b¬#]�lT}�lT��zQÍ­¬­�jlvuL} дz:¬­kml�� uatD��¬­��lT� t:gj�a}:z:Í­{_z:gjlT}�lvuc©>z�lT}Q¬­�3F �aw�}:��¬¨lT}¸uL�j�^E+uL��¬­�: nv���L�@;JÀ05 �az:g��L©éz:gjl&t�lZuLÍ­���a}:¬­z:g�kmt+uL}�l/ͨlT����z:g¼{-uL�j�yÌ�w�¬­z�lC�Sg9uLͭͨlT����¬­���¸z��Qw��j��lT}�t:zSuL�j�GÀ

νlo��lT}:¬­��l¸uy�&�a�j�&lT��z:w9uLÍ­Í­{¯t�¬­kQ��ͨlT}CÍ­¬­�jlvuL} дz:¬­kmlotSl&̼wjlT��z:¬ÏuLͪuLÍ­���a}:¬­z:g�k z:g9uL�P� uatC��}�lT�A¬¨�awjt:Í­{½u&��uL¬­ÍÏuL��ͨl�À_Ý�-ua���b¬­z:¬¨�a�XØ)� l/��¬­��l`.m²+aNb,YX ²,�µ�+4·:· ��uL}:¬ÏuL�¼z�z:g9uLz*�ªl/��lTÍ­¬¨lT��l/¬¨t't:¬­kQ��ͨlClT�j�aw���g-z��Q��l��Sg9uL}Sua�Tz�lT}:¬�Lvl&�«uat��}Sua�Tz:¬¨�vuLÍ´À»Î½lCt���lTz���gjl&��z:g�¬¨t���lT}�t:¬¨�a��¬­�7c/uLg�Í­g9uLwjt�lTz�uLÍ´À�: nv���<D�;JÀ

d ������S�S"$��� '���S�#

fhg�¬¨tot�l&�Tz:¬¨�a� ��¬­��l&tmz:gjl �9uat�¬¨����l �j��¬­z:¬¨�a�jto�L©�z:gjl��a�b�:l&�Tz�t¸uL�j� uLÍ­���a}:¬­z:g�kQ¬¨�-©elvuLz:w�}�l&tmz:g9uLzO�ªl«uL}�l zSuLÍ���¬­���uL���aw�zvÀG_Ýz ¬¨t't���Í­¬­z ¬­��z��Dze� �Q�9uL}:z�tvÀ~fhgjl �j}�t:z'�9uL}:z'¬­��z:}����bwj�&l&t'z:gjl��9uat�¬¨���j�azSuLz:¬¨�a�«uL�j�8©Êua�Tz�t�uL���aw�zhkm���bw�ÍÏuL}��l&�&�akQ����t:¬­z:¬¨�a�XÀ»fhgjl/tSl&�&�a�j����l&t��T}:¬­��l&t�t��akml/l&t�t�lT��z:¬ÏuLÍ��juLzSuDt:z:}:wj�Tz:w�}�l&t5z:g9uLz5uL}�l��jl&l&��l&�yz��Qg9uL�j�bͨl���}SuL��gjtlgf¸�T¬¨lT��z:Í­{�À

hUikj lnmUo0p2qer�stoGuwvxm?y{z|m?}�~��<~�m?��m��`p0�2o0~�s<u>v���u>o6��s�r�zW�2}_Ý��z:g�¬¨t/�9uL��lT}vØXz:gjlQz�lT}:k �aÄ:ÂïÆ>� �'¬­Í­Í~��lT�j�az�l¸uL�½w��j�b¬­}�l&�Tz�l&�¯��}SuL��gXØ��'g�¬­Í¨lQz:gjlDz�lT}:k ¤aÈ#��Ä ÂïÆw� �'¬­Í­Í»��lT�j�az�l¸u�b¬­}�l&�Tz�l&�-��}SuL��gXÀνl/ͨlTz|+½��lT�j�az�l/z:gjl/��w�kZ��lT}��L©é��lT}:z:¬¨�&l&t�uL�j��, z:gjl���w�kZ��lT}��L©l&�b��l&tvÀ

��lTzh®2° ²´³µ�¶D·é��l�u���}SuL��gV�'¬­z:gm��lT}:z�lJxmt�lTzh³�uL�j�¸l&�b��l�t�lTz»¶oÀ�_ß©�¹ ¬¨t»uCt:w��jt�lTz»�L©�³mؼz:gjlT�8®O� ¹$¬¨t~z:gjlÇJ¥I����Ä ÂïÆw�8È>Ë�¤a¥�ÅS§:¤V�%� ®DÀfhgjl Å�£a¡�Æ�¦¨§J¡¸§JË9É �L©�u���}SuL��gy® ¬¨t~z:gjl5��}SuL��g ®R�'¬­z:g¸z:gjl�tSuLkml5t�lTzª�L©���lT}:z:¬¨�&l&t�ؼ��w�z�'gjlT}�lO����µ��)�D¬¨t�uL��l&�b��lC¬#]�¬­z�¬¨t��j�az�uL�-l&�b��lZ�L©ª®DÀ   Å�£aËjË�§:Å&Éߧ:¤�Å�£a¡�Æb£aË�§TË9É ¬¨t+uDk¸ucxA¬­k¸uLÍ�t�lTz5�L©��lT}:z:¬¨�&l&tz:g9uLzCuL}�lDuLÍ­Í��&�a���jl&�Tz�l&� ��{��9uLz:gjtvÀ   Å�£�� Å�£a¡�Æb£aË�§TË9É ¬¨t/u¸�&�a���jl&�Tz�l&� �&�akQ���a�jlT�¼zC�L©~z:gjlD�&�akQ��ͨlTkmlT�¼zvÀ/fhgjlË�§TÈ#���I�:£aÄ��A£v£v¤ �L©~u���lT}:z�lJxC�8¬¨t'z:gjl/t�lTz

��²�b·»°����������9µ��K��¬¨t5uL�-l&�b��l/�L©»®��<�

  �9uL¬­}K���4µ��)�D�L©é��lT}:z:¬¨�&l&t�z:g9uLz�¬¨t5�j�az�uL��l&�b��l/¬¨t+u Ë�£aË>� §:¤���§ ÀhfhgjlZt�lTz5�L©é��lT}:z:¬¨�&l&t�¬­�«³ ¾����)��z:g9uLz+uL}�lC�j�az�jlT¬­��g����a}�t+�L©�� uL}�l/¬­z�t Ë�£�Ë)�ßË�§TÈ#���I�:£aÄïÇ Ø9uL�j�y¬¨t5��lT�j�az�l&� �Á²��b·JÀ

 7ÇTÉeÄ £�Ë@�«Ë�§JÈ����x�:£�Ä �L©5u�t�lTz0¹�� ³$�L©'��lT}:z:¬¨�&l&t�¬¨t�u���lT}:z�lJxZ�3� ³ ¾�¹ z:g9uLz0¬¨t�u��jlT¬­��g����a}D�L©5lT��lT}:{��lT}:z�lJx��L©¹_À   Ë�£aË>�ßË�§JÈ����x��£aÄ �L©�¹ ¬¨t5uD��lT}:z�lJx7 a��³]¾h¹ z:g9uLz�¬¨t��j�az�uD�jlT¬­��g¼���a}��L©~uL�¼{ ��lT}:z�lJxy¬­��¹ÁÀ  ¡m£v¤a¥A¦¨§ �L©4uL�ow��j�b¬­}�l&�Tz�l&�¸��}SuL��g�® °±²´³~µ�¶�·~¬¨thu/t�lTz»¹ �L©G��lT}:z:¬¨�&l&t z:g9uLz uLÍ­Í�g9u&��l�z:gjl5tSuLkml�t�lTzª�L©G�jlT¬­��g¼���a}�t¬­��³ ¾h¹_Ø9tSl&lK¡�¬­��w�}�l¸n�À fhg9uLz5¬¨tvØ�¹ ¬¨t+uDkm���bw�ͨlZ¬#]½lT��lT}:{8kmlTk0��lT}��L©ª³ ¾ ¹ ¬¨t5lT¬­z:gjlT}+umt:z:}��a���¸�jlT¬­��g����a}�a}5u��j�a�bд�jlT¬­��g����a}vÀªfhgj�aw���g�z:gjlT}�l/¬¨t�u���lT�jlT}SuLÍ­¬�L�uLz:¬¨�a���L©ékm�A�bw�ͨl&t'z��Q�b¬­��}SuL��gjtvØx�ªlK�'¬­Í­ÍG�j�az'}�lJ©elT}'z��Dz:gjlTk¬­��z:g�¬¨t��9uL��lT}vÀ

��� ù0¢2G ? ���

Page 6: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

¡�¬­��w�}�lmn��   ��}SuL��g«uL�j�y¬­z�t'km�A�bw�ͨl&tvÀ

f?�ª�mtSlTz�tª¹ uL�j�� £��L§JÄJ¦­ÂïÆ ¬Ã©�¹ ¾� 8Ø� ¾~¹_Ø�uL�j�� ��m¹ uL}�l/uLÍ­ÍG�j�a�jlTkQ��zß{�À¹��� 3��lT�j�az�l&t0²>¹ ¾� Q·��²� ¿¾ª¹ ·JØ�z:gjl Çg�a¡�¡¸§TÉeÄJÈÊÅ0¤aÈ ��§JÄ�§JË�ÅS§ �L©� uL�j�y¹ÁÀfhgjl�©Ê�aͭͨ���'¬­���mzk�ª�m�a�jt�lT}:��uLz:¬¨�a�jt�uL}�l��j�az5g9uL}�� z��D��lT}:¬Ã©>{�À

�»Ò � �¯���! �"$# �&%(' ��)+*-,�#."0/�1(2!343�5 ¹ ÂaË�¤ ÂaÄ�§o£��L§JÄJ¦ÃÂJÆ�Æ�È>Ë@��¡Q£�¤a¥�¦¨§ïÇ � É�b§JË ¹7¾6 � �¾5¹ � &��¹ � 7�o¹ � ÂaË�¤ 8�D¹ ÂaÄ�§DÂa¦ ÇT£¸¡m£v¤a¥A¦­§ïÇ9�  �9uL¬­}�¹_Ø: �L©�b¬¨tÊ� �a¬­��z�km�A�bw�ͨl&t5¬¨t Â�¤<;&Â�ÅS§TË9É ¬Ã©~lT��lT}:{ykmlTk0��lT}��L©� ¢¬¨t�ua�L��ua�&lT�¼z+z��olT��lT}:{ykmlTk0��lT}��L©~¹_ØuL�j� Ë�£aË�Â�¤<;&Â�ÅS§TË9É ¬Ã©é�j�DkmlTkZ��lT}��L©= 3¬¨t5ua�L��ua�&lT�¼z5z��¸uL�¼{ kmlTkZ��lT}5�L©é¹_À�»Ò � �¯���! >�?# �&%(' ��)+*-,�#."0/�1(2!343t��Ë)�+ÆbÂ�ÈFÄ ¹ ÂaË�¤ £ C¤aÈFÇ@;&£aÈ>Ë9É¡m£v¤a¥A¦¨§ïÇCÈÏÇZ§JÈeÉ�b§JÄ0¼¤<;T¼ÅS§JË9Éh£aÄ/Ë�£aË�Â�¤��;&Â�ÅS§JË�ÉA�

  Å�£aË@�aÄJ¥j§JË�ÅS§/ÆbÂaÄTÉeÈeÉeÈe£aË ¬¨t�um�9uL}:z:¬­z:¬¨�a�½�L©'³ �'gjlT}�lDlvua��g �9uL}:z:¬­z:¬¨�a���TÍÏuat�t+¬¨t�u¸km�A�bw�ͨl�À+fhgjlDua�L��ua�&lT�j�T{}�lTÍÏuLz:¬¨�a� �a� kmlTk0��lT}�t �L©Zu½�&�a����}:wjlT�j�&l«�9uL}:z:¬­z:¬¨�a� ¬¨tm¬­z�t�lTÍé0u½��}SuL��gXØ �'gjlT}�l�lvua�Sg t�lTz8�L©/z:gjl-�&�a����}:wjlT�j�&l�9uL}:z:¬­z:¬¨�a�Á¬¨tCz:}�lvuLz�l&�Áuat0uyt:¬­����ͨl¸��lT}:z�lJx�À fhg�¬¨tZ��}SuL��g¯¬¨tZ��lT�j�az�l&� ®CBEDmÀ�®CBED$�&�akQ��ͨlTz�lTÍ­{¯t:��l&�T¬��9l&tCz:gj��t�ll&�b��l&tD�L©+® z:g9uLzDuL}�l¸�j�az�t:w��jtSlTz�t0�L©5u�t:¬­����ͨlm�9uL}:z:¬­z:¬¨�a� �TÍÏuat�t�:et�l&la¡�¬­��w�}�l8�<;JÀ  6T¼Å&É´£aÄ ¬­��u��&�a����}:wjlT�j�&l

C

D

B

IH

G

AG:

G/P:

{A..I}: Members of congruence partition P

G

F

I

H

F

E

A E

D

C

B

¡4¬­��w�}�l0�0�   ��}SuL��g7�'¬­z:g��&�a����}:wjlT�j�&lC�9uL}:z:¬­z:¬¨�a�«uL�j��z:gjl/�&�a}:}�l&t:���a�j�b¬­���8̼wj�az:¬¨lT��zvÀ

�9uL}:z:¬­z:¬¨�a�FD ¬¨t uL�8¬­�j�bwj�&l&� t:w�����}SuL��gy®V� ¹ ©e�a} t��akml�¹ ��DmÀfhgjl+t:w�����}SuL��gjt ¬­�j�bwj�&l&�o�¼{mz:gjl��9uL}:z�tht:��l&�T¬Ã©e{� ����� �

Page 7: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË H

{a,b}

{i,j,k,l}

{i,j,k}

(1)

{a,b,c}(0) (0)

(1)

(1){g,h}

V(R) (0)

(P){a,b,c,d,e,f,g,h}

{d,e}

gedc

lf

h

b

(0)

a

i j k

a g

f

e

d

c

b

h

l

k

j

i

¡�¬­��w�}�lZ�0�~fhgjl/km�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a��z:}�l&lC�L©éz:gjl/��}SuL��g-�L©�¡�¬­��w�}�lmn�À

��}�l&�T¬¨t�lTÍ­{-z:gj��t�l0l&�b��l&t��L© ® z:g9uLzCuL}�lC�j�az�t:��l&�T¬��9l&���¼{-z:gjl0Ì�wj�az:¬¨lT�¼zvÀ+fhg¼wjtvØ���¬­��lT��um�&�a����}:wjlT�j�&lD�9uL}:z:¬­z:¬¨�a�DmØI� l/k¸u&{yt:��l&�T¬Ã©e{y® ��{ ��¬­��¬­���Qz:gjlCÌ�wj�az:¬¨lT�¼z/®CBED uL�j�-®V� ¹ ©Ê�a}�lvua��g-�9uL}:z:¬­z:¬¨�a���TÍÏuat�t'¹ ��DmÀ

³oØ��ouL�j�yz:gjlZt�¬­����ͨlTz��a��t:w��jt�lTz�t��L©~³ tïuLz:¬¨t ©>{8z:gjlC��l �j��¬­z:¬¨�a� �L©»uDkm���bw�ͨl�Ø�uL�j��uL}�l/z:gjl ÉeÄïÈ+�vÈÊÂa¦X¡m£v¤a¥A¦¨§ïÇ�L©h®QÀ   t�uo}�l&t:w�Í­zvØGlT��lT}:{���}SuL��g�g9uat/uLz�ͨlvuat:z+ze�ª�y�&�a����}:wjlT�j�&lD�9uL}:z:¬­z:¬¨�a�jt�ØX�a�jl��'¬­z:g�uot:¬­����ͨl��9uL}:zZ³mØGuL�j��a�jl`�'¬­z:g^� ³��b�9uL}:z�tvØ>�'gjlT}�l0lvua�Sg��9uL}:z�¬¨t�u¸t:¬­����ͨlTz��a�«t�w��jt�lTz5�L© ³¸À   ��}SuL��g�¬¨t Æ�ÄJÈ>¡m§ ¬Ã©¬­z5g9uat+�a��Í­{ z:}:¬­��¬ÏuLÍkm�A�bw�ͨl&tvØ�gjlT�j�&lC�a��Í­{8z:gjl/zk�ª�mz:}:¬­�A¬ÏuLÍ��&�a����}:wjlT�j�&lZ�9uL}:z:¬­z:¬¨�a�jtvÀ

fhgjl�Ì�wj�az:¬¨lT�¼z�uL�j� t:w��jt:z:}:wj�Tz:w�}�l&t'uLͨt��0��}�l&t�lT}:��l+¬­�b©e�a}:k¸uLz:¬¨�a��uL���aw�zhz:gjl+uL}:}SuL����lTkmlT��z��L©4�az:gjlT}hkm���bw�ͨl&t�L©»® ��»Ò � �¯���! ���# �&%(' ��)+*-,�#."0/�1(2!343�5 D ÈFÇ ÂÁÅ�£aË@��Äï¥j§JË�ÅS§mÆ�ÂaÄJÉeÈeÉeÈÊ£aË £aË ® � Â�Ë�¤�� � D � É�b§JË�� ÈFÇy¡m£v¤a¥A¦¨§0ÈFË ®CBED È ���� ÈFÇ/Âo¡m£v¤a¥A¦­§�£ ® ��»Ò � �¯���! � # �&%(' ��)+*-,�#."0/�1(2!343�5 ¹ ÈÏÇ�Â8¡m£v¤�¥�¦¨§m£ ®±°¢²´³µ�¶D· ÂaË�¤ ÈFÇ�ÂoÇJ¥I�ïÇv§TÉ5£ ¹ � É�b§JË ÈÏÇÂo¡m£v¤a¥A¦­§�£ ® È �2ÈeÉ~ÈFÇZÂm¡m£v¤a¥A¦¨§D£ ®O� ¹ �

5 {/z:gjl&t�l»ze� ��ͨlTkQk¸uat�Øa¬­zX©Ê�aͭͨ����t4z:g9uLz4¬Ã©j�j�5km���bw�ͨl �v��lT}:ÍÏuL�jtu'�9uL}:z:¬­z:¬¨�a�Q�TÍÏuat�t��L©�u��&�a����}:wjlT�j�&l �9uL}:z:¬­z:¬¨�a�DmØ�z:gjlCkm�A�bw�ͨl&t5�L©z:gjlZÌ�wj�az:¬¨lT�¼z/uL�j��z:gjl0t:w��jt:z:}:wj�Tz:w�}�l&t���¬­��lCz:gjlZkm���bw�ͨl&t+�L©ª®DÀhfhg�¬¨t5k¸u���l&t5¬­z���l&t:¬­}SuL��ͨlz��mt���l&�T¬Ã©e{ ® wjt�¬­���m�&�a����}:wjlT�j�&lC�9uL}:z:¬­z:¬¨�a�jt&�'gj��tSl��9uL}:z�t5���D�j�az��v��lT}:ÍÏuL� uL�¼{8km���bw�ͨl&t�À   ÇTÉeÄ £�Ë@�m¡m£v¤�¥�¦¨§ ¬¨tu8km�A�bw�ͨlQz:g9uLzZ����lT}:ÍÏuL�jtC�j�y�az:gjlT}Ckm�A�bw�ͨlQ�L©�®QØ��'g�¬­Í¨l¸u ÇTÉeÄ £aË�� �T¥�£�ÉeÈ´§JË9É ¬¨tC�a�jl-�'gj��tSlQ�9uL}:z�tZuL}�lQt:z:}��a���km�A�bw�ͨl&tvÀ«fhgjl ¡m£v¤�¥�¦­ÂaÄy¤¼§:Å�£a¡�Æb£aÇïÈeÉeÈe£aË �L©�® ¬¨t0z:gjl¸w���¬¨Ì�wjl ��l&�&�akQ����t�¬­z:¬¨�a���L©+® �a��zSuL¬­�jl&�Á�¼{ �j�j�b¬­����u�&��uL}�t�l&t�z�t:z:}��a���Q̼wj�az:¬¨lT��z6D±z:g9uLzhg9uathkm�a}�l�z:g9uL���a�jl��9uL}:zvØjuL�j�oz:gjlT����l&�&�akQ����t:¬­���Q¬­z�t ©´ua�Tz��a}�t'}�l&�Tw�}�t:¬­��lTÍ­{�Øt�l&l�¡�¬­��w�}�l��bÀéfhgjl5��l&�&�akQ����t:¬­z:¬¨�a� �L©Glvua�Sg¸©´ua�Tz��a}ªt���l&�T¬��9l&t»z:gjl'©´ua�Tz��a}vÀ   t~�a�jtSlT}:��l&�8uL���v��l�Ø�z:gjl'©´ua�Tz��a}�tªuL�j�z:gjlCÌ�wj�az:¬¨lT�¼z�z:gjlT�-t:��l&�T¬Ã©>{y®DÀ��� ù0¢2G ? ���

Page 8: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

³ ¬¨t�z:gjlZ�a��Í­{yk¸ucxb¬­k¸uLÍ�km���bw�ͨl�Àhνl`�'¬­Í­Í4ͨlTz�z:gjl �¼È����b§ïÇTÉ Çï¥)�J¡m£v¤a¥A¦­§JÇ ��lZz:gj��t�lCkm�A�bw�ͨl&t5z:g9uLz�uL}�lZ�j�az�&�a��zSuL¬­�jl&�-¬­��uL��{y�az:gjlT}5km�A�bw�ͨlClJxb�&lT��z�³mÀªf��¸��l&tS�T}:¬­��l/gj��� z:gjlZ��l&�&�akQ����t:¬­z:¬¨�a��¬¨t���lT}:¬­��l&�GØ)� lCwjt�l�z:g�}�l&l�vuat�l&t�À

� � ) � Ò ����� Ò � ® ÂaË�¤ ® Â�Ä:§�§:Â�Åg��Å�£aË9Ë�§:Å&Éߧ:¤ ��� �jl/�&�a�jt�l&̼wjlT�j�&l��L© �XlTkQk¸uat5�bÀ­n/uL�j�y�bÀø�Z¬¨thz:g9uLzh¬Ã©�u0��}SuL��guL�j�D¬­z�t~�&�akQ��ͨlTkmlT�¼z'uL}�l'���az:gm�&�a���jl&�Tz�l&�Gؼz:gjl5�&��uL}�t�l&t:z~t:z:}��a���Z�&�a����}:wjlT�j�&l5�9uL}:z:¬­z:¬¨�a�¸�L©�® ¬¨t��¬­��lT�m��{z:gjlhg�¬­��gjl&t:z~t:w���km�A�bw�ͨl&tvÀéf��/t�l&lhz:g�¬¨t�Ø��j�az�lhz:g9uLz �XlTkQk¸uC�bÀ­nh¬­kQ��Í­¬¨l&t»z:g9uLz¬Ã©�ze� �/g�¬­��gjl&t:z»t�w���km���bw�ͨl&t¹ uL�j� �v��lT}:ÍÏuL�XØ�z:gjlT¬­}éw���¬¨�a�D¬¨tu�km�A�bw�ͨl�Ø�uL�j�Dt:¬­�j�&l ¹ uL�j� 2uL}�l»g�¬­��gjl&t�zét�w���km���bw�ͨl&tvØaz:gjlT¬­}éw���¬¨�a�kZwjt:z���l�³mÀ|_Ýz�uLͨt��Q¬­kQ��Í­¬¨l&t+z:g9uLz�¹ uL�j� ¾ ¹ºuL}�lCkm�A�bw�ͨl&tvÀ'fhg¼wjtvØ9¹ �� 3° ³mØ�uL�j�(�v¹-µ ¾ ¹Z�¬¨t�uD�9uL}:z:¬­z:¬¨�a� �L©ª³3¬­�¼z��mzk�ª�okm���bw�ͨl&t�À fhgjlT� �XlTkQk¸u8�bÀø�D¬­kQ��Í­¬¨l&t5z:g9uLz+lT¬­z:gjlT}�®3�a}�¬­z�t+�&�akQ��ͨlTkmlT�¼z¬¨t��b¬¨t��&�a���jl&�Tz�l&�GØ�uD�&�a�¼z:}Sua�b¬¨�Tz:¬¨�a�XÀνl~�&�a�j�TÍ­wj��l~z:g9uLz�¬Ã©b�jlT¬­z:gjlT}®��j�a} ® ¬¨t��b¬¨t��&�a���jl&�Tz�l&�GØ�� l~k¸u&{�ͨlTz D]��léz:gjl~g�¬­��gjl&t:z4t:w���km�A�bw�ͨl&t��L©9®DÀ� z:gjlT}éz:g9uL�o³mØa�j�+w���¬¨�a�Q�L©�kmlTk0��lT}�t�L©:D ¬¨tu�km�A�bw�ͨl'�L©�®DØ�t:¬­�j�&lªz:gjlT{�uL}�l g�¬­��gjl&t:z~t�w���km���bw�ͨl&tvÀ�5 {�XlTkQk¸u��bÀø�bؼ¬­z©Ê�aͭͨ����t z:g9uLz'®CBED�g9uatª�j�C�j�a��z:}:¬­��¬ÏuLÍ�km�A�bw�ͨl&tvؼgjlT�j�&l*�ªl�g9u&��l��aua�Twj�awjt�Í­{mlT�¼w�kmlT}SuLz�l&�¬­z�t'km�A�bw�ͨl&tvÀ

� ���c� � � Ò � ����� Ò � ® ÈFÇ«¤aÈFÇTÅS£aËjË�§:ÅvÉߧ:¤ À3fhgjl«g�¬­��gjl&t:zyt:w���km�A�bw�ͨl&t�uL}�l«�j�az�u½�9uL}:z:¬­z:¬¨�a� �L©m³mØ�t��P� l«kZwjt:z}�l&t��a}:z5z�� uL�j�az:gjlT}+kmlTz:gj���«�L© �j�j�b¬­���oz:gjl��&��uL}�t�l&t:z��&�a����}:wjlT�j�&l��9uL}:z:¬­z:¬¨�a�XÀ   �&�a���jl&�Tz�l&� �&�akQ���a�jlT�¼z¬¨t0u km�A�bw�ͨloz:g9uLzZk¸u&{��j�az��v��lT}:ÍÏuL� uL�¼{½�az:gjlT}�km���bw�ͨl�Ø�t��-¬­zCkZwjt:z���lmt:z:}��a���jÀ�_Ý�_ua���b¬­z:¬¨�a�XØlT��lT}:{�j�a�¼z:}:¬­�A¬ÏuLͪw���¬¨�a���L©��&�akQ���a�jlT��z�tD�az:gjlT}0z:g9uL� ³ ¬¨t�u km�A�bw�ͨl�Øé��w�z0t�¬­�j�&l¸lvua��g�t:wj�Sg¯w���¬¨�a���v��lT}:ÍÏuL�jtuL�j�az:gjlT}vØ��j� t:wj��g km�A�bw�ͨl��a��zSuL¬­�jl&� ¬­�«z:g�¬¨t � uv{-¬¨t�t:z:}��a���jÀK_Ýz�©e�aͭͨ����t�z:g9uLz+z:gjlDt�lTz D �L©ª�&�a���jl&�Tz�l&��&�akQ���a�jlT��z�t��L© ®±¬¨t�z:gjl0�&��uL}�tSl&t:z�t:z:}��a���8�&�a����}:wjlT�j�&l��9uL}:z:¬­z:¬¨�a� �L© ®QÀ&���a}�l&����lT}vØX®CBED ¬¨t+lTkQ��zß{�Ø�t��z:gjl�km���bw�ͨl&t��L©~® BED uL}�l � wjt:z'z:gjl/�����ªlT}�t�lTz5�L©�DmÀ

� Ò � ) Ò �m����� Ò � ® ÈFÇD¤aÈFÇTÅS£aËjË�§:ÅvÉߧ:¤ À�J�¬­�j�&lDz:gjl�km�A�bw�ͨl&t/�L©h®¢uL�j� ¬­z�t��&�akQ��ͨlTkmlT�¼z0uL}�l�¬¨��lT��z:¬¨�vuLÍ´ØX}�lvuat��a��¬­���t:{AkQkmlTz:}:¬¨�0z��8z:g9uLz/�L©ªz:gjl��9uL}SuLͭͨlTÍ~�vuat�lD�b¬¨�TzSuLz�l&t�z:g9uLz�z:gjlQ�&��uL}�t�l&t:z/t:z:}��a���y�&�a����}:wjlT�j�&lD�9uL}:z:¬­z:¬¨�a� D�L©h®3¬¨t+��¬­��lT�«��{-z:gjlD�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z�t/�L© ®mÀ+fhgjlDÌ�wj�az:¬¨lT�¼z0®CBED¢¬¨t��&�akQ��ͨlTz�l�ØXt:¬­�j�&l ® BED¢¬¨tlTkQ��zß{�Ø9uL�j��z:gjl/km���bw�ͨl&t5�L© ® BEDmØ�gjlT�j�&l/z:gjl/km�A�bw�ͨl&t5�L©»®CBEDmØ�uL}�lCuL�¼uL¬­�-z:gjl/�����ªlT}+t�lTz��L©�DmÀ

J�w�kQk¸uL}:¬�L&¬­���jØG��¬­��lT��t��akml0��}��A�&l&�bw�}�lZ©Ê�a} �j�j�b¬­���8z:gjl0g�¬­��gjl&t:z/t:w���km�A�bw�ͨl&t��L© um��}SuL��gXØ�� l0k¸uv{��&�akQ��w�z�lz:gjl�km�A�bw�ÍÏuL}C��l&�&�akQ����t:¬­z:¬¨�a�½uat+©Ê�aͭͨ����tvÀ _ß©h® ¬¨t��b¬¨t��&�a���jl&�Tz�l&�GØGͨlTz D¢��lZz:gjlD�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼zvØ�uL�j�¬Ã© ®�¬¨t �b¬¨tS�&�a���jl&�Tz�l&�GØ�ͨlTz D±��l�¬­z�t'�&�akQ���a�jlT��z�tvÀ � z:gjlT}��'¬¨t�l�ØjͨlTz D���l+z:gjl�g�¬­��gjl&t�zht�w���km���bw�ͨl&t��L©®DÀ»®CBED¬¨tmu«t:z:}��a���½Ì�wj�az:¬¨lT�¼zvÀÁq5l&�Tw�}�t:¬­��lTÍ­{��&�akQ��w�z�l z:gjl }�l&t:zQ�L©�z:gjl ��l&�&�akQ����t:¬­z:¬¨�a� �a� ®O� ¹r©e�a}Qlvua�Sg ¹ � DmÀ5 { ��lTkQk¸uatZ�bÀø�yuL�j�½�bÀ �jØXu t:w��jt�lTz�¹��L©�³ ¬¨tCuokm�A�bw�ͨlD¬#]_¬­z�¬¨t ���7©Ê�a}Ct��akml�km���bw�ͨl����L© �a�jlQ�L©ªz:gjl}�l&�Tw�}�t:¬­��lTÍ­{Á�&�akQ��w�z�l&��̼wj�az:¬¨lT��z�tvÀ =ªua��g t:wj��g�Ì�wj�az:¬¨lT�¼zQlT¬­z:gjlT}Dg9uatZ�j���j�a�¼z:}:¬­�A¬ÏuLÍhkm���bw�ͨl&tvØ�a}DlTͨt�l8lT��lT}:{t:w��jtSlTz5�L©�¬­z�t���lT}:z:¬¨�&l&t�¬¨t5u�km�A�bw�ͨl�À

�X¬¨t:z:¬­���mz:gjl�kmlTkZ��lT}�t5�L©uLÍ­ÍX�L©4z:gjl�}�l&�Tw�}�t:¬­��lTÍ­{y�&�akQ��w�z�l&���&�a����}:wjlT�j�&lC�9uL}:z:¬­z:¬¨�a�jt�zSu���l&t��C²+ 4 · t:�9ua�&l�¬­�z:gjl-� �a}�t:zC�vuat�l�À`E+��� lT��lT}vØ�� l0k¸uv{�}�lT��}�l&tSlT�¼zCz:gjlD��l&�&�akQ����t:¬­z:¬¨�a�½z:}�l&l�¬­�P.m²+4·�t:�9ua�&lD�¼{�wjt:¬­���-uL�P.m²�·ßÐt:�9ua�&lo}�lT��}�l&t�lT��zSuLz:¬¨�a� ©e�a}�lvua��g��j�A��lo�L©5z:gjloz:}�l&l�Ø»uL�j�_ͨlTz:z:¬­��� z:g�¬¨t0�j�A��l8�vuL}:}:{_u����a¬­�¼z�lT}Dz�� u�Í­¬¨t�z��L©�¬­z�t�Sg�¬­Í¨�b}�lT�XÀ�_ß��z:g�¬¨tZz:}�l&l�Ø�z:gjl¸Í¨lvuc©'��l&tS�&lT�j�juL�¼z�t��L©'lvua�Sg¯�j�A��l��Á��¬­��lQz:gjl¸�&�a}:}�l&t:���a�j�b¬­���-km�A�bw�ͨlQ¹ÁÀCJ�¬­�j�&llvua�Sg �j����l �L©+z:gjl z:}�l&l g9uatmuLzQͨlvuat:zDze�ª�¯��g�¬­Í¨�b}�lT�XØW�ªl k¸uv{_}�l&�&����lT}m¹ ©e}��ak � ¬­� .m² � ¹^� ·0z:¬­kml�تt��«z:g�¬¨t}�lT��}�l&t�lT��zSuLz:¬¨�a�«�L©é¹ ¬¨t�uat�lgf¸�T¬¨lT�¼z�uat5uL�¼{�À

¡9�a}5�j�azSuLz:¬¨�a�9uLÍ4�&�a����lT��¬¨lT�j�&l�Ø>� l �'¬­Í­ÍX¬¨��lT��z:¬Ã©>{�� uL�j�y¹_ÀG¡9�a}5¬­�jt:zSuL�j�&l�ØI� lCk¸uv{ t���lvu��8�L©ézk�ª�¸km���bw�ͨl&t¹ uL�j�F 3uat ��lT¬­����t�¬­��Í­¬­����th¬­�oz:gjl+z:}�l&l��Az:g�¬¨thkmlvuL�jt z:g9uLz ¹ ¬¨tªz:gjl�ͨlvuc©4��l&t��&lT�j�juL��z�t��L©�uZz:}�l&l+�j����l��4Ø� ±¬¨tz:gjl�ͨlvuc©4��l&t��&lT�j�juL��z�t'�L©4uCz:}�l&l��j����l���ØAuL�j���-uL�j�V�¸uL}�l+t:¬­��Í­¬­����tvÀ�¡9�a}huL��{m¬­�¼z�lT}:�9uLÍG�j�A��l�¹ÁØ�z:gjl+t:w�����}SuL��g�L©h®3¬­�j�bwj�&l&� ��{�u¸tSlTz+�&�a�jt:¬¨t�z:¬­��� �L©»�a�jl�kmlTk0��lT}/�L©ªlvua�Sg �Sg�¬­Í¨� �L© �L©»¹ ��¬­��l&t+z:gjlD̼wj�az:¬¨lT��z���}����bwj�&l&���{8z:gjlC}�l&�Tw�}�t:¬­��lC�vuLÍ­Í��a�y¹ÁÀ

  ¤¼§��¼§JË�§JÄ Â�ÉߧCË�£v¤¼§ ¬¨t uZ�9uL}SuLͭͨlTÍG�a}ªt�lT}:¬¨l&t �j�A��l5¬­�oz:gjl���l&�&�akQ����t:¬­z:¬¨�a� z:}�l&l�À�=��lT}:{okm���bw�ͨl+�L©�®2z:g9uLzª¬¨t�j�az�uLÍ­}�lvua�b{-uQ�j�A��lC�L©z:gjlCz:}�l&lC¬¨t+uDw���¬¨�a���L©~�Sg�¬­Í¨�b}�lT� �L©»uQ��lT��lT�jlT}SuLz�l0�j�A��l�ÀW=~��lT}:{yw���¬¨�a���L©~��g�¬­Í¨�b}�lT� �L©uQ��lT��lT�jlT}SuLz�lZ�j����l�¬¨t�u�km���bw�ͨl�À~fhg�wjtvØjz:gjl/��l&�&�akQ����t:¬­z:¬¨�a� uLͨt��D}�lT��}�l&t�lT�¼z�t�¬­kQ��Í­¬¨�T¬­z:Í­{�uLÍ­ÍGkm���bw�ͨl&t��L©~®QÀ

� ����� �

Page 9: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË D

���+, Ö ��)F� ' � "9����� ²´®C·)�� ® �bÂLÇZ£aËj¦ �o£aË�§C�L§JÄTÉߧ�� � � '�Ò * � Ò ���4��* � �ͨlTz����lCu��jl%� ¬­��z�lT}:�9uLÍ�z:}�l&l/�j�A��l��)�� ® ÈFÇC¤aÈÏÇ&Å�£aËjË�§:Å&Éߧ:¤-� '�Ò *

ÍÏuL��lTÍX}� ÆbÂaÄ:Âa¦>¦­§T¦ ��©Ê�a}5lvua�Sg-�&�a���jl&�Tz�l&�-�&�akQ���a�jlT�¼z/® �ÊØbk¸u���l ��� ²´® �Ï·huQ��g�¬­Í¨���L©é}Ò �>� Ò )�� ® ÈFÇZ¤aÈÏÇ&Å�£aËjË�§:Å&Éߧ:¤-� '�Ò *

ÍÏuL��lTÍX}� Ç&§TÄïÈ´§ïÇ �b©Ê�a}5lvua��g-�&�a���jl&�Tz�l&���&�akQ���a�jlT��z�® �G�L© ®DØ�k¸u���l ��� ²ß® �Ï·huD�Sg�¬­Í¨�-�L©��Ò �>� Ò

ÍÏuL��lTÍ��� Æ�ÄJÈF¡¸§ �A©e�a}�lvua�Sg�g�¬­��gjl&t:z5t:w���km�A�bw�ͨl/¹ÁØ�k¸u���l ��� ²´®O� ¹ ·5uD��g�¬­Í¨���L©��� Ò ���4��* �

hUih �Cr�}�~evZz2s<m�vxu>o0pGs<u>}�+���[� l ��}�l&t�lT�¼zmz:gjl �9uat�¬¨�8�juLzSu t�z:}:wj�Tz:w�}�l&t¸uL�j� uLÍ­���a}:¬­z:g�kmtQz:g9uLz��ªl �jl&l&��z���lgf¸�T¬¨lT��z:Í­{�}�lT��}�l&t�lT�¼z¸uL�j�g9uL�j�bͨlZ��}SuL��gjt5uL�j�yz:gjlT¬­}�km�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a�jtvÀ

� � �" � ��×�)�� � ���c� )F� ) Ö *-)�*-,F �a�jt:¬¨��lT}+z:gjl/©Ê�aͭͨ���'¬­��� ��}��a��ͨlTk�À�A�¬­��lT�«umtSlTz��L©»t:z:}:¬­����t��L©~¬­�¼z�lT��lT}�t�©>}��ak nCz��V+_uL�j�«uL� uL}:}Su&{��L©»lTkQ��zÝ{��wj� ��lTz�t�:>Í­¬¨t:z�t0�L©ht�z:}:¬­����t�;JØ�� lO� uL��zZz��y�9uL}:z:¬­z:¬¨�a�_z:gjlmt:z:}:¬­����t/¬­��z�����}��aw��jt0t��yz:g9uLzCzk�ª�-t�z:}:¬­����t0uL}�lQ¬­�½z:gjltSuLkmlo��}��aw���¬Ã©�uL�j�_�a��Í­{¯¬Ã©�z:gjlT{_uL}�l¸¬¨��lT��z:¬¨�vuLÍ´Ø»uL�j�_z:gjlT�Áͨlvuv��l8z:gjl¸��wj� ��lTz�t0¬­�Áz:gjlT¬­}D¬­��¬­z:¬ÏuLÍ t�zSuLz�l�À�fhg�¬¨t�vuL����l/ua�&�&�akQ��Í­¬¨t:gjl&�y��{o}Sua�b¬Ãxyt��a}:z:¬­���m¬­� .¸²+ON ,�·»z:¬­kml�ØI�'gjlT}�lK,7¬¨thz:gjlZt:w�k �L©�ͨlT����z:gjt��L©éz:gjl/t:z:}:¬­����tvÀE+��� lT��lT}vØ<�ªlW�'¬­Í­Í �j�j��¬­zéwjt�lJ©>w�ÍAz����a�jt�lT}:��l z:g9uLzU�ªlh�vuL�mua�&�&�akQ��Í­¬¨t:gD¬­z�¬­�+� wjt:zG.m²,�·Xz:¬­kml t�¬­�j�&lªz:gjl ��}��a��ͨlTk���Al&t'�j�az���ÍÏua�&lZuL��{ }�l&t:z:}:¬¨�Tz:¬¨�a�-�a��z:gjl/�a}���lT}�¬­�Y�'g�¬¨��g-z:gjl��9uL}:z:¬­z:¬¨�a�«�TÍÏuat�t�l&t'k0wjt:z���l���}�l&t�lT�¼z�l&�GÀ

�X¬���lC}Sua�b¬Ãx�t��a}:z:¬­���jØjz:gjl0uLÍ­���a}:¬­z:g�k ��}��A�&l&l&��t5¬­��}��aw��j��t�À2_Ý��}��aw��j���SØ)�ªlZuat�t�w�kml/�¼{ ¬­�j�bwj�Tz:¬¨�a�«z:g9uLz&�ªlg9uv��l+uLÍ­}�lvua�b{Q©Ê�aw��j�mz:gjl5�9uL}:z:¬­z:¬¨�a�y�TÍÏuat�t�l&t»�L©Xz:gjl�t:z:}:¬­����tªz:g9uLz»g9u&��l�ͨlT����z:g8ͨl&t�t~z:g9uL����� �Ø�uL�j�¸z:g9uLz»��}��aw��jt�L©t:z:}:¬­����t�z:g9uLz5g9u&��l/¬¨��lT��z:¬¨�vuLÍ ²���� �·ßд¬­�¼z�lT��lT}+��}�l ��xbl&t5uL����lvuL}�z��a��lTz:gjlT}�¬­���aw�}�Í­¬¨t:zvÀ

fhgjl0uLÍ­���a}:¬­z:g�k �b¬¨t:z:}:¬­��w�z�l&t'z:gjl/t:z:}:¬­����t�z��Dz:gjl/�a}���lT}�l&�y��wj� ��lTz�t+ua�&�&�a}��b¬­���mz��Dz:gjl�¬­�¼z�lT��lT}�z:g9uLz+uL����lvuL}�t¬­�8t��akml�����t:¬­z:¬¨�a���ïÀ   t�z:}:¬­���Z¬¨tªuLÍ��hu&{bt ua����l&�oz��Cz:gjl�©>}��a��zª�L©4u���wj� ��lTz � t Í­¬¨t:zª�L©Xt:z:}:¬­����t0�'gjlT�¸¬­zª¬¨t~¬­�jt�lT}:z�l&�GÀ_ß©�u-t�z:}:¬­����¬¨t�z:gjl �j}�t:z�t�z:}:¬­����z�����lo¬­�jt�lT}:z�l&�_¬­��u-��wj� ��lTzvØ~z:g9uLz���wj� ��lTzD¬¨tDua����l&�_z�� u�Í­¬¨t:zD�L©��j�a�jlTkQ��zß{��wj� ��lTz�tvÀ

� �j�&lDz:gjlDt:z:}:¬­����t/uL}�lD�b¬¨t:z:}:¬­��w�z�l&�GØG¬­z��&�a�j�vuLz�lT�9uLz�l&t/z:gjl�Í­¬¨t:z�t��L©ªt:z:}:¬­����t�z:g9uLzCuL����lvuL}�¬­�«z:gjl0�j�a�jlTkQ��zß{��wj� ��lTz�tvØ4lTkQ��zÝ{�¬­����z:gjlQ��wj� ��lTz�t0¬­��z:gjlQ��}��A�&l&t�tvÀO_Ýz/wjt�l&t/z:gjlDÍ­¬¨t�zZ�L© �j�a�jlTkQ��zÝ{���wj� ��lTz�tZz��-u&���a¬¨�¯��¬¨t:¬­z:¬­���uL��{ lTkQ��zÝ{ ��wj� ��lTz�t+�bw�}:¬­���mz:g�¬¨t�t:z�lT�XÀ

  ©ez�lT}¸}��aw��j���ïØ�u½t�lTz8�L©/t�z:}:¬­����t¸z:g9uLzog9u&��l�¬¨��lT�¼z:¬¨�vuLÍ �Êд¬­�¼z�lT��lT} ��}�l ��x�l&t8uL}�l��&�a�jt�l&�Tw�z:¬­��l-¬­�]z:gjl�Í­¬¨t:zvÀ|éuL}:z:¬­z:¬¨�a� �TÍÏuat�t�l&t0�&�a}:}�l&t:���a�j�b¬­��� z����Êд¬­�¼z�lT��lT}Qt:z:}:¬­����tD�vuL�¯��lm}�lTkm����l&�_©>}��ak z:gjl¸Í­¬¨t:zDuLzZz:g�¬¨t����a¬­�¼zvÀ�fhgjl}:w�����¬­��� z:¬­kml8�vuL�_��l¸�Sg9uL}:��l&��uLzO.m²�·Zz��-z:gjl�� �"!�¬­�¼z�lT��lT}�tQ�L©�z:gjl8t:z:}:¬­����t�lJx�uLkQ¬­�jl&�GتuL�j� .¸²�·/z��«lvua��g¬­��z�lT��lT}~�L©jz:gjl#�Êд¬­��z�lT��lT}~t:z:}:¬­����t�z:g9uLzéuL}�lª}�lTkm�v��l&��©e}��ak z:gjl»Í­¬¨t:zvÀ�5hl&�vuLwjt�l �L©jz:gjlªt:¬­kQ¬­ÍÏuL}:¬­zÝ{0�L©9z:g�¬¨téuLÍ­���a}:¬­z:g�kz��Q}Sua�b¬Ãxyt��a}:z:¬­���jØI� l �'¬­Í­Í��vuLÍ­ÍX¬­z Ä:Â�¤aÈ$�CÆbÂaÄTÉeÈ>ÉeÈÊ£aËjÈ>Ë@� À

� � >� % Ö ��� )�*-,½�D×�&&��� Ò *4� Ó � )>�&�v�A+¬­��lT� uy��w�kZ��lT}:¬­�����L©���lT}:z:¬¨�&l&t��L©�u���}SuL��gXØU�ªl¸k¸u&{¯t��a}:zDuLÍ­Íhua�L��ua�&lT�j�T{�Í­¬¨t:z�t�t���z:g9uLz0�jlT¬­��g¼���a}�tQuL����lvuL}¬­��uatS�&lT�j�b¬­���o�a}���lT}vØj¬­�t.m²+aNb,�· z:¬­kml�Ø9�¼{y�&�a�j�vuLz�lT�9uLz:¬­���yuLÍ­Í4ua�L�:ua�&lT�j�T{ Í­¬¨t:z�t�uL�j�yz:gjlT�-}Sua�b¬ÃxAÐßt��a}:z:¬­���¸z:gjl}�l&t:w�Í­z:¬­���oÍ­¬¨t:z'wjt:¬­���m��lT}:z�lJx��L©�a}:¬­��¬­��uat'��}:¬­k¸uL}:{�t��a}:z|��lT{yuL�j��lT�j�b¬­���o��lT}:z�lJx�uat5t�l&�&�a�j�juL}:{yt��a}:z|��lT{�À»fhg�¬¨t¬¨t�ua�&�&�akQ��Í­¬¨t:gjl&�7�'¬­z:gyze�ª�¸t:zSuL��ͨlK.m²+ONZ,�·ªt��a}:z�t�Øju �j}�t�zh�a�jl/�a�yz:gjl�t�l&�&�a�j�juL}:{���lT{8uL�j��u�t�l&�&�a�j�y�a�jl/�a�z:gjl���}:¬­k¸uL}:{Y��lT{�Ø�t�l&l�F �a}:kmlT�½lTz/uLÍ´À�: nv�����<;JÀ�νlD�a��zSuL¬­� z:gjlQua�L�:ua�&lT�j�T{�Í­¬¨t�z�t���{-�Tw�z:z:¬­���yz:g�¬¨t �j�9uLÍéÍ­¬¨t:z�uLz���a¬­�¼z�t*�'gjlT}�l/z:gjl/��}:¬­k¸uL}:{yt��a}:z|��lT{ ��g9uL����l&tvÀ

��� ù0¢2G ? ���

Page 10: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

� � �� � *-) Ö * Ô�� )�*4×νl��jl&l&�8u ��¬¨�a�bÐe¡4¬­�j� �juLzSu0t:z:}:wj�Tz:w�}�l5©e�a}ht��akml5�L©��aw�} �a��lT}SuLz:¬¨�a�jtvÀfhg�¬¨t ¬¨t uL��{¸t:z:}:wj�Tz:w�}�l�z:g9uLzªk¸uL¬­��zSuL¬­�jtuD�9uL}:z:¬­z:¬¨�a� D3�L©umt�lTz�³mØjuL�j�-t:w������a}:z�t�z:gjl�©e�aͭͨ���'¬­���¸�a��lT}SuLz:¬¨�a�jt9����������� ������ � JAlTz6D�� °6�����)��� �C��³��¼À������ � ¡4¬­�j��u����a¬­��z�lT}5z��Dz:gjl/kmlTk0��lT}5�L©�D z:g9uLz5�&�a��zSuL¬­�jt+u���¬­��lT�Y���-³oÀ��������� � A+¬­��lT�Z��µ�� � ³¸ØGkmlT}:��lmz:gjlDz:gjlQzk�ª��tSlTz�tC�L© D z:g9uLzZ�&�a�¼zSuL¬­� �«uL�j�(� �¼{ }�lT��ÍÏua�T¬­����z:gjlTk �'¬­z:g

z:gjlT¬­}�w���¬¨�a��¬­��DmÀ  � uL}:��¬­z:}SuL}:{ÁtSl&̼wjlT�j�&lyt:zSuL}:z:¬­���t�'¬­z:g �a�jl �������� ������ Ø~�a� u t�lTzD�L©+t:¬�Lvl�+~Ø»uL�j� u�kQ¬Ãx�l&� t�l&̼wjlT�j�&l

�L© + � ��������� uL�j�^, ������� �a��lT}SuLz:¬¨�a�jtozSu���l&t�.¸²+?Xª²,�µ�+4·:·Qz:¬­kml�Ø't�l&l�f�uL}e�:uL� : nv�����<;JØ*F �a}:kmlT�]lTz8uLÍ´À: nv�����<;JÀQfhgjlOX]¬¨tCuL�½lJxbz:}�lTkmlTÍ­{«t:ͨ���hд��}����'¬­���«��w�z/w��¼���aw��j��l&� ©ew��j�Tz:¬¨�a�½z:g9uLz/g9uat/��uLÍ­wjlmuLz/km��t:z�©e�aw�}/¬­�uL��{8��}Sua�Tz:¬¨�vuLÍéuL����Í­¬¨�vuLz:¬¨�a���L©éz:gjlC�juLzSuDt:z:}:wj�Tz:w�}�l�À

f4�yua��g�¬¨lT��lQuQÍ­¬­�jlvuL}+z:¬­kmlZ���aw��j�-©e�a}�km���bw�ÍÏuL}���l&�&�akQ����t�¬­z:¬¨�a�XØ�uo�&lT}:zSuL¬­�«}�l&t:z:}:¬¨�Tz:¬¨�a���L©~z:gjlCw���¬¨�a�bÐ �j�j���}��a��ͨlTk �'¬­Í­Íé��l0}�lTͨlT��uL�¼zvÀCνl�tïu&{-z:g9uLzCuot�l&Ì�wjlT�j�&l��L© �������� �a��lT}SuLz:¬¨�a�jtC�a�«z:gjlDt�lTzZ³ Ä�§ïÇ´Æj§:Å&É>Ç uo��}SuL��g® °$²´³~µ�¶�·�¬Ã© ØX©e�a}ZuL��{ �L©'z:gjlQt:w��jtSlTz�t�� z:g9uLz0uL}�l¸�T}�lvuLz�l&�½�A¬Ïu8z:gjl �������� tvØ4®O���¿¬¨tZ�&�a���jl&�Tz�l&�GÀ   �j�az:gjlT}©Ê�a}:kZw�ÍÏuLz:¬¨�a���L©�z:g�¬¨th��}��a��lT}:zß{o¬¨t z:gjl+uat�tSlT}:z:¬¨�a�8z:g9uLz �'gjlT�jlT��lT}'zk�ª�Dt�lTz�t �"!�uL�j�#� 4 uL}�l�kmlT}:��l&� ��¬Ïu�u ����������a��lT}SuLz:¬¨�a�_z:gjlT}�lV�'¬­Í­Í uLÍ��hu&{bt0��l¸uL�Ál&�b��l¸¬­�Á® z:g9uLz�� �a¬­�jt�uykmlTkZ��lT}0�L©$�%!Cz���uykmlTk0��lT}D�L©&� 4 ÀC¡j�a}DuL�����lT}:��¬¨l%� �L©�z:gjl&tSl+�&�a�j�&lT��z�t'tSl&l�l�À �VA+wjt:z�l&�bz : nv�����<;JÀ~fhgjl�©Ê�aͭͨ���'¬­���Dz:gjl&�a}�lTk ¬¨th�T}:wj�T¬ÏuLÍ�©Ê�a} z:gjl�Í­¬­�jlvuL}hz:¬­kml���aw��j���L©�aw�}�uLÍ­���a}:¬­z:g�k�À

' '�Ò�Ö � Ò � "$#�(8��� ÖAÕ � *4× ' ����&&� * #."0/�1:2:343*)4§TÉ"+ ° ²´³µ�¶D· ��§DÂ8ÉeÄ:§�§ � ��Ë>�QÇ&§��&¥�§JË�ÅS§¸£� +-,./10�. ÂaË�¤,324/5.6 £ïÆ�§JÄ:Â�ÉeÈÊ£aË�Ç0£aË ³ É�AÂ�ɪÄ�§ïÇ´Æ�§�Å&É>Ç7+ Å�ÂaËP��§'Æ�§JÄ&£aÄ§:¤8È>Ë .m²+aNb,�· ÉeÈF¡¸§9�

� � � 8����c� )>� � � Ó:9ZÖ � � � Ò � Ò *�� Ò × �oÒ � � Ò � Ò *��v�b� ) Ö *4�_Ý� u¯�&�a�¼��lT�¼z:¬¨�a�9uLÍCua�L��ua�&lT�j�T{¼Ð´Í­¬¨t�zo}�lT��}�l&t�lT��zSuLz:¬¨�a�¿�L©0u��b¬­}�l&�Tz�l&� ��}SuL��g ®DØ'lvua�Sg]��lT}:z�lJx �vuL}:}:¬¨l&tyu�Í­¬¨t�z¸�L©¬­z�tZ�jlT¬­��g¼���a}�tvÀYJ�wj��g�uy}�lT��}�l&t�lT��zSuLz:¬¨�a�Á�TͨlvuL}:Í­{�g9uat�u��b¬¨tSua�b��uL�¼zSuL��lV�'gjlT�P�ªl uLͨt��yg9uv��lmz�����lvuLÍ2�'¬­z:g¯z:gjl�&�akQ��ͨlTkmlT��z0�L©ªz:gjl���}SuL��gXØX�a}/km�a}�lD��lT�jlT}SuLÍ­Í­{�Ø��'¬­z:g��j�a�bÐßl&�b��l&t9�+�9uat�t:¬­���8©>}��ak u8t:zSuL�j�juL}���lT�j�&�A�b¬­�����L©hu��}SuL��g«z��mt:wj�Sg«uL��lT�j�&�A�b¬­���o�L©¬­z�t��&�akQ��ͨlTkmlT�¼z�}:w�ͨl&t��aw�z+uL�¼{y����t�t:¬­��¬­Í­¬­zß{��L©»uDÍ­¬­�jlvuL}�z:¬­kml0uLÍ­���a}:¬­z:g�k�À'νl�T¬­}��Tw�k0��lT�¼z+z:g�¬¨t&�A¬­�j�y�L©���}��a��ͨlTk �'¬­z:gyz:gjl�©e�aͭͨ���'¬­���o�juLzSuQt�z:}:wj�Tz:w�}�l�À;yÒ�< *�)F� ) Ö * � 2 5ïË Â �9uL}:z:¬ÏuLÍ­Í­{ �&�akQ��ͨlTkmlT�¼z�l&� � £�Ä kQ¬Ãx�l&� Ä:§ßÆ�Ä:§ïÇv§JË9É´Â�ÉeÈÊ£aË £ ® °�²´³~µ�¶�· §.�L§TÄ�� �L§JÄTÉߧ��¡mÂaÈ>Ë9É´ÂaÈ>Ë�Ç+Âm¦ ÈFÇJÉ~£ Z§JÈeÉ�b§JÄ ��²�b· £�Ä ��²�b· ÂaË�¤mÂm¦ÃÂ@��§J¦XÉ�AÂ�É~ÈÏÇC§JÈ>É��§JÄ t:zSuL�j�juL}�� £aÄ �&�akQ��ͨlTkmlT�¼z�l&� É�AÂ�É»ÉߧJ¦F¦ Ç= �¼ÈÊÅg�-£ CÉ�b§ZÉ = £8Å�ÂLÇ&§JÇ�ÂïÆ�Æ�¦ È´§ïÇ�T£aÄ � �?>D§�¤¼§A@~Ë�§ZÉ�b§ t:¬�Lvl £ ZÂm¡DÈ$�¼§:¤oÄ�§ßÆ�Ä�§ïÇ&§JË�É´Â�ÉeÈe£�˽ɴ£���§ +VNb, � � = �b§JÄ�§+ ÈÏÇCÉ�b§ZËj¥A¡V��§JÄ0£ �a§JÄJÉeÈÊÅS§ïÇ � ÂaË�¤ , � ÈFÇCÉ�b§CÇï¥A¡ £ ZÅ�Â�Ä ¤aÈ>Ë�Âa¦ ÈeÉeÈʧïÇ0£�CÉ�b§JÈ>ÄZÂLÇSÇ&£vÅTÈÊÂ�Éߧ:¤8¦ ÈÏÇTÉ>Ç4�;yÒ�< *�)F� ) Ö * � CB�# ; � ' � ' � ��� Ò �m� � #."0/�/ED�343F)4§TÉXÉ�b§ ��¬­�9uL}:z:¬­z�l'�&�akQ��ͨlTkmlT��z £�'Â0¤aÈ>Ä:§:ÅvÉߧ:¤O�JÈ ÆbÂaÄTÉeÈ>Éߧ0�aÄ:ÂïÆ>�²´³E!vµï³ 4 µHG/· ��§�É�b§ �aÄ:ÂïÆ>� I

³ ! µï³ 4 µ�J�Ke³ ! L ³ 45M ��²´³ 4 L ³ ! ·ONm¾PG&QY� : n�;

5ïË Â kQ¬Ãx�l&�½��¬­�9uL}:z:¬­z�lo}�lT��}�l&t�lT�¼zSuLz:¬¨�a� � §�Â�Åg���L§TÄJÉߧ���ÈFË ³E!*R Ä:§ïÇßÆ � ³ 4OS �AÂLÇ8§JÈeÉ�b§JÄ Â�¦ ÈFÇJÉ�£ ¸É�A£aÇ&§o¡¸§J¡V��§JÄïÇ£ ³ 4 R Ä:§JÇ´Æ � ³E! S É�AÂ�É�ÂaÄ�§o£�¥AÉ �ÝË�§JÈ����x�:£aÄïÇ � £aÄ8§T¦ Çv§oÂ�¦ ÈÏÇTÉ�£ mÉ�b£LÇ&§¸¡¸§J¡V��§JÄïÇ8£ ³ 4 R Ä:§ïÇßÆ � ³E! S É�bÂ�É+Â�Ä:§¸Ë�£�É£a¥bÉ �ßË�§JÈ#���x�:£aÄïÇ4�

fhg�¬¨tª}�lT��}�l&t�lT��zSuLz:¬¨�a�yg9uat~k¸uL�¼{¸¬­�¼z�lT}�l&t:z:¬­������}��a��lT}:z:¬¨l&tW�'g�¬¨�Sg uL}�l5��lT{��a�j�oz:gjl5t��&�a��l��L©Xz:g�¬¨tª�9uL��lT}vÀéfhgjl}�lTͨlT��uL�¼z�}�l&t:w�Í­z'©Ê�a}��aw�}5uLÍ­���a}:¬­z:g�k ¬¨t'z:gjl�©Ê�aͭͨ���'¬­��� �' '�Ò�Ö � Ò � �?# ; � ' � ' � ��� Ò �m� � #."0/�/ED�343 �hÈ+�L§JË ÂD¡DÈ ��§:¤mÄ�§ßÆ�Ä�§ïÇv§JË9É´Â�ÉeÈÊ£a˽£ /¸¤aÈ>Ä:§�Å&Éߧ:¤��aÄ:ÂïÆ>��£aÄC¤aÈ>Ä:§:ÅvÉߧ:¤�JÈ ÆbÂaÄTÉeÈeÉߧ&�aÄ:ÂïÆ>� � @~Ë�¤aÈFË��QÉ�b§�ÇTÉeÄ £aË��a¦ ��� Å�£aËjË�§�Å&Éߧ:¤yÅ�£a¡�Æb£�Ë�§JË9É>Ç/É´ÂUTa§ïÇ/ÉeÈ>¡m§�¦ È>Ë�§:ÂaÄ�È>Ë�É�b§+ÇJÈCVc§/£ �É��§�Ä�§ßÆ�Ä�§ïÇ&§JË>�É´Â�ÉeÈÊ£aË:�

� ����� �

Page 11: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË �

� � ( �9�� h������ %&�'� ,.�/�0*��� '�213��4�5"768��#9�:������

� w�}�uL����}���ua��g wjt�l&thlTͨlTkmlT��z�t ©>}��ak u/��}�lT�A¬¨�awjtht:z:}SuLz�lT��{�Ø�t�l&l*=~g�}�lT�b©elTwj�Sg¼z5lTz'uLÍ´À�: nv���L�@;JØ��'g�¬¨�Sg�� l���l&t��T}:¬­��l�j}�t:zvÀ*_´©�� ¬¨t+um��lT}:z�lJx-�L© ®DØ�ͨlTz � ²´®�µ��b·���lT�j�az�lV���)�muL�j�-z:gjl0t�lTz��L©~k¸ucxb¬­k¸uLÍ�km�A�bw�ͨl&t��L©ª® z:g9uLz+���¸�j�az�&�a��zSuL¬­�(��À�fhg9uLz�¬¨tvØ9¹ � � ²´®�µ��b·�¬Ã© lT¬­z:gjlT}+¹ ° ���)�¼ØG�a}+¹ ¬¨t/umkm�A�bw�ͨlD�L©h®3z:g9uLz/����l&t��j�az/�&�a�¼zSuL¬­�(��ØuL�j�mlT��lT}:{D��}��a��lT}»t:w���lT}�t�lTzz:g9uLz~¬¨t»u+km�A�bw�ͨl��L©4® �&�a��zSuL¬­�jt2��À � ²´®�µ���·é¬¨t»u/�9uL}:z:¬­z:¬¨�a�o�L©�³mÀéfhgjlh©Ê�aͭͨ���'¬­���uL}�lClvuat�{8�&�a�jt�l&Ì�wjlT�j�&l&t5�L© �XlTkQk¸um�bÀø�bÀ

�»Ò � �¯� � �"$#�� ' � Ò * � Ò �4� ' � Ò �m� �A 6#."0/�/ 343t��¦>¦~Ë�£aË9ÉeÄJÈ+�vÈeÂ�¦¡Q£�¤a¥�¦¨§ïÇ8£� ®CB � ²´®�µ����L· Å�£aË9É´Â�ÈFË ������� �7�ÇJ¥I�ïÇv§TÉ � £ � ²´®�µ�� � · ÈÏÇDÂaË�ÈFË�ÉߧJÄïË�Âa¦XË�£v¤�§Q£ 0É��§0¡Q£�¤a¥�¦­ÂaÄQ¤¼§:Å�£�¡5Æ�£LÇïÈeÉeÈe£�ËÁ£ ® B � ²´®Dµ�� � · È � � � ÈFÇ0ÂaËÂaË�ÅS§ïÇJÉ´£�Ä-£ ��� � � ÈFË É��§ ¡m£v¤a¥A¦ÃÂ�Ä-¤¼§:Å�£a¡�Æb£aÇïÈeÉeÈe£aË¿£ ® �^��¦>¦h£�É��§JÄ ÇJÉeÄ:£aË@�½¡Q£�¤a¥�¦¨§ïÇ-ÂaÄ�§ ÇJ¥I�JÇ&§TÉ>Ç�£ 8ÇT£a¡¸§¡¸§J¡V��§JÄD£ � ²´®�µ�� � · ��»Ò � �¯� � >�?#�� ' � Ò * � Ò �4� ' �m� *4× �8Ö���Ò *�� Ò � ,�#."0/�/� 343 5 ¹ ÈFÇ�Â8¡m£v¤a¥A¦­§¸£ ® � É�b§T˽É�b§0ÇJÉeÄ:£aË@�y¡Q£�¤��¥A¦­§JÇ0£ ® É�AÂ�ÉhÂaÄ:§'Æ�Ä:£ïÆ�§TÄCÇï¥)�ïÇv§TÉ>Ç0£ ¹ ÂaÄ�§0É�b§CÇJÉeÄ:£aË@�o¡m£v¤a¥A¦­§ïÇ0£� ®O� ¹ É�AÂ�É ÂaÄ�§�Æ�Ä £JÆ�§JÄ/ÇJ¥I�ïÇv§TÉ>Ç�£ ¹ �

fhgjl¸©Ê�a}�lT���a¬­����zk�ª��ͨlTkQk¸uatZ��¬­��l uyt:z:}SuLz�lT��{�©Ê�a} �j�j�b¬­���«z:gjlmkm�A�bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a���L©�® ©e}��ak z:gjlkm�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a�jt��L©t:w�����}SuL��gjtvÀ

���+, Ö ��)F� ' � ��� =g�}�lT�b©ÊlTwj��g¼z�lTz5uLÍ´À?: nv���L�@; * � ��� � �XlTz&� � ��lCuL�-uL}:��¬­z:}SuL}:{ ��lT}:z�lJx�À ��� ²ß® B � ²´®Dµ�� � ·:·'uL�j�t� ��� ²´®O� ¹�·*�c¹ � � ²´®�µ�� � · �¼Ø����� � ��� � ��� ²´®C·��lTz + ��l�z:gjl/km���bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a�«�L©~® B � ²´®Dµ�� � · �� Ö � Ò ��� ' ¦­§:Â� ¹ ��� +�× Ö

� Ö � � Ò *���� ¹ � � ²´®Dµ�� � · ��XlTz +�� ��l�z:gjlCkm���bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a�-�L©»®O� ¹ �q�lT��ÍÏua�&l/¹ �'¬­z:g +�� ¬­� + �)�� ¹ ÂaË�¤¸ÈeÉ>ǪÆbÂaÄ�§JË9É�� ²>¹ · ÂaÄ�§-�:£�É�DÆbÂaÄ:Âa¦>¦­§T¦4£aÄ��:£�É�8Çv§JÄJÈʧïÇCË�£v¤¼§ïÇCÈ>Ë +2� '4Ò *

q5lTkm����l/¹ ©>}��ak + uL�j��uLz:zSua��g-¬­z�t5��g�¬­Í¨�b}�lT�«uat5��g�¬­Í¨�b}�lT���L© � ²>¹�·

fhgjlD��}��A�L© �L©h�&�a}:}�l&�Tz:�jl&t�t�©Ê�aͭͨ����t+©e}��ak �XlTkQk¸uatZ�bÀ­nmuL�j���bÀø�bØ�uL�j�½uL���a�jt�lT}:��uLz:¬¨�a��z:g9uLz+z:gjlD¬­���jlT} )�� Ðt:zSuLz�lTkmlT��zD}�lTkm�v��l&tD¹ ¬Ã©�uL�j�¯�a��Í­{¯¬Ã©�¹ ©ÊuL¬­Í¨t0z�����l¸t:z:}��a���«¬­� ®DÀ-fhgjlo¬­kQ��ͨlTkmlT�¼zSuLz:¬¨�a� �L©*=g�}�lT�b©ÊlTwj��g�zlTzªuLÍ´Àw: nv���L�@;é�&�akQ��w�z�l&t»z:gjlhkm�A�bw�ÍÏuL}ª��l&�&�akQ����t:¬­z:¬¨�a�jtª�L©X®O� ¹$��{0}�l&�Tw�}�t:¬¨�a�XØbuL�j�Qwjt�l&tªu�t�lT�9uL}SuLz�l'z�l&��g���¬¨Ì�wjlz�� �j�j��z:gjl0km�A�bw�ÍÏuL}/��l&�&�akQ����t�¬­z:¬¨�a� �L©'®CB � ²´®�µ�� � ·�z:g9uLz+zSu���l&t/ua�b��uL�¼zSuL��lQ�L©»¬­z�t�t:¬­kQ��ͨl�t:z:}:wj�Tz:w�}�l�À/fhgjl��l&t:z'���aw��j�yt��0©ÊuL}h©Ê�a}'z:g�¬¨t�uL����}���ua��g�¬¨t*.m², ��+»²:²+�4�·ïµ�,RQ�S�T0+4·:·JØ�t�l&l �«��F �a���jlTÍ­Í4uL�j�YJ���¬­��}Sua�P: nv�����<;JÀ~fhgjl���az:z:ͨlT�jl&� �y¬¨t5uDt�z�lT���vuLͭͨl&�-�vuLͭͨl&� �L§TÄJÉߧ��ZÆ�ÂaÄJÉeÈeÉeÈÊ£aËjÈ>Ë@� Øx�'g�¬¨��g-¬¨t'wjt�l&�yz��Q�&�akQ��w�z�l � ²´®Dµ�� � ·JÀ

νl��j���2��l&t��T}:¬­��l/uDt:¬­����¬��9�vuL�¼z�¬­kQ��}�����lTkmlT�¼z+�a�yz:gjl+t�z:}SuLz�lT��{�ØI�'g�¬¨��gy¬¨t'�bwjl�z���c/uLg�Í­g9uLwjt`: nv����H<;JØx�'gj�wjt�l&�½¬­z/z����a��zSuL¬­�½z:gjlD�9uL}SuLͭͨlTÍ»z:¬­kmlm���aw��j�«©Ê�a}Ckm�A�bw�ÍÏuL}Z��l&�&�akQ����t�¬­z:¬¨�a�½z:g9uLzK�ªlm��¬­��lQ��lTͨ���CÀQνlQwjt�lD¬­zz��y�a��zSuL¬­�¯uoÍ­¬­�jlvuL} дz:¬­kmlQt�l&Ì�wjlT�¼z:¬ÏuLͪuLÍ­���a}:¬­z:g�k�Ø4uL�j�½u¸�9uL}SuLͭͨlTÍ»uLÍ­���a}:¬­z:g�k�©e�a}+z:}SuL�jt:¬­z:¬­��lQ�a}:¬¨lT�¼zSuLz:¬¨�a�XÀDfhgjlt:z:}SuLz�lT��{m�vuLͭͨt»©Ê�a}ª�&�akQ��w�z:¬­��� ��� ²´®V� �Á² � � ·:·»uL�j� ��� J ®O� ��² � � · N �¼{Q}�l&�Tw�}�t:¬¨�a�XØbuL�j�mz:gjlT� �j�j�b¬­���O� ��� ²´®V� ¹ ·*�¹ � � ²´®�µ�� � · �Z��{ uDz�l&�Sg���¬¨Ì¼wjlZ�vuLͭͨl&� Ä�§ïÇJÉeÄJÈeÅvÉeÈe£aË À

_ß© � ³¸Ø'ͨlTz z:gjl Ä:§ïÇTÉeÄïÈÊÅ&ÉeÈÊ£aË É´£ º�L©Zz:gjl�km�A�bw�ÍÏuL} ��l&�&�akQ����t:¬­z:¬¨�a�¿�L©�® ��l�z:gjl t�lTz �L©Ckm���bw�ÍÏuL}��l&�&�akQ����t:¬­z:¬¨�a�jt��L©�z:gjl�kmlTkZ��lT}�t��L©2��²´®O� ¹�·*�L¹ ¬¨t�uZk¸ucxb¬­k¸uLÍGkm�A�bw�ͨl/�L©®�z:g9uLz'¬¨th�&�a�¼zSuL¬­�jl&�-¬­�F V�¼À�¡j�a}� µ�� � 8Ø�ͨlTz ��� ����l/z:gjl/}�lTÍÏuLz:¬¨�a�9�'gjlT}�l � uL�j���+g9u&��lZz:gjlCtSuLkmlCtSlTz5�L©�jlT¬­��g¼���a}�t5¬­�«³ ¾ 8À|J�¬­�j�&l/z:g�¬¨t�¬¨tuL�-l&Ì�w�¬­��uLͨlT�j�&l/}�lTÍÏuLz:¬¨�a�XØjz:g�¬¨t'��¬­��l&t�uD�9uL}:z:¬­z:¬¨�a�-�L©= oØI�'g�¬¨�Sg7�ªlK�'¬­Í­Í��vuLÍ­ÍXz:gjl Ç&Âa¡¸§g�ßË�§TÈ#���I�:£aÄ'Æ�ÂaÄJÉeÈeÉeÈÊ£aË À

¡9�a}+z:gjl��&�a}:}�l&�Tz:�jl&t�tvØ��j�az�l0z:g9uLz�u¸¹ � ¢¬¨t/uQkm�A�bw�ͨl��L©h®±¬#]�uLÍ­Í��L©~¬­z�t�kmlTkZ��lT}�t�g9u&��l0z:gjl0tïuLkml0t�lTz�L© �jlT¬­��g¼���a}�t/¬­�¯³ ¾'¹ÁÀCfhg�¬¨t�g9uL����lT�jt�¬#]_¬­z�¬¨tCumkm�A�bw�ͨlD¬­�¯®O� uL�j�½uLÍ­ÍékmlTk0��lT}�tC�L©ª¹ºg9u&��lDz:gjlQtSuLkml�jlT¬­��g����a}�th¬­�y³¯¾- oÀfhgjl5��}����&l&�bw�}�l �j�j��thlvua��g8k¸ucxA¬­k¸uLÍ�¹$z:g9uLzª¬¨thu/km�A�bw�ͨl+�L©�®O� 3uL�j�mg9uat �a��Í­{mt:z:}��a����jlT¬­��g����a}�t�uL�j�8�j�a�bд�jlT¬­��g¼���a}�t'¬­��®¯¾� 8Ø�uL�j�¸k¸u���l&th¬­z z:gjl�}����azh�L©�uZz:}�l&l�¬­�yuC©e�a}�l&t:zvÀfhg9uLz z:g�¬¨t z:}�l&l+¬¨tªz:gjlkm�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a���L©»®O� ¹ ©e�aͭͨ����t'©e}��ak �XlTkQk¸um�bÀø�bÀ��� ù0¢2G ? ���

Page 12: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

nv� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

���+, Ö ��)F� ' � �G� fhgjlCq5l&t�z:}:¬¨�Tz�J�z�lT� * � ��� �= �]³ uL�j� z:gjl/km���bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a� + �L©»®O� ����� � ��� �z:gjl�}�l&t:z:}:¬¨�Tz:¬¨�a��z��� �L©éz:gjl/km�A�bw�ÍÏuL}5��l&�&�akQ����t�¬­z:¬¨�a���L©»®c�lTͨlTz�l�©e}��ak + uLÍ­ÍG�j�A��l&t'z:g9uLz�}�lT��}�l&tSlT�¼z�tSlTz�t'z:g9uLz�¬­�¼z�lT}�tSl&�Tz�km�a}�l/z:g9uL�-�a�jlCtïuLkmlJд�jlT¬­��g¼���a}��TÍÏuat�t �_Ý� z:gjl�}�lTk¸uL¬­��¬­���D©e�a}�l&t�zvØ���}��aw��yz��a��lTz:gjlT}5uL��{o}��A�az�thz:g9uLz�uL}�l+©e�a}:kmlT}'�Sg�¬­Í¨�b}�lT�-�L©éu��&�akQkm�a����lT��lT�bÐlT}SuLz�l��j�A��l�uL�j��uL}�l��&�a�¼zSuL¬­�jl&� ¬­��u¸�&�akQkm�a��tSuLkmlJд�jlT¬­��g����a}C�TÍÏuatStvØGuL�j�-k¸u���l�z:gjlTk$��g�¬­Í¨�b}�lT�½�L© u�jl%����lT��lT�jlT}SuLz�lC�j�A��l�À

�»Ò � �¯� � �� �?�b§D¡m£v¤a¥A¦­Âaĸ¤¼§:Å�£�¡5Æ�£LÇïÈeÉeÈe£�Ë�Ǹ£ �É��§Q¡¸§J¡V��§JÄïǸ£ ��²´®O� ¹ ·-�b¹ � � ²´®Dµ�� � ·4¾K����� � ����� ÂaÄ�§�aÈ+�L§JËP�g�8É�b§ZÄ:§JÇJÉeÄJÈeÅ&ÉeÈÊ£aË�ÇZ£ CÉ��§C¡m£v¤�¥�¦­ÂaÄ�¤�§:Å�£a¡�Æb£LÇJÈ>ÉeÈÊ£a˽£ ® É´£ �Á² � � · ÂaË�¤ �Á²� � · �8C� ÖGÖ �S� fhgjl«}�l&t:z:}:¬¨�Tz:¬¨�a�jt ��¬­��l«z:gjl«km���bw�ÍÏuL}y��l&�&�akQ����t:¬­z:¬¨�a� �L©D®O� ¹ ©e�a} lvua��g¿k¸ucxb¬­k¸uLÍ�km���bw�ͨl �L©D®�&�a��zSuL¬­�jl&�¸¬­���Á²� � ·~uL�j� ��²� � ·JÀé���Ct�wj��gmkm�A�bw�ͨl5�&�a��zSuL¬­�jt2� � ÀU_´©�¹ � � ²´®Dµ�� � ·¼¾������ � ���¼Ø¼¬­z~¬¨tªu/k¸ucxA¬­k¸uLÍkm�A�bw�ͨlQ�L©'® z:g9uLzC���Al&t+�j�azC�&�a�¼zSuL¬­�Z� � ØXuL�j��t:¬­�j�&l�� � ¬¨t/lT¬­z:gjlT}Zu8t:z:}��a���y�jlT¬­��g����a}C�a}Cu8�j�a�bд�jlT¬­��g¼���a}vØX¬­zk0wjt:z���l/�&�a�¼zSuL¬­�jl&��¬­�7�Á²����· �a} ��²����·JÀ

  Í­���a}:¬­z:g�k �Q��¬­��l&t+u�g�¬­��gbдͨlT��lTÍ4����lT}:��¬¨l%�2�L©éz:gjlCt:z:}SuLz�lT��{�À~νl/k¸u���lC�a�jl �j�9uLÍ��a�jtSlT}:�auLz:¬¨�a�XÀ

���+, Ö ��)F� ' � X� c/uLg�Í­g9uLwjt�: nv����H<;)�� ® �bÂLÇZ£aËj¦ �o£aË�§C�L§JÄTÉߧ�� � � '�Ò *

� Ò ���4� * �8uat�uDz:}�l&l��j����lÒ �>� Ò

����������� � JAlTͨl&�Tzmu-��lT}:z�lJxP� � Ø~uL�j�_}�l&�Tw�}�t:¬­��lTÍ­{Á�&�akQ��w�z�loz:gjlokm�A�bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a�jtQ�L©�®O� ��² � � ·ZuL�j�®O� � ²� � ·JÀ

���������������������� t:¬­���mz:gjl&t�lCz:}�l&l&tvØ   Í­���a}:¬­z:g�k7�bØ9uL�j� �XlTkQk¸uo�bÀø�bØ �j�j�GØ�©Ê�a}5lvua��g�¹ � � ²´®�µ�� � ·X¾������ � ���¼Øjz:gjlkm�A�bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a���L©»®O� ¹_À

�������������! "��#�������� F �akQ��w�z�lZz:gjl/km���bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a���L©~®CB � ²´®�µ�� � · �$%&��'�)(� *�+�������� t:¬­���   Í­���a}:¬­z:g�k7�bØbuatSt�lTkZ��ͨl�z:gjl&t�l+km�A�bw�ÍÏuL}���l&�&�akQ����t�¬­z:¬¨�a�jt'z���©e�a}:k¢z:gjl�km�A�bw�ÍÏuL}'��l&�&�akQ���LÐ

t:¬­z:¬¨�a���L©~®DÀ

�»Ò � �¯� � � # ; � ' � ' � �4� #."�/(/(2!343 5 ��� ÈFÇDÂa˯Â�Ë_ÂaÄ��JÈ>ÉeÄ:ÂaÄg���L§JÄTÉߧ��«£ ® � É��§JËV&£aÄQÂaË)�8¡Q£�¤a¥�¦¨§-, Å�£�Ë)�É´ÂaÈ>ËjÈ>Ë@� � � � , � ��²� � · ÈÏÇZ¸¥�Ë9Èe£aË�£ &Vc§JÄ £ £aÄ/¡m£aÄ�§�Å�£a¡�Æb£aË�§JË9É>Ç�£ ®V� �Á² � � · � ÂaË�¤., ���Á²� � · ÈÏÇZÂo¥�Ë9Èe£aË£ &Vc§JÄ:£y£aÄ/¡m£aÄ:§�Å�£��ôÅ�£a¡�Æb£�Ë�§JË9É>ÇQ£ ®O� ��² � � · �8C� ÖGÖ �S� _ß©~uDkm���bw�ͨl , �&�a��zSuL¬­�jt&� � Ø�z:gjlT�-t�¬­�j�&l � � ¬¨t'�j�a�9ua�L�:ua�&lT��z�z��¸lT��lT}:{o��lT}:z�lJx�¬­� , ¾ ��²� � ·JØjlT��lT}:{��lT}:z�lJx�¬­� , � �Á²� � · ¬¨t5�j�a�9ua�L�:ua�&lT��z�z��¸��lT}:{8��lT}:z�lJx�¬­� , ¾ �Á² � � ·JÀhfhg¼wjt , � �Á²� � · ¬¨t+uDw���¬¨�a���L©�LvlT}���a}�km�a}�l/�&�a���jl&�Tz�l&���&�akQ���a�jlT��z�t5�L©»®O� ��² ���&·&Àfhg9uLz , ����² ���v·ª¬¨t5u�w���¬¨�a�-�L©�&�LÐß�&�akQ���a�jlT��z�t��L©~®O� �Á²�� ��·¬¨t'��}�����l&��¬­��uDt�{�kQkmlTz:}:¬¨� �hu&{�Ø9�¼{8}�lT��lT}�t:¬­���mz:gjl/}��aͨl&t��L©él&�b��l&t5uL�j���j�a�bÐßl&�b��l&tvÀ

/ ( �)�*������� ���,.�/��*��� '�21 ����5"768��#9� ��������S10 � '�32 ���� '�54 �S"$�

νlZ�j���2��l&t��T}:¬­��l0uL�«uLÍ­���a}:¬­z:g�k z:g9uLz�¬¨t5�9uat�l&���a�   Í­���a}:¬­z:g�k �ouL�j��z:g9uLz5}:w��jt�¬­�t.m²+aNb,YX ²,�µ�+4·:·hz:¬­kml�Àfhgjl5w���¬¨�a�bÐ �j�j�8�a��lT}SuLz:¬¨�a�jthuL}�l�z:gjl5�a��Í­{Q�a�jt:zSua�Tͨl�z��ZuCÍ­¬­�jlvuL}»z:¬­kml����aw��j�GÀéfhg�¬¨thuLÍ­���a}:¬­z:g�k ¬¨tªt:¬­kQ��ͨlT}»z:g9uL�uL��{8��}�lT�A¬¨�awjt�uLÍ­���a}:¬­z:g�k �'¬­z:gyz:g�¬¨t����aw��j�GÀ

6Gikj 7 oGr���r } ��s�p2v��<p2s<uP� m�s9�<�Gu oGu>vxm!y z|m!}�~��<~�m?� ��s<u)uνl/wjtSl�z:gjl�©e�aͭͨ���'¬­���o�juLzSuDt:z:}:wj�Tz:w�}�lZz��D}�lT��}�l&t�lT�¼z�km���bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a��z:}�l&l&tvÀ

� ����� �

Page 13: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË n�n

; �b�v� %��v� �������4� Ò �" # �8Ò � � Ò � Ò *��v�b� ) Ö * Ö �5� � Ö ×��>���m× Ò � Ö � � Ö �E)Ï� ) Ö * �v� Ò¼Ò +3� Ò � � Ò � Ò *��v�b� )�� Ò � � ¼Åg�«Ë�£v¤¼§o£ QÉ�b§mÉeÄ�§�§8Å�ÂaÄJÄJÈʧïÇoÂDÆb£aÈ>Ë9ÉߧJÄmÉ´£� �L§TÄJÉߧ�� £ QÉ�b§��aÄ:ÂïÆ>� = �¼ÈÊÅg�«ÈFÇm£aË�§m£�QÈ>É>Ç

¦­§� 0¤¼§ïÇ&ÅS§JË�¤¼ÂaË9É>Ç � Å�Âa¦>¦­§:¤8ÈeÉ>Ç }�lT��}�l&t�lT�¼zSuLz:¬­��l �� )>� ��)+*-,9��� � ¼Åg��Ë�£v¤¼§�£ CÉ��§ZÉeÄ�§�§0ÈFÇ/Âo¡¸§J¡V��§JÄZÈFË�Âm¦ ÈFÇJÉ»É�bÂ�É»¡mÂaÈ>Ë9É´ÂaÈ>Ë�Ç/É�b§0Åg�¼È>¦Ã¤aÄ�§J˯£ /ÈeÉ>Ç ÆbÂaÄ�§JË9ÉA�� ��� Ò *���� � Â�Åg�mË�£v¤¼§ � £ 5É�b§�ÉeÄ�§�§�Å�ÂaÄJÄJÈʧïÇ�Â'Æ�£aÈFË�ÉߧJÄ5É´£0È>É>Ç�ÆbÂ�Ä:§JË�ÉA�.�?��ÈÏÇ�Æb£aÈ>Ë9ÉߧJÄ5ÈÏÇ�¥�Æw�ÝÉ´£��ô¤�Â�Éߧ/§ ��ÅS§ßÆ�Éé¥AË�¤¼§TÄ

Ç´Æ�§�ÅTÈeÂ�¦4ÅTÈ>Ä ÅT¥A¡�ÇJÉ´ÂaË�ÅS§ïÇ�É�bÂ�ɪÂaÄ�§D¤¼§ïÇ&ÅTÄJÈ���§�¤7��§J¦­£ = �� Ö � � Ö * Ò *��v��� � Â�Åg� �L§JÄTÉߧ��-£ �É�b§0ÇJ¥I����Ä ÂïÆw�«¥AË�¤¼§JÄQÅ�£aËjÇïÈʤ¼§JÄ:Â�ÉeÈe£�˽Ä:§JÇïÈʤ¼§ïÇ0È>˯Â8¥AËjÈe£�Ë)� @~Ë�¤oÇJÉeÄJ¥bÅ&Ée¥AÄ�§QÉ�AÂ�É

Ä:§��'§:ÅvÉ É��§0ÅS£aËjË�§:ÅvÉߧ:¤�Å�£a¡�Æb£�Ë�§JË9É>Ç0£ ® �� £aÄ-§:¼Åg� �L§JÄTÉߧ�� £ yÉ�b§a�aÄ ÂJÆ>� É�AÂ�É0¼Å&Ée¥bÂ�¦F¦ �½ÈFÇy¯Ä:§ÝÆ�Ä:§JÇ&§JË9É´Â�ÉeÈ@�L§ £�yÂÁÅ�£a¡�Æb£�Ë�§JË9É � = §y¡mÂaÈ>Ë9É´ÂaÈ>Ë]ÂÆb£aÈ>Ë9ÉߧJÄCÉ´£8É�b§�Å�£aÄJÄ�§ïÇ´Æb£�Ë�¤aÈ>Ë@�8Ë�£v¤¼§�£ CÉ��§0¤�§:Å�£a¡�Æb£LÇJÈ>ÉeÈÊ£aË�ÉeÄ�§�§9�

fhgjl/Í­¬¨t�z5�L©ét�¬­��Í­¬­����t�k¸uL¬­�¼zSuL¬­�jt�z:gjlC�a��lT}SuLz:¬¨�a�jt5�L©��lTͨlTz:¬­���olTͨlTkmlT��z�t�uL�j���L©~�&�a�j�vuLz�lT�9uLz:¬¨�a���L©~t:wj�Sg�Í­¬¨t:z�t'¬­�.¸²�·hz:¬­kml�À~fhg�¬¨t��vuL�-��lCua�Sg�¬¨lT��l&���¼{yuQ�T¬­}��Tw�ÍÏuL}5���aw���Í­{�дͭ¬­�I��l&��Í­¬¨t�zvÀ

fhgjlC�9uL}�lT�¼z5���a¬­�¼z�lT}5kQ¬­��g¼z5�j�az5��l�w��bдz��LÐß�juLz�lZ¬­�yz:gjl�©e�aͭͨ���'¬­���o�vuat�l&t �n�À�fhgjl��9uL}�lT�¼z�¬¨t�³ ��bÀ�fhgjl��9uL}�lT�¼z�¬¨t5uD�&�a���jl&�Tz�l&�-�&�akQ���a�jlT�¼zvÀνlZt:w�kQk¸uL}:¬�LvlCz:gjl���}��a��lT}:z:¬¨l&t5�L©�aw�}5�juLzSuQt:z:}:wj�Tz:w�}�lC¬­�yz:gjl�©e�aͭͨ���'¬­���¸Í¨lTkQk¸u0�

�»Ò � �¯� >�6� ¤¼Â�É´Â_ÇJÉeÄJ¥bÅvÉe¥�Ä�§�&£aÄy¡m£v¤a¥A¦ÃÂaÄ ¤¼§:Å�£�¡5Æ�£LÇïÈeÉeÈe£�Ë ÉeÄ:§�§JÇ Å�ÂaË��S§�¡QÂ�ÈFË9É´Â�ÈFË�§�¤½È>Ë Â = Â��¯É�AÂ�É0È>ÉÂa¦>¦­£ = ǸÉ�b§*T£�¦F¦­£ = È>Ë@��£JÆ�§JÄ:Â�ÉeÈe£�Ë�ǸÈ>Ë .¸²��²,�µ�+4·:· Âa¡m£aÄTÉeÈ V�§:¤«ÉeÈ>¡m§ � = �b§JÄ�§ �²,�µ�+4· ÈFÇmÉ�b§ Âa¡m£aÄTÉeÈCVc§:¤(�:£a¥AË�¤£aË É�b§Z¥AËjÈe£�Ë)� @~Ë�¤o¤�Â�É´ÂQÇTÉeÄï¥�Å&Ée¥AÄ:§DÉ�bÂ�É~ÈFÇ/¥¼Çv§:¤��

��������� � � £aÄZÂaË)�¸Ë�£v¤¼§ = §�Å�Âa˽Â�Å�Åï§ïÇSÇZÈeÉ>Ç ÆbÂaÄ�§JË9É È>Ë�É�b§0ÉeÄ�§�§ R È /È>Éh§��aÈFÇJÉ>Ç S �� �� ������ �K��Ë>�mË�£v¤¼§�£ ZÂ8¤¼§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£a˽ÉeÄ:§S§DÅ�ÂaËP��§�¤¼§J¦¨§TÉߧ:¤�Â�ɪÂaË)�oÉeÈF¡¸§9����� �� �`��Ë)�¸É = £¸Ë�£v¤¼§ïÇ � ÂaË�¤ � É�AÂ�ɪÂaÄ�§�É�b§�Åg�¼È>¦Ã¤�Ä:§J˽£ CÉ��§/Ç&Âa¡¸§ Æ�ÂaÄ:§TË9É�� Å�Â�ËZ�S§GJ¥�Ç&§:¤8È>Ë9É´£¸£�Ë�§9�8C� ÖGÖ �S�

��������� � fhgjlmuL�jtSlT�j�&lm�L©ªw��bдz��LÐß�juLz�l¸�9uL}�lT�¼zC���a¬­�¼z�lT}�t/¬­�½tS�akmlQ�L© z:gjlD�j�A��l&t/����l&tC�j�az/��}�lT��lT�¼zCz:gjlQ�juLzSut:z:}:wj�Tz:w�}�lQ©e}��ak �j�j�b¬­���«z:gjlm�9uL}�lT��z��L©5uL��uL}:��¬­z:}SuL}:{��j����l¸¹ �° ³7lgf¸�T¬¨lT�¼z:Í­{�À7_ß©'¹ ���Al&tC�j�az0g9u&��luL�yw��bдz��LÐß�juLz�lC�9uL}�lT��z5���a¬­�¼z�lT}vØ�u ������ �a��lT}SuLz:¬¨�a�-�a��¬­z�t'}�lT��}�l&tSlT�¼zSuLz:¬­��l0��}����bwj�&l&t5z:gjl/�&�akQ���a�jlT��z��z:g9uLz/�&�a�¼zSuL¬­�jtC¹ÁÀ�_ß©»¹��°��QØ�z:gjl��9uL}�lT�¼zC¬¨t�� �G�az:gjlT}��'¬¨t�lQ¬­z�¬¨tZ³¸ÀZfh¬­kml&t:zSuLkQ��¬­�����9uL}�lT��zC���a¬­�¼z�lT}�tuL�j�¸�j�A��l&th�L©�z:gjl�z:}�l&l��'gjlT� z:gjlT{ouL}�l+�T}�lvuLz�l&�8��¬­��l&t�uK�hu&{8�L©4��lTz�l&�Tz:¬­���-�'gjlTz:gjlT}'u0�9uL}�lT��z ���a¬­�¼z�lT}h¬¨t�aw�z��L©é�juLz�l�Øj¬­�9.m²�· z:¬­kml�À

� �� ������ � c�lTͨlTz:¬¨�a�¸�L©Xu+�j�A��l|�o�vuL�Q��l�ua��g�¬¨lT��l&�¸�¼{Qt:��Í­¬¨�T¬­���C¬­z~�aw�z~�L©�z:gjl����aw���Í­{�дͭ¬­�I��l&�mÍ­¬¨t:z~�L©�¬­z�t~t:¬­��Í­¬­����tvØw����juLz:¬­���¸z:gjlC��g�¬­Í¨�-���a¬­��z�lT}5�L©z:gjl/�9uL}�lT�¼z � ¬Ã©��jl&�&l&tStSuL}:{�Ø9uL�j��t:��Í­¬¨�T¬­���¸z:gjl/Í­¬¨t:z5�L©��g�¬­Í¨�b}�lT�«�L©��«¬­�¼z��z:gjl�Í­¬¨t:z5�L©é��g�¬­Í¨�b}�lT��uLz � À

���� �� � νl��j}�t:zQzSu���l z:gjlyÍ­¬¨t:z�tQ�L©���g�¬­Í¨�b}�lT�]�L© � uL�j� ��uL�j� �&�a�j�vuLz�lT�9uLz�l�z:gjlTk�À�νl ��¬­��l z:g�¬¨tQ�jl%� Í­¬¨t:zz��9�4ØtSuv{�تuat�Í­¬¨t:zQ�L©���g�¬­Í¨�b}�lT�Xت�¼{_w����juLz:¬­��� z:gjl ��g�¬­Í¨�����a¬­�¼z�lT}¸uL�j�Áz:gjl8w���¬¨�a�bÐ �j�j� �juLzSu«t:z:}:wj�Tz:w�}�lua�&�&�a}��b¬­����Í­{�À»fhgjlT�7� l/k¸u&{y��lTͨlTz�l/z:gjlCz:}�l&l/�j����l ��À

��� ù0¢2G ? ���

Page 14: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

nv� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

6Gih ���Gu v��2rBs<�?~e�2�{o0~e}�vI~�zWqe~k�GufhgjlC©e�aͭͨ���'¬­���o¬¨t+�aw�} Åg�bÂaÄ ��ÈFË@�-¤aÈÏÇ&ÅTÈ Æ�¦ È>Ë�§ ©e�a}����aw��j�b¬­���8z:gjlZ�&��t:z+�L©�a��lT}SuLz:¬¨�a�jt+�bw�}:¬­���oz:gjlZ¬­�j�bwj�Tz:¬­��l0t:z�lT��L©   Í­���a}:¬­z:g�k �jÀ

n�À�fhgjlT}�l�uL}�lY.m²+4· ��������� �a��lT}SuLz:¬¨�a�jt¸�v��lT}¸uL� lT��z:¬­}�ly}:w�� �L©+z:gjl�uLÍ­���a}:¬­z:g�k�Ø tS��z:gjly�&��t�zm�L©�z:gjl&t�l ¬¨tua�&�&�aw��¼z�l&��©e�a}�t�lT�9uL}SuLz�lTÍ­{�Ø�uL�j� zSu���l�.m²+aN ,YXª²,�µ�+4·:· z:¬­kml�Ø9t�l&l�F �a}:kmlT�-lTz�uLÍ´À�: nv�����<;JÀ

�bÀ   Åg�AÂ�Ä �¼§ �&�a�jt:¬¨t:z�t��L© u ��x�l&����w�kZ��lT}C�L©|.¸²�·5�a��lT}SuLz:¬¨�a�jtZuL�j��u ��xbl&�«��w�kZ��lT}C�L© ������ �a��lT}SuLz:¬¨�a�jt¬­�yw���¬¨�a�bÐ �j�j�-�juLzSuQt�z:}:wj�Tz:w�}�l&tvÀ

�bÀ'νl�k¸uL¬­��zSuL¬­�oz:gjl��T}�l&�b¬­z»¬­�¼��uL}:¬ÏuL�¼zhz:g9uLz lvua��g8�j����l��L©4uC��l&�&�akQ����t:¬­z:¬¨�a�8z:}�l&l5�vuL}:}:¬¨l&thuC�T}�l&�b¬­zªz:g9uLzª�vuL��9u&{o©e�a}5um��g9uL}:��l�À   �j�A��l � t��T}�l&�b¬­z5�vuL�-�a��Í­{8��l�}�lTͨlvuatSl&�yz��Q�9u&{o©Ê�a}5�a��lT}SuLz:¬¨�a�jt*�'gjlT��¬­z'¬¨t���lTͨlTz�l&�GÀ

�jÀ �XlTz5z:gjl Â�Å&ÉeÈ+�L§m§:¤���§ïÇ �L©ª®±��l/z:gj��t�l0l&�b��l&t�z:g9uLz+uL}�lZ�j�az�t:w��jtSlTz�t5�L©G�Á²����·��a} �Á²����·JÀ �XlTz�, ����lCz:gjl�¼w�k0��lT}Z�L©'ua�Tz:¬­��lQl&�b��l&tvÀ�νlmua�bgjlT}�lDz�� z:gjlQ�b¬¨t��T¬­��Í­¬­�jlDz:g9uLz �ªlm¬­�j�Tw�}�.m², �¨·+��g9uL}:��l&tCz:g9uLzZuL}�lD�j�az�9uL¬¨� ©Ê�a}'�¼{8}�lTͨlvuat�l&���T}�l&�b¬­z�tvØ9uL�j�7�ªlC��lT����t:¬­z�.m², �Ï·ª�jl%�2�T}�l&�b¬­z�t5¬­�yz:gjl/z:}�l&l�À

=ªua��g-l&�b��l/¬¨t�ua�Tz:¬­��l�¬­��lJxjua�Tz:Í­{ �a�jl�}�l&�Tw�}�t:¬­��l/�vuLÍ­ÍG¬­�yz:gjl�}�l&�Tw�}�t:¬¨�a�yz:}�l&l�ÀG_ß©!� lCua�bgjlT}�l�z��Dz:gjl/��g9uL}:��¬­����b¬¨t��T¬­��Í­¬­�jl�Øb¬­zh}�l&�&lT¬­��l&t*.¸²�· ��g9uL}:��l&th¬­� z:g�¬¨t }�l&�Tw�}�t:¬­��l/�vuLÍ­Í´Ø�uL�j�o�j�D�Sg9uL}:��l&t�lTͨt�l%�'gjlT}�l�À~fhg�¬¨t ��¬­��l&t�u0���aw��j��L©2.m²,�· �Sg9uL}:��l&t5�v��lT}5z:gjlClT�¼z:¬­}�l/}�l&�Tw�}�t�¬¨�a��z:}�l&l�À~fhg�¬¨t'��¬­��l&t5u����aw��j���L©G.m²,�· ������� �a��lT}SuLz:¬¨�a�jtvØ)�'g�¬¨��g�¬¨t.¸²+aNb,YXª²,�µ�+4·:·JØ�uL�j�Y.m²,�·h�&�a�jt�zSuL�¼z дz:¬­kmlZ�a��lT}SuLz:¬¨�a�jtvÀ

_Ý��ua���b¬­z:¬¨�a�XØ�� l»g9uv��l uL��.m²+GN`,YXª²,�µ�+4·:·���}�lT��}��A�&l&t�t:¬­����t�z�lT�XÀUJ�wj�&�&l&t�t:©>w�Í­Í­{/k¸uL¬­��zSuL¬­��¬­���/z:gjl»�b¬¨t��T¬­��Í­¬­�jl�'¬­Í­Í�¬­kQ��Í­{�z:gjl�.¸²:²+CN^,�· Xª²,�µ�+4·:·5���aw��j�GØ�t:¬­�j�&l�uL�«l&�b��l0�a��Í­{���l&�&�akml&t+ua�Tz:¬­��l��a�j�&l�Ø�gjlT�j�&lZz:gjlZ�¼w�kZ��lT}�L©���x�l&�AÐß�&��t:zª�a��lT}SuLz:¬¨�a�jthuL�j� ������� �a��lT}SuLz:¬¨�a�jtW�'¬­Í­Í���l'���aw��j��l&�o�¼{Dz:gjl .¸²,�·�T}�l&�b¬­z�tª¬­��¬­z:¬ÏuLÍ­Í­{ou&��uL¬­ÍÏuL��ͨl��a�z:gjl�l&�b��l&t�À�E+��� lT��lT}vØAuL�a.¸²:²+*Na,�· Xª²,�µ�+4·:·���aw��j��©Ê�a}»uL�¼{Dkm�A�bw�ÍÏuL}~��l&�&�akQ����t:¬­z:¬¨�a� uLÍ­���a}:¬­z:g�k ¬­kQ��Í­¬¨l&tªuL�.¸²+ NO,YXª²,�µ�+4·:·4���aw��j�GØ�t:¬­�j�&l�Øa¬Ã©�+��3, N h¬­��z:gjlh�a}:¬­��¬­�9uLÍ���}SuL��gXØ�� l'�vuL�Qt:��lT�j��.m²+|N�,�·���}�lT��}��A�&l&t�t:¬­���z:¬­kml��b¬­��¬¨�b¬­���oz:gjlC¬­��¬­z:¬ÏuLÍ���}SuL��g-¬­��z��¸¬­z�t��&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z�tvØ�t��aÍ­��lZz:gjlC��}��a��ͨlTk7�a��lvua��g �&�akQ���a�jlT�¼zvØ�'gjlT}�l +«¬¨t|.¸²,�·JØ�uL�j�oz:gjlT� k¸u���l+z:gjl�}����az'�L©Xz:gjl&t�l+z:}�l&l&t ��l�z:gjl���g�¬­Í¨�b}�lT���L©�uZ�jl%� �9uL}SuLͭͨlTÍ�}����azh�j�A��l�Ø�z���a��zSuL¬­��z:gjl/��l&�&�akQ����t:¬­z:¬¨�a�«�L©~®QÀ

��lTz+z:gjl Â�ÅvÉeÈ@�L§QË�§JÈ����x�:£�ÄSÇ �L©huotSlTz�¹º�L©»��lT}:z:¬¨�&l&t���l0z:gj��t�lZkmlTk0��lT}�t/�L©'³ ¾h¹ºz:g9uLz/uL}�lQua�L�:ua�&lT��z�z�� ukmlTk0��lT}+�L©�¹ �a��uL�-ua�Tz:¬­��lZl&�b��l�À

�»Ò � �¯� �� 5 ¹ ÈFÇ Â�Ë�£v¤¼§�£����� ²´®O� ��² � � ·:· £aÄ���� J ®O� ��² � � · N É�AÂ�É/ÈFÇ Â½¡m£v¤a¥A¦­§�£ ® � É�b§�Åg�AÂaÄ��aÈ>Ë@�¤aÈFÇTÅ&ÈøÆ�¦ È>Ë�§ZÂ�¦F¦­£ = Ç/¥¼ÇZÉ´£¸È>Ë�ÅT¥AÄ0£aË�§�Åg�AÂ�Ä �¼§DÂ�É ¹ &£aÄ0§:Â�Åg��¼Å&ÉeÈ@�a§CË�§JÈ#���x�:£aÄD£ ¹ �8C� ÖGÖ �S� νlQt�gj���±z:g9uLzCz:gjlQt:w�k��L© ͨlT����z:gjtZ�L©'ua�Tz:¬­��lQ�jlT¬­��g¼���a}CÍ­¬¨t:z�t/�L©'t:wj��g��j�A��l&t/¬¨t`.¸², � ·JÀQfhgjlQt:w�k�L©~ͨlT����z:gjt/�L©~z:gjlDua�Tz:¬­��lZ�jlT¬­��g����a}�Í­¬¨t:z�t+�L©»�L©ªuLÍ­Í�ͨlvu&��l&t��L© ��� ²´®O� �Á²�� � ·:·�uL�j� ��� JT®O� � ²� � · N�¬¨t+lJxjua�Tz:Í­{7, � À  �¼{�¬­��z�lT}:�9uLÍé�j����lZz:g9uLz�}�lTk¸uL¬­�jt/uQkm�A�bw�ͨl��L© ®±g9uat/uL�«ua�Tz:¬­��l0�jlT¬­��g¼���a}�Í­¬¨t:z5z:g9uLz+¬¨t�¬¨��lT�¼z:¬¨�vuLÍéz��mz:g9uLz��L©lvua�Sg½�L©»¬­z�t���g�¬­Í¨�b}�lT�XÀZfhg�wjtvØ�z:gjl0ͨlT����z:g½�L©»¬­z�t/ua�Tz:¬­��l0�jlT¬­��g����a}/Í­¬¨t�z�¬¨t/uLz�km��t:z�g9uLÍéª�L©»z:gjlDt:w�k$�L©»Í¨lT����z:gjt�L©ªua�Tz:¬­��l��jlT¬­��g¼���a}+Í­¬¨t:z�t��L©~¬­z�t+��g�¬­Í¨�b}�lT�XÀ �XlTz��²>¹�·'��lCz:gjl0ͨlT����z:g �L©~z:gjl0t:gj�a}:z�l&t�z��9uLz:g�©>}��ak ¹ z��ou¸Í¨lvuc© Àfhgjl/t:w�k �L©�ͨlT����z:gjt��L©4�jlT¬­��g����a}'Í­¬¨t:z�t'�L©0�v¹ ��4²>¹ ·ª°\' uL�j� ¹�¬¨th�j����l��L© ��� ²´®V� �Á²�� �v·:· �a} ��� J&®O� ��² ���&· N¬¨t5u0km�A�bw�ͨlC�L©®��C¬¨t5uLz'km��t:z|, � B���"�À2J�w�kQkQ¬­���mz:g�¬¨t5�v��lT}�¬­�j�T}�lvuat:¬­���¸�auLÍ­wjl&t'�L©G'o��¬­��l&t5u���l&�akmlTz:}:¬¨�Ct�lT}:¬¨l&tz:g9uLz5t�w�kmt'z��Qͨl&t�thz:g9uL� ��, �ÊÀ

9ZÖ � Ö � �>��� Ó � �?�b§0Åg�AÂ�Ä �aÈ>Ë@� ¤aÈFÇTÅTÈ Æ�¦ ÈFË�§ZÂa¦>¦­£ = Ç/¥¼ÇCÉ´£¸ÈFË�Å&¥�ÄZÂ8Åg�AÂaÄ��¼§DÂ�É §.�L§JÄg�oË�£v¤¼§0£ ���� ²´®O� �Á²�� ��·:· ÂaË�¤��� J&®O� � ²� � · N É�AÂ�É~ÈFÇ�Ë�£�ɪ¸¡m£v¤a¥A¦­§D£ ® �J¥bÉ2�AÂLÇZÂaË«ÈFË�Å&Èe¤¼§TË9ɪÂ�ÅvÉeÈ@�L§Q§�¤��¼§9�

8C� ÖGÖ �S� _ß©ªz:gjlm�j����lD}�lTk¸uL¬­�jtZu8km�A�bw�ͨlm�L©5®DØGz:g�¬¨t�©Ê�aͭͨ����tC©>}��ak ��lTkQk¸u �jÀø�bÀ � z:gjlT}��'¬¨t�l�ØXz:gjl¸��g9uL}:��lQ¬¨t�9uL¬¨��©Ê�a}+��{�z:gjlD�T}�l&�b¬­z�z:g9uLz+¬¨t�©e}�l&l&�9�'gjlT�«¬­z�¬¨t � �� ������ �«�bw�}:¬­��� z:gjl�}�l&t:z:}:¬¨�Tz/t:z�lT���L©   Í­���a}:¬­z:g�k$�jÀO:efhgjl� �� ������ �a��lT}SuLz:¬¨�a��}�lJ©elT}�t'z��Dz:gjlZ�a�jl/��}��v�A¬¨��l&�y�¼{�c/uLzSuVJ�z:}:wj�Tz:w�}�l/�jÀ­n�;JÀ

� ����� �

Page 15: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË nv�

�+�az�lyz:g9uLz�� lyk¸u&{�¬­�j�Tw�}ou �Sg9uL}:��l-uLz¸lT��lT}:{Á�j����ly�L© ��� ²´®O� �Á²�� � ·:·muL�j� ��� J&®V� �Á² � � · N�z:g9uLzmg9uat¸uL�¬­�j�T¬¨��lT��z'ua�Tz:¬­��l�l&�b��l��A¬Ã©Xz:gjl��j����l5¬¨t uZkm���bw�ͨl+�L©�®DØ�z:gjl��Sg9uL}:��l+¬¨tª�&����lT}�l&�o��{ �XlTkQk¸uZ�jÀø�bØ�uL�j�¸¬Ã©X¬­zª¬¨t»�j�azvØz:gjl���g9uL}:��l��vuL�8��l��9uL¬¨�¸©e�a} �¼{mz:gjl+�T}�l&�b¬­z z:g9uLz ¬¨tª}�lTͨlvuat�l&�8¬Ã©X¬­z ¬¨t � �� ������ �o�bw�}:¬­���Dz:gjl�}�l&t:z:}:¬¨�Tz t�z�lT�XÀfhg¼wjtvØz:gjlC�a��Í­{8zß{A��lC�L©��j�A��lCuLz&�'g�¬¨�Sg7�ªl/k¸uv{ �j�az5t:��lT�j�Y.m²�· z:¬­kml/¬¨t5�a�jl/z:g9uLz�g9uat'�j�Q¬­�j�T¬¨��lT��z+ua�Tz:¬­��lCl&�b��l�À

��lTzC¹���lQt:wj��gÁu8�j����l�À�¹�k0wjt:zC��lmu8�j����lQ�L© ��� J ®O� � ²� � · N Ø�t:¬­�j�&lmuLÍ­Í�j����l&tZ�L© ��� ²ß®O� ��² � � ·�·�g9u&��l� � uat0uL�Áua�Tz:¬­��lm�jlT¬­��g¼���a}vÀo¹ ¬¨t/}�lT��}�l&tSlT�¼z�l&�_�¼{½u �j�A��lm�L© c/uLzSuYJ�z:}:wj�Tz:w�}�lo�jÀ­nQz:g9uLzC¬¨tC}�lTz:w�}:�jl&�¯��{«z:gjl}�l&�Tw�}�t:¬­��l �vuLÍ­Í z��«®O� � ²����·JÀ�¹ }�lTk¸uL¬­�jtQu�km���bw�ͨl8�L©+®DØt��-¬­z0k0wjt:z0��lmwjtSl&�¯¬­�_z:gjl8uat�tSlTkZ��Í­{¯�L© ��� ²´®C·�bw�}:¬­����z:gjlD¬­�j�bwj�Tz:¬­��l¸t:z�lT�XÀ�_ß©ªz:gjlD�9uL}�lT��z �L©h¹º¬­� ��� J ®O� ��² � � · N g9uat�¬­�j�T¬¨��lT�¼z0ua�Tz:¬­��lml&�b��l&t�Ø ¬¨t/�j�azu-km���bw�ͨl8�L©�®DØtS� $¬¨t��j�azDu��j����l8�L© ��� ²ß®C·TÀ _ß�_z:g�¬¨tD�vuat�l�Øé¹ g9uat�u��b¬#]GlT}�lT�¼zD�9uL}�lT�¼zD¬­� ��� ²´®Z·/z:g9uL�¬­z8����l&to¬­� ��� J&®O� ��² ���v· N�Øh��w�zo�aw�} ��g9uL}:��¬­�����b¬¨t��T¬­��Í­¬­�jl����Al&to�j�az8uLͭͨ���7wjtoz��½¬­�j�Tw�}yu½��g9uL}:��l uLz¸¹.��{}�l&t�lTz:z:¬­���¸¬­z�t'�9uL}�lT�¼z����a¬­��z�lT}vØ9�a}'�¼{ t:��Í­¬¨�T¬­���m¬­z5¬­�¼z��Qz:gjlC�Sg�¬­Í¨�yÍ­¬¨t:z��L©é¬­z�t'�jl%� �9uL}�lT��zvÀ �XlTz'wjt��vuLÍ­Í�uL�¼{ �j����l�L© ��� J&®O� � ²� � · N8z:g9uLz�g9uat5�j�D¬­�j�T¬¨��lT�¼z�ua�Tz:¬­��lCl&�b��l&t�uL�j�-u��jl%�2�9uL}�lT�¼z5¬­� ��� ²´®C·�u Æ�Ä:£��J¦¨§J¡ Ë�£�¤¼§ À

fhgjlT}�lyuL}�l¸z:g�}�l&loz:}:¬¨� �At0z:g9uLz��ªlowjt�lm©Ê�a}Z���aw��j�b¬­���«z:gjlo�&��t:z��L©5t:��Í­¬¨�T¬­��� z:gjlm��}��a��ͨlTk �j�A��l&tZ¬­��z���z:gjlc/uLzSuVJ�z:}:wj�Tz:w�}�lZ�jÀ­n�}�lT��}�l&t�lT�¼zSuLz:¬¨�a� �L© ��� ²´®C·4�f�}:¬¨� �½n��   Í­¬¨t:zC�L© ��}��a��ͨlTk ��g�¬­Í¨�b}�lT�¯�L©�u8�j�A��lQ¹ �vuL�½��lQ�a��zSuL¬­�jl&�(�'¬­z:gj�aw�zC�A¬¨�aÍÏuLz:¬­����z:gjlm�Sg9uL}:��¬­�����b¬¨t Ð

�T¬­��Í­¬­�jl�Øuat/©Ê�aͭͨ����t�À7_´©h¹ }�lTk¸uL¬­�jtQu km���bw�ͨl8�L©�®DØ��j�a�jl¸�L©'¬­z�t���g�¬­Í¨�b}�lT� uL}�lm��}��a��ͨlTk ��g�¬­Í¨�b}�lT�XÀ� z:gjlT}��'¬¨t�l�Ø�¹ kZwjt�z»g9u&��l�¬­�j�T¬¨��lT�¼z�ua�Tz:¬­��l�l&�b��l&tvÀéνl�k¸uv{Q¬­�j�Tw�}huZ��g9uL}:��l�uLz»¹�uL�j�8uLz lvua��g ��g�¬­Í¨��L©�¹ z:g9uLz�g9uat5uL�y¬­�j�T¬¨��lT��z�ua�Tz:¬­��lCl&�b��l�Ø���{�F �a}��aÍ­ÍÏuL}:{ �jÀ �jÀ � �jlJд��{¼Ðß�a�jl�Ø>�ªlCt:��Í­¬¨�&l�z:gjl&tSl/��g�¬­Í¨�b}�lT��aw�z �L©�z:gjl����aw���Í­{¼Ð´Í­¬­�I��l&�8��g�¬­Í¨�oÍ­¬¨t:z»�L©G¹ÁÀéÎ]g9uLz~}�lTk¸uL¬­�jt ¬¨t»z:gjl�Í­¬¨t:z»�L©G¹ � t»��}��a��ͨlTk¢��g�¬­Í¨�b}�lT�XÀνlk¸uv{8t:��lT�j�7.m²�· z:¬­kml�g9uL�j�bÍ­¬­���¸z:g�¬¨thÍ­¬¨t:zvØ�t:¬­�j�&l�z:gjlT}�l�¬¨t'�a��Í­{ �a�jl/�L©4z:gjlTk uLz'¹_Àfhgjl&tSl/��g�¬­Í¨�b}�lT���l&�&�akml¸�Sg�¬­Í¨�b}�lT���L©5u-t:¬­����ͨl¸�j�A��l�� ¬­� ��� ²´®C·TØuL�j�Z� lo�vuL�Át:��Í­¬¨�&l¸z:gjlTk uat�u���}��aw��_¬­��z��-z:gjl�Sg�¬­Í¨�yÍ­¬¨t:z��L©�� ¬­� .¸²�·hz:¬­kml�À

f�}:¬¨� �-�0�&_ß©X¹ ¬¨t uZ�j�A��l��L© ��� J ®O� � ²� � · N z:g9uLz ��lTz�t � �� ������ �GØ�uL�j�¸¬­z�t ��g�¬­Í¨�b}�lT�-uL}�l+�&�a���jl&�Tz�l&�y�&�akQ���a�jlT�¼z�t�L©C®DØ»�a}Dz:gjl �jl%� �9uL}�lT�¼zm�L©+¬­z�tD��}��a��ͨlTk���g�¬­Í¨�b}�lT� ¬­� ��� ²ß®Z·C¬¨tmu �&�a���jl&�Tz�l&� �&�akQ���a�jlT��z¸�L©�®DØz:gjlT�9�ªl����¸�j�az+�jl&l&��z��¸w����juLz�l0lJxb��Í­¬¨�T¬­z:Í­{�z:gjl0�9uL}�lT�¼z����a¬­�¼z�lT}�t��L©»¬­z�t���}��a��ͨlTk$�Sg�¬­Í¨�b}�lT�XØ��¼{�z:gjllJx��&lT��z:¬¨�a�jt&� l/k¸ua��l�©Ê�a}&c�uLzSuaJ�z:}:wj�Tz:w�}�l/�jÀ­n�À

f�}:¬¨� �-�0�&_ß©4z:gjl/w���¬¨�a���3�L©éz:gjl/��}��a��ͨlTk �Sg�¬­Í¨�b}�lT�-¬¨t'z:gjlT¬­}��jl%�2�9uL}�lT�¼z5¬­� ��� ²´®Z·TØbz:gjlT��z:gjl/�j�A��lCt:z:}:wj�Tz:w�}�l� z:g9uLzª©e�a}:kmlT}:Í­{Q}�lT��}�l&t�lT��z�l&� ¹$k¸u&{m��l5t:��Í­¬¨�&l&�o¬­�o�b¬­}�l&�Tz:Í­{ uat»z:gjl5�j����l+t:z:}:wj�Tz:w�}�l � }�lT��}�l&tSlT�¼z:¬­�����¬­��z:gjl�}�lT��}�l&tSlT�¼zSuLz:¬¨�a���L© ��� ²´®C·TÀfhgjl/�9uL}�lT��z����a¬­�¼z�lT}�t5�a�yz:gjl���}��a��ͨlTk ��g�¬­Í¨�b}�lT��t�z:¬­Í­ÍX�&�a}:}�l&�Tz:Í­{���a¬­�¼z�z��Dz:gjlC�9uL}�lT�¼z5¬­� ��� ²´®C·&Ø���w�z|�ªlC���m�j�az�g9u&��l/z��¸t:��lT�j��uL��{oz:¬­kmlCw����juLz:¬­���oz:gjlTk�À

�»Ò � �¯� >2 �?�b§�É�¼Ä�§�§ZÉeÄïÈÊÅOTvÇCÂa¦ = Â��LÇ�ÂïÆ�Æ�¦ �QÉ´£QÉ�b§ Æ�Ä:£��J¦¨§J¡ Åg�¼È>¦­¤aÄ:§TË«£ /ÂaË)�DË�£v¤�§�É�AÂ�ÉÈFÇ ����������¤m¤a¥AÄïÈ>Ë@�É�b§ZÈ>Ë�¤a¥�Å&ÉeÈ+�L§CÇJÉߧßÆ �8C� ÖGÖ �S�¹ )F�¸� � ���L� � � Ò � * Ö × Ò fhgjl'��}��a��ͨlTk¢��g�¬­Í¨�b}�lT� uL}�l5�&�a���jl&�Tz�l&�¸�&�akQ���a�jlT�¼z�t �L©4®O� ¹_ØbuL�j�GØAt:¬­�j�&l'z:gjlT{Dg9u&��l

�j�8¬­�j�T¬¨��lT�¼z0ua�Tz:¬­��lml&�b��l&t�ØGz:gjlT{«uL}�lmuLͨt��y�&�a���jl&�Tz�l&�¯�&�akQ���a�jlT��z�tC�L©'®DØXgjlT�j�&lm��g�¬­Í¨�b}�lT�¯�L© z:gjlD}����azvÀfX}:¬¨� �At0nCuL�j�-�Dt:wxf¸�&l/z��mt���Í­¬¨�&l�z:gjlTk ¬­�¼z�� ,� ²´®C·JÀ

¹ )F�¸��� Ò ��) Ò ��* Ö × Ò�Õ )Ï� ' � Ò � Ò �L� � � � Ö � � Ò � � ' ) �>×�� Ò *� fhgjlT� z:gjl�w���¬¨�a��� �L©C��}��a��ͨlTk ��g�¬­Í¨�b}�lT� ��lJÐ�&�akml&t+z:gjl0�jl%���9uL}�lT��z��L©ªz:gj��t�lDtSuLkmlZ��}��a��ͨlTk��Sg�¬­Í¨�b}�lT��¬­� ��� ²ß®Z·JØ�t�¬­�j�&l�� ¬¨t/u¸km���bw�ͨlD�L©h®DØGuL�j��j�DÍÏuL}:��lT}�w���¬¨�a�-�L©��g�¬­Í¨�b}�lT�«�L©�¹ ¬¨t5u�km�A�bw�ͨlC�L©~®QÀ~fX}:¬¨� �y�mt:wxf¸�&l&t'z��mt:��Í­¬¨�&l�z:gjlTk ¬­�¼z�� ,� ²´®C·JÀ

¹ )F�¸��� Ò ��) Ò ��* Ö × Ò�Õ )Ï� ']Ö * Ò � � Ö �� Ò � � ' ) �>×�� �a}¹ )F�¸� � ��) � Ò * Ö × Ò fhgjlT� ®O� ¹�¬¨tª�&�a���jl&�Tz�l&�GØjuL�j�m¹$¬¨t �&�a�¼zSuL¬­�jl&�8¬­�8u0t:¬­����ͨl5�&�a���jl&�Tz�l&� �&�akQ���a�jlT�¼zh�L©

®O� ��² � � ·&À�_Ý� ��� ²´®C·TØAlT��lT}:{Q��lT}:z�lJxm¬­�¸z:g�¬¨t �&�akQ���a�jlT�¼z g9uat»z:gjl�tSuLkml�ͨlvuat:z �&�akQkm�a��uL�j�&l&t:z��a}�� �'¬­z:g��� ù0¢2G ? ���

Page 16: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

nT� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

������������ ���������

��������� ��������� ��� � ������� � ��� ����� ��� � �!�"���

¡4¬­��w�}�lC� �2J���Í­¬­z:z:¬­���¸z:gjl/��}SuL��g�ua�&�&�a}��b¬­���oz��muD��¬­���az

���LØ���{ �XlTkQk¸u �bÀ �jÀ/fhgjl���}��a��ͨlTk ��g�¬­Í¨�b}�lT�½�L©»¹ uL}�l0k¸ucxb¬­k¸uLÍét�w��jt�lTz�t��L©»¹ºz:g9uLz+}�lTk¸uL¬­��km���bw�ͨl&t�L©'®DÀ�5 {   Í­���a}:¬­z:g�k �bØ�z:gjlD��}��a��ͨlTk ��g�¬­Í¨�b}�lT��uLÍ­Í��l&�&�akml¸��g�¬­Í¨�b}�lT�¯�L© � Ø�uL�j� z:gjl��j}�t:zC�&�a�j�b¬­z:¬¨�a�uL����Í­¬¨l&tvÀ�fX}:¬¨� � n k¸u&{0��lªwjt�l&�0z��+t���Í­¬¨�&lªz:gjlTk�¬­��z�� , ²´®C·JØa��w�zé�j�azéz���w����juLz�l z:gjlT¬­}é�9uL}�lT�¼z���a¬­�¼z�lT}�tvÀ_ßz }�lTk¸uL¬­�jthz���t:gj���¿z:g9uLz'fX}:¬¨� �8�DuL����Í­¬¨l&tvØAz:g9uLzh¬¨tvØAz:g9uLzhz:gjl��9uL}�lT��zh���a¬­��z�lT}�t��L©�z:gjl+��}��a��ͨlTk ��g�¬­Í¨�b}�lT����Q�j�az5�jl&l&�-z��m��l/w����juLz�l&�GÀ|J�w�������t�l� m¬¨t�uD�jlT¬­��g¼���a}+�L© � ¬­� ³ ¾ �3ÀW_Ýz5�vuL���j�az5��lZuQkmlTk0��lT}+�L©�Á²� � ·TØGt:¬­�j�&l�z:gjlT��¬­z � �aw�ͨ� ��l�¬­��z:gjlDtSuLkmlQ�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼zZ�L© �Á²� � ·+uat�¹_Ø�uL�j�t� �aw�ͨ� ��l�&�a�¼zSuL¬­�jl&��¬­� � ØXu �&�a��z:}Sua�b¬¨�Tz:¬¨�a�XÀO5 w�zK ��vuL���j�az/��lmuokmlTk0��lT}Z�L© �Á²�� �v·.� _Ýz/¬¨tCua�L�:ua�&lT��zCz��ylT��lT}:{��lT}:z�lJx�¬­� � ØGt:¬­�j�&l � ¬¨tCuokm�A�bw�ͨlD¬­�½®QØ�gjlT�j�&lD¬­z+¬¨tCuL�½ua�Tz:¬­��lD�jlT¬­��g����a}Z�L©ªz:gjlD��}��a��ͨlTk���g�¬­Í¨�b}�lT��L©»¹_ØB�'g�¬¨�SgXØG�¼{ uat�t:w�kQ��z:¬¨�a�XØ�g9uv��lD�j�o¬­�j�T¬¨��lT��z0ua�Tz:¬­��lDl&�b��l&tvÀ�J�¬­�j�&l � g9uat��j�8�jlT¬­��g����a}�t/¬­�¯®DØ�¬­z¬¨t/lT¬­z:gjlT}0u �&�a���jl&�Tz�l&�¯�&�akQ���a�jlT�¼z��L©�® �a}Zu w���¬¨�a�½�L©h�&�a���jl&�Tz�l&�_�&�akQ���a�jlT��z�tZ�L©5®DÀ�_ÝzC�vuL���j�azC��luow���¬¨�a�½�L©ª�&�a���jl&�Tz�l&�½�&�akQ���a�jlT�¼z�tC�L©�®DØXt:¬­�j�&l�z:gjlT}�lQl&�b��l&t���lTze�ªl&lT�_t��akmlD�L©ª¬­z�t���g�¬­Í¨�b}�lT�XØ��9uLkmlTÍ­{��lTzk�ªl&lT�ouL��{0��}��a��ͨlTk ��g�¬­Í¨�Q�L©�¹$uL�j�muL�¼{D��g�¬­Í¨�m�L©�¹ z:g9uLz¬¨té�j�az»u���}��a��ͨlTk3�Sg�¬­Í¨�GÀ�� ¬¨t~u+�&�a���jl&�Tz�l&��&�akQ���a�jlT��zvØjgjlT�j�&lCf�}:¬¨� ���muL����Í­¬¨l&tvÀ

6Gi$# %as<u)z0s<m�vxu>}�}�~k�G��XlTzQuL� ¼Å&ÉeÈ@�a§-�aÄ £�¥vÆ �L©5l&�b��l&t0��lQz:gjl¸��}��aw��Á�L©�l&�b��l&t0z:g9uLzZ��l&�&�akml&t�ua�Tz:¬­��la�'gjlT��t��akmlQ}�l&�Tw�}�t:¬­��l8�vuLÍ­Í»¬¨tua�Tz:¬­��uLz�l&�GÀ-fhgjl ua�Tz:¬­��l¸��}��aw��jtQuL}�l8u��9uL}:z:¬­z:¬¨�a���L©�z:gjlol&�b��l&t�À FªÍ¨lvuL}:Í­{�ØU�ªlok¸u&{Z�A�j��� z:gjloua�Tz:¬­��lo��}��aw��jtuc©ez�lT}   Í­���a}:¬­z:g�k ��g9uatC}:w��XØ���w�z`� lV�'¬­Í­Í �j�j�½¬­z0wjt�lJ©>w�ͪz��7�A�j��� z:gjlTk ¬­�Áua�b��uL�j�&l�À�fhg�¬¨tZ}�l&̼w�¬­}�l&t0wjtZz����}�l&�&�akQ��w�z�l0z:gjl/}�l&�Tw�}�t:¬¨�a��z:}�l&lC��lJ©Ê�a}�l �ªlZua�Tz:w9uLÍ­Í­{y}:w��   Í­���a}:¬­z:g�k �jÀ2¡j�a}�z:g�¬¨tvØI� l/}:w��   Í­���a}:¬­z:g�k HbÀ0JAl&l¡�¬­��w�}�l/�Q©Ê�a}5uL�y¬­Í­Í­wjt:z:}SuLz:¬¨�a�XÀ

���+, Ö ��)F� ' � 2�� q5l&�Tw�}�t:¬¨�a�jf�}�l&l<:�AK;)�� ® �bÂLÇ/Ë�£mË�£v¤¼§ïÇC� '4Ò *Á� Ò ���4� * ��¬­Í �Ò �>� Ò

t�lTͨl&�Tz*� � wjt�¬­�����'g9uLz�lT��lT}��T}:¬­z�lT}:¬¨�a�-¬¨thwjtSl&�y¬­�   Í­���a}:¬­z:g�k � ���! ° �'&!( � ��) �+*�+ + � &�& ²´®V� �Á² � � ·�· �� 4 ° �'&!( � ��) �+*�+ + � &�& ²´®V� ��²� � ·�· �k¸u���l��1!vµ�� 4 ��l�z:gjlCͨlJ©>z5uL�j�y}:¬­��g�z5��g�¬­Í¨�b}�lT�«�L©�� � Øb}�l&t:��l&�Tz:¬­��lTÍ­{ �� Ò ���4� * � �

FhuLÍ­Í9z:g�¬¨tz:}�l&lhz:gjl Ä�§:ÅT¥AÄSÇJÈe£aËyÉeÄ�§�§ � ÀUJ�¬­�j�&l'z:gjlh��lT}:z:¬¨�&l&tª�L©�z:gjl'z:}�l&l�uL}�lh��lT}:z:¬¨�&l&tª�L©G®Dؼz:gjlhͨlvuat:z Ðß�&�akQkm�a�uL�j�&l&t:z��a}h�L©4uL�ol&�b��l��L©�®¿¬¨t2�ªlTÍ­ÍÃÐß��l �j�jl&�GÀªf��ª�Dl&�b��l&thuL}�l�¬­�oz:gjl5tSuLkml+ua�Tz:¬­��l5��}��aw��8¬#]yz:gjlT{Qg9u&��l�z:gjl�tSuLkmlͨlvuat:z Ðß�&�akQkm�a�yuL�j�&l&t:z��a}~¬­�¸z:gjl'}�l&�Tw�}�t:¬¨�a�mz:}�l&l�ÀfhgjlT}�l5uL}�l5t:¬­kQ��ͨl5uLÍ­���a}:¬­z:g�kmt»©e�a} �j�j�b¬­���Zz:gjl'ͨlvuat:z Ðß�&�akQkm�a�

� ����� �

Page 17: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË n�H

uL�j�&l&t:z��a}�¬­� � �L©»lvua�Sg�l&�b��l ²�4µ��b·��L© ®DÀ � �jl�t:¬­kQ��ͨlQuL����}���ua��gXØGt�l&l�F �a}:kmlT��lTz�uLÍ´ÀU: nv�����<;JØXua��g�¬¨lT��l&t�z:g�¬¨t.¸²+ NY,YX ²,�µ�+4·:·JØ���{Qk¸uL¬­�¼zSuL¬­��¬­���mu/w���¬¨�a�bÐ �j�j� t:z:}:wj�Tz:w�}�l+�bw�}:¬­����uCz:}Su&��lT}�tSuLÍG�L©Gz:gjl�z:}�l&l�À��«�a}�l&�v��lT}vØb�a�j�&llvua�Sg�l&�b��lC¬¨t�ÍÏuL��lTͨl&�Y�'¬­z:g�¬­z�t�ͨlvuat:z Ðß�&�akQkm�a� uL�j�&l&t:z��a}vØ)� lCk¸uv{8}Sua�b¬Ãx-tS�a}:z�uLÍ­Í�l&�b��l&t&�'¬­z:g-z:gjl0ua�Tz:¬­��lCl&�b��l��}��aw�� uat~��}:¬­k¸uL}:{Qt��a}:zG��lT{�Ø�t:zSuL}:z:¬­���Z��lT}:z�lJx¸uat»tSl&�&�a�j�juL}:{DtS�a}:zG��lT{�Ø�uL�j�mlT�j�b¬­���0��lT}:z�lJx¸uat~z�lT}:z:¬ÏuL}:{mt��a}:z���lT{�Àfhg�¬¨t���¬­��l&tvØ9©e�a}+lvua��g«��lT}:z�lJx�uL�j�y©Ê�a}�lvua��g�ua�Tz:¬­��l0��}��aw��XØ�u¸t��a}:z�l&��Í­¬¨t:z+�L©�jlT¬­��g¼���a}�t+�L©~z:gjlZ��lT}:z�lJx�z:g9uLz�uL}�l}�lvua�Sg9uL��ͨlC�a�-uL�yl&�b��l�¬­�8z:gjl/ua�Tz:¬­��l���}��aw��XÀ~fhg�¬¨t'lT��z:¬­}�l/��}�lT��}��A�&l&t�t:¬­���mt:z�lT� zSu���l&t*.m²+ONZ,YXª²,�µ�+4·:·ªz:¬­kml�À

νlZk¸u&{�z:g�wjt+uatSt:w�kmlCz:g9uLz�uQt��a}:z�l&� ua�L�:ua�&lT�j�T{yÍ­¬¨t:z��L©»�Tw�}:}�lT�¼z:Í­{�ua�Tz:¬­��lZ¬­�j�T¬¨��lT�¼z/l&�b��l&t�¬¨t�u&��uL¬­ÍÏuL��ͨl�uLzlvua�Sg-��lT}:z�lJx ¬­��uD�vuLÍ­ÍXz��   Í­���a}:¬­z:g�k �jÀ

6Gi 6 ���Gubs<u>} ��s�~kv��Y}���u)z�i

νl�uat�t:w�kml��¼{¸¬­�j�bwj�Tz:¬¨�a�8z:g9uLz z:gjl�ze�ª��}�l&�Tw�}�t:¬­��l��vuLͭͨtªg9u&��l+}�lTz:w�}:�jl&�Cc/uLzSu�J�z:}:wj�Tz:w�}�l+�jÀ­n5©e�a}'®O� �Á²�� � ·uL�j�½®O� ��² � � ·JÀ5νl0kZwjt:z/�vuL}:}:{-�aw�z+z:gjl0ze�ª� t:z�lT�jt��L©   Í­���a}:¬­z:g�k��bÀ�fhgjl �j}�t:z�t:z�lT��}�l&Ì�w�¬­}�l&t+wjt5z�� � �� �������j�A��l&tD�L© ��� J ®V� ��²� � · N z:g9uLzDuL}�l8�j�az�km���bw�ͨl&t�¬­� ®DÀ«Î½l ���-z:g�¬¨t��¼{P�ª�a}��A¬­���«¬­�j�bwj�Tz:¬­��lTÍ­{Áw��Áz:gjl¸z:}�l&l�Øt:zSuL}:z:¬­����uLzCz:gjl¸Í¨lvu&��l&t�z:g9uLzZg9uv��l¸¬­�j�T¬¨��lT��zDua�Tz:¬­��lol&�b��l&tvÀ   t�u �9uat�lo�vuat�l�Ø�ͨlvuv��l&tQuL}�lQ��lT}:z:¬¨�&l&tD�L©5®QØ�uL�j�©Ê�a}+lvua�Sg��L©~z:gjlTk�Ø�z:gjl0�jlT¬­��g¼���a}�t�¬­� �Á²� � ·�uL}�lZ��¬­��lT� �¼{-z:gjl0t��a}:z�l&��Í­¬¨t:z�t+�L© ua�Tz:¬­��l�l&�b��l&tvÀ�_ß©~z:gjl���g�¬­Í¨�b}�lT��L©/uL� ¬­�¼z�lT}:�9uLÍ'�j�A��l ¹ruL}�l km���bw�ͨl&tmuL�j� �vuL}:}:{_¬¨��lT�¼z:¬¨�vuLÍ�ua�Tz:¬­��l �jlT¬­��g����a}QÍ­¬¨t�z�tvØk¸uL}��Á¹ruatmu�km���bw�ͨl�ØuL�j����¬­��lC¹ºum�&�a��{-�L©z:gjl0ua�Tz:¬­��l0�jlT¬­��g¼���a}�Í­¬¨t�z��L©~�a�jl0�L©z:gjl0��g�¬­Í¨�b}�lT�XÀ � z:gjlT}��'¬¨tSl�Ø�k¸uL}��y¹ ©Ê�a}���lTͨlTz:¬¨�a�XÀfhg�¬¨t���}����&l&�bw�}�lZk¸uL}��At�©e�a}���lTͨlTz:¬¨�a�«lJxjua�Tz:Í­{ z:gj��t�lCkmlTk0��lT}�t+�L© ��� J ®O� � ²� � · N z:g9uLz�uL}�l/�j�az5km�A�bw�ͨl&t��L©ª®uL�j�-z:g9uLz�g9u&��lZ¬­�j�T¬¨��lT�¼z+ua�Tz:¬­��l0l&�b��l&tvÀ~νlCz��awj��g��a��Í­{ �j�A��l&t��L©z:gjl/z:}�l&l/z:g9uLz�g9u&��l/ͨlvuc©~��l&t��&lT�j�juL��z�t*�'¬­z:g¬­�j�T¬¨��lT��z¸ua�Tz:¬­��l8l&�b��l&t�À   ͭͪw���z��awj��gjl&���j�A��l&tQuL}�l¸km�A�bw�ͨl&tD�L©+®DÀ fhg¼wjtvØ~uLÍ­Í �j����l&tD�L©5z:gjloz:}�l&loz:g9uLzQuL}�l�j�az�km���bw�ͨl&t5¬­� ® uL}�l0lJxA��Í­¬¨�T¬­z:Í­{ � �� ������ �GÀW_ß��z:gjlC��}����&l&t�t�Ø9lvua��g��j����lC��lTz�t+uDÍ­¬¨t�z5�L©¬­z�t5��g�¬­Í¨�b}�lT�«z:g9uLz+uL}�l��lTͨlTz�l&�GØ�uL�j��u�Í­¬¨t:z5�L©¬­z�t��Sg�¬­Í¨�b}�lT��z:g9uLz+uL}�l/�j�az5��lTͨlTz�l&��uL�j��g9u&��lC¬­�j�T¬¨��lT��z�ua�Tz:¬­��lZl&�b��l&tvÀ»fhgjl0�&�a�b©e�a}:kQ¬­zÝ{�'¬­z:g�z:gjlZ��g9uL}:��¬­���o�b¬¨tS�T¬­��Í­¬­�jl/¬¨t'¬­kQkml&�b¬ÏuLz�l�Øj��{ �XlTkQk¸uD�jÀø�bÀ

¡9�a}Cz:gjlQt�l&�&�a�j��t:z�lT���L©   Í­���a}:¬­z:g�k �bØG¬­z/t:wxf¸�&l&tCz�� ¬¨��lT�¼z:¬Ã©e{-©Ê�a}/lvua��g¯��lTͨlTz�l&�½��lT��lT�jlT}SuLz�lm�j����lDz:gj��t�l��}��aw��jtD�L©���g�¬­Í¨�b}�lT�Áz:g9uLzDuL}�l¸�j�az0��lTͨlTz�l&�GØ���w�z�z:g9uLzZg9u&��lo¬¨��lT�¼z:¬¨�vuLÍ �jlT¬­��g¼���a}�Í­¬¨t:z�tvÀ�_ß© 7¬¨t�u���lT��lT�jlT}SuLz�l�j�A��l�Ø��ªl�}�lTkm�v��l/��lTͨlTz�l&� �Sg�¬­Í¨�b}�lT��uL�j� ��g�¬­Í¨�b}�lT���'¬­z:g ¬­�j�T¬¨��lT�¼z5ua�Tz:¬­��l/l&�b��l&t ©e}��ak ¬­z�t'���aw���Í­{�дͭ¬­�I��l&� Í­¬¨t:zh�L©�Sg�¬­Í¨�b}�lT�XÀ+fhgjl0}�lTk¸uL¬­��¬­��� Í­¬¨t:z5��¬­��l&t+z:gj��t�l���g�¬­Í¨�b}�lT�t�'¬­z:g«�j�¸¬­�j�T¬¨��lT��zCua�Tz:¬­��l0l&�b��l&tvÀ�_´©~z:gjlT}�l0¬¨t�km�a}�lZz:g9uL��a�jlC�Sg�¬­Í¨�y¬­�yz:g�¬¨t'Í­¬¨t:zvØbz:gjlT�yz:gjlCw���¬¨�a�-�L©4z:gjl&t�l/�Sg�¬­Í¨�b}�lT�-¬¨t5u�kmlTk0��lT}����L© � ²´®�µ����c·JØ�tS�-�ªlC�T}�lvuLz�lCu��jl%��j�A��lZ}�lT��}�l&tSlT�¼z:¬­������Ø�uL�j�-k¸u���l0z:g�¬¨t�Í­¬¨t:z5��lC¬­z�t�Í­¬¨t:z��L©»�Sg�¬­Í¨�b}�lT�XÀ�fhg�¬¨t+zSu���l&t .m²�·�z:¬­kml�Ø��'g�¬¨��gt�ªl��&�aw��¼zuatª�9uL}:zª�L©Xz��awj�Sg�¬­��� 8À�νl���}��aw��8z:gj��t�l�w��j��lTͨlTz�l&�8��g�¬­Í¨�b}�lT� z:g9uLzªg9u&��l��j�a�jlTkQ��zß{¸�jlT¬­��g����a} Í­¬¨t:z�t»wjt:¬­���0z:gjl}Sua�b¬Ãx-�9uL}:z:¬­z:¬¨�a��¬­���8��}��A�&l&�bw�}�l�:�JAl&�Tz:¬¨�a�½�bÀø�<;JØ��a�-z:gjlC�jlT¬­��g����a}+Í­¬¨t:z�t+�vuL}:}:¬¨l&�-��{yz:gj��t�lZ�Sg�¬­Í¨�b}�lT�XØXuL�j�-�T}�lvuLz�lu��9uL}�lT�¼zm©e�a}Qlvua��g ��}��aw��^�'¬­z:g�km�a}�l z:g9uL� �a�jly��g�¬­Í¨�GÀÁfhg�¬¨tDzSu���l&tDz:¬­kml ��}��a���a}:z:¬¨�a�9uLÍ'z���z:gjl ��w�kZ��lT}m�L©lTͨlTkmlT��z�t5¬­�yz:gj��t�lC�jlT¬­��g¼���a}5Í­¬¨t:z�tvØjuL�j� z:g�wjt5���Al&t'�j�az�u ]Gl&�Tz�z:gjl�z:¬­kml/���aw��j���L©éz:gjlCuLÍ­���a}:¬­z:g�k�À

fhg�wjtvØAz:gjl5��w�kZ��lT}'�L©�kmlTkZ��lT}�th�L© � ²´®Dµ�� � ·z:g9uLz'uL}�l5z:gjl�w���¬¨�a� �L©���g�¬­Í¨�b}�lT�y�L©4uZ��lT��lT�jlT}SuLz�l+�j�A��l*� ¬¨tuLzªkm��t:zª�a�jl5��Í­wjt»z:gjl5�¼w�kZ��lT}h�L©4ua�Tz:¬­��l5l&�b��l&tª¬­�j�T¬¨��lT�¼zhz��Cz:gjl�t�lTz»�L©X��lT}:z:¬¨�&l&tªz:g9uLz0�8}�lT��}�l&t�lT�¼z�t�À~fhgjl5tSuLkml�vuL���j�az'��l�tïuL¬¨�o�L©�u0��}:¬­kml+�j�A��l�Àfhg¼wjtvØb¬Ã©�uZ��}:¬­kml+�j����l+¬¨th��lTͨlTz�l&�GØx�ªl+k¸uv{¸�j�azh¬­�j�b¬­��¬¨�bw9uLÍ­Í­{oz��awj��gyz:gj��t�lkmlTk0��lT}�t'�L© � ²´®�µ�� � ·~z:g9uLzª¬­zh�&�a�¼zSuL¬­�jt�uL�j�oz:g9uLz g9u&��l+�j�Z¬­�j�T¬¨��lT��z�ua�Tz:¬­��l+l&�b��l&t�À�E����ªlT��lT}vØx�ªl�k¸u&{oz��awj��guLÍ­Í~�az:gjlT}C�Sg�¬­Í¨�b}�lT�XÀ�¡j�a}ZuL�¼{ ��lTͨlTz�l&����}:¬­kmlD�j�A��l � ØB�ªlDk¸uv{«�a��zSuL¬­�_u8���aw���Í­{�дͭ¬­�I��l&�½Í­¬¨t:z/�L© z:gjlQ��g�¬­Í¨�b}�lT��'¬­z:g��j��¬­�j�T¬¨��lT�¼z¸ua�Tz:¬­��l l&�b��l&tD�¼{½}�lTkm����¬­���½uLÍ­Í'�az:gjlT}Q��g�¬­Í¨�b}�lT�Á©e}��ak ¬­z�tD���aw���Í­{�дͭ¬­�I��l&�ÁÍ­¬¨t:z��L©+��g�¬­Í¨�b}�lT�XÀFhuLÍ­ÍXz:g�¬¨t5���aw���Í­{¼Ð´Í­¬­�I��l&�-Í­¬¨t:z5�L©é��}��a��ͨlTk ��g�¬­Í¨�b}�lT� uL� ¥�Ë�É´£a¥bÅ%�AÂ��J¦¨§0¦ ÈFÇJÉ À

fhgjlZ}�l&t:z:}:¬¨�Tz:¬¨�a� �L©�z:gjl0��l&�&�akQ����t:¬­z:¬¨�a�«z��O��²� � · ¬¨t'©Ê�aw��j��¬­��z:gjl0tSuLkml �hu&{�À'fhgjl/�jlJxbz5}�lTk¸uL}��8©Ê�aͭͨ����t©e}��ak z:gjl�©´ua�Tzªz:g9uLzW�ªl+��lTz'uLzhkm��t:z �a�jl�w��¼z��awj�Sg9uL��ͨl/Í­¬¨t:zª©e�a}hlvua��gy��lTͨlTz�l&�8�j�A��l�ØbuL�j�oz:gjl��a�jt�lT}:��uLz:¬¨�a�oz:g9uLzuD�j�A��l�¬¨t��a��Í­{ ��lTͨlTz�l&��¬Ã©é¬­z'g9uat5uL��¬­�j�T¬¨��lT��z+ua�Tz:¬­��lCl&�b��l�À

�8Ò �¯������ B �+É/É�b§«§JË�¤_£ yÉ�b§ Ä�§ïÇJÉeÄJÈÊÅ&É/ÇTÉߧßÆ � É��§�£aËj¦ � �L§JÄTÉeÈÊÅS§ïÇ = ÈeÉ�A£a¥bÉZÅT¥AÄïÄ�§JË9É�ÆbÂaÄ�§JË9É'Æb£aÈ>Ë9ÉߧJÄïÇyÂaÄ�§¡¸§J¡V��§JÄïÇo£ � ²´®�µ�� � · � �?�b§TÄ:§oÂaÄ�§ .¸² N , � · ¥AË9É´£a¥�Åg�AÂ��T¦­§o¦ ÈÏÇTÉ>Ç � ÂaË�¤ .m²&N , � · ¡m§T¡O�S§JÄSÇo£� � ²´®�µ�� � ·É�AÂ�É ÂaÄ�§ZË�£�É»ÈFË«¥�Ë�É´£a¥bÅ%�AÂ��J¦¨§0¦ ÈFÇJÉ>Ç � = �b§JÄ�§ , � ÈÏÇCÉ�b§ZËj¥A¡V��§JÄ0£�ZÂ�Å&ÉeÈ+�L§Q§:¤��¼§JÇ4�

��� ù0¢2G ? ���

Page 18: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

n � � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

6Gi�� ���Gu�� � y{mUo0p2qeu>}7} ��u)z*iνlC��l �j�jl0uL�-l&̼w�¬­��uLͨlT�j�&l/}�lTÍÏuLz:¬¨�a�-�a�-��lT}:z:¬¨�&l&t5�L© �Á²� � ·JØx�'gjlT}�l�©Ê�a} ��µ��C��³¸Ø ���9�8¬Ã©uL�j���a��Í­{8¬Ã© �

� �«uL�j�C�yuL}�lC�&�a�¼zSuL¬­�jl&�-¬­��z:gjlCtSuLkml/�&�a���jl&�Tz�l&���&�akQ���a�jlT��z��L©»®�����Á² � � ·AØj�a}

� z:gjlT¬­}5�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z�t�uL}�l/���az:g��&�a�¼zSuL¬­�jl&��¬­�-u�km���bw�ͨlC�L©»®�z:g9uLz5¬¨t5uDt:w��jt�lTz��L© �Á²� � ·JÀFªÍ¨lvuL}:Í­{�Ø� ¬¨t uL�yl&Ì�w�¬­�auLͨlT�j�&l+}�lTÍÏuLz:¬¨�a�XØbt��0z:g�¬¨tª��¬­��l&t�uZ�9uL}:z:¬­z:¬¨�a�y�L© ��²� � · �AͨlTz wjth�vuLÍ­Í�z:gjl+�9uL}:z:¬­z:¬¨�a�y�TÍÏuat�t�l&tz:gjl ��ÂLÇïÈÊÅ��J¦Ã£�ÅOTvÇ �L© ��² � � ·TÀ

fhgjlC�9uat:¬¨�/��ͨ�A� �bt��L©U��²� � ·huL}�lC�&�akQ��w�z�l&��¬­�-um�&�akQ��ͨlTkmlT�¼zSuL}:{�� uv{:� ���9�¸¬Ã©~uL�j���a��Í­{8¬Ã©<�� �«uL�j�C�yuL}�lC�&�a�¼zSuL¬­�jl&�-¬­��z:gjlCtSuLkml ÅS£��ôÅ�£�¡5Æ�£aË�§JË9É �a}

� z:gjlT¬­}5�&�LÐß�&�akQ���a�jlT�¼z�t�uL}�l����az:g-�&�a��zSuL¬­�jl&��¬­��u�km�A�bw�ͨlC�L©»® z:g9uLz�¬¨t5umt:w��jt�lTz��L©��Á²� � ·JØ�XlTz�]��lT�j�az�l/z:gjlC�9uat:¬¨����ͨ��� �At��L©U��²� � ·'uL�j�yͨlTz� � ��lT�j�az�l/z:gjlC�9uat:¬¨�/��ͨ�A� �bt��L© ��²� � ·JÀ

fhgjl-}�l&�Tw�}�t:¬­��l«�vuLÍ­Í5z�� ��² � � ·Dg9uat¸�&�akQ��w�z�l&�]z:gjl-�&�a���jl&�Tz�l&� �&�akQ���a�jlT�¼z�t �L© �Á²� � ·muat8u w���¬¨�a�bÐ �j�j�t:z:}:wj�Tz:w�}�l�Àéfhgjl»}�l&t:z:}:¬¨�Tz�t�z�lT�Z¬­�Zz:gjl»k¸uL¬­���vuLÍ­ÍAg9uat��&�akQ��w�z�l&��z:gjl»k¸ucxA¬­k¸uLÍAkm���bw�ͨl&t��L©�® z:g9uLzéuL}�lª�&�a�¼zSuL¬­�jl&�¬­� ��²� � ·JÀfhgjl&t�l/zk�ª�¸t�z:}:wj�Tz:w�}�l&t���¬­��lCuL��¬­kQ��Í­¬¨�T¬­z�}�lT��}�l&tSlT�¼zSuLz:¬¨�a� �L©�z:gjl/�9uat�¬¨����ͨ��� �At��L© ��² � � ·&À

fhgjl��9uat:¬¨�5��ͨ��� �At �L©?�Á²�� � ·»uL}�l���l �j�jl&�o¬­�8uZt:¬­kQ¬­ÍÏuL}0�hu&{�ØA��w�z ¬­z»¬¨tª�j�az»�jl&�&l&t�tSuL}:{Qz��0k¸uL¬­�¼zSuL¬­��uZt:��l&�T¬ÏuLÍw���¬¨�a�bÐ �j�j�y�juLzSu�t:z:}:wj�Tz:w�}�l�©Ê�a}ªz:gjlTk�Àνl+g9uv��l�uL�yua�Tz:¬­��l/l&�b��l�z���lvua�Sg ��lT}:z�lJxo¬­�C�Á² � � ·TÀU_ß©�z:gjlT}�l/uL}�l5km�a}�lz:g9uL�8�a�jl'�9uat:¬¨����ͨ�A� ��Ø�z:gjlT{muL}�l'z:gjl���g�¬­Í¨�b}�lT� �L©�z:gjl5}����azvØAuL�j�Gؼt:¬­�j�&l�z:gjl&t�l�g9uv��l�¬­�j�T¬¨��lT�¼z'ua�Tz:¬­��l�l&�b��l&t»©>}��ak� � Øx�ªlCk¸u&{ z��awj��g uL�j� w����juLz�lZuQ��ͨ��� �oÍÏuL��lTÍG©Ê�a}�lT��lT}:{okmlTk0��lT}+�L©�z:gjlTk �bw�}:¬­���¸z:gjl � � дkm�A�bw�ͨl&t5t�z�lT�XÀ

f4� �j�j�½t:z:}��a���ykm�A�bw�ͨl&tC�&�a��zSuL¬­��¬­���Y� � Ø��ªl¸��l �j�jl¸u8�b¬­}�l&�Tz�l&�½��¬­�9uL}:z:¬­z�lQ��}SuL��g� �°7²�5µ� �Fµ�¶��'·JÀ-¡j�a}� �� uL�j��� �K��#�eØ+²��8µ�� �¨·0¬¨t¸uL� l&�b��l��L©� ¬Ã©CuL�j� �a��Í­{Á¬Ã©+z:gjlT}�l�¬¨tmuL� l&�b��l��L©Z®7��lTzk�ªl&lT��� uL�j�� � ØéuL�j� ²�� � µ��Q·/¬¨t�uL��l&�b��l8�L©� ¬Ã©�uL�j�Á�a��Í­{½¬Ã©�z:gjlT}�l¸¬¨tDu-�j�a�bÐßl&�b��lo��lTze� l&lT��� uL�j��� � À�νl��j�j�_z:gjlt:z:}��a����Í­{�Ðß�&�a���jl&�Tz�l&�Á�&�akQ���a�jlT��z�tZ�L©� yØ�uL�j��z:gjl Å�£�¡5Æ�£aË�§JË9É �aÄ:ÂïÆ>� Ø!�'g�¬¨��g¯g9uatZ�a�jlD�j�A��lD©e�a}Clvua�Sg¯t:z:}��a����&�akQ���a�jlT�¼z¸�L©� �تuL�j��l&�b��l&tQz�lTÍ­Í­¬­���(�'g�¬¨��g^�'g�¬¨�Sg t:z:}��a����Í­{�Ðß�&�a���jl&�Tz�l&� �&�akQ���a�jlT�¼z�t¸uL}�l }�lvua�Sg9uL��ͨl ©>}��ak�'g�¬¨�Sg��a��uDt:¬­����ͨlZl&�b��l/�L©� �Øjt�l&l`F �a}:kmlT�«lTz5uLÍ´À?: nv�����<;JÀ

�»Ò � �¯� D �?�b§JÄ�§�ÈÏÇD ¥�Ë9È �T¥j§DÉ´£ïÆ�£a¦Ã£ �aÈeÅSÂa¦éÇT£aÄTÉ'£ �`� Ç&§TÉ�ÅS£aË9É´ÂaÈ>ËjÈ>Ë@� � � ÈFÇ�ÂaËÁÂaË�ÅS§ïÇTÉ´£aÄQ£ � � È DÂaË�¤£aË9¦ �mÈ /ÈeÉ~ÈFÇC¸¥AËjÈe£�Ë £ ��� � � ÂaË�¤oÉ�b§Z¡¸§J¡V��§JÄïÇ0£ �ÇJÉeÄ:£aË@� Å�£a¡�Æb£�Ë�§JË9É>ÇCÈ>Ë ÂmÇï¥ � � £ CÉ�AÂ�É~ÇT£aÄTÉA�8C� ÖGÖ �S� νlmt:gj���3z:g9uLz0uyt�lTz/¹ ¬¨tZuyt:z:}��a���-km���bw�ͨlm�&�a��zSuL¬­��¬­���Y� � ¬Ã©�uL�j�½�a��Í­{ ¬Ã© ¬­zC¬¨t0u8w���¬¨�a�_�L© ��� � �uL�j�y�9uat�¬¨����ͨ��� �AtvØ�uL�j�yz:gjlT}�l�¬¨t��j�Ql&�b��l/�L©� ¢©>}��ak u��9uat�¬¨����ͨ��� �o¹ z��mu��9uat:¬¨����ͨ��� �o¬­��³]¾»¹_À0J�¬­�j�&l/z:gjlt:z:}��a���¸km���bw�ͨl&t��&�a�¼zSuL¬­��¬­���V���CuL}�l�z��azSuLÍ­Í­{y�a}���lT}�l&����{oz:gjlC¬­�j�TÍ­wjt:¬¨�a��}�lTÍÏuLz:¬¨�a�XØjz:g�¬¨t5l&t:zSuL��Í­¬¨t:gjl&t�z:gjl/�TÍÏuL¬­k�À

�Q�4� Ò � �b�b� ) Ö * 1 )�§TÉ ¹ ��§/Â�Çv§TÉéÉ�AÂ�É~Å�£�Ë9É´ÂaÈ>Ë�Ç � � �|��ÈFË�Åï§ � � ÈFÇ+Â�¤<;&Â�ÅS§TË9ÉÉ´£m§.�L§JÄg�Q¡¸§J¡V��§JÄC£ �Á²� � · ÂaË�¤Ë�£�Ë�Â�¤<;&Â�ÅS§JË�É�É´£�§.�a§JÄ��-¡m§T¡O�S§Jĸ£ �Á²����· � ¹ ÈFÇDÂy¡Q£�¤a¥�¦¨§mÈ QÂaË�¤�£�Ëj¦ � È Q§ �L§JÄg��¡¸§J¡V��§Jĸ£ ¹ ÈFÇD¼¤<;T¼ÅS§JË9ÉÉ´£y§.�L§TÄ��o¡m§T¡O�S§JÄ�£ ��²� � ·G¾ ¹ ÂaË�¤¸Ë�£aË�Â�¤<;&Â�ÅS§JË9É É´£ §.�L§JÄg�o¡¸§J¡V��§TÄ0£ ��² � � ·�¾ª¹ ��Q�4� Ò � �b�b� ) Ö * / � �L§JÄg�C¡m£v¤a¥A¦¨§�ÅS£aË9É´ÂaÈ>ËjÈ>Ë@� � � ÈFÇ'ÂC¥�ËjÈÊ£aË £ � � Â�Ë�¤&Vc§JÄ £�£aÄh¡m£aÄ�§5Å�£�ËjË�§:Å&Éߧ�¤QÅ�£a¡�Æb£aË�§TË9É>Ç£ ��² � � · Â�Ë�¤oÅS£��ôÅ�£�¡5Æ�£aË�§JË9É>ÇQ£ ��²� � · �fhg�¬¨t�©e�aͭͨ����t�¬­kQkml&�b¬ÏuLz�lTÍ­{8©>}��ak � �jt�lT}:�auLz:¬¨�a�-�jÀø�bÀ�Q�4� Ò � �b�b� ) Ö * +" 5 ¹ ÈÏÇ�Â�¥AËjÈÊ£aË�£ � � ÂaË�¤��:ÂLÇJÈeÅ��J¦Ã£�ÅOTvÇ � ÈeÉ�ÈÏÇ�Â�¡m£v¤a¥A¦¨§+È +ÂaË�¤Q£aËj¦ �ZÈ 5É�b§TÄ:§+ÈFÇ5Ë�£Q§:¤��¼§£ ïÄ:£a¡ Â��:ÂaÇïÈÊÅ-�T¦Ã£vÅ TmÈFË ¹ É´£oÂC�:ÂLÇJÈeÅO�T¦Ã£vÅ TmË�£�É~È>Ë ¹ �f4�½t�l&l z:g�¬¨tvØ»�j�az�lyz:g9uLzm¬Ã©�¹r¬¨t¸u«w���¬¨�a� �L© � � uL�j���9uat�¬¨� ��ͨ��� �Atvتz:gjlT� ¬­zQ¬¨t¸uLͨt��½u«w���¬¨�a�]�L©+�&�a���jl&�Tz�l&��&�akQ���a�jlT�¼z�t��L©é® � ��

�Á²� � ·�uL�j�y�&�LÐß�&�akQ���a�jlT��z�t5�L©é® � �Á²�� � ·AÀ�=��lT}:{8kmlTkZ��lT}��L© ��² � � ·(�m¹�¬¨th�j�a�9ua�L��ua�&lT�¼zz��ÁlT��lT}:{ kmlTkZ��lT}y�L© �Á²� � ·h¾Z¹_Ø5uL�j� lT��lT}:{ kmlTk0��lT}y�L©`��²� � · �¯¹.¬¨t ua�L�:ua�&lT��z8z��ÁlT��lT}:{ kmlTkZ��lT}y�L©��² � � ·X¾ª¹_À~fhgjlZ�a�jt�lT}:�auLz:¬¨�a�-z:gjlT�y©e�aͭͨ����t'©e}��ak � �jt�lT}:��uLz:¬¨�a�¯nZuL�j�yz:gjl/��l �j��¬­z:¬¨�a�«�L©� �À

� ����� �

Page 19: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË n�D

_Ýz+}�lTk¸uL¬­�jt�z��8t:gj��� z:g9uLz/u¸km���bw�ͨl0¹ z:g9uLz+�&�a��zSuL¬­�jt � � ¬¨t/umt:z:}��a���y�a��Í­{y¬Ã©»¬­z�¬¨t/umw���¬¨�a� �L©|��� � �QuL�j�LvlT}��0�a}km�a}�l��9uat:¬¨�'��ͨ�A� �btvؼuL�j�ou �ªlvu��m�a��Í­{�¬Ã©G¬­z¬¨t~�j�az»t:wj�Sgou�w���¬¨�a�XÀGJ�w�������t�l'¹7¬¨t~�j�az»t:wj�Sgou�w���¬¨�a�XÀU5 {� �jt�lT}:��uLz:¬¨�a�-�DuL�j� z:gjl���l �j��¬­z:¬¨�a���L©4�9uat:¬¨�+��ͨ�A� �btvØA¹ �v��lT}:ÍÏuL�jt�uZk¸ucxb¬­k¸uLÍ�km���bw�ͨl/�L©~®�z:g9uLzh¬¨t'�&�a�¼zSuL¬­�jl&�¬­� �Á²���v·Z�a} ��²� ��·JØuL�j�¯¬¨tZz:gjlT}�lJ©Ê�a}�l¸�j�azDt:z:}��a���jÀ J�w�������t�l¸¹ ¬¨tZt:wj�Sg�u�w���¬¨�a�XÀ9J�¬­�j�&l uLÍ­Í»km�A�bw�ͨl&t0�j�az�&�a��zSuL¬­��¬­���(� � uL}�l �&�a�¼zSuL¬­�jl&�Á¬­�b��² � � ·C�a} ��² � � ·JØéz:gjlT{ÁuL}�lo�&�a��zSuL¬­�jl&��¬­�_�9uat:¬¨�o��ͨ�A� �btvÀ   �¼{¯km���bw�ͨl z:g9uLz/�v��lT}:ÍÏuL�jt�¹�kZwjt:z��&�a�¼zSuL¬­�(����À 5ª{ �XlTkQk¸u �bÀ­n�Øé²>¹ ¾ Q· �_²� 2¾'¹�·'¬¨t/u¸km���bw�ͨl�z:g9uLz��v��lT}:ÍÏuL�jt/¹ÁØ�u�&�a��z:}Sua�b¬¨�Tz:¬¨�a�XØ�t��� ¢���Al&th�j�az5lJxb¬¨t:zvØjuL�j�y¹ ¬¨t�t:z:}��a���jÀ

¡9�a}»z:gjlhz:¬­kml'���aw��j�GØ��j�az�l'z:g9uLz2�ªl�k¸uv{Dz��awj��g¸lvua�Sgo�&�akQ���a�jlT�¼zª�a}ª�&�LÐß�&�akQ���a�jlT��z»z:g9uLz~g9uat uL�Q¬­�j�T¬¨��lT�¼zua�Tz:¬­��l�l&�b��l�À�νl k¸u&{QuLͨt��+z��awj��g¸z:gj��t�lhz:g9uLzékmlT}:��l|�'¬­z:gQ�az:gjlT}�tvؼt:¬­�j�&l z:gjlT}�l�uL}�l*.¸²+4·XkmlT}:��l&t»�v��lT}»u+}:w��Q�L©z:gjl/uLÍ­���a}:¬­z:g�k�ÀWF �LÐß�&�akQ���a�jlT�¼z�t&�'¬­z:g8�j��¬­�j�T¬¨��lT��z5ua�Tz:¬­��l/l&�b��l&thkmlT}:��l��'¬­z:gyt��akml��az:gjlT}'�&�LÐß�&�akQ���a�jlT�¼zvØjt��uLÍ­Í9�&�LÐß�&�akQ���a�jlT�¼z�tªk¸u&{���lhz��awj��gjl&�GÀ0F �akQ���a�jlT�¼z�t»z:g9uLz~g9u&��l'�j�C¬­�j�T¬¨��lT�¼z ua�Tz:¬­��l5l&�b��l&t~�vuL���j�az~��lhz��awj��gjl&�GØ��w�z�z:gjlT¬­}�w���¬¨�a�-¬¨t5uQkm���bw�ͨl/z:g9uLz+����l&t��j�az��&�a�¼zSuL¬­� � � ØjgjlT�j�&lCz:gjlT{�uL}�l/g�¬¨����lT�-¬­�jt�¬¨��lZuD�9uat:¬¨�/��ͨ�A� ��Ø9uL�j�}�l&t:¬¨��l�w��j��lT}'uZt:¬­����ͨl5}����azh�L©Xz:gjl5©e�a}�l&t:zª��}����bwj�&l&�o�¼{¸z:gjl5}�l&t:z:}:¬¨�Tzht:z�lT�XÀGJ�¬­�j�&l�z:g�¬¨tª��ͨ�A� �Q¬¨tªw���¬¨Ì�wjl5¬­� lvua��g�L©éz:gjl�.m²+4·ª¬­�j�vuL}:�9uLz:¬¨�a�jt+�L©éz:gjl/}�l&�Tw�}�t:¬­��lZuLÍ­���a}:¬­z:g�k�Ø9¬­z'k¸u&{ ��l�z��awj�Sgjl&�GÀ

º¬¨tmz�����ÍÏuL}:��l�z��½�&�akQ��w�z�l-lJxb��Í­¬¨�T¬­z:Í­{�Øht��t�ªlY�ª�a}�� �'¬­z:g]u kQ¬Ãx�l&� }�lT��}�l&t�lT��zSuLz:¬¨�a� �'gjlT}�l�z:gjl-t�lTzm�L©�jlT¬­��g����a}�tm�L©� ¬¨t�}�lT��}�l&t�lT�¼z�l&��¬­� u«t:zSuL�j�juL}�� �hu&{ uL�j�_z:gjl t�lTzD�L©��jlT¬­��g����a}�t¸�L©� � ¬¨tD}�lT��}�l&t�lT�¼z�l&� �'¬­z:g¬­z�t��&�akQ��ͨlTkmlT�¼zvÀ0fhgjl��¼w�kZ��lT}C�L©ªl&�b��l&t/uL�j�«�j�a�bÐßl&�b��l&t���¬­��lT��lJxA��Í­¬¨�T¬­z:Í­{�¬­�«z:g�¬¨t+}�lT��}�l&t�lT��zSuLz:¬¨�a�½¬¨t+�TͨlvuL}:Í­{���aw��j��l&�¸��{mz:gjl���w�kZ��lT}'�L©G�Tw�}:}�lT��z:Í­{8ua�Tz:¬­��l+l&�b��l&tvÀ2J�¬­�j�&l5lvua�Sg8�&�akQ���a�jlT�¼z�uL�j�8�&�LÐß�&�akQ���a�jlT��z ���j����tªz:gjl��w�kZ��lT}��L©éua�Tz:¬­��l�l&�b��l&thz��D�az:gjlT}h�&�akQ���a�jlT�¼z�t�Øx�ªl�k¸u&{8�&�akQ��w�z�l�z:gjl5kQ¬Ãx�l&�8}�lT��}�l&t�lT�¼zSuLz:¬¨�a���L©�z:gjl�©e�a}��T¬­�����}SuL��g �'¬­z:g_u �¼w�k0��lT}D�L© ������� uL�j�½�&�a�jt:zSuL�¼z дz:¬­kml8�a��lT}SuLz:¬¨�a�jtZ��}��a���a}:z:¬¨�a�9uLÍ»z��yz:gjlm�¼w�kZ��lT}D�L©'ua�Tz:¬­��ll&�b��l&tvÀWFª}�lvuLz:¬­���mz:gjl/t:z:}��a���m�&�akQ���a�jlT�¼z�t�uL�j� z:gjlT¬­}'z��a���aͨ�a��¬¨�vuLÍ�t��a}:zhz:gjlT�y©Ê�aͭͨ����t�¬­�8z:¬­kmlC��}��a���a}:z:¬¨�a�9uLÍXz��z:gjl t:¬�Lvl»�L©jz:g�¬¨t�kQ¬Ãx�l&�C}�lT��}�l&t�lT��zSuLz:¬¨�a�Q�¼{Zfhgjl&�a}�lTk±�bÀ�fhg�¬¨t��&�akQ��ͨlTz�l&t�z:gjl0� � дkm�A�bw�ͨl&tét:z�lT�XØL�¼{ ��lTkQk¸u��jÀ DAÀ

6Gi�� ���Gubr�}�} uwy��0qeub} ��u)z*i

�»Ò � �¯� �"�" 5 ¹ ÈFÇ�Â�Å%�¼ÈF¦­¤�£ DÂa˯ÂaË�Åï§ïÇJÉ´£aÄ G £ � � ÈFË ��� ²´®C· � ÂaË�¤ ¹ �AÂLÇ�Ë�£8ÈFË�ÅTÈe¤�§JË9É'Â�Å&ÉeÈ+�L§¸§�¤��¼§ïÇ �É�b§T˽§TÈ>É�b§TÄ G £aÄ ¹ ÈFÇ/ Å�£�ËjË�§:Å&Éߧ�¤�Å�£a¡�Æb£aË�§JË9Éh£ ® �8C� ÖGÖ �S� νl��TÍÏuL¬­k z:g9uLz�¬Ã© , ¬¨t+uL� uL�j�&l&t:z��a}+�L©G���/z:g9uLz+g9uat�uQkmlTkZ��lT}��9�'¬­z:g��j�¸¬­�j�T¬¨��lT��zCua�Tz:¬­��l0l&�b��l&tvØz:gjlT� , g9uat �j���jlT¬­��g¼���a}�t'¬­�-³ ¾ , Àéfhg�¬¨th¬¨th�a�¼�A¬¨�awjt'¬Ã© , ° ³mÀ2J�w�������t�l , �° ³mÀ   Í­Í���lT}:z:¬¨�&l&t��L©!��²� � ·g9uv��l8ua�Tz:¬­��lol&�b��l&tCz���� � À¸fhg�wjtvØ , kZwjt:z0¬­�¼z�lT}�t�l&�Tz �Á²� � ·JØéuL�j�¯uLÍ­Í»��lT}:z:¬¨�&l&t��L© , �'¬­z:gj�aw�zQua�Tz:¬­��l¸l&�b��l&t}�l&t:¬¨��l+¬­� ��²� � ·TÀU_ß©!��²� � ·9¾ , ¬¨t �j�a�jlTkQ��zß{�ØjͨlTzW�¸��l�u0kmlTkZ��lT}h�L©��Á²� � ·9¾ , À2J�¬­�j�&l �m¬¨t'ua�L��ua�&lT�¼z'z���� � ج­zh¬¨t'ua�L�:ua�&lT��z'z��DlT��lT}:{mkmlTkZ��lT}��L© , � ��² � � ·»��¬Ïu�ua�Tz:¬­��l/l&�b��l&tvØ�uZ�&�a��z:}Sua�b¬¨�Tz:¬¨�a�XÀ JA�-��²� � ·»¬¨t'u�t:w��jt�lTzh�L©, À�fhgjlT}�l�¬¨t��j�ol&�b��l0©e}��ak � � z�� ��²� � ·JØ�uL�j�«t:¬­�j�&l�� � � , uL�j� , ¬¨t�u¸km���bw�ͨl�Ø�z:gjlT}�l�¬¨t5�j�8l&�b��l0©>}��ak, z�� ��²� � ·G¾ , À0J�¬­�j�&l`� �&�a��zSuL¬­�jt&� � uL�j�7��²� � ·TØ ��²� � ·G¾ , ° ³ ¾ , Ø���}�����¬­���¸z:gjlC�TÍÏuL¬­k�À

fhg�wjtvØ�G2kZwjt:z�g9uv��l��j�D�jlT¬­��g¼���a}�t�¬­�-³ ¾ G�À�_´© G �°2³¸Ø�G�¬¨t�uD�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼zvØ9uL�j� z:gjl�ͨlTkQk¸u¬¨tCtïuLz:¬¨t �9l&�GÀ-_´© G °¢³ uL�j�¯® ¬¨tZ�&�a���jl&�Tz�l&�GØ�z:gjlT��¬­z/¬¨tC�a�j�&lmuL�¼uL¬­�_tSuLz:¬¨t �9l&�GÀ � z:gjlT}��'¬¨t�l�Ø G ¬¨tZu8�9uL}SuLͭͨlTÍ�j�A��l�Ø9uL�j�y¹ ¬¨t5uQ�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼zvØ�uL�j�GØjuL�¼uL¬­�XØjz:gjl�ͨlTkQk¸um¬¨t'tSuLz:¬¨t �9l&�GÀ

fhgjl�� � дkm���bw�ͨl&t+t:z�lT�-��¬­��l&t5©Ê�a}�lvua��g«�9uat:¬¨�Z��ͨ��� � �¢z:gjlZt:k¸uLͭͨl&t:z�uL�j�&l&t:z��a} G��]�L©�� � z:g9uLz+�&�a��zSuL¬­�jt�¬­zvÀ� ¬¨tCu8w���¬¨�a�½�L©'u8t�lTz ���L©ªkmlTk0��lT}�t0�L© � ²´®�µ�� � ·JØXlvua�Sg½�L©W�'g�¬¨�Sg¯uL����lvuL}�tZuat�z:gjlD}��A�azC�L©hu8z:}�l&lD¬­��z:gjl©Ê�a}�l&t:z���}����bwj�&l&�Á�¼{¯z:gjl¸}�l&t:z:}:¬¨�TzQt:z�lT�XÀ«f��«¬­kQ��ͨlTkmlT��z   Í­���a}:¬­z:g�kº�Y�'¬­z:gj�aw�z�¬­z�t�¬­���jlT} )�� Ðßt:zSuLz�lTkmlT��zvØ��ªlk0wjt:z+l&t�zSuL��Í­¬¨t:g-z:gjlZkmlTkZ��lT}�t+�L© �7uat+��g�¬­Í¨�b}�lT� �L© G���À|_´© � g9uat5�a��Í­{��a�jlZkmlTkZ��lT}vØ�z:gjlT�-z:g�¬¨t�kmlTkZ��lT}�¬¨t� Ø4uL�j�Z�ªl¸t�¬­kQ��Í­{ uLz:zSua�Sg_¬­z0uat0uy��g�¬­Í¨�_�L© G � ¬­�b.¸²�·+z:¬­kml�À � z:gjlT}��'¬¨t�l�Ø�z:gjlm��ͨ��� � �&�a��zSuL¬­��¬­����¹ � �¬¨t�� wjt:zQz:gjly�&�a���jl&�Tz�l&�]�&�akQ���a�jlT��zo�L©/®V� ��²� � ·��a}mz:gjly�&�LÐß�&�akQ���a�jlT�¼z8�L© ��² � � ·0z:g9uLz¸�&�a�¼zSuL¬­�jt¸¬­zvÀ^_´©�¹g9uat/uL�-¬­�j�T¬¨��lT��zCua�Tz:¬­��l�l&�b��l�Ø�z:gjlT�«�¼{Yc/uLzSuCJ�z:}:wj�Tz:w�}�l0�jÀ­n�Øw�ªl��vuL�«��lT} ©Ê�a}:k$u ������� �a��lT}SuLz:¬¨�a� z����j�j�-¬­z�t��ͨ�A� ��ØAuL�j�oz:gjl�kQ¬­��¬­k0w�k uL�j�&l&t:z��a}ª�L©�� � z:g9uLz �&�a�¼zSuL¬­�jthz:gjl���ͨ�A� �Q��¬­��l&tªz:gjl��9uL}�lT��zª¬­��.m²�·~z:¬­kml�Àéνl�z:gjlT�l&t:zSuL��Í­¬¨t�g«¹�uat/uo��g�¬­Í¨���L©»z:g�¬¨t/uL�j�&l&t:z��a}vØG�&�akQ��ͨlTz�l��'¬­z:g½uL� w����juLz�l&���9uL}�lT�¼z/���a¬­�¼z�lT}vØ�¬­�P.¸²�·�z:¬­kml�À�fhgjl}�lTk¸uL¬­�j��lT}C�L©~kmlTk0��lT}�t/�L© � ²´®�µ����c·�}�l&t:¬¨��l0¬­��w���z��awj��g9uL��ͨlDÍ­¬¨t:z�tvÀ J�¬­�j�&l�lvua��g w��¼z��awj�Sg9uL��ͨlDÍ­¬¨t:z�}�l&t�¬¨��l&t�¬­���� ù0¢2G ? ���

Page 20: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

nv� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

�a�jlQ�&�a���jl&�Tz�l&�½�&�akQ���a�jlT�¼z0�L©�®O� ��² � � ·JØ��ªlD��lT} ©Ê�a}:k uot�¬­����ͨl � ��+�� ©Ê�a}�lvua�Sg½w��¼z��awj�Sg9uL��ͨlmÍ­¬¨t:z/z�� �j�j� ¬­z�t�9uat:¬¨�+��ͨ�A� ��À~fhg�¬¨t ¬¨��lT�¼z:¬��9l&t5z:gjl+t�¬­����ͨl��9uat�¬¨����ͨ�A� �oz:g9uLz'�&�a�¼zSuL¬­�jt'z:gjlTk�ÀG_ß©�z:gjl+Í­¬¨t:z'�a��Í­{¸g9uath�a�jl�kmlTk0��lT}vØ� l�ua���0¬­zuat~u���g�¬­Í¨�Q�L©9z:gjl �9uL}�lT�¼zvØ��&�akQ��ͨlTz�l �'¬­z:gmu��9uL}�lT��z���a¬­��z�lT}vÀ � z:gjlT}��'¬¨t�l�Ø��ªlht:��Í­¬¨�&l z:gjl w��¼z��awj�Sg9uL��ͨlÍ­¬¨t:z5¬­�¼z��m¬­z�t��9uL}�lT�¼z � t+��g�¬­Í¨��Í­¬¨t�z'¬­�9.m²�·'z:¬­kml�À~νl/�jl&l&�b� � z�w����juLz�l/z:gjlC�9uL}�lT�¼z5���a¬­�¼z�lT}�t+�L©éz:gjlClTͨlTkmlT�¼z�t+�L©z:gjl+w��¼z��awj�Sg9uL��ͨl/Í­¬¨t:zvØbt:¬­�j�&l�ØA�¼{��XlTkQk¸u��jÀ­n�n�ØAz:gjl+�9uL}�lT�¼z'¬¨t z:gjl��&�a���jl&�Tz�l&�y�&�akQ���a�jlT��z��L©é®�z:g9uLz'�&�a�¼zSuL¬­�jtz:gjlTk�Ø9uL�j���j�Q�9uL}�lT�¼z����a¬­��z�lT}�¬¨t'}�l&Ì�w�¬­}�l&�yw��j��lT}*c/uLzSuVJ�z:}:wj�Tz:w�}�lC�jÀ­n�À

�+lJxbzvØ@� l+�jl&l&�¸z��Dt:¬­k0w�ÍÏuLz�l�z:gjl+lg]Gl&�Tzh�L©Xz:gjl+¬­���jlT} )�� Ðßt:zSuLz�lTkmlT��z'�L©   Í­���a}:¬­z:g�k �bÀéνl+k¸u&{¸t���lT�j��.m²�·z:¬­kml-uLzmlvua�Sg uL�j�&l&t:z��a}m�L© � � Ø»t:¬­�j�&l�z:gjl&t�lC�ªlT}�l��T}�lvuLz�l&� �bw�}:¬­���½z:gjly¬­�j�bwj�Tz:¬­��l�t:z�lT�XÀ^_ß©�¹ ¬¨t¸u«�9uL}SuLͭͨlTÍuL�j�&l&t:z��a}0�L©W� � �'¬­z:g¯uLzCͨlvuat:z/z:g�}�l&l¸��g�¬­Í¨�b}�lT�XØéuL�j� ¬¨t�¬­z�tZ��g�¬­Í¨�½z:g9uLzC�&�a��zSuL¬­�jt`� � ØXz:gjlT��z:gjlmw���¬¨�a�¯�L©'uLÍ­Ít:¬­��Í­¬­����t��L©� ¢¬¨t�umt:¬­����ͨl0kmlTkZ��lT}�� �L© � ²´®�µ�� � ·JÀ5fhg�wjtvØ�¬­�-z:gjl��Tw�}:}�lT�¼z�z:}�l&l�Ø9¹ �'¬­Í­Í�g9u&��l0ze� � ��g�¬­Í¨�b}�lT�XØ uL�j���0Àéνl�kZwjt:z'��lTͨlTz�l�� ©e}��ak¢z:gjl5z:}�l&l�ØjuL�j�8ͨlTz ¹�¬­��gjlT}:¬­zh¬­z�t �Sg�¬­Í¨�b}�lT�XÀ2_ß©X¹ °2³mØ��ªlª� wjt:zht:��Í­¬¨�&l�� � t���aw���Í­{�дͭ¬­�I��l&��Í­¬¨t:z��L©��g�¬­Í¨�b}�lT�-¬­�¼z��Qz:gj��t�lZ�L©4¹ � t�À~fhgjl/��g�¬­Í¨�b}�lT�«�L©�¹ uL}�l/�&�a���jl&�Tz�l&�-�&�akQ���a�jlT�¼z�t5�L©»®DØ�t��c/uLzSu7J�z:}:wj�Tz:w�}�lQ�jÀ­nQ���Al&t/�j�az/}�l&Ì�w�¬­}�lD�9uL}�lT�¼zC���a¬­�¼z�lT}�tvÀ � z:gjlT}��'¬¨t�l�Ø��¼{ �XlTkQk¸u �jÀ­n�n�ØXlvua�Sg¯��g�¬­Í¨�¯�L©ª¹�¬­���� ²ß®C·hg9uat�uL��¬­�j�T¬¨��lT��z+ua�Tz:¬­��lCl&�b��l�Ø�t��-�ªlZk¸u&{ z��awj��g«lvua��g��L©éz:gjlTk�Ø�uL�j��k¸u���l/z:gjl/�9uL}�lT�¼z����a¬­��z�lT}5���a¬­�¼zz��Q¹ÁÀ~fhgjl���}��A�&l&�bw�}�lK�'gjlT��¹ ¬¨t5uQt�lT}:¬¨l&t'�j�A��l�¬¨t�t:¬­kQ¬­ÍÏuL}vÀ

f4� �&�akQ��ͨlTz�l�c/uLzSuCJ�z:}:wj�Tz:w�}�l0�jÀ­nC©e�a} ��� ²ß®C·TØ>�ªl�kZwjt:z/uLͨt��¸��}����bwj�&l0z:gjlZw���¬¨�a�bÐ �j�j�«t:z:}:wj�Tz:w�}�l&t�©Ê�a}�®DÀfhgjlª}�l&�Tw�}�t:¬­��l �vuLͭͨt4}�lTz:w�}:��w���¬¨�a�bÐ �j�j�Dt:z:}:wj�Tz:w�}�l&t�z:g9uLz4��¬­��lªz:gjl �&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z�tuL�j���&�LÐß�&�akQ���a�jlT��z�t�L©0®O� ��² ����·muL�j�]®V� ��²����·JÀRJ�¬­�j�&l(������� �Z��² ���v·D¬¨t8u��&�a���jl&�Tz�l&�]�&�akQ���a�jlT�¼zvØ�uL�j�]uLÍ­Í+ua�Tz:¬­��l�l&�b��l&t8uL}�l¬­�j�T¬¨��lT��z¸z�� ¬­zvØG� lykZwjt�zm���½u«w���¬¨�a� �a� uLÍ­Í��&�akQ���a�jlT��z�t¸�L©Z®O� �Á²�� � ·DuL�j�]®V� �Á² � � ·0z:g9uLzQg9u&��l�¬­�j�T¬¨��lT�¼zua�Tz:¬­��l8l&�b��l&tvÀ7_ô��lT�¼z:¬Ã©e{�¬­����z:gjl&t�l¸�&�akQ���a�jlT�¼z�t�}�l&Ì�w�¬­}�l&tQu � ��+�� �a�¯z:gjlolT�j�½��lT}:z:¬¨�&l&tD�L©�lvua��g ua�Tz:¬­��lol&�b��l�À�+� �az:gjlT}D�&�akQ���a�jlT�¼z�touL}�l8u ]Gl&�Tz�l&�GÀZ¡j�a}Dz:gjl �&�LÐß�&�akQ���a�jlT��z�tvØ t:{�kQkmlTz:}:¬¨�8}�lvuat��a��¬­����t:gj����tDz:g9uLzmuLÍ­Íh�&�LÐ�&�akQ���a�jlT�¼z�t'�L©G��� � ��� ��²� � ·k0wjt:z ��l+t:w��jt�lTz�t �L©�uCt�¬­����ͨl+�&�LÐß�&�akQ���a�jlT�¼zvÀ   �&�LÐß�&�akQ���a�jlT��z ���L©�®O� �Á²�� � ·k0wjt:z���lCkmlT}:��l&��z��a��lTz:gjlT} �'¬­z:g�z:g�¬¨t+t�lTz5¬Ã©~z:gjlZ�¼w�k0��lT}/�L©¬­�j�T¬¨��lT�¼zCua�Tz:¬­��l�l&�b��l&t�z:g9uLz����mz����Á²����·'¬¨t�ͨl&t�tz:g9uL�b� ��² � � ·�� À

� �� �� 5��0�S0� 2 ���� '� 4 �S"$�� ��*���

fhgjlD�a��Í­{����az:z:ͨlT�jl&� �«��}�lT��lT�¼z:¬­���-u¸Í­¬­�jlvuL}�z:¬­kml0���aw��j��z:g9uLz+}�lTk¸uL¬­�jt�¬¨t+z:gjl-Xª²,�µ�+4·5z�lT}:k�À�νl��'¬­Í­Íét:gj���gj���2z��Q�T¬­}��Tw�kZ��lT��z�z:g�¬¨t'©Êua�Tz��a}���{yt��akml���}�lT��}����&l&tSt:¬­���¸�a��z:gjl�}�l&�Tw�}�t:¬¨�a��z:}�l&l�À

� �jlD��ÍÏua�&l��'gjlT}�l��ªlD¬­�j�Tw�}:}�l&� z:gjl�z�lT}:k � uat�¬­� �j�j�b¬­��� z:gjl�ͨlvuat:z Ðß�&�akQkm�a�_uL�j�&l&t:z��a}�t/�L© l&�b��l&t�¬­�«z:gjl}�l&�Tw�}�t:¬¨�a��z:}�l&l�Àhfhg�¬¨t5�vuL����lClTÍ­¬­kQ¬­�9uLz�l&��gjlT}�lC�¼{ wjt:¬­���¸z:gjlZ��]�дͭ¬­�jl/ͨlvuat:z Ðß�&�akQkm�a��uL�j�&l&t:z��a}�t+uLÍ­���a}:¬­z:g�k7�L©E�uL}�lTÍ uL�j�½f�uL}e�:uL� : nv���L�@;JÀ fhgjlm�az:gjlT}Z��ÍÏua�&lV� uatC¬­�_k¸u���¬­���-z:gjl¸�&�a���jl&�Tz�l&�¯�&�akQ���a�jlT�¼z�tD�L©hz:gjlm¬­�j�bwj�&l&�t:w�����}SuL��g_�9uat�t�l&��z���u8}�l&�Tw�}�t:¬­��l¸�vuLͭͪuv�auL¬­ÍÏuL��ͨl¸¬­�½z:g9uLz0�vuLÍ­Í´Àofhgjl&t�lm�&�akQ���a�jlT�¼z�tDuL}�lQ}�lT��}�l&t�lT��z�l&�P�'¬­z:g_uw���¬¨�a�bÐ �j�j��t:z:}:wj�Tz:w�}�l8¬­�¯z:gjloz:}�l&l8�juLzSu-t:z:}:wj�Tz:w�}�lo¬­� ��lTkQk¸u-�jÀø�bÀ-νlotS�aÍ­��lmz:g�¬¨t���}��a��ͨlTk ��{½}�l&t:z:}:¬¨�Tz:¬­���z:gjl0����tSt:¬­��ͨl0�Sgj�a¬¨�&l&t�©e�a}+z:gjlZ}�l&�Tw�}�t�¬¨�a��z:}�l&l�Ø�t��¸z:g9uLz�z:gjl0w���¬¨�a�bÐ �j�j� t:z:}:wj�Tz:w�}�l&t/�vuL����lCk¸uL�9uL��l&�9�'¬­z:gj�aw�z¬­�j�Tw�}:}:¬­���oz:gjl`Xª²,�µ�+4· ©Êua�Tz��a}vÀ

Î gjlT��}:w�����¬­���   Í­���a}:¬­z:g�k H-�a�Á��lT}:z�lJxÁtSlTzZ¹_ØéͨlTz�wjt��¼w�k0��lT}Dz:gjl8��lT}:z:¬¨�&l&t�z:g9uLz0g9uv��l uLÍ­}�lvua�b{_��l&lT�t�lTͨl&�Tz�l&�_uat/z:gjlQ¬­��¬­z:¬ÏuLÍ~��¬­���azC¬­�¯��}:¬¨�a}Z�vuLͭͨtvØG¬­�½z:gjlQ�a}���lT}Z¬­�Z�'g�¬¨��g_z:gjlT{ � lT}�lmt�lTͨl&�Tz�l&�GÀmfhg�¬¨tZ�¼w�kZ��lT}:¬­����&�a�jt:z:¬­z:w�z�l&t~u���}�l&�a}���lT}é�¼w�k0��lT}:¬­���C�L©�z:gjlª���a}:z:¬¨�a�Q�L©9z:gjlª}�l&�Tw�}�t:¬¨�a�Dz:}�l&lhuLÍ­}�lvua�b{��&�akQ��w�z�l&�GÀ������lT}:z�lJxZ¬­�D¹g9uat'{�lTz5��l&lT�y��w�kZ��lT}�l&�GÀ �XlTz�z:gjl Ä:§:Åï§JË�Å%� �L©~u���lT}:z�lJx��-¬­�y¹ ��l�z:gjl�k¸ucxb¬­kZw�k7�L©�z:gjl���}�l&�a}���lT}��¼w�kZ��lT}�tuLkm�a�����jlT¬­��g����a}�to�L©�� z:g9uLzQg9u&��l�uLÍ­}�lvua�b{Á��l&lT�Á��w�kZ��lT}�l&�GÀ _´©�� g9uatD�j� t:wj��g �jlT¬­��g����a}vت¬­z�tD}�l&�&lT�j�T{¯¬¨t��l �j�jl&�¸z��C��l ÐSn�À � w�}ªkm���b¬��9�vuLz:¬¨�a�oz��   Í­���a}:¬­z:g�k H�¬¨t~z��Ct�lTͨl&�Tz2� � z��/��l'z:gjl'��lT}:z�lJx��'¬­z:go��}�lvuLz�l&t:z»}�l&�&lT�j�T{�Ø}SuLz:gjlT}�z:g9uL��t�lTͨl&�Tz:¬­���¸¬­z5uL}:��¬­z:}SuL}:¬­Í­{�À

  Í­���a}:¬­z:g�k � ��¬­��l&t5z:gjlCuLÍ­���a}:¬­z:g�k ©Ê�a}��&�akQ��w�z:¬­���8uD}�l&�Tw�}�t�¬¨�a��z:}�l&l � tSuLz:¬¨t ©e{A¬­����z:g�¬¨t��&�a�jt:z:}SuL¬­�¼zvÀfhgjl��&�a}:}�l&�Tz:�jl&t�t»©e�aͭͨ����t~©e}��ak3z:gjl�©e�aͭͨ���'¬­���Z���a¬­�¼z�tvØ@�'g�¬¨��gyuL}�l'z:}:¬­�A¬ÏuLÍ9z��Zl&t�zSuL��Í­¬¨t:gm�¼{Q¬­�j�bwj�Tz:¬¨�a� �a�¸z:gjl

��w�kZ��lT}��L©é��lT}:z:¬¨�&l&t�uLÍ­}�lvua�b{yt�lTͨl&�Tz�l&� �n�À�fhgjlT}�l/¬¨t��a�jl/���aw���Í­{�дͭ¬­�I��l&��Í­¬¨t:zh©Ê�a}�lvua��g�}�l&�Tw�}�t:¬­��lC�vuLÍ­ÍXz:g9uLz'g9uat5�j�az'{�lTz���l&lT�y¬­��¬­z:¬ÏuLz�l&�GØj��w�z&�'gj��t�l

�9uL}�lT�¼zhg9uat'uLÍ­}�lvua�b{¸��l&lT�o¬­��¬­z:¬ÏuLz�l&�GØjuL�j�oz:g�¬¨t Í­¬¨t:z ��¬­��l&thz:gjl�t�lTzh�L©���lT}:z:¬¨�&l&t z:g9uLz �'¬­Í­Í���l��9uat�t�l&�oz��0z:gjl�vuLÍ­Í´À

� ����� �

Page 21: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË nv�

���+, Ö ��)F� ' � BG� * � ��� �   ���aw���Í­{�дͭ¬­�I��l&� Í­¬¨t:z���}�lT��}�l&t�lT�¼z:¬­���mt:w��jtSlTzª¹��L©���lT}:z:¬¨�&l&t'�L©®DØb�a}���lT}�l&�o��{¸z:gjlT¬­}h}�l&�&lT�j�T{�À

=»ua�Sg-kmlTkZ��lT}5�L©�� �vuL}:}:¬¨l&t+u�ÍÏuL��lTÍXz:g9uLz5¬¨��lT�¼z:¬��9l&t�¬­z+uat5u�kmlTkZ��lT}��L©������� � ��� �»u�}�l&�Tw�}�t:¬¨�a��z:}�l&l � Àq�lTkm�v��lZz:gjl �j}�t:z���lT}:z�lJxC� � ©e}��ak�� �Fª}�lvuLz�l�uL��lTkQ��zß{yÍ­¬¨t:z�� ! z:g9uLz&�'¬­Í­ÍXgj�aͨ�yz:gjl��jlT¬­��g����a}�t+�L©���� �� Ö � Ò ��� ' ���7��²� � · × Ö

)�� � �bÂLÇ/Ë�£�ÉW�S§�§JË Ç&§T¦­§:ÅvÉߧ:¤�� '�Ò *)�� � ÈFÇZÂm¡¸§J¡V��§JÄD£ � � '4Ò *

}�lTkm�v��l`�¸©e}��ak��5Ø���w�z'¬­z5¬­�¼z���� !&Ø�uL�j� ÍÏuL��lTÍX¬­z5ua�&�&�a}��b¬­����Í­{Ò �>� Ò km�v��l`�oz��Dz:gjl/©>}��a��z5�L©�z:gjlCÍ­¬¨t:z�� � z:g9uLz5�Tw�}:}�lT�¼z:Í­{��&�a��zSuL¬­�jt�¬­z

��lTz � !���l�z:gjl/z:}�l&lZ�a��zSuL¬­�jl&�y�¼{ }�l&�Tw�}�t:¬¨�a�-�a��� ! ��� fhgjlC�vuLÍ­ÍXz�� � ! k¸u&{ g9u&��lCkm���b¬��9l&�yz:gjlZ�a}���lT}5�L©���lT}:z:¬¨�&l&t5¬­������lTz � 4 ��l�z:gjl/z:}�l&lZ�a��zSuL¬­�jl&�y�¼{ }�l&�Tw�}�t:¬¨�a�-�a������ u���l � ! uL�j� � 4 ��g�¬­Í¨�b}�lT�«�L©���� �� Ò ���4��* � � À

�bÀ�fhgjl/�a}���lT}5�L©lvua��g-Í­¬­�I��l&��Í­¬¨t:z'¬¨t���l&t��&lT�j�b¬­���o¬­���a}���lT}5�L©�}�l&�&lT�j�T{�À

f4�a��lTz:gjlT}vØ�z:gjl&t�lCzk�ª�o¬­�¼��uL}:¬ÏuL�¼z�t�¬­kQ��Í­{yz:g9uLz�z:gjlC��lT}:z�lJxY���Ct�lTͨl&�Tz�l&�«uat5z:gjlZ¬­��¬­z:¬ÏuLÍ4��¬­���az �'gjlT�«u¸�vuLÍ­Í�¬¨t¬­��¬­z:¬ÏuLz�l&�-g9uat'��}�lvuLz�l&t:z�}�l&�&lT�j�T{�uLkm�a���mz:gj��t�lC��lT}:z:¬¨�&l&t��9uat�t�l&� z��mz:gjl�}�l&�Tw�}�t:¬­��lZ�vuLÍ­Í´À

�»Ò � �¯��2! �"R��¦ ��£�ÄïÈeÉ�¼¡��mÄï¥AË�Ç/È>Ë�¦ ÈFË�§:ÂaÄZÉeÈ>¡¸§9�8C� ÖGÖ �S�   Í­ÍX���aw���Í­{�дͭ¬­�I��l&�-Í­¬¨t:z��a��lT}SuLz:¬¨�a�jt�zSu���l�.m²�·hz:¬­kml�Ø�uL�j���vuL����l/��g9uL}:��l&��z��muL�-l&�b��lC�L©�� � À   t:¬¨��l©e}��ak$z:gjlDz:¬­kmlD}�l&̼w�¬­}�l&�«�¼{«z:gjl�}�l&�Tw�}�t:¬­��lQ�vuLͭͨt�¬­z���lT�jlT}SuLz�l&tvØGz:gjlQ�vuLÍ­Íé}�l&Ì�w�¬­}�l&t`.m²&N3��²����·:·�z:¬­kml�ÀZfhg�¬¨t��¬­��l&t/uL�Z.m²+CN^,�·'���aw��j�-©e�a}/uLÍ­Í4}�l&�Tw�}�t�¬­��l0�vuLͭͨtvØ�t:¬­�j�&l�lvua��g ��lT}:z�lJx��L© ®3g9uat+z:gjl0}��aͨl0�L©2� � ¬­�«�a��Í­{��a�jl�vuLÍ­Í´À

��lTz+³ � ��lT�j�az�l/z:gjlCt:w��jt�lTz��L©é��lT}:z:¬¨�&l&t�z:g9uLz5uL}�lCͨlvuc©é��l&t��&lT�j�juL��z�t5�L©éz:g9uLz��j�A��l�Øb¬­�j�TÍ­wj�b¬­���V�¸¬­z�tSlTÍé Àfhgjl}�l&�Tw�}�t:¬¨�a��z:}�l&l � g9uat5uL��¬­kQ���a}:zSuL�¼z���}��a��lT}:zÝ{��'¬­z:g-}�l&t:��l&�Tz'z��mz:gjlC�&�a���jl&�Tz:¬­��¬­zÝ{yt:z:}:wj�Tz:w�}�lZ�L©~® ��»Ò � �¯��2! >� )�§TÉ ����³ �S§C �L§TÄJÉߧ��8ÂaË�¤ � ÂmÅ�£aËjË�§:Å&Éߧ:¤ Å�£a¡�Æb£�Ë�§JË9É~£ ®O� ³ � �.�?�b§JË�É�b§TÄ:§CÈÏÇ/§��aÈFÇJÉ>Ç � � �É�AÂ�É»ÈFÇZÂaË�ÅS§ïÇTÉ´£aÄZÉ´£8Âa¦>¦�£�É�b§JÄC¡¸§J¡V��§TÄSÇ�£� � È>Ë � �8C� ÖGÖ �S� νlo��}��A�&l&l&�¯��{½¬­�j�bwj�Tz:¬¨�a�¯©e}��ak z:gjlm���az:z��ak �L© � z��-z:gjl¸z��a�XÀY_ß© �3� �QØ�z:gjl¸�TÍÏuL¬­k ¬¨t0�TͨlvuL}:Í­{tSuLz:¬¨t �9l&�½�¼{(��À � z:gjlT}��'¬¨t�l�Ø � � ��²�b·TØ � ¬¨tDuy�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼zD�L© �Á²��·TØ�uL�j� �Á²��b·�¬¨tZz:gjlotSlTzZ�L©�j�A��l&t+¬­� +� Øw�'gjlT}�l � ¬¨t��a�jl��L©~z:gjlZzk�ª� ��g�¬­Í¨�b}�lT���L©2��À*5 {yz:gjl0¬­�j�bwj�Tz:¬­��l0g�{����az:gjl&t:¬¨tvØ�uL����Í­¬¨l&�«z�� +� Ø9z:gjl�TÍÏuL¬­k gj�aͨ��t�©e�a} �mÀ

;yÒ�< *�)F� ) Ö * 2 �� )�§TÉ � ��§m ÉeÄ:§�§m¤¼§ @~Ë�§�¤y£a˽É�b§ �L§JÄTÉߧ��yÇ&§TÉ ³ ÂaË�¤ � Â8Ä�§:ÅT¥AÄSÇJÈe£�˯ÉeÄ:§S§9� >D§ZÇ&Â��oÉ�bÂ�É � ÈÏÇ�&�agjlT}�lT��z = ÈeÉ� � Â�Ë�¤ ® È &£aĸ§.�L§JÄg� �Z� ³ ÂaË�¤ §.�L§JÄg��Å�£aË9Ë�§:Å&Éߧ:¤�ÅS£a¡�Æb£aË�§JË�É � £ ®O� ³ � É�b§mÄ:§JÇJÉeÄJÈeÅ&ÉeÈÊ£aË�&� � ÈFÇZÅ�£aËjË�§�Å&Éߧ:¤�� �jt�lT}:��lCz:g9uLz � ¬¨th�j�az'�jl&�&l&t�tSuL}:¬­Í­{ �&�agjlT}�lT�¼z5z��D¬­z�tSlTÍé À�_ß©?�ªl0uL}�lCuL��ͨl/z��D�&�akQ��w�z�lZt:wj�Sg-uD�&�agjlT}�lT�¼z�z:}�l&l�Ø�¬­z©ew�Í��jͭͨt5z:gjlZ�jl&�&l&t�tSuL}:{y�&�a�j�b¬­z:¬¨�a�jt+z��¸uL����Í­{�fhgjl&�a}�lTk n�Ø�uL�j��z:g�wjt5z��m��lT} ©Ê�a}:k7uLÍ­Í4w���¬¨�a�bÐ �j�j���a��lT}SuLz:¬¨�a�jt5¬­�uLkm�a}:z:¬�Lvl&� Í­¬­�jlvuL}'z:¬­kml�Àfhg�wjtvØbz:gjl5��}��a��ͨlTk �L©�lTÍ­¬­kQ¬­�9uLz:¬­���Qz:gjl X�©Êua�Tz��a}h}�l&�bwj�&l&thz��D�&�akQ��w�z:¬­���¸u0�&�agjlT}�lT�¼zz:}�l&l�©Ê�a} � uL�j��®DÀ

fhgjlT}�l�uL}�lCk¸uL��{yz:}�l&l&t5�a�«® z:g9uLz�uL}�lZ�&�agjlT}�lT��z��'¬­z:g«uQ��¬­��lT� � ÀªÎ½l �'¬­Í­Í4�&�akQ��w�z�l�uDw���¬¨Ì�wjlZ�a�jl/z:g9uLz¬¨t���l �j�jl&�«uath©Ê�aͭͨ����tvÀ��� ù0¢2G ? ���

Page 22: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

;yÒ�< *�)F� ) Ö * 2 )�§TÉ � �S§/É�b§/Ä:§:Å&¥�ÄïÇïÈÊ£aË�ÉeÄ�§�§0£ |�aÄ ÂJÆ>� ÂaË�¤ ����³ � >D§�ÇTÂ��DÉ�AÂ�É � ÈÏÇ t:w����a}��b¬­�9uLz�l É´£�ÇT£a¡¸§£�É��§JÄC�L§JÄTÉߧ�� �6��³ È �

� � ÈÏÇZÂaË�ÂaË�ÅS§JÇJÉ´£aÄD£ � ÈFË �� � ÂaË�¤ � ÂaÄ:§�;&£aÈ>Ë�§:¤��g�oÂCÆbÂ�É�yÈ>Ë ®O� ³�� �¡9�a}»lvua��gO����³ z:g9uLz~¬¨té�j�azz:gjlh}����az~�L© � z:gjlT}�l'¬¨t~u+w���¬¨Ì�wjlhkQ¬­��¬­k¸uLÍ)�6�«³ t:wj�SgQz:g9uLzG�Z¬¨tt�w����a}��b¬­�9uLz�l

z��-��Àéνl/�vuLÍ­ÍGz:g�¬¨thw���¬¨Ì¼wjl��j�A��l � z:gjl �:£LÇïÇ �L©���À~fhgjl+����t�th}�lTÍÏuLz:¬¨�a�y¬¨thz:gjl��9uL}�lT�¼z�}�lTÍÏuLz:¬¨�a�-¬­�yu�z:}�l&l ���a�³oØx�'g�¬¨��gY�ªlZ�vuLÍ­ÍGz:gjl Çï¥)�:£aÄ:¤aÈFË�Â�Éߧ�ÉeÄ:§S§ �L© � À

�»Ò � �¯��2! >2 �?�b§�ÇJ¥I��£aÄ ¤�ÈFË�Â�Éߧ�ÉeÄ:§�§ � ÈFÇCÅS£��b§JÄ�§JË9É = ÈeÉ� � �8C� ÖGÖ �S� νl���}����&l&l&� ��{o¬­�j�bwj�Tz:¬¨�a��©>}��ak¢z:gjl����az:z��ak �L© � z��Dz:gjl�z��a�XÀ �XlTz&�m��l�t��akml���lT}:z�lJxy�L©®±uL�j� �u¸�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z��L©ª®O� ³ � uL�j�-uat�t�w�kmlCz:g9uLz5z:gjl0�TÍÏuL¬­k ¬¨t5��¬­��lT��©Ê�a}�uLÍ­Íé³��¯©Ê�a}&����³ � Ø)� �°\��ÀªÎ½lg9uv��lCz��Qt:gj��� z:g9uLz$�&� �±¬¨t��&�a���jl&�Tz�l&�GÀ

��lTz*� �K� ����l/z:gjl/��lT}:z�lJx�¬­� ���'¬­z:g�lvuL}:Í­¬¨l&t:z���}�l&�a}���lT}+�¼w�k0��lT}vÀ 5 { �XlTkQk¸uVHbÀø�bØ>� ��¬¨t5uL��uL�j�&l&t:z��a}+�L©z:gjlC�az:gjlT}5kmlTkZ��lT}�t5�L© �QÀ�_ß©!� � �°{�oz:gjl/¬­�j�bwj�Tz:¬¨�a�-g�{����az:gjl&t:¬¨t�t:gj����t�z:gjl/�TÍÏuL¬­k�À

JA�ot:w�������t�l��j����z:g9uLz*� � ° ��ÀWJAlTz/® ��z��Q��lCz:gjl/��}SuL��g-z:g9uLz�¬¨t��a��zSuL¬­�jl&����{y}�lTkm�v�A¬­���V�yuL�j�-uLÍ­Í4l&�b��l&tz:g9uLzZuL}�lmua�Tz:¬­��lmuLz �y©e}��ak ®O� �mÀD® � kQ¬­��g¼z/�j�az/��lD�&�a���jl&�Tz�l&�GÀ��XlTz �7!vµ � � �vµ � " ��l�z:gjlD�jl%�±�&�akQ���a�jlT��z�tvÀ=»ua�Sg ���4¬¨t��&�akQ��ͨlTz�lTÍ­{y�&�a�¼zSuL¬­�jl&�y¬­��lT¬­z:gjlT}*��²�b· �a} ��² ��·JÀG5 { �XlTkQk¸uOHbÀø�bØ ����g9uat5u0��lT}:z�lJx�����z:g9uLz'¬¨t5uL�uL�j�&l&t:z��a}C�L©'uLÍ­Í~�az:gjlT}/kmlTkZ��lT}�tC�L© � � À � � ¬¨tCu �&�a���jl&�Tz�l&���&�akQ���a�jlT��zC¬­�_®O� ³�����ÀZfhg�wjtvØG��{�z:gjl�¬­�j�bwj�Tz:¬¨�a�g�{����az:gjl&t:¬¨t��ªlCk¸uv{-uat�t:w�kml/z:g9uLz �&� ���鬨t5�&�a���jl&�Tz�l&�GÀ|5 {y��l �j��¬­z:¬¨�a� ���鬨t5��lTͨ��� �8¬­� � Ø9uL�j�Y���éuL�j�Y�yuL}�l�:�a¬­�jl&�y�¼{�u��9uLz:g�¬­� �3uL�j�yz:g¼wjt5¬­��®O� ³ � À2JA�����4¬¨t�t:w����a}��b¬­�9uLz�lCz��-��À

νl0t:gj����z:g9uLz*�8¬¨tvØj¬­��©Êua�TzvØ9z:gjlC����t�t��L©����ôÀ|J�w�������t�lCz:g�¬¨t5¬¨t��j�az�z:gjl0�vuat�l�ÀhfhgjlT�-z:gjlT}�lCk0wjt:z5��lCt��akml��lT}:z�lJx7� z:g9uLz5¬¨t+uQ��l&t��&lT�j�juL�¼z��L©����~uL�j�-uL�«uL�j�&l&t:z��a}+�L©��oz����'g�¬¨�Sg ����¬¨t5t�w����a}��b¬­�9uLz�l�À 5 w�z5z:gjlT�7� � ���uL�j������¬¨t'�j�az�z:gjl�g�¬­��gjl&t�z��j����lC¬­� ���ôØ9uD�&�a��z:}Sua�b¬¨�Tz:¬¨�a�XÀ

JA�m¬­� �hØjuLÍ­ÍG��lT}:z:¬¨�&l&t&���uL}�lC�b¬­}�l&�Tz5�Sg�¬­Í¨�b}�lT���L©U��Ø9uL�j�C�8¬¨t��&�a���jl&�Tz:¬­��� uLÍ­ÍGz:gjl �&� ��� À���+, Ö ��)F� ' � D�� F �akQ��w�z�l0u�}�l&�Tw�}�t:¬¨�a�-z:}�l&l � uL�j�y¬­z�t�t:w����a}��b¬­�9uLz�l/z:}�l&l�©Ê�a}5uD�&�a���jl&�Tz�l&����}SuL��g ®DÀ

��lTz � ��lCuQ}�l&�Tw�}�t:¬¨�a��z:}�l&lC�&�akQ��w�z�l&�Y�'¬­z:g   Í­���a}:¬­z:g�k � ��4uL��lTÍ�lvua�Sg�l&�b��l¸² �4µ���·'�L©~®6�'¬­z:g-z:gjl/ͨlvuat:z Ðß�&�akQkm�a�«uL�j�&l&t�z��a}� ( � ² ��µ��j· ¬­� � �� Ö � Ò ��� ' �L§JÄTÉߧ�� � £ ® × Ö

�XlTz �Á��lCz:gjl��jlT¬­��g¼���a}�t��L©��a�'¬­z:g-lvuL}:Í­¬¨lT}5��}�l&�a}���lT}5�¼w�k0��lT}�z:g9uL�7� �  t�t:¬­��� �'°� ���B��� ( � ²��µ�)L· � )`� �0�0uat'����t�t��L©U� �

�»Ò � �¯��2! B ��¦ ��£�ÄïÈeÉ�¼¡��0ÈÏÇZÅ�£�ÄïÄ�§:Å&ÉA�8C� ÖGÖ �S�

_ß© � uL�j� ��uL}�l/zk�ª�Q��lT}:z:¬¨�&l&t�Ø)�ªlZtSu&{8z:g9uLz ��� ��¬Ã©�z:gjl/��}�l&�a}���lT}5�¼w�kZ��lT}��L© � ¬¨t'ͨl&t�t'z:g9uL��z:g9uLz��L© �LÀ��lTzW�y��l�uL�yuL}:��¬­z:}SuL}:{o��lT}:z�lJx�À2J�w�������t�l�z:gjl/uLÍ­���a}:¬­z:g�k uat�t:¬­���jtª�j�0����t�tªz��`��ÀfhgjlT�C�yg9uatª�j���jlT¬­��g����a}

� t:wj��g-z:g9uLz�� � �4À2_ßz'©Ê�aͭͨ����t5z:g9uLz*��¬¨t��j�az�¬­��z:gjlCͨlJ©>z5t�w���z:}�l&lC�L©»uL�¼{�uL�j�&l&t:z��a} � �L©��4Øjt�¬­�j�&l/z:gjlT��¬­z�g9uat��� � uatªu��jlT¬­��g����a}vÀfhg�wjtvØ�lT��lT}:{D��lT}:z�lJxm�L©4® ¬¨t~lT¬­z:gjlT}hu/��l&t��&lT�j�juL��zª�L©��8�a}»g9uat~lvuL}:Í­¬¨lT}»��}�l&�a}���lT}ª�¼w�kZ��lT}z:g9uL� �4À�_´©�uy��l&t��&lT�j�juL��z �«�L©|�_¬¨t�u �jlT¬­��g¼���a}0�L© (�� �4Ø4z:gjlT� �9�ª�aw�ͨ�¯g9u&��l¸��l&lT�Át�lTͨl&�Tz�l&����lJ©e�a}�l���Ø�u�&�a��z:}Sua�b¬¨�Tz:¬¨�a�XÀ»fhgjlh��lT}:z:¬¨�&l&tz:g9uLz»uL}�l'��l&t��&lT�j�juL��z�t»�L©B�ouL}�l�u+w���¬¨�a�¸�L©��a�jl'�a}km�a}�l'�&�a���jl&�Tz�l&�¸�&�akQ���a�jlT�¼z�t�L©ª®DÀ|J�¬­�j�&l�®±¬¨t�uat�t:w�kml&��z��m��l/�&�a���jl&�Tz�l&�GØ>�«kZwjt:z���lCz:gjl/}����azvØ�t���©´uL¬­Í­w�}�lZ�L©z:gjlZuLÍ­���a}:¬­z:g�k7z��¸uat�t:¬­����u����t�t'z��D¬­z�¬¨t5�&�a}:}�l&�TzvÀ

��lTz�wjt'�j��� uat�t:w�kml/z:g9uLz|� ¬¨t'z:gjl�����t�t'�L©�� uL�j� ��¬¨t'z:gjl���lT}:z�lJxyz:gjlCuLÍ­���a}:¬­z:g�k7uat�t�¬­���jt5uathz:gjl/����t�t'�L©��Àνl/kZwjt�z5t:gj��� �h° �0ÀG_Ýz�t:wxf¸�&l&t'z��mt�gj��� z:g9uLz ���^��uL�j� ���^�0À

f4�Dt:gj��� ��� �0Ø��ªl+ͨlTz/²��µ �G·��l+uL��{¸l&�b��l5©>}��ak � z���uZ��lT}:z�lJxV�'¬­z:g8ͨ���ªlT}'��}�l&�a}���lT}h��w�kZ��lT} z:g9uL���4À�_ß©�¸°�� ( � ²��µ ��· �3� z:gjlT�7���a}���¬¨t'¬­�-z:gjl/ͨlJ©>z�t:w���z:}�l&lC�L©U��Ø9uL�j��uD�jlT¬­��g¼���a}��L©U��À2=¬­z:gjlT}�²��µ���·��a}0²��µ �4µ���·

� ����� �

Page 23: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË �bn

¬¨t5u��9uLz:gXØjuL�j�C� t:w���lT}�t�l&��l&t ��t:zSuLz:wjt�uat'����t�t'�L©!��ØjuD�&�a�¼z:}Sua�b¬¨�Tz:¬¨�a�XÀ fhg¼wjtvØI���^��ØjuL�j�GØ�t:¬­�j�&l ��¬¨t't:wj��g�u��Ø ���^�0À

νlm�j��� t�gj��� z:g9uLzK� � �LÀO_´©W�_¬¨t0u �jlT¬­��g¼���a}0�L© �0Ø � ( � ²��µ��/·C° �0Ø4l&t:zSuL��Í­¬¨t:g�¬­���-z:g9uLz`� � �LÀO_´© �¬¨t+�j�az/um�jlT¬­��g����a}/�L©2��Øw� ¬¨t+¬­�9� � t+}:¬­��g¼z/t:w���z:}�l&l�À �XlTz � ��l�um�9uLz:g ¬­� + � ©e}��ak �±z��a�4À J�¬­�j�&l��±¬¨t�z:gjl����t�t��L©U�4Ø �3kZwjt:z5lJxb¬¨t:zvÀfhgjlCtSl&�&�a�j�y��lT}:z�lJx��L© �3¬¨t'¬­��� � t5ͨlJ©>z�t:w���z:}�l&l�Ø9t:¬­�j�&l/¬­z�¬¨t5u��jlT¬­��g����a}+�L©���À �XlTz( ��lQz:gjl¸ÍÏuat:zZ��lT}:z�lJx¯�L© � z:g9uLzZ¬¨tC�j�azDuykmlTk0��lT} +

ØéuL�j�½Í¨lTz �«��lQ¬­z�t0t:wj�&�&l&t�t��a}��a�¯z:gjl¸�9uLz:gXÀYJ�¬­�j�&lO��huat/t�lTͨl&�Tz�l&�«��lJ©Ê�a}�l ��ØB��g9uat/um�jlT¬­��g����a}K yt:wj�Sg�z:g9uLz ( �R  � �4À�fhg�¬¨t�l&t:zSuL��Í­¬¨t:gjl&t�z:g9uLz  8¬¨t+¬­� + �»ØGuL�j�� � � ( � ²�0µ  �· � �cÀ

νl-g9u&��l-uLÍ­}�lvua�b{ t�gj���'� z:g9uLzoz:gjl �j}�t:z¸t:z�lT� �L©   Í­���a}:¬­z:g�k D�¬¨t¸Í­¬­�jlvuL}vÀ νlywjt�l�z:gjlyͨlvuat:z Ðß�&�akQkm�a�uL�j�&l&t:z��a}�tCuLÍ­���a}:¬­z:g�k��L©WE+uL}�lTÍ~uL�j� f4uL}e��uL� : nv���L�@;5z��8�&�akQ��w�z�l�z:gjl�ͨlvuat:z/�&�akQkm�a�¯uL�j�&l&t:z��a}�t/�L©»z:gjlDl&�b��l&t¬­� � À � �j�&l�z:g�¬¨t'¬¨t�ua�&�&�akQ��Í­¬¨t�gjl&�GØjlvua��g���lT}:z�lJx ��k¸uv{o��l+ÍÏuL��lTͨl&�C�'¬­z:g�¬­z�th����t�t ¬­� .m² � ��² ��· � ·hz:¬­kml�ØI�'g�¬¨��g��¬­��l&t+uL� .m²+aNb,�· ���aw��j�8©Ê�a}��&�akQ��w�z:¬­���¸z:gjlCt:w����a}��b¬­�9uLz�l/z:}�l&l�À

fhg�¬¨t8�&�a�j�TÍ­wj��l&toz:gjl���}��A�L©/�L©/z:gjl-Í­¬­�jlvuL}¸z:¬­kml����aw��j� �L©/z:gjl�km�A�bw�ÍÏuL}o��l&�&�akQ����t:¬­z:¬¨�a�¿uLÍ­���a}:¬­z:g�k �a��&�a���jl&�Tz�l&� ��}SuL��gjtvÀ _ß© ® ¬¨t+�j�az/�&�a���jl&�Tz�l&�GØ�¬­z�t+�&�a���jl&�Tz�l&���&�akQ���a�jlT�¼z�tZ�vuL�«��lZ©e�aw��j��¬­� Í­¬­�jlvuL}+z:¬­kml�ØGuL�j�z:gjl+km�A�bw�ÍÏuL}���l&�&�akQ����t:¬­z:¬¨�a�-uLÍ­���a}:¬­z:g�k uL����Í­¬¨l&� z��Dlvua�Sg �&�akQ���a�jlT�¼zvÀ0�«u��A¬­���Dz:gjl+}�l&t:w�Í­z:¬­���Dz:}�l&l&t'��g�¬­Í¨�b}�lT��L©~uD�9uL}SuLͭͨlTÍX�j�A��lC�&�akQ��ͨlTz�l&t�z:gjlCkm���bw�ÍÏuL}5��l&�&�akQ����t:¬­z:¬¨�a�-�L©»®DÀ

  t ��l&t��T}:¬­��l&�GØAz:gjl�uLÍ­���a}:¬­z:g�k¢z��Z�&�akQ��w�z�l � uL�j� ��}�l&Ì�w�¬­}�l&t ze� �0�9uatSt�l&tvØ �j}�t:z»z��0}:w��   Í­���a}:¬­z:g�k � ØAuL�j�z:gjlT� z��0wjt�l�E�uL}�lTÍGuL�j�8f�uL}e�:uL�8z�� �j�j�¸z:gjl�ͨlvuat:z Ðß�&�akQkm�a�-uL�j�&l&t:z��a}'�L©�lvua�Sg�l&�b��l5¬­�8z:gjl+}�l&t:w�Í­z:¬­���0z:}�l&l�Ø�uL�j��j�9uLÍ­Í­{�Ø9z��Qwjt�l/z:gjl/ͨlvuat�z Ðß�&�akQkm�a� uL�j�&l&t:z��a}�t��L©�z:gjlZl&�b��l&t�z��m�&�akQ��w�z�l �hÀ~fhgjlZͨlvuat:z Ðß�&�akQkm�a�«uL�j�&l&t:z��a}�t��L©z:gjl+l&�b��l&t uL}�l+uLͨt��C}�l&Ì�w�¬­}�l&�o�¼{Qz:gjl�km�A�bw�ÍÏuL}h��l&�&�akQ����t:¬­z:¬¨�a�yuLÍ­���a}:¬­z:g�k�Øbz��Z�a}:�¼uL��¬�Lvl�z:gjl5l&�b��l&tª¬­�¼z��0��}��aw��jtz:g9uLz�uL}�lZua�Tz:¬­��lZuLz�z:gjlCtïuLkml/z:¬­kml�À   Í­���a}:¬­z:g�k � �vuL����l/km�A�b¬��9l&�yz��Q��}��A�bwj�&lZuLÍ­Í��L©éz:g�¬¨t�¬­�b©Ê�a}:k¸uLz:¬¨�a�-¬­��ut:¬­����ͨlC�9uat�tvÀfhgjl/km�A�b¬��9l&�y��lT}�t:¬¨�a��¬¨t���¬­��lT��uat   Í­���a}:¬­z:g�k �bÀ

���+, Ö ��)F� ' � 1G� * � ��� �   ���aw���Í­{�дͭ¬­�I��l&�DÍ­¬¨t:z �«}�lT��}�l&t�lT��z:¬­���Zu5t�lTz4¹ �L©j��lT}:z:¬¨�&l&t�L©�® z:g9uLzéuL}�lª�9uat�t�l&�Zz��+u5}�l&�Tw�}�t:¬­��l

�vuLÍ­Í´Ø��a}���lT}�l&����{yz:gjlT¬­}�}�l&�&lT�j�T{�À =»ua�Sg«kmlTk0��lT}��L© � �vuL}:}:¬¨l&t/uQÍÏuL��lTÍ4z:g9uLz+¬¨��lT�¼z:¬��9l&t+¬­z/uat+ukmlTkZ��lT}'�L© �5Ø�uL�j� ��¬¨t ÍÏuL��lTͨl&���'¬­z:gyu �¼§TË�§JÄ:Â�É´£aÄ ØI�'g�¬¨��g ¬¨t z:gjl+¬­��¬­z:¬ÏuLÍ���¬­���az ¬­� z:gjl5�9uL}�lT��z�L©éz:gjl/}�l&�Tw�}�t:¬­��lC�vuLÍ­ÍXz:g9uLz5¬­z'}�lT��}�l&t�lT�¼z�tvÀ

����� � ��� �  ¢Æ�ÂaÄ:§TË9É ÍÏuL��lTÍ­¬­���½��¬­��¬­���Áu«}�l&�Tw�}�t:¬¨�a� z:}�l&l � Øhu«ÍÏuL��lTÍ­¬­���½�L©/lvua��g l&�b��l7�'¬­z:g ¬­z�tQͨlvuat:z�&�akQkm�a�¯uL�j�&l&t:z��a}C¬­� � ØXuL�j�½u �:£aÇSÇ ÍÏuL��lTÍ­¬­���y��¬­�A¬­���-u¸z:}�l&l � z:g9uLz/¬¨t/�&�agjlT}�lT�¼zK�'¬­z:g �uL�j�-®

q�lTkm�v��lZz:gjl �j}�t:z���lT}:z�lJxC� � ©e}��ak z:gjl+©e}��a�¼z5�L©����Fª}�lvuLz�l�uL��lTkQ��zß{yÍ­¬¨t:z�� !'z:g9uLz&�'¬­Í­ÍXgj�aͨ�yz:gjl��jlT¬­��g����a}�t+�L©�� � �  t�t:¬­����� � uat'z:gjlC��lT�jlT}SuLz��a}5�L© � !�uL�j���L©����� Ö � Ò ��� ' ���7��²� � · × Ö

)�� � �bÂLÇ/Ë�£�ÉW�S§�§JË Ç&§T¦­§:ÅvÉߧ:¤�� '�Ò *)�� � ÈFÇZÂm¡¸§J¡V��§JÄD£ � � '4Ò *

}�lTkm�v��l`�¸©e}��ak������w�z&�8¬­�¼z���� !�uL�j� }�lTÍÏuL��lTÍX¬­z+uat5u�kmlTkZ��lT}��L© � ! ���}��v�A¬¨t:¬¨�a�9uLÍ­Í­{-uat�t:¬­����� � uat'z:gjl/����t�t��L©�� �uat�t:¬­����� � uat'z:gjl/ͨlvuat�z Ðß�&�akQkm�a�«uL�j�&l&t:z��a}5�L©�²� � µ���·

Ò �>� Ò km�v��l`�oz��Dz:gjl/©>}��a��z5�L©�z:gjlCÍ­¬¨t:z�� ��z:g9uLz5�Tw�}:}�lT�¼z:Í­{��&�a��zSuL¬­�jt�¬­z ���}��v�A¬¨t:¬¨�a�9uLÍ­Í­{-uat�t�¬­���yz:gjl/��lT�jlT}SuLz��a}5�L©�� �Xuat'z:gjl�����t�t��L©U� �uat�t:¬­���yz:gjlC��lT�jlT}SuLz��a}��L©�� � uat'z:gjl/ͨlvuat:z Ðß�&�akQkm�a� uL�j�&l&t:z��a}5�L©�²� � µ��j·

��lTz � !���l�z:gjl/z:}�l&lZ�a��zSuL¬­�jl&�y�¼{ }�l&�Tw�}�t:¬¨�a�-�a��� ! ��� fhgjlC�vuLÍ­ÍXz�� � ! k¸u&{ g9u&��lCkm���b¬��9l&�yz:gjlZ�a}���lT}5�L©���lT}:z:¬¨�&l&t5¬­������lTz � 4 ��l�z:gjl/z:}�l&lZ�a��zSuL¬­�jl&�y�¼{ }�l&�Tw�}�t:¬¨�a�-�a�Y�'g9uLz�}�lTk¸uL¬­�jt5�L©����� u���l � !�uL�j� � 4 ��g�¬­Í¨�b}�lT�«�L©�� � �� Ò ���4��* � � À

��� ù0¢2G ? ���

Page 24: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� � ¦ ÈÊÂLÇ��ZÂ��¼¦ �bÂa¥¼Ç � � §TË�Ç��h¥¼ÇTÉߧ:¤�É ��� £aÇSÇ ���?�«Å£aËjË�§J¦>¦

�»Ò � �¯��2! D ��¦ ��£�ÄïÈeÉ�¼¡��oÈÏÇZÅ�£�ÄïÄ�§:Å&ÉA�8C� ÖGÖ �S� fhgjl0}�l&�Tw�}�t:¬¨�a� z:}�l&lZ}�lTz:w�}:�jl&��¬¨t�z:gjl0tSuLkmlDuat�z:gjl0�a�jl0}�lTz:w�}:�jl&�-��{   Í­���a}:¬­z:g�k � À�=»ua��g«��lT}:z�lJx-¬¨t�Sgj��t�lT�¯uat+z:gjl0¬­��¬­z:¬ÏuLÍé��¬­���az ���Z¬­��t��akml0}�l&�Tw�}�t:¬­��lD�vuLÍ­Í´À �XlTzm² �4µ���·5��l�uL�½uL}:��¬­z:}SuL}:{�l&�b��l�ØXuL�j�«Í¨lTz ����lZ¬­z�tͨlvuat:z Ðß�&�akQkm�a�½uL�j�&l&t:z��a}vÀ   t�t�w�kmlK�'¬­z:gj�aw�z+ͨ��t�t��L©��lT�jlT}SuLÍ­¬­zÝ{-z:g9uLz���¬¨t+��gj��t�lT�«��lJ©e�a}�l`��À|_Ýz5k0wjt:z���l/z:g9uLz�P�t��²��·�uL�j�Y�9� �Á²�G·TÀ �XlTz �¿uL�j��� !5��l0z:gjlCÍ­¬¨t�z�t*�'¬­z:g«z:gj��t�l0�9uLkml&t�¬­��z:gjlZ}�l&�Tw�}�t�¬­��l0�vuLÍ­Í��'gjlT}�l`� ¬¨tz:gjl+¬­��¬­z:¬ÏuLÍ���¬­���azvÀ~fhgjlT���y¬¨tªkm�v��l&�8z�� � !vØ��Q}�lTk¸uL¬­�jth¬­� ��ØAuL�j�a�y¬¨tªz:gjl5��lT�jlT}SuLz��a}'�L© ��À ��¬¨tª�j�az z��awj��gjl&���lTze� l&lT��z:gjl+z:¬­kml��'gjlT�yz:gjl+}�l&�Tw�}�t:¬­��l��vuLÍ­ÍG�a� � !h¬¨t ��lT�jlT}SuLz�l&�-uL�j���'gjlT�8¬­z'�&�akQ��ͨlTz�l&tvÀ2c�w�}:¬­���Qz:g�¬¨t z:¬­kml�Ø� ¬¨t t�lTͨl&�Tz�l&�8uatªz:gjl�¬­��¬­z:¬ÏuLÍ���¬­���azW� � ¬­�otS�akml5ͨ���ªlT}h}�l&�Tw�}�t:¬­��l+�vuLÍ­Í´À   zªz:g�¬¨tªz:¬­kml�ؼz:gjl���lT�jlT}SuLz��a}'�L© �5Ø@�'g�¬¨��g¬¨t`�4Ø�¬¨tZ�&�a}:}�l&�Tz:Í­{¯uat�t�¬­���jl&�_uatCz:gjl¸Í¨lvuat:z Ðß�&�akQkm�a� uL�j�&l&t:z��a}��L©Z² �4µ���·JÀ Î gjlT�P� ¬¨tZtSlTͨl&�Tz�l&�½ÍÏuLz�lT}vØ�z:gjlT}�lm¬¨t�j�y}�lvuatSt:¬­����kmlT�¼z��L©'u8ͨlvuat:z Ðß�&�akQkm�a��uL�j�&l&t:z��a}Cz��½² �4µ���· �ét:¬­�j�&l �_g9uatZuLÍ­}�lvua�b{ ��l&lT�¯tSlTͨl&�Tz�l&�GØXz:gjlQl&�b��lQ¬¨t¬­���j�a}�l&�GÀ»Î½lC�&�a�j�TÍ­wj��l/z:g9uLz5lvua�Sg-l&�b��l/��lTz�t�uatSt:¬­���jl&� ¬­z�t��&�a}:}�l&�Tz5ͨlvuat�z Ðß�&�akQkm�a�«uL�j�&l&t:z��a}vÀ

fhgjlZ����tSt5�L©�� ¬¨t�z:gjl/k¸ucxb¬­kZw�k7��}�l&�a}���lT}���w�kZ��lT}+�L©z:gjlZͨlvuat:z Ðß�&�akQkm�a� uL�j�&l&t�z��a}�t��L©z:gjlCt�lTzK��² ��µ��j· �² �4µ���·0¬¨tQuL��l&�b��l8�L©/®$uL�j� � g9uatQlvuL}:Í­¬¨lT}D��}�l&�a}���lT}Q��w�kZ��lT}Qz:g9uL�b�w�¼À(5 {½z:gjlo©e�a}�lT���a¬­���jØ~z:gjl&t�lyuL}�loz:gjll&�b��l&t�¬­�j�T¬¨��lT�¼z�z������'gj��t�lZͨlvuat:z Ðß�&�akQkm�a��uL�j�&l&t:z��a}5g9uat�uLÍ­}�lvua�b{y��l&lT� uat�t:¬­���jl&�Y�'gjlT�Y� ¬¨t5t�lTͨl&�Tz�l&�GÀ*=»ua��gz:¬­kmlQt:wj�Sg½uL� l&�b��l-�huatCuat�t:¬­���jl&��uoͨlvuat:z Ðß�&�akQkm�a�_uL�j�&l&t:z��a} ��ØB�7� uat/��¬­��lT�t�_uat�¬­z�t+����t�tvÀ/fhgjlD��}�l&�a}���lT}��w�kZ��lT}5�L©�z:gjl+��lT�jlT}SuLz��a}�t��L©�z:gjl�Í­¬¨t:z�th�&�a��zSuL¬­��¬­���O�ouL}�l+¬­�j�T}�lvuat:¬­���¸uathz:gjl/uLÍ­���a}:¬­z:g�k ��}����&l&l&��t�ØAtS�0z:gjl+ÍÏuat:z����t�t�ÍÏuL��lTÍuat�t:¬­���jl&��z��V�y¬¨t5z:gjl0�a�jl`�'¬­z:g�ÍÏuL}:��l&t:z+��}�l&�a}���lT}���w�kZ��lT}vØXuat5��l&t:¬­}�l&�GÀ����¸����t�t5¬¨t5t:w��jtSl&̼wjlT��z:Í­{uat�t�¬­���jl&�yz��-��Ø9t:¬­�j�&l�¬­z'g9uat�uLÍ­}�lvua�b{ ��l&lT��t�lTͨl&�Tz�l&�GÀ

� �%&�������#

� ��� � � � � ����� � � � ���� �������� ��� � � � ��� � � ����� � � � ����� ������� �����"! �$# � ���+�'��%�� ��� � ��'& � ��� �� � ����� � &)( � �+� �� ��� �!�� � ��(�� � � � �*( � ���� � �,+.-�/10�2435-6/87:9<;>=@?5ACB'2EDF-�/G7:=�H';�I�24;E2J-�DF9K0 �'�'LM ON�POQ� ���R �

�TS ��� & � � ��� ��% � �U �����P6� S ��� & � � �,V ���W� �YX ������ � � ��� � S���ZY� � � � ��� ������+���,� � X ��� �����P�! �\[�] ^�=�D<71/10_3�;<� �`aVcb � � ����S ��& � � � �������_� ��� ���� � � �+���+� � ��TS ��� � ���M% ��� ��% � �U �����d6� S ��� � ���e%8�Cf �hgi� �jb"� � %8�Ck � � � � �ml ���n � � � �hX �po)�q� �����d�! �*# %�� � ��� � � � � � � � � �� ��� ��%�� ��� � ��'& � ���

� � � � ��(�� � ��r_sF[t+vu_w�xy=63�B{z_/|w ���'L ��}�~OQ_����R ��TS � � � � ��� � � � � � �!�������� �����6� S � � �� � � � � #�� � � � � ����� �p�Y� ��� �������! ��# � �E� ��� � � � ��%�� ��� � ��'& � � � � � � � � �M����( � �e& �� � ��� � � � � ��� � � ��(�� � � ` � � �O� � � ZY�'��� ������� � ��� ��� DF-KB'0_�|/10�2J=�DF2E/G7:9)9K=6H�9J2|B{/G;t71H�9K=63�B{z_/|24Dt;E9E7:24H�9J24�q����/1071H�/|24D<H�-6/87:=6H�-6]

� =6DF�6;@0�=JB{�>� ��� ����� � % � & �t~"d�N �!�y� 2K9E/Gz_DF2���=�/�24;�71H�xy=63�B{z_/|24DtrU947:2EH{9K2 �'( �!�����t � ER6Q� }�} � l_( � � � ��� ��� � � % �!� ��TS � � � � ��� � � � � � �!�������� ����R�� S � � �� � � � � #W� � � � � ����� �p�'� �{� �����R"! ��# � �En�%�� � ��� � ��%�� ��� � ���& � ��� & � � � % � � ����� � &)( �!�

�+� �� ��� � ` � V ��� ��� ��l � � ����� � ��� �Y� DF2K24;>71H�[�] ^"2��EDF--�H{� ��DF=J^�DF-63i3>71H¡^6��xp[�[�� � �E¢ ��� � /£0s<H¤/�2ED<H{-�/G7:=�H{-�]�xy=�]1]e=�¥Ez_71z_3 � � % � & ��N6�¡N ����� 2J9E/8z_DF2��>=6/|2<;�71H¦xy=�3YB�z_/�2EDtr�9E7:24H�9K2 �'( �!��� �,~��OQ_��} � l¡( � � � ��� ��� � � %������ Z ��� � � � � �����¤§ o)�

� f ���'%�� � � ���� ����d6�>f �!�'%�� � � �E� ZY�U� ����d�! ��Z �'����� � ��( � � ��%M% ��%�& � � � % � � ����� � &)( � �+� �� ��� �U¨'© ª�«�s<H�/ � ]��i=6DF�6;F0'=KB¬=6H � DF-JB�0� 0�2K=6DF2E/G7:9­xy=6H�9K2�B�/:;�71H¦xy=�3YB�whr�9E7:w,®O� �¯��°K±6��}_ �

� f ���'%�� � � � ��� ��% � �p ����¡N �>f �!��% � � � ��� ZY� � g�� � � ����� ��² � ��� � �W� � S ��� � �E%M%G��� � � �¡� ���¡N�! ��Z �'����� � � � � ��( � ������ ����%�& � � � % � ������ � &)( � �+� �� ��� �y�,DF=�9K2K2K�671H¡^6;�=a?W/10�2>2E7³^ 0_/10�-6H¤H�z�-�]¤[>xC+��r_sF[�+v;<´63�B'=6;K71z_3�=6H¬�67µ;E9EDF24/|2i-6] ^"=6D<71/£0¡3>; �{��L }�~OQ¡�"d �

� Z � � � ��� � � � � � ��� ��% � �U �����R�� Z � � � ��� � � � � ��� #W� � g ��� � n�� �W�p¶�� �j� � S ����� ��%e%8�C� � � � �"� � �ml � %e%�� �� � �jl � ² �y� �����R"! � # �·)¸£¹�ºO» ���������� � � � � � � ����¼ � � � ��%�� ��� � ��'& � ��� �����( � �e& � � � ��� ����� � &)( � �+� �� ��� ��� �an ��� �+� � � ��� � � ��� � � �5& � � � % � � ����� � & �( � �+� �� ��� �!� � � ��(�� � �,u_=6z_D<H{-�]C=a?�[�] ^�=�D<71/10_3�; �p ~'L½}����OQ_}���R �

� Z � � � ��� � � � � � � � ��� � ��� � � � � ���� ����P�� Z � � � ��� � � � � ��� #W� � � �­� � ��� � � � � ��� gi��� �����P�! � V ��� ��� � �!� } � � � � � ��� � � �����"( � � �� �LS %�� � �E��� ������ � � � �E%������������ � �¾& ��� (���� �F& � � � 0'2K=6DF2E/87:9J-�]�xy=63�B{z_/|2EDtr�9E7:24H�9J2 �{N�P'L½}�N�N Q¡��P�� ��½g �!� � n � � ��V�� �:¿ � � �� ����"dO� g �!� � n�� �W��¶W� � � �V�� �:¿ � � �¤� �¤ZY�j� �����d�! ��# %�� � ��� �� ��M& � ��% � ��� � � �'& � ��� �'�F( ����� ��%"�����+�

�!� ����� ¿ � � � � �+��� � � � ��� ��u_=6z_D<H�-6]C=@?)x�=�3YB�z_/�2ED>-�H{�)r{´O;</�2E3ÀrU947:2EH{9K24; �¤��P'L½}�P��OQ'}�}_ ��½g � % � & � ������ ��¡N�N � g � % � & ��� ���_� �¤S���� �"N�N�! � V ��� � � &)('%��4Á�� �a� ��� � � &)( � � �!���M%�� �a� � � ��(�� � � � � � � � �� ��� � � � � � % ��� � � � �yu¡wx�=�3)�E71H{w � 0�2K=6D<´5rU24DEw� �¤}�}'L ~��OQ_��P ��½g�� �+��������p �����6� g�� �+�� �����t² �t� �������! �ÃZ � ��� � � � � � � �����8Ä � � � ��� (�%�� � � � � � ��(�� � � � � � � ��� � �F( � � �+� � � ��(�� �E%�������� �

� 0�2K=6DF2E/�w�x�=�3YB�z_/�wCrU947:w ��}�P�� � O!<LM }��OQ{ �R� �� ����� �

Page 25: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

��� Å&ÈʧJË�É ÂaË�¤�� Ä:Â�Å&ÉeÈÊÅ�Âa¦���¦ �¼£aÄïÈeÉ�¼¡�Ç�T£�Ä ��§��T¥j§JË9ÉeÈÊÂa¦�� £v¤a¥A¦ÃÂaÄ �0§:ÅS£a¡�Æb£LÇïÈeÉeÈÊ£aË ���

� � ����� � � � �­� � � � � � �U ��¡N6�6� � �!� � ����� � � � � � � � � � � ��� �pS��j� ��¡N6�"! � � � ������ �M¿ � � � ��� � � &)( � �� � � ��� � ��� � � ��� � ��� � ���� � ��(�� � ���W7µ;E9EDF24/|2t[qB�B�]T7:2K�5+.-�/10�2435-6/87:9<; �C �L½}�P' 4Q_}�P"N �

� � � � ��% � � ��V"� �G¿ � � �� �����R�� � � � ��%G�Of � � � �tV�� �:¿ � � � � �"� �����R"! ��� ���+����%�� ��� � ���& � � ���UÄ � ��� � � � ��� � � � � � � &)& ��� � � ��� � � ��� � �r¡sF[t+vu_=�z_D<H{-6]h=6H¦xy=�3YB�z_/G71H¡^ �� �'L �����OQ¡�"d�d �� � � S ����� ��%e%8�� �����d6�>� � S ����� ��%e%8�C� � � ��� ���"d�! �5# � � � ¹�º ! � � � � ��& � � � ��% ��% � ��� � � �'& � ��� & � � � %�� � ����� � &)( � �+� �� ��� �!�

� � ��(�� � � � �¾} � �+� � � ��� � � ��� ��[�] ^�=�D<71/10_3>7:9K- �p ER¤L }�}��6Q_}6R"� �� � � S ����� ��%e% � � ��l¡(�� ��� ���U�U �����R��i� � S ����� ��%e%8��� � � � � � �l¡(�� ��� ���U��² � b �C� �����R"! � X � � ��� �+� ��M& ��& � � � %�� � ����� � &)( � �+� �

�� ��� � � � �4�'����� � � � � � � �� � � �� ��� � � � � ��� � ��� �!� � � &)( � � �!���M%�� �a� � � ��(�� � ����DF= 9J2K2K�671H¡^6;�=a?�/10�2��7 ?K/105[�H¤H�z�-6]_[>xh+���r_sF[t+r¤´63�B'=6;K71z_3�=6H�W7µ;E9EDF2E/�2�[�] ^�=�D<71/10_3�; ��d_L½d���~6Q_d�R"d �

� � � S ����� ��%e% � � ��l¡(�� ��� ���U�U ��������i� � S ����� ��%e%8��� � � � � � �\l¡( � ��� ������² � b ��� �����"! � � � ��� � ��� �� � � �4Á$( � � � � � � ��� � � � �� � ��& � �+� ��� �

� � � S ����� ��%e% � � ��l¡(�� ��� ���U�U ��������i� � S ����� ��%e%8�t� � � � � � � l¡(�� ��� ���U�t² � b �W� ������! � � � � � %�� � ����� � &)( � �+� �� ��� � � �� � � � �� � � �� ��� ��� � � ���� ��� ����7µ;E94DJ24/|2t+.-6/£0'2E35-6/87:9<; �U}�P' � � �"!<LM ����6Q_}�R� �

� ��� � � � ���U ���"dO�i� �� � � � ���¡� �_�W�U� ����d�! �p# % � ��� � � �'& ��� ���F( � � � � ��� ���� � � � �+�� � � � � ��� ��� � � &)( � �� � � ��� � � � (���M& �e� ���� ���� �� � � �E% ��� � ��� �E������ �F��� � �E& � � � � � � � % � � � � � � ���� ��� � �q[�H�H�-6]½;W=a?5ACB�24DJ-6/87:=6H';�I�2<;E2K-6DF9K0 ��R�LM ���dOQ_}�}�d �

� � � %M% � � � � �¾l¡( � ��� �����U ������6�)� � %M% � � �¡² �¡�W� � � � l¡(�� ��� ���U�¡² � b �{� �������! � ` � � � �E& � � � ��%'& � � � % � � ����� � &)( � �+� �� ��� �yu_=�z_D<�H{-6]h=a?t/£0'2t[>xC+ ����~'LM <Q� �� �� l¡(�� ��� ���U�� ���"}O�il¡( � ��� ������² � b �h� ����}�! ���� � � ��� � � � � � � � �+�� � � � � ��� ����� � &)( � �� � � ��� ���W7µ;E9EDF2E/�2)-JB�B{]T7:2K�­35-6/£0'2E3)-�/G7:94; �

���'L½}�~��OQ'}��' �� l ���� � � � �� �����}O�)l ���� � � � � gi�y� �����}�! �)+.-�9K0¡71H�25;E9K0'2K�6z_]T71H¡^ � 71/£0*B{DJ2J9K2K��2EH{9K2 9K=�H';</GDJ-671H�/G;<� b ��f � ������ �E�p§ � ���� � �� �a� �!�� �!�� � % � � � � �!�� � % ��� � � � � �� V�� �:¿ � � �U ��������iV"� �G¿ � � ��� ��ZY�'� �����"! ���i-6/|-t;</8D<z�94/8z_DF24;,-6H���H�24/ � =6DF�>-6] ^"=6D<71/£0¡3>;<� l � ��� ���a� � ��� ` � � � �+� � ����%�� � � # ('('%������

� ��� � � ��b ���M%������E%M(���� � �� � ��% ����� ��� ��% � �U ����"}O� � ��%��������y² � �yV�� �:¿ � � ��� �jZY� �Y� � � �¦X �On�%�� � � Z�� X ��� �����}�! � V ��� � � � � � � � �� ��� �!� �� � ����� � ( � � ��%e%��E%

��� � � ��(�� � ��r{7:-63�u_wqx�=�3YB�z_/�w �C � �L }����OQ_�' � �

��� ù0¢2G ? ���

Page 26: Efficient and Practical Algorithms for Sequential Modular ... · HAL Id: inria-00072855 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY

Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

ÉditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ������ ������� �� ���

ISSN 0249-6399