Transcript
Page 1: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

HAL Id: inria-00072492https://hal.inria.fr/inria-00072492

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Protocol Insecurity with Finite Number of Sessions isNP-complete

Michaël Rusinowitch, Mathieu Turuani

To cite this version:Michaël Rusinowitch, Mathieu Turuani. Protocol Insecurity with Finite Number of Sessions is NP-complete. [Research Report] RR-4134, INRIA. 2001, pp.18. �inria-00072492�

Page 2: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--41

34--

FR

+E

NG

ap por t de r ech er ch e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Protocol Insecurity with Finite Number of Sessionsis NP-complete

Michaël Rusinowitch and Mathieu Turuani

N° 4134

mars 2001

Page 3: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy
Page 4: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois, Campus scientifique,

615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy (France)Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19

�������������� ������������������������� "!#�$�������&%'�)(+*,�-�.��/102�����3�4���5�6�4�.%'�178��9(;:5�$����

<1=?>A@CBEDGFIHKJ�LM=ON�PRQK=?ST>A@UBEN�VW<WB�SX@�=OYGJ[Z\J�]TJ�BEN�=

^-_a`Mb2ced9fhgjiMkalmcenporqslut4lmc$nc$v\t$wsnmt$xanzy|{}b�~�ornplu�Gxac����o���c4v�����o�v�_ac$o

�Iws�a��or�|vI�Rce��cMt�_8c$��t�_ac�kE�����M���,f bw���y�d��s�8��f��M���8wsqscTy

�����M�$�A�a�s�s��� c�lpk}�rcMy|v lmqrwAv�c�v _ac�t4osb2�anmc4¡Rlpv�{�os¢�v _ac��8� osv oRt4ornzlpk�y|cTt4xa��l£v�{5�a��os~anmc$b¤¢¥os�jw2¦8kalpv c�k}xab�~�cM�\os¢Cy cMy�y lpork8y§ ¦a¡RcM�¨k}xab�~�cM�Ko�¢�lpkGv�c$��npcTwX�scM�#��xak8y�©�ª � c«y _aoA¬¤v�_8wAvv�_aluyK�a��os~anmc$b­lmy�®I�C¯°t4orb��8npc$v c«lmk±w³²\ornpcM�r¯O´µw�o�b2oR�Rc$n¶o�¢lmkGv ��x8�RcM��yMª·^-_acK��cMy xan£v��Ro}cMy�k8o�v�wry y xab2cKw·nmlpb2lpv�osk�v�_ac�y lp¸McKo�¢�b2cMy�y�w�qscTyew�k8�³y xa�a�Eos� v�y9kaoska¯¹w�v osb2luty|{}b2b2c4v�� lutcMk8t4��{}�Rv lmosk·ºrc${RyMª � c�wsnmy o�a��oA�scjv _8w�v¶lmk·os���Rc$�Iv oK~ax8lpnu�«w�k·w�v|v�wrt�º�¬�lpv _«w¦a¡RcM�)k}xab9~Ec$�¶o�¢�y cMy�y|lmosk8y�v _ac�lmkGv ��x8�RcM�k8c$cM�8y�oskanm{v�o�¢¥or� qrc�b�cTy y�w�qrcMy�o�¢���ornp{}kaorb2lmwsnzy|lm¸$cr»R�a� oA�}lu�RcM�Kv _8w�v�v _acM{�ws� c���c$�a��cMy c$kGv cT��wry��aw�qGy$ª¼·½G¾�¿�À9Á �AÂ��s� y|cTt4xa��l£v�{r»Rt4osb2�anmc4¡Rlpv�{s»8w�v|v�wst�º�»}�a� osv oRt4orn?»a®\�C¯°t4osb2�anmc4v�c

Page 5: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

���­������ ��-���� ������|���5���e�z���2�5�������5�6����:������������4�� �5�9(+*5�����)�����5������3���4���5����� ��% �17a��9(;:)�4��

��� ����� � ��� k«i$v x8�alpc�nuwKt$osb2�anmc4¡Rl£v�i9�Rc�nmw��cMt�_8c$��t�_ace��� x8kac�w�v|v�wr�Gxac9�awsk8yInmcMy\�a� osv oRt4ornpcTy�ws�Rb2c$v|v�wskGv¶xakac9~�or� kacy xa�\nmc9kaorb9~a��c�bwA¡Rlmbw�n��ac�y cMy�y|lmosk8y\�rx�� lmnmy¶�Ec$xa�rc$kGv¶c$¡}iTt4xRv�c$�Tª\®Iorx8yIb2orkrv�� ork8y\�Gxac�t4c��8� or~anp`Mb2c�cMy|vj®\�C¯¹t$osb2�anmc4v�Eosxa�Cnmc�b2o}�a`$nmcI��� lmkGv ��x8yC�RcI²¶osnmc$�G¯O´µw�o8ª���c�� iTy|x8n£v�wAvCkac-kaiMt$cMy�y|lpv cIwsx8t4x8kac-nplmb2l£v�cIy|xa�µnmw¶v�wslpnmnmcI�RcMy�b2cMy�y�w�qscTy�� c��4x8y�orxc$k}�roA{siMy��8w���npcTyµ�8ws�|v�lmt$lp��w�kGv�yM»sc$v��Ec$��b�c$v-��� xRv�lpnmluy|cM��RcTy�v�c$��b2cMyµ�GxacMnmt$osk8�GxacTy�cMkv�wskrv-�Gxac¶t$npc$¢ y��awsk8y�npcTyµcMk8t4��{}�Rv lmosk8yy {Gb2i$v ��lm�GxacTy$ª�®\osx8yj�a��osxa�rosk8y\i$qGw�nmc$b2c$kGvj�Gxac�n � lmkGv ��x8yj��cMxRv�y|c2t4orkrv�c$kGv cM�j��� iMb�c$v|v�� c�RcTy¶b2cTy y�w�qrcMy\�ac9v�w�lmnpnmc�²"!Ig�Eosnm{Gk8osb2lmwsnpcr»R��orxa�It4ork8y|v ��xalp��cjxakac�wAv v�wr�rx8c��8w�k8y-xak)�a��o�v�oRt4osnmc$#�k8osb9~8� ce�Rc�y cMy�y|lmosk�yµ¦8¡}irª% Á �M� ¿ �'& � ��� y|cTt4xa��l£v�is»at$osb2�anmc4¡Rl£v�is»aw�v|v�wr�Gxacs»R�a��o�v�oRt4osnmcs»R�ri$��l£¦Et$wAv�lporkz»R®I�C¯(��osb2�anmc4v

Page 6: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� �

����3��� ���5�����4���

<µ�sc$k)wsy�y xab2lpkaq��Ec$� ¢¥cMt�vIt$� {}�Rv�osqs��w��8_G{r»sv�_ace�RcMy lmqsk5o�¢3�a��o�v�oRt4osnuy�¢¥os��y cMt4x8� cjc$nmcMt4v ��oskalutIv���wsk8y�wst�v�lpork8ylmy_almqs_8np{KcM� ��os� ¯�8� orkacw�k8��t4ork}�sc$kGv�lpork8w�n��Aw�nmlm�8wAv lmosk,v cTt�_akalu�GxacMye~�wsy cM�,osk lpkR¢¥or� bwsnµw���qsxab2cMkrv�y�w�k8�$=Aos�jv cTy�v�lpk8q«w���c�k8o�v9y x�>Kt4lmc$kGv¢¥or��b2c$c4v�lpk8q2v _ace��cM�Gxalm� cT��y cMt$xa� lpv�{Knmc$�rc$nOª

� k,v _8c�osv _acM�e_8w�k�� »3�rc$��l£¦�tMwAv�lpork·v o}osnuy�~8wry|cT� ork ¢¥or� bw�nCb2c$v _aoR�ay�_8wX�rc�~Ec$cMk³�rx8l£v�cKy|x8tMt4cTy y|¢¥xan�lpk³�Rluy�t4oA�scM� lmkaqk8c$¬@?EwX¬Iy�lmk5¬�cMnpnp¯Oº}kaoA¬�k5y|cTt4xa��lpv�{�a� osv oRt4ornmyMª�^-_acMy c¶b2c$v _aoR�ay-lmk8t4nmx8�Rc�y|v�w�v cjc4¡R�anmos��wAv�lpork�x8y lpk8q�b2oR�Rc$np¯°t�_acMt�º}lmkaq�wrylmkBAp�T�R»zdR��»C�T�R»���C¹» nporqslut��a��osqr��wsb�b2lmkaqDAm�E�CO»�v�c$��b � cM¬�� lpv lmkaqFAHGR»��XdICO»zv ��c$c2w�xRv�osbwAv�wDAm�M�JC�or�jt$osb9~8lpk8w�v lmosk«os¢µv�_acMy cv�cMt�_ak8lm�GxacTy$ª � v _8c$��w��a�8� oGwst�_acTy�wslpb�w�v-�8� oA�}lmkaq�v�_acet4os����cMt�v�kacMy�y�os¢�w��a��o�v oRt$osnOª�^-_8c${Kw���cj~8wsy cM��osk�wsxRv _8c$kGv lut$w�v lmosknmosqrlmtMy-os�-�a��oA�}lpkaq2y cMt4x8� lpv�{K�a��os�Ec$� v�{~}{Klpk��Rx8t�v�lpork)x8y|lmkaqlmkGv c$��wst4v lm�scj�a��oGos¢ ¯¹wry y luy�v�w�kGv�y § y c$cKA dR»z�ILJC ©4ª

!\n£v�_aosx8qs_¨v _8c«qscMkac$��w�nI�rc$��l£¦Et$wAv�lpork¨�a� or~anmc$b;lmy�xak8�RcTt4lu�aw�~anmc«cM�sc$k±lmk¨v _8c ��cMy|v ��lmt4v cM�¨tMwsy c«¬�_ac$��c)v _8c,y lp¸Mc«o�¢b2cTy y�w�qrcMy-luy�~�orxak8�RcT�MA L�CO»alpv\luy�lmkrv�c$��cMy|v lmkaq�v olmk}�scMy|v lmqrw�v ce�RcTt4lu�aw�~anmc�¢¥��w�qsb2cMkrv�y-o�¢�v _acex8k8�Rc$��nm{GlmkaqnmosqrlmtMy�w�k8��v _8c$lm�t$osb2�anmc4¡Rlpv�{sªC^-_ac�y x8t$t$cMy�y�os¢�v�_ace�a��wst4v lut$w�n��rc$��l£¦�tMwAv�lporkKv o}ornmy-lmk8�Rlut$w�v cMyv�_8wAv�v�_ac$��c�bwX{Kc4¡Rluy�v�lmkrv�c$��cMy|v lmkaq�RcMt$lm�8w�~anmc¢¥��w�qrb2c$kGv�y9v _8w�vKt$w��av xa��c�bwskG{ t$osk8t$� c$v c)y|cTt4xa��lpv�{ �8� or~anpcMby$ªON8os�2lmk8y�v�w�k8t$cs»6A LJC\y _aoA¬Iy9v _8w�v¬�_ac$k b2cMy�y wsqscMy�w���c~Eosx8k8�RcM�5w�k��K¬�_acMk�kao�kaork8t4cTy § l?ª csªCk8c$¬1�8wAv�wG©µws� cjt4��cMw�v cM��~}{2v _ac��a��o�v�o}t$osn w�k��v _8cjlmkGv ��x8�RcM�M»Gv _8c$k�v _ac�c4¡Rluy�v�c$k8t$cos¢jw y|cTt4��cMt${P?EwX¬�lmy2�RcTt4lu�aw�~8npc)w�k8�#²5<#Q\�µ^�R/ST<�¯¹t$osb2�anmc4v crª±^-_ac5t$osb2�anmc4¡Rlpv�{�¢¥os��v _ac·tMwsy c�o�¢¶¦8kalpv c·y cMy�y|lmosk8y�luyb2cMkrv�lporkacM�)wry�or��cMk)lpkUA LJCOª

!�� cMnmw�v cM���RcTt4lu�aw�~8lpnml£v�{ ��cMy xan£v�lmy�w�nuy|o·qrlp�rc$k�lpkVAp��C¹ª«^-_ac�wsxRv _8os��yeqslm�scKw·�8� oRt4cT�Rxa��c2¢¥os�9t�_8cMt�º}lpk8q·¬�_ac4v�_ac$��w�kx8k8y w�¢¥cey�v�wAv c�luy-� cTwst�_8ws~anmc\~}{v _8ce�a� osv oRt4orn?ª�^-_acMlp����cMy xanpv-_8osnu�ay¢¥os�v _8c�t$wsy cjo�¢3¦�kal£v�c�y|cTy y lmosk8y�~axRvI¬�lpv _)kao2~Eosxak8�8york v�_ac)lpkGv ��x8�RcM��b2cTy y�w�qrcMyMª�^-_8c${ �Ro kao�vw�nmnmoA¬6qsc$k8c$��w�nb�cTy y�w�qrcMy § kaosv¶��x�y�vk8wsb�cTy�©�wsy�c$k�t4��{G�av lmosk ºrc${RyMª ^-_acwsxRv _8os��y-_8wX�scjkao�vIwsk8w�nm{}¸$cM��v _8c�t4osb2�anmc4¡Rlpv�{Ko�¢�v _acMlp�\�a� oRt$cM�Rxa��csª

� x8��� cTy|xanpvIy�v�wAv�cMy-v�_8wAv�¢¥or�Iw�¦a¡RcM�5k}xab9~Ec$��os¢�lmkGv c$��nmcMwX�scT�K�a��o�v�oRt4osn ��xak8y-~8xRv\¬�lpv _)kao~Eosxak8�8y�osk�v _acelmkGv ��x8�RcM�b2cTy y�w�qrcMyµv _acec$¡}luy|v c$k�t4c�o�¢�w�k)wAv|v�wst�ºluy�®I�C¯°t4orb2�anpc$v crª � cew�nmnmoA¬[�axa~anmlut�ºsc${cMk8t4��{}�Rv lmosk)wsy¬c$nmnzwsyv�_ac���oGy y lp~8lpnml£v�{os¢9y|{}b2b2c4v�� lut«cMk8t4��{}�Rv lmosk±¬�lpv _Ww�k}{#b2cTy y�w�qrcsªWR°k[v _aluy���w��Ec$�5¬�c,oskanm{±t4osk�y|lu�Rc$�F������������·�a� or��cM�|v�lpcTy$ªYXIoA¬c$�rc$�Z �������,�$�[�2� Z �[�2����t$wsk·~�c�_8w�k��RnpcT�)lpk,wKy|lmb2lpnuw��\¬wX{rª�X\c$k8t$cs»E_ac$��c9w�8� osv oRt4ornzlmy¶t4ork8y|lu�RcM� cT�)lpk8y cMt$xa��cel£¢�lpv¶luyI�Eory�y|lm~anmcv�o� cTwst�_5w�y|v�w�v c�¬�_acM� cewy cMt4��c4vv cM� b.qsc$v�y-�Eory�y|cTy y cM��~G{�v _acelmkGv ��x8�RcM�Mª\ � ¾EÁ ��� Á�] �^ ½P_ � _µ½ �}��� c�¦8��y|v�lpkGv ��oR�Rx8t$c9lmkU`}cTt�v�lpork ��osxa�jb2o}�ac$n�o�¢µ�8� osv oRt4ornmyjw�k8� lpkGv�� x8�ac$�ew�k8�«v _ac�kaosv lmoskos¢a������) Z � Z ��� Z ��b�ª-^-_ac$k·lmkF`}cMt4v lmosk«d2¬c�y�v�x8�R{��a� or��cM�|v�lpcTy�o�¢��RcM� lm�AwAv�lpork8y-¬�l£v�_·lpkGv�� x8�ac$�I��xanmcMyMª^-_aluyIwsnpnmoA¬Iy-x8y�v�o�ac$��lp�rc\�Eosnm{Gk8osb2lmwsn�~�orxak8�ay�¢¥os�-kaor� bwsnEw�v|v�wrt�ºRy�wsk8�Kv�_ac$k�v o2y|_8oA¬±v _8w�v�v�_ac��a��os~anmc$b6os¢z¦8k8�Rlmkaq2w9k8os��b2wsnEw�v|v�wst�ºluylmk ®\��ª � c�y _aoA¬�lmkP`}cMt4v lmosk �v _8w�v¶v�_ac9c$¡}luy|v c$k�t4c9os¢�w�k w�v|v�wrt�º�luyj®I�C¯¹_8w���� ª6R°k !I�a�Ec$k��Rl£¡«¬c9y _aoA¬Wv _8w�v¶v�_ac�®\��8� oRt4cT�Rxa��c�os¢8`}cTt�v lmosk�d5t$w�k,~�c2c$¡Gv�c$k8�acM�·v�o)_8w�k8�anpc�¬�cTw�ºrc$�\lmkGv ��x8�RcM�jb2oR�RcMn § !I�8��cMk8�Rlp¡,�8ª ds©¶w�k�� w�nuy o��a��o�v�o}t$osnuy¬�lpv _·t�_aorlmt$c���orlpkGv�y § !I�a�Ec$k��Rl£¡)�8ª �r©4ª

c d 5� �������������fe � �5���

� c�t$osk8y lm�ac$��w)b2oR�Rc$n�o�¢-�a� osv oRt4ornmy�lmk v _ac�y�v�{}nmco�¢(A ��COª)^-_acKwst4v lmosk8y�o�¢�w�k}{«_aorkacMy|v9�a��lpk�t4lm�8w�n�ws� cKy ��cTt4lp¦8cM��wry�w��w�� v luw�nmnp{�os���RcM� cT�enpluy|v�v�_8wAvCwsy�y o}t$lmw�v cTy v o § v�_ac¢¥os��bwAv�o�¢E©�wI��cMt$c$lm�scT�eb2cMy�y wsqscµlpv�yCt$os��� cTy|�Eosk��Rlpk8qI� cM�anm{sª�^-_8cwrt�v lm�}l£v�{os¢�v _8c�lmkGv ��x8�RcM�jluyjb2o}�ac$nmcM�«~}{·��c$¬���l£v�c9��xanmcMyjosk y|c$v�yjo�¢µb2cTy y�w�qrcMyMª � ct4ork8y|lu�RcM�\v�_8wAv�v _8c�lmkalpv luw�nmlp¸TwAv lmosk,�a_8wry|c�o�¢�almy|v ��lp~8xRv lmkaq�ºsc${Ryw�k8�Ko�v _8c$�-lpka¢¥os��b2w�v lmoskK~Ec4v�¬c$cMk��a��lmk8t4lm�8w�nuy�lmy�lpb2�anmlut4lpvMª�^-_ac�w��a�8� oGwst�_luy�Gxalpv cjk8w�v xa��w�n�w�k8��l£v-luyy lmb��8npcev�o�t4orb2�alpnmc9w¬�lu�Rc���w�k8qsceo�¢C�a��o�v�oRt4osnz�acMy�t4��lp�Rv�lpork8y�v�oKosx8��¢¥os��bw�nmlmy b)ª8N8os�\lmk8y|v�wsk8t4c�c$¡Rlmy|v lmkaq2v o}ornmy¶y|x8t�_·wry� !\�g`�hiA Ea»�jIC3os� � !'`R��k5hiAm�TdICz¬osxanu����cM�|¢¥or� b6v�_aluy-v ��w�k8y nuwAv lmosk5¬�lpv _5¢¥c$¬1b�oR�Rlp¦�tMwAv lmosk�y$ª� ce�a��cMy c$kGv-osxa��b2oR�Rc$n3b�or� cj¢¥or� bw�nmnm{kaoA¬eª

l � � ½ ���3m� % ½ �T�M�3n ½ �^-_8c�b�cTy y�w�qrcMy�c4¡at�_8wskaqscT���axa� lmkaqjv _8c��a� osv oRt4ornRc4¡RcMt$xRv lmosk2w���c-~axalmn£v�x8y|lmkaq��8wslp��lpk8qKo9p(q�p5r�wsk8��c$k8t$� {}�Rv�lpork�or��cM��w�v os��ys p'tJuv�» s p't�w v�ª � cews�8��w2y xa��cM��y�t4��lm�Rv-v o�Rluy�v�lpkaqrxaluy|_5~�c$v�¬�cMc$k5�axa~anmlut�ºsc${ § wr©-w�k8�5y|{}b2b2c4v�� lutjºsc${ § u4©c$k�t4��{G�av lmosk8yMª^-_8c9y|c$v\o�¢C~8wry|luteb2cMy�y�w�qscTy-lmyI¦8kalpv c�w�k8�·�RcMkao�v�cM�)~}{yx5z/{J|~}rªaR¹v¶t$oskGv�wslpk�yIk8wsb�cTy-¢¥os�¶�a��lpk8t$lp��w�nuyIw�k8�·w�v orb�luteºrc${Ry¢¥��osb v�_acjy c4v��F�I�3}rª�`}lmk8t$c\¬c\_8wX�rc\we¦8k8l£v�c¶k}xab�~�cM��os¢3y|cTy y lpork8y�¬c\wsnmy o�wry y xab2c\w�k}{�kaosk�t4c\luyw�~8wry|lutIb2cMy�y�w�qsc��3¬ct$osk8y lu�Rc$�-v�_8wAvIlpvI_8wsy-~Ec$cMk·t4��cMwAv�cM��~�c$¢¥os��cjv _ac�y cMy�y|lmosk)w�k���~Ec$nmoskaqGyv o�v _8c�lpk8l£v�lmwsn ºGk8oA¬�npcT�Rqsc�o�¢�v _ace�8� lmk8t4lm�8wsn�v _8w�vqrc$kacM��w�v cTy�lpvMª

!\k}{9b2cMy�y wsqsc�t$wsk2~�cIx�y|cT��wry�w�ºscM{�¢¥or�µy|{}b2b2c4v�� lut-c$k8t$� {}�Rv�lporkzª � kanp{�c$nmc$b2c$kGv�y�¢¥� orbY�P�I��} ws� c�x8y cM��¢¥or���axa~anmlutºrc${2c$k�t4��{G�av lmoskzª�gjlm�sc$k5w��ax8~anplut¶ºsc${ § � cTy|�zª��a��lm�Xw�v cjºscM{R©g��»��������RcMkao�v�cMy�v _ac�wsy�y o}t$lmw�v cT��a��lp�AwAv�c\ºrc${ § ��cMy �zª��axa~anmlutºrc${a©w�k8�5lpvIlmy�wsk�cMnpcMb2c$kGvIo�¢��F�I�3}rª�gjlm�scMk)w2y {}b�b2c$v ��lmtjºscM{��Kv _acMkz»$�����¶¬�lpnmn3�RcMkao�v�c¶v�_ac�y�w�b2c�ºsc${rª^-_8cjb2cTy y�w�qrcMy-w���cjv _acMk5qrc$kacM��w�v cT�K~}{v _8c�¢¥osnmnpoA¬�lmkaq § v�� cMcT©qs��w�b2bw��I�

|T}��������Vx�z/{J|T}���o[|~}��1q9|T}���r�� s |T}��$t w ����� u � s |T}��$t u� u%��������3�J�9��

Page 7: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

� �6� ����1���������������3����� Z �$�

Naor��t$osk8t$lmy lpork±¬�c �Rc$k8o�v c ~G{i| � q�|�� q������q�|� v�_ac,y c4v)o�¢eb2cMy�y wsqscTy s | � q�|�� q������q�|��tGª gjlm�sc$k[v�¬�o y c4v�y�os¢b2cMy�y wsqscTy�� wsk8���� �¬c��ac$kaosv c2~}{�� q��� �v _ac2xak8lpork o�¢�v _acMlp��cMnpcMb�cMkGv�y�w�k8�,qslm�scMk w5y|c$v�os¢µb2cMy�y wsqscTy�� wsk8� wb2cMy�y wsqsc'z�»a¬c��ac$kaosv c�~}{�� q9zµv _ac�y c4v���� s ztGª� � Á � Á � Á &�� _µ½ �������s�R��� Á m� c�y _8w�nmn��RcTy t$� lm~Ec9�a��o�v�o}t$osnuy�~}{5w�npluy|v¶o�¢µwrt�v�lpork8y-¢¥os�jcMwrt�_·�a��lpk8t$lp��w�nOªaR°k«os���Rc$�\v o��acMy�t4��lp~Ec�v�_ac9�a��o�v�oRt4osn�y�v�c$�8y¶¬�clmkrv�� oR�Rx�t4ceb2cMy�y wsqsc�v�c$��b2y § os�Iv c$��byIlmk«y|_8os� v�©�ª � c9wsy�y|x8b�c�v�_8wAv¶¬�ce_8wX�rc�w�¦8kalpv c9y c4v¶o�¢C�Aw���lmws~anpc����� }ª�^-_acMk)v _acy c4vIo�¢�v cM� by-luy�qscMkac$��wAv�cM�K~}{v�_ac�¢¥osnmnmoA¬�lpkaq�v�� cMcjqr��wsb2b2ws��

z/�� J| �;�H�!���" ���x�z/{J|T} ��o2z/�� I|Mq9z/�� J|Tr�� s z/�� J|Pt w ����� u � s z/� J|Pt u# �%$ �hzc$v����" '& z)(-~Ec�v _8c9y|c$v�o�¢��Xws� luw�~8npcTyv _8w�vIo}tMt4xa��lmk·w�v cM� b z�ª !Y���*,���[���������2���wsy�y|lmqsk�yv c$��byv o�Aw���luw�~anmcMyMª�!+*��������-,����*��������[�����2����wsy�y|lmqsk8y\b2cMy�y wsqscMy�v o��Aws� luw�~anmcMyMª\^-_ac�w��a�8nplut$w�v lmosk·o�¢µw�y|x8~8y�v�l£v�xRv lmosk/. v o5wv cM� b z�luy\¬���lpv|v cMkDz0.�ª � cw�nuy o�¬���l£v�c21 3547698Cv�_acKy|x8~8y�v�l£v�xRv lmosk:.#�Rc4¦8k8cM� ~}{�.;&�3<( �=6 wsk8�/.;& �>((� �·¢¥os�(�@?�A3�ª2^-_8c2y c4v�os¢y xa~Rv�c$��b2y�os¢z�lmy��Rc$k8o�v cT��~}{CBD6FEG&2z)(�ª«^-_acTy|ckaosv�w�v lmosk8y�w���cc4¡}v cMk8�RcM� v�o y c4v�y9o�¢-v cM� byIH lpk w y�v�w�k8�aws��� ¬-wX{sª~Naor�9lmk8y�v�w�k8t$cs»H�.D� s z0.B��zKJLHKtrª

!6�a��lmk8t4lm�8w�n § c$¡Rt$c$�Rv9v _ac�lpkalpv luwAv�os��©���c$�anmlpcTy�wA¢ v�c$��� cTt4c$lm�}lpk8q«w·b2cMy�y wsqscbwAv�t�_alpk8q w·y|�EcMt$l£¦�cM��v�c$��b wry y oRt4luwAv cT�v o)lpv�yet$xa� ��c$kGvey�v�wAv�csª2^-_ac$k,¢¥� orb v _8c��8� cM�Glmosx�y|nm{·� cTt4cMlp�rcM�·b2cMy�y�w�qscTy § w�k�� lmkalpv luw�nCº}kaoA¬�nmcM�RqrcT©\_ac2~axalmnm�ayjv�_ac2kac4¡}vb2cMy�y wsqsc2_acK¬�lmnmn�y c$k8� ª)^-_aluy�t4ork8y|v ��x8t�v�lpork�wsb2osxakGv�yev o«y xa~8y|v lpv xRv�c��Aw�nmxacMye¢¥os�ev�_acK�Aw���lmws~anmcMy�o�¢Iwskao�v�_ac$��y ��cTt4lp¦8cM�v cM� b)ª

! �a� osv oRt4orn�lmyjqslm�scMk ¬�lpv _�wK¦�kal£v�cy|c$vjos¢��a��lpk�t4lm�8w�n�k8wsb�cTy�MON x5z/{J|~}r»3wsk8� w��8w�� v luw�nmnm{·os���Rc$��cM�·nmluy�v�o�¢-y�v�c$�8y¢¥os�cMwrt�_�a��lpk8t$lp��w�nEk8w�b2csªC^-_aluy��8w�� v luw�n�os���Rc$��lmyµv o�c$k8y xa��cIv _�wAvµv�_ac�wst�v�lpork8y�os¢3cMwrt�_�a��lpk8t$lp��w�n�w���cI��cM�|¢¥or� b2cT�lpkKv _ac��lpqr_rv�os���RcM�Mª�S«os��c9¢¥or� bw�nmnm{·¬�c2wsy�y|oRt$lmw�v c�v�o)cMwst�_,�a��lpk8t$lp��w�n�x w���w�� v luw�nmnp{·or���RcM� cT�)¦8kalpv cy c4vP&RQ�S�q�T�UWVX(�ª�<µwrt�_�a��o�v�o}t$osn�y�v�c$�«luy¶y �EcMt4lp¦8cT�«~}{·w�t4orxa�anmc9os¢Cv c$��by¶�ac$kaosv cM��Y=Z[B±wsk8�·luy\lmkGv c$k��RcM�)v�o���c$�8� cTy|cMkrv\y osb2c�b�cTy y�w�qrc�Yc4¡R�EcMt4v cM�5~}{�w��a��lpk�t4lm�8w�n�x�wsk8�5_almy-��c$�8np{\B v�o�v�_aluy�b2cMy�y wsqscsª�XIcMk8t4c�w��a��o�v�oRt4osnzy �EcMt4lp¦�tMwAv lmoskL] luy�qrlp�rc$k�~}{1�

s &_^9q`YbacZdBXaR((�e^;Jgf�t¬�_acM� chf � s &2x-q`i0(+� xjJkM w�k8�\i�JlQ�Sgtrª � c¬���l£v�cF� f6��¢¥os��v�_acKy lp¸Mco�¢;f�ª\m�nFi%z¶w�k��/H�nXo«w���c�¦a¡RcM� b2cMy�y wsqscMyx8y cM�Kv o2lpk8l£v�lmw�v cew�k8��t4nmory cjw��a��o�v�o}t$osn�y|cTy y lmoskzª !Ik)½ m<pX�¥� Á m � ½ m�� ¢¥os��w9�8� osv oRt4orn�luy�w�y c4v-o�¢�b2cMy�y wsqscMyMª ! � Á �X� ½ ���½"qz½ � ���� Á m Á �X ½ ��r lmy\w�orkac4¯Ov os¯Oorkac�bw��a�almkaq r �f5s s"t q�u�;qI� f5��tey x8t�_�v _8w�vI¢¥os�Iwsnpn�xvJ\M6w�k��wixT�UyVgz¬ce_8wX�rcr &2x�q)i0({T r &2x-q|z}(�ª R°kKosv _acM��¬os���ay r �Rc4¦8k8cMyw�kKc$¡}cTt4xRv�lporkKos���RcM��¢¥os��v _8c¶�a��o�v�oRt4osnEy�v�c$�8yMªC^-_aluyµor���RcM�µluy�t$osb2�8w�v lm~anpc¬�lpv _v�_ac¶��w�� v luw�naor���RcM�µos¢zcMwst�_�8� lmk8t4lm�8wsn?ª !@_ � Á � Á � Á & ½"qz½ �'�3��� Á m luy�qslm�sc$k~}{2w�qr� orxak8�2y|x8~8y�v�l£v�xRv lmosk�.�»Rw�t4or� ��cMt4vc4¡RcTt4xRv�lpork or���ac$� r w�k8��w)y|cT�Gxac$k8t$co�¢-c$k}�}lp��osk8b�cMkGv�ybH�~ q�����q�H�� �<� �rc$��l£¢¥{}lmkaq���m�nFi%z�J�H�~s»DHhnXoCJ�H�� �<�p»�wsk8� ¢¥os�9wsnpnth� � � � f6� »9Y��}�"�������0.�J2H � � � w�k8�LBF�}�"�������0.�J2H � ª<µwrt�_ y�v�c$�@^¶os¢Iv _8c��a��o�v�o}t$osn�c4¡}v cMk8�ay�v _ac)t4x8� ��c$kGv9cMkG�}lm� orkab2c$kGv9~}{ ws�8�Rlpk8q·v _ac)t4or� ��cMy �Eosk8�Rlmkaq«b2cTy y�w�qrcwBXa_.¬�_acMkLYbaR.,luy-�a��cMy c$kGvMª

��q � � _ & ½ � l·½}½ Â�^�� ���3�J^3� Áz½  ½ � _ � Á � Á � Á &R°kv�_ac\kaosv�w�v lmosko�¢fA LJC¹»rv�_ac¶®\c$cT�R_8w�b `Rt�_a��o}cM�RcM���a��o�v�o}t$osn�� xanmcMy�w���cIwsy�¢¥osnmnpoA¬IyM»s¬�_acMkK¬c\wsy�y|x8b�cIv _8w�v�c$�rc$��{9kaork8t4cluy-lpk8t$npx��RcM�5lmk�v�_acelmkal£v�lmwsnzº}kaoA¬�nmcM�Rqrc¶os¢�v�_ace�a��lpk8t$lp��w�n�v _�wAvI¬�lmnpnzt$� cTwAv c�lpv�

!9� � xy~}&[��SD( s x � &[��S�q�����q���S;(�� �I� � & s oR��S8q���S�rt ��� (!jd�� x � &[��Sgq���6q)3 � (�� �I� � & s o�3 � q�� � r,t � V ( s x � &[��S�q�����q`3 � q9� � (�� �I�"��& s � � t � � (�j����� ~ &[� S (�� � � � & s oR3e� q�� S rt ��� ( s�� � &[� S q�� � q`3e��q`� � (�� � � �"& s oR3e� q`� � rt � V (��d3��� � &[��S�q�����q)3-� q����(�� �I�F��& s ����t � � ( s�� � &[��S�q�����q`3e��q9���(

� cjtMw�kKor~Rv�wslpk�v _acj�a��o�v oRt$osn�y|v cM�8y�¢¥os�osxa�-y c4v|v�lpk8q9¢¥��osb v _8c¶��xanmcMyo�¢ A L�C ~}{y|lmb2�anp{x8kal£¢¥{}lmkaq2w�nmn�v�c$��byµ¬�lpv _�v _acy�w�b2c9��o}o�v¶y {}b9~Eosn�xW� »�¢¥or�jwsnpn��a��lpk�t4lm�8w�nuy5x9ª X\c$��cs»�v _aluyjb2cMw�k�yIv�_8wAv x � &[��Sgq���6q`�IS�(\xakalp¦8cTy¶¬�lpv _Px � &[��Sgq���6q)3 � (w�k��\� � &���S8q���5q)3 � q`�I�K(xak8l£¦8cTy-¬�l£v�_�� � &[��S8q����5q`3e��q�� � (4ª Naor�I®Ic$cT�R_8wsb�`Rt�_a��o}cM�RcM�-�a� osv oRt4orn?»}¬ceos~Rv�w�lmk��

� § § !�»m�T©4»�m�nFi%z Z s o��IS8q���S�r,t � � ©§ § !�» ds©4» s o�� S q�� � r�t � V Z s � � t � � ©§ § �j»p�X©�» s o�3 � q��S�r,t � � Z s o�3 � q����8rt � V ©§ § �j» dr©�» s � � t � � Z HhnXo ©  

¡�¢ �X¡¤£

Page 8: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� G

^-_8c�os���Rc$��lmkaqry�ork y|v c$��y\w���cev�_ac9orkacMy\v _8w�vjws� c�c$¡R��cTt�v cT� �yQ S �vQ � � s�t q�� t�¬�lpv _ t T U V��a» t T U � �aª6hzc$vx�y�c$b2�a_8wry|lm¸$c�v _8w�vexakanmlpºrc A LJC�lmk osxa�ey ��cTt4lp¦�t$w�v lmosk ¬c2�Ro)kao�vet4ork8y|lu�RcM�¶v�_8wAv�v�_ac2�a� osv oRt4orn�y �EcMt4lp¦�tMwAv lmosk wsy�w)y|c$vos¢µ��xanpcTy\¬�_8c$��c § ¢¥��c$cX©I�Aws� luw�~anmcMyjt$wsk«~Ec���c$�anuwst$cM�·~}{«wskG{5v�c$��b2yMª R°k orxa�jtMwsy c�v�_ac�y�t4or��c�o�¢µ�Aws� luw�~anmcMy\bwX{5lmk8t$npx8�acy c$�rc$��w�n�nmlpkacTy-lpk5v�_ac�y|�EcMt$l£¦Et$wAv�lporkzª#`}cMc¶¢¥or��c4¡aw�b2�anmcjv _ace�8� osv oRt4orn � v�¬-wX{G¯¹�Ic$cMyqrlp�rc$k�lmk !I�a�Ec$k��Rl£¡)�8ª �8ª������� ����R°k A L�CEkaosv�w�v lmoskKy osb2c�v cM� by8xynFn�� &[������ (�ws� cIx�y|cT�2¢¥os���axa~anmlmtJ=X�a��lm�Xw�v c¶ºsc${RyMª��xRv¬�c¶�Ro�kaosv�kacMcM�2v�_ac$b_8c$��c �Cy|lmk8t$c�v _ace�Eory�y|cTy y lmosk�os¢�w2ºrc${os�Iw�kaork8t4c�~}{�w��a��lmk8t4lm�8w�n luy-lmb��8nplut4lpvIv _8� orxaqs_�v _acewrt�v lmosk�y-_ac�t$wsk��Ec$� ¢¥os��b)ª� m��$� �� ½ �R°k�v�_ac�²¶osnmc$�,´µw�o·b2oR�Rc$n6A ��Cµv _8cKlpkGv�� x8�ac$��_8wry�v _8c�w�~almnml£v�{,v o·cTwX�scMy��R��os�z»zv o �Rlm�scM�|v9wsk8� b2c$b2os��lp¸Mcb2cMy�y wsqscMyM»zv�ot$osb2�Eory cKw�k8���RcTt4osb2�Eory cs»3v o·c$k�t4��{G�av9w�k8���RcTt4��{G�av�¬�_ac$k³_ac_�wsy�v�_acºscM{s»�v o·qscMkac$��wAv�c2kac$¬ b2cMy�y wsqscMyewsk8��y|cMk8�v�_ac$b v�oKo�v�_ac$�¶�8w�� v lut4lm�8w�kGv�y�¬�l£v�_«w2¢ w�nuy c�lm�ac$kGv lpv�{sª � c9wsy�y xab2c�_ac$��c�¬�lpv _aorxRv\nmory�yIos¢�qrc$kacM��wsnplpv�{v�_8wAv¶v _ac�lmkGv ��x8�RcM�y {Ry�v�c$bwAv�lmtMw�nmnp{��Rlp�rc$� v�y-b2cMy�y�w�qscTy$»R�Eory�y lp~anm{Kb2oR�Rlp¦8cMy-v�_ac$b w�k��K¢¥or� ¬-w����ay�v _acMb¤v�o2v _ace��cMt$c$lm�scM�x8k8�Rc$�Iv _ac�lu�RcMkrv�l£v�{os¢�v _8c�o >t$lmwsn�y c$k��Rc$�Tª R°k o�v�_ac$�j¬os���ay\wsnpnCt4orb2b9xakalut$w�v lmosk8y¶w���c9b2cM�almw�v cM� ~}{)w�_aoGy�v�lpnmc9cMkG�}lm� orkab2c$kGv¶��c$�8� cTy|cMkrv�cM�~}{,v _ac�lpkGv ��x8�RcM�Mª5^-_acKlmkGv ��x8�Rc$��wst�v�lpork8y�¢¥or��b2o}�al£¢¥{}lmkaq·v _8cKb2cMy�y wsqscMyews� cKy lmb9xanuwAv�cM� ~}{,��c$¬���l£v�c� x8npcTyeosk³y c4v�y9o�¢b2cTy y�w�qrcMyMªµ^-_ace��c$¬���l£v�ce� cMnmw�v lmosk5lmyI�ac4¦8kacT�)~}{�� s �� zl£¢�v _8c$��cec4¡Rlmy|v�yIoskaceos¢�v�_ac���xanpc���s� 2lpk5v�_ac9w�����wX{~�cMnpoA¬y x8t�_5v _�wAv���luyIwy|x8~8y|c$vIo�¢ ��wsk8�2�� luy�os~Rv�w�lmkacM��~}{���c$�anuwst$lpk8q���~}{\ �lmk�� ª � c9¬�� lpv cIs��¶¢¥os�Iv _ace��c,?8c$¡Rlp�rc�wsk8�v���wsk8y l£v�lp�rc�t$npoGy|x8� cjo�¢ s±ª

^-_8cjy c4v-o�¢3b2cTy y�w�qrcMyKBF~j��c$�8� cTy|cMkrv�yCv�_ac�lpkalpv luw�n�º}kaoA¬�nmcM�Rqrc\os¢zv _acjlmkrv�� x��Rc$�Tª � c�wsy�y|x8b�c¶v _�wAv�wAv-npcTwsy|vµv _8cjk�w�b2cos¢�v�_acelmkrv�� x��Rc$�����e�" ��2�K~Ec$nmoskaqGyv o�v _aluyIy c4vTª

R°kGv�� x8�ac$����xanpcTyIw���ce�Rlm�Glu�RcT��lmk·y|cM�scM��wsn qs��osxa�8y��� x8npcTy-¢¥os�It$osb2��oGy|lmkaqor�I�RcMt$osb2��oGy|lmkaq2b2cMy�y wsqscTy$ª�^-_acTy|ce��c$¬���lpv c��xanmcMy�ws� c-v�_acIorkanp{�oskac�¬cIt4ork8y lm�RcM��lmk2v _aluy��8ws��cM�µw�k���wsk}{9b2c$kGv lmoskos¢��|��xanpcTy�����c4¢¥c$��v o+� �3�,�,��� x8npcTy$ª R°k2v _acI¢¥osnmnpoA¬�lmkaq�E» E2wsk8� ���c$�a��cMy c$kGv9w�k}{ b2cMy�y wsqscKw�k�� � ��c$�a��cMy c$kGv�y9w�k}{,cMnpcMb2c$kGv�os¢a�P�I�EªDN8os��lpk8y|v�wsk8t4cr»�v _ac�� xanmc�¬�lpv _ nmws~�cMn!�"�&�oR��q�E,r)(��c$�anuwst$cMy-w2y|c$vIo�¢�b�cTy y�w�qrcMy{��q�E\~}{v _ac�¢¥ornpnmoA¬�lmkaqy|c$v-os¢�b2cTy y�w�qrcMy{��q�EJqIo���q�E,r4ª

²¶cMt$osb2�Eory l£v�lpork���xanmcMy ��osb2�Eory l£v�lpork���xanmcMy!�#�&�o���q�E,r`('� oR��q�E,r s���q�EJqo���q�E,r ! " &9o_��q�E,r`('� ��q�E�s���q�E�qo���q�E,r! # & s �$t w � ('� s �$t w � q��f���{s s �$t w � q�� � ��q`� !$"�& s �$t w � ('� ��q�� s���q��Dq s �$t w&%�!$#�& s ��t u' ('� s �$t u' q�E�s s ��t u' q�EJq�� ! " & s �$t u' ('� ��q�E�s���q�E�q s �$t u'

! u & s s �$t u' t u' ('� s�s �$t u' t u' s���q s s �$t u' t u' q ! $ & s�s �$t u' t u' ('� � s �$q s s ��t u' t u'! u & s s �$t w � t w � �"� ('� s�s �$t w � t w � �"� s���q s�s �$t w � t w � �"� ! $ & s�s �$t w � t w � �"� ('� � s �$q s s ��t w � t w � �"�� c)�Rc$kaosv c�v _ac·ws�a�anmlmtMwAv lmosk#o�¢¶w,��xanpcLY v o�w y|c$vgH'o�¢\b2cTy y�w�qrcMy�¬�l£v�_ ��cMy xan£v H� -~}{5Hds � H� ?ª � c)¬���l£v�c

! " � s ! " &��'(�G¢¥os�Iwsnpn b2cMy�y�w�qscTy{��tG»rwsk8��!$#s»(! $ »(! u lmk�v�_ac\y�w�b2c-¬wX{r»sw�k�� �9lmyµt$w�nmnmcM�9v _aca_ �G�2m���� _ � &�� ½ � � o�¢Ew���xanmc! " &R�'(�»)!$#�&��'(�»)! u &R�}(4»sor�*! $ &R�}(4ª � cItMw�nmn  ½ �G��pa�a�� Á m wey cM�Gxac$k�t4c-o�¢�� x8npcIws�a�anmlmtMwAv lmosk�y�H ~ s � � H � s � + ���"s � , Hyzª^-_8c���xanmcMybY � ¢¥or��i�� t q����;q)n w���c�tMw�nmnpcT�·v�_ac���xanmcMyjo�¢µv _aluy��Rc$��lp�Aw�v lmosk�-«ª � c�¬���l£v�c Y�J - § ws~ax8y lp�rc$nm{a©�v�o5�RcMkao�v�cv�_8wAvWY lmy�orkac�o�¢�v _ace��xanmcMy�Y � »R¢¥or��i � t q����;q)nC»}v _�wAvI_8wry-~�cMc$k)x8y cM��lpk5v _8c��Rc$��lp�Aw�v lmosk.-·ª� c�wry y xab2cev�_8wAvjkao\BF�t$wsk·~�c2y|lmb2�anml£¦8cT�·~}{/! u ª5R°k«osv _acM�\¬os���ay-v�_ac�b2cTy y�w�qrcMyIy ��cTt4lp¦8cM�«~G{�v _8c9�a��o�v�o}t$osn��Ro

k8o�v¶ws�a�anm{�x8y c$nmcMy�y�c$k�t4��{G�av lmosk8yMª � v�_ac$��¬�lmy c�¬�cek8c$cM�5v�oKt4ork8y|lu�RcM�\w�kaosv _acM���a��cMy c$kGv�w�v lmosk)o�¢�v _ac9�a� osv oRt4orn ¬�_ac$��c�v _8cb2cTy y�w�qrcMy-w���cey|lmb2�anplp¦8cT� ª�^-_ac�lpb2�Eos� v�wsk8t4c�o�¢���xanpcTy0! $ luy�y|_aoA¬�k5~}{�w�k)c$¡Rwsb2�anpc�lmk !I�a�Ec$k8�al£¡5�8ªm�

���M�M�a�21z�^-_8c$��c·lmyKwsk±wAv|v�wst�º lmkl�h�a� osv oRt4ornIy|cTy y lmosk8y2lp¢�v _ac·lmkGv ��x8�Rc$��t$w�k os~av�w�lmk v _ac,y|cTt4��c4v2v cM� b­lpk±lpv�yº}kaoA¬�nmcM�aqsc·y|c$vw�¢ v cM��t$osb2�anmc4v�lpkaq)wAveb2ory|vh� y cMy�y|lmosk8yMª � ct4osk�y|lu�Rc$�¶¦8��y�vjv _8c2tMwsy c�os¢�w5y|lmkaqsnmc2y|cTy y lporkzª9^-_acMk,¬c2y|_8wsnpnCy ºsc$v�t�_«lmk`Rxa~8y cMt�v�lpork«dRª ��_aoA¬Uv o��cM�Rx�t4cjv _ac�tMwsy c¶os¢�y c$�rc$��w�n y cMy�y|lmosk8y�v�o�v�_acexakalu�Gxac�y cMy�y|lmosk5t$wsy csª

`Rlpk8t$c� cTt4cMlp�rcM�«b2cTy y�w�qrcMyew���c�¦8n£v�c$��cM� ~}{ �a��lmk8t4lm�8w�nuy�¬�lpv _ v _ac�npc$¢ v|¯¹_8w�k8��y lm�aco�¢��8� osv oRt4ornC� xanmcMyM»zv�_acc4¡Rluy�v�c$k8t$cos¢�w�k,wAv|v�wst�º·t$wsk«~Ec�c$¡}�8� cTy y cM�«wry¶w�t4ork8y�v���wslpkGv¶y osnm�}lpk8q��a��os~anmc$b~��luy¶v _acM� c�w�¬-wX{�¢¥or�\v�_ac�lpkGv ��x8�RcM�\v�o�~8xalpnu�·¢¥��osblpv�y\lpk8l£v�lmwsn�º}kaoA¬�nmcM�Rqrc�w�k8�·wsnp��cMwr�R{�y c$kGv\b2cMy�y�w�qscTyIwkacM¬�b�cTy y�w�qrc § �Rc$¦8kacT�)~G{)wKy xa~8y|v lpv xRv�lpork·¢¥os�Iv _ac��Xws� luw�~8npcTy�o�¢�8� osv oRt4orn���xanmcMy�©�v _�wAv�¬�lpnmn�~�c�wstMt4cM�Rv cT�K~}{2v _acj��cMt4lm�almc$kGvM»Rwsk8��y o9orkz»}xakGv lmn�v�_ac�c$k8��o�¢zv�_ac�y|cTy y lporkz»}w�k8��y|x8t�_�v _8w�v-w�vv�_acec$k���v�_ac�y cMt4��c4v-v�c$��b�lmyIºGk8oA¬�k�~}{Kv�_acelpkGv�� x8�ac$�Tª� c9lmkGv ��oR�Rx8t4c�kaoA¬ wK�8� cT�RlmtMwAv�c�3({ I� ��¢¥or�jt�_acTt�º}lpkaq�¬�_ac4v�_ac$��wKb2cMy�y�w�qsc�tMw�k«~Ec2t4osk�y�v�� x8t4v cT�·~}{�v�_ac9lmkGv ��x8�RcM�

¢¥��osb y osb2c�º}kaoA¬�k)b2cMy�y wsqscMyMª-^-_almy\�a��cM�Rlut$w�v c9tMw�k)~�c��}lpcM¬�cT�·wsy-v�_ac�t$osb2��oGy|lpv lmosk)os¢��a� cT�Rlut$wAv�cMy5}�}n z4�·wsk8�2�"nX�5�76¢¥��osb h�ªa�Cw�x8nmy oskMAp�ILJC¹ª85½ � m��¥��� Á m:9 ;4< Á ��n ½�=?> ��� H *� Z ���,�5�9.��%����)�� Z �<, ���,�#zK*� Z ������) �����P� � Z �5�����,���+�4�bH� ��������LH s��hH� Z �<,zxJ2H� A@g�1���,�P�#�(� Z �+� � Z ��z(�4�-¢¥or� qrcM� .����) H Z �-,+�#� , �������%�����a*,�(zxJB3({ ���}&�H�(C@

�������3�J�9��

Page 9: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

E �6� ����1���������������3����� Z �$�

� c�wry y xab2c9v�_8wAv�v�_ac$��c�luyew5y|�EcMt$lmwsn�b2cTy y�w�qrc B����� ��Iz\lmk v�_ac��a� osv oRt4ornCy|�EcMt$l£¦�tMwAv�lporkzª�hzc$v��·~Ec�v�_act$ws���Rlmk8wsn�os¢f�» l?ª csª)v�_acv�o�v�wsnµk}xab�~�cM�9o�¢Iy|v cM�8y�os¢v�_acK�a��o�v�oRt4osnOª !Ik w�v|v�wrt�º luy9w)�a��o�v�oRt4osnµc4¡RcMt$xRv lmosk³¬�_acM� c2v�_ac�lmkGv ��x8�RcM�9t$wskb2oR�Rl£¢¥{�cMwst�_�lpkGv�c$��b�cT�RluwAv cjc$k}�}lm� orkab2c$kGvwsk8��¬�_8c$��c\v�_ac�b�cTy y�w�qrc�B����� ��Iz�~Ec$nmoskaqGy�v o�v _acj¦8k8wsn c$k}�}lp��osk8b�cMkGvMª R°k5wskwAv v�wrt�º«v _8cKlpkGv�� x8�ac$��luy�w�~8npcv�o)¢¥os��qscKwskG{,b2cMy�y wsqsc2c4¡R�EcMt�v�cM��~}{,w«�a� lmk8t$lp�8wsn�~}{ x8y lmkaq«lpv�y�lmkalpv luw�n�ºGk8oA¬�npcT�Rqscw�k��w�nm��cMws�a{,y c$kGv�b2cMy�y�w�qscTy § y|�almcM� lpk³v�_ac5c$k}�}lp��oskab2cMkrv�y�©4ª ^-_aluy9b2cTw�k8yM»�¢¥os��b2wsnpnm{s»�v�_8wAv2w«qslm�scMk³�a��o�v oRt$osnµc$¡RcMt4xav lmoskz»¬�lpv _,y cM�GxacMk8t4c�o�¢�cMk}�Glm��oskab2c$kGv�y�H�~�q�����q�H � » luy�w�k,w�v|v�wst�º5l£¢�¢¥os��w�nmn t � i � �·¬c9_�wX�scIHW� ��� q�B � �"� � � � .5s��hHy�-w�k��H � q�B � �"� ����� .!s��wH ��� � ¬�l£v�_ B����� ��Iz J H ��� � ªMXIoA¬c$�scM��~}{ �ac4¦8kalpv lmoskOzPJ 3({ I� �}&�HI(�l��#v�_ac$��c�luy H� �y x8t�_�v�_8wAvHvs ��z�q`H� Oª XIcMk8t4c�¬cet$w�k)��c4¢¥or� b�xanmw�v c¶v _ac��ac4¦8kalpv lmosk)os¢�w�k)wAv v�wrt�ºKx8y lpkaq�v�_ace�a��cM�Rlut$w�v c�3({ I� � �85½ ��m��¥�� Á m���;��a�$�M�8�C1 =��8������ Z :������������>] � s Yb a Z B; a �e^DJ f�t� Z �,�������� )K�,�� Z * �vB����� ��Iz Z �<, Z �����) ���}*K� �3����������9, ����� Z �K���$���[� Z ��b������8�;��,*�� B<~� Z � wAv|v�wst�º �4�P,����������2*��,P*,� Z *��������-,~���*,���[���������2����. Z �<, Z ���������,� � � ��,�����2�������, ��� r �f��<s t q������q��M������~� � Z ���

.,��� Z ���9i�� t q������q��U�#� � Z �J��Y���./J/3({ I���}&RB<~ q�B � .1q����u�;q�BX� ��� .X( Z �<,�B����� ��IzKJ.3({ I� �}&_B<~�q�B � .1q�u���;q�B � .X(���3������Y � � Yb � �"� � � � Z �-, B � � B � �"� � � � @� c�lmkrv�� oR�Rx�t4cKkaoA¬6w·b2cTwsy xa� c�osk w�v|v�wrt�ºRy�w�k8� w·kao�v�lpork³o�¢Ib2lmkalpbwsnw�v|v�wrt�º wsb2oskaq wsnpnwAv v�wst�ºRyM»�tMw�nmnpcT� ������7

) Z � Z �[� Z �bAª � cKy _8w�nmn��a��oA�sc�lmk,v�_ackac$¡Gv9y cMt4v lmosk8yjv�_8wAv�k8os��b2wsn�wAv v�wst�ºRy�_8wX�rc��Eosnm{}kaosb2luw�nC~Eosxak��ay�¢¥os��w·y xalpv�ws~anpc��c$�a��cMy c$kGv�w�v lmosk�o�¢3v�c$��by$ª^-_ac������I�-o�¢Cw�b�cTy y�w�qrc\v�c$��b z�lmyI�ac$kaosv cM�U� z��Rw�k8�5�Rc$¦8kacM�)wry��� � ���e�" ��2��� ���� � z�� � t ¢¥os�IwskG{osv _acM��c$nmc$b2c$kGv�zxJ x5z/{J|~}� w�k8��� cTt4xa��y lp�rc$nm{~G{M� o�3 q9�3rI� � � 3 ���V� ����� t » � s 3 t � � � � 3���� � ����� t®Iosv cev�_8wAv����9�" &� �5luy�v _8c9b2lpk8lpbw�n�y lp¸Mc�b2cMy�y wsqscsª � c9��cMt$wsnpn3v _8w�v¶w2¦8k8l£v�c9b9x8n£v�lmy c4vjoA�sc$��k�wAv x8��wsn3k}xab9~Ec$��yIluy\w

¢¥xak8t4v lmosk�� ¢¥��osb�m � v�o m �'¬�lpv _5¦8k8l£v�c��Rosbw�lmkzª � c�y|_8wsnpnzt$osb2�8ws� c¶¦8kalpv ceb�xan£v�lmy c4v�y�o�¢�k8wAv�xa��w�nuy-~G{�c4¡}v cMk8�Rlmkaq�v _acos���RcM� lmkaq�oskLm � wry¢¥osnmnpoA¬Iy� �����@�'lp¢Di0({��?� �+w�k��wi|i0(�¬�_ac$kacM�scM�y�5&�3<(���� &�3-(�v�_ac$k�� & �>(����5& �>(�¢¥or�Iy|orb�c��� 33ª85½ ��m��¥�� Á m�� ;/m Á � � � &��a�$�M�8�C1 =! 6� Z �[� Z �b��4�µkaor� bwsn�� .5� �3�')��������4�,���#�/.6�������1�_* Z ������ �������|*�����6o�� Y � . �,q����u�;qI� Y � . � r�4�() ���$��) Z �

��nmcMw���nm{�lp¢3v�_ac$��cjlmy�wsk�wAv v�wrt�º2v _acM� cjlmy-w�kaor� bw�n�wAv v�wrt�ºKy|lmk8t4c¶v _8cjb2cTwsy xa� c¶lmy�w�¬c$nmn£¯O¢¥osxak��RcM��os���Rc$��lpk8q9osk�¦8kalpv cb9x8n£v�lmy c4v�y¶os¢�kaorkakacMqrwAv�lp�rc9lmkGv c$qrc$��y$ª � c2kaoA¬ �a��cMy c$kGvjwsk,®\�1�a��oRt4cT�Rxa��c9¢¥or�j�Rc$v cTt�v lmkaq5v _8c�c$¡Rlmy|v cMk8t4c�o�¢�w�kaor� bw�nwAv v�wrt�ºEª

" #�$ �$� �����5���&�9/ � % �e�C( ���&% �z�����(' �4�¤�4��% �

� c�¦8��y�v�y _aoA¬ y|orb�c�~�wsy lmt¢ wrt�v�yµosk�v _8c\��c$�a��cMy c$kGv�w�v lmosk�o�¢ b2cMy�y wsqscv cM� byµwry*)�������,����,+ 6�,� �����2�,��� Z :3� § ² !Ige©�ªG^-_ac$k¬c�y _8w�nmn�y _aoA¬�_aoA¬Wv o�os~Rv�w�lmk·¢¥� orb wsk}{5�RcM� lm�AwAv�lpork wb2os��c9t4orb2�8wst4v¶osk8csª � c9¬�lmnpn�v�_ac$k«~Ec�ws~anmc�v o��a��oA�sc�v _�wAvjwkaor� bw�nzw�v|v�wst�ºK_8wryIw���ornp{}kaorb2lmwsn3y|lm¸$c�¬eª �Mª vMªµv _ac�y lm¸$ceo�¢�v _8ce�a� osv oRt4ornzw�k8�5lmkGv ��x8�Rc$�\ºGk8oA¬�npcT�Rqscr»G¬�_8c$k·x8y lpk8q2²"!Ig��c$�a��cMy c$kGv�w�v lmosk8yMª

-�.0/ 132547608 9:8 ;=<>258047?^-_ac�8 �A@ ¿ � ½ _ � ½ � ½ m��M�R��� Á m o�¢Cw2y|c$v�H os¢�b2cMy�y�w�qsc¶v cM� by-luyv _aceqr��ws�a_C&CB6q0Dc(¬�lpv _)nuw�~Ec$nmcM��cM�RqrcMyM»R¬�_ac$��c �� v _ac�y c4v�o�¢��scM�|v�lmt$cMy�B � BD6<E &�H�(4»Rv _ac�y c4v�os¢�y xa~Rv�c$��b2y-os¢�Hª

� v _ac�y c4v�o�¢�cM�aqscMy�DP�FE s�G uIH �KJ #�-s G � �¹v�_ac$��c�c4¡Rluy�v�yyEI¬�lpv _ G u � s�G � t ' os� G u �Wo G � q�E,rt� s�G u $ � ��L #�-s G � �¹v�_ac$��c�c4¡Rluy�v�yyEI¬�lpv _ G u � s EJt�MONIor� G u �WoREJq G � r�t

��½ � �a�21 9 �1���+)P Q� ����:������������ Z ���2���F�4�(���$�CR,���2@

¡�¢ �X¡¤£

Page 10: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� j

R¹¢ n³luy\v _ac�kGx8b9~Ec$�\os¢�c$nmc$b2c$kGv�y\lmk�BD6FEG& z)(4»�orkac�tMw�k·��c$bw���º�v _8w�v &CB6q0D�(�_8wry\wAvjb2ory|v�n³kaoR�RcTy\w�k�� �'� n�cM�RqrcMyMªX\c$k8t$c¶lpv�y�y lm¸$cjlmynmlpkacTw��lpk�nC»Rw�k8�K¢¥os��t4ork}�sc$k8lpcMk8t4c¶¬�c�y|_�w�nmn��ac4¦8kac¶v _8c�²"!Ig¶¯¹y lp¸Mc\os¢�H»R�RcMkao�v�cM��~G{F� H"� � S�� »Gv o�~Ecv�_ac�kGx8b9~Ec$��o�¢��almy|v lmk8t�vey xa~Rv cM� byjo�¢xH» l?ª c�v _ac�k}xab9~Ec$��o�¢�c$nmc$b2c$kGv�y¶lmk/BD6FEG&�HI(�ª-Naor�jw�v cM� b z�»�¬c�y lmb��8np{·¬���l£v�c� z�� � S��#¢¥os��� s zt � � S��-ª\�½ � � � 9 !���� Z ��� �������/.����,��) ��H .,��� Z ��� � Z ��� Z *,�;� 3 Z �-, .,��� Z ���3)+�,��� Z *�� zO �#�g� Z ������� HP1 3�4 z|8/� � S�� � � H+q9z�� � S��� � ÁzÁ�] � y c$c!\kakac$¡��8ªHGRª �

� Á � Á & & �R� ¾ 9 !���� Z �����������9.����,��) � H .,��� Z ���"* �������<,(����*,����������[�2�����( �#�a� Z ���(� H���� � S�� � � H+q��D&�3 � (�q����;q�;&�3 � (�� � S�������,��� s 3 � q�����q`3 � t"�4�(� �3�-���,���/.I� Z ��� Z *,���,�����L�h�" ������� Z ������� Z �;���" ~�4� � �������� ������~� � Z ���;&�3e�|(�?� 3e��@� � ÁzÁ�] ��� c-��x8y|v\w��a�8np{ hzc$b2bw5�ew�~EoA�sc¶¢¥os��cMwrt�_\3 � ª �

��½ � �R��1 � ���P�������bF� � Z � Z �����;����s �/q` G Z : :�� �2�������PH Z �-,D�%�P����)�:1���%�"� �3�"���������������"* ) Q��H� 5���3���O�#� Z ���* ����,� Z ) Q��7�����:����,���,�$� Z �[�2���i�9.IH 0� Z �-,\ G C��������":3����������) � Z �a�[��)+� �4�"���OR���������,F����1��T������� ����-�������b���� � Z � Z ���������)�� ��� � Z ��� Z �H���"���\H Z �-,+� �3���P����� Z ,�,� G Z �-,����)5:������-� �3�+) Q��7�����:����,���,�$� Z �[�2���wH� ��/.�H+q` �@� c�w���cjqsoslmkaq�v o2�a��cMy c$kGvIw�k·®\�¨�acMt4luy lpork5�8� oRt4cT�Rxa��c\¢¥or�-¦8k8�RlmkaqKwsk)w�v|v�wst�º�ªC^-_ace�a��o}t$cM�Rx8� c�w�b2osx8krv�yv oqrxacMy�y

w,t4or� ��cMt4v9c4¡RcMt$xRv lmosk os���Rc$� r »�w«�Eory�y|lm~anmc�qs��osx8k8�³y|x8~8y�v�l£v�xRv lmosk�y .�»�¢¥or���Aws� luw�~anmcMy ���" }»�¬�lpv _#w,² !Ig¶¯°y|lm¸$cv�_8wAv2luy�Eosnm{}kaosb2luw�nmnp{)~Eosxak��RcM� »Ev�_ac$k v o�qrxacMy�y¶º����9npluy�v�yjo�¢���xanpcTy\os¢�npcMkaq�v�_�n � w�k8�)¦�k8w�nmnp{5v�o5t�_acTt�º�v�_8wAv�¬�_acMk w��a�8np{}lmkaqv�_acMy cenmlmy|v�y-os¢���xanmcMyv _8celpkGv ��x8�RcM�It$wsk�~ax8lpnu�)w�nmn c4¡R��cTt�v�cM�5b2cMy�y wsqscMy-wry-¬�cMnpnzwryv _ac�y cMt$� c$vMª� c·wsy�y|x8b�c)qrlp�rc$k¨w,�8� osv oRt4orn-y �EcMt4lp¦�tMwAv lmosk s &|^9q`Yb a Z B; a (T� ^PJ fatrª hzc$vw] � s Y� a q�B; a � ^wJ f�tr»w y cMt$� c$v

b2cTy y�w�qrc B����� ��Iz\w�k�� w�¦8kalpv cy|c$vjos¢µb2cMy�y wsqscTy�B<~�¢¥os��lmkalpv luw�n�lpkGv�� x8�ac$��º}kaoA¬�npcT�Rqscrª(R¹¢K]�q�BF~�luy�kaosvjqrlp�rc$k«lmk ²"!Ig¶¯��c$�8� cTy|cMkrv�wAv�lporkz»Gv _acM{�w���c\¦���y|vIt4osk}�rc$� v cM�Kv o�v _8lmy¢¥or� bwAv § lpk)�Eosnm{Gk8osb2lmwsnEv lmb�cX©�ª � cewsy�y|x8b�cjv�_8wAv-v _8c�² !Ig¶¯¹y lm¸$c�o�¢]�q�B<~eluy�nC»}v _ac�¦�kal£v�c9y c4v�o�¢��Aw���lmws~anpcTylpk2] luyW�K»aw�k8�f� f5� � ��ª^-_8c�®\�±�a��o}t$cM�Rx8� c¶¢¥os�It�_acTt�º}lpkaq�v�_ac�c4¡Rlmy|v cMk8t4ceos¢�wsk5w�v|v�wrt�ºKluyIwsy�¢¥osnmnmoA¬Iy��

�sª\gjx8cMy�y�w�t$os��� cTt�v-c4¡RcMt$xRv lmosk5os���RcM� r ��f �-s s�t q�u���;q���tGªh3c4vyY���� Yb � �"� � � � w�k���BF� � B; � �"� � � � ¢¥or��i J s�t q����u�;q���t

dRª\gjx8cMy�y�w�qr� orxak8��y|xa~�y�v�l£v�xRv lmosk«y x8t�_�v _8w�v�¢¥os�Iw�nmnF32J��»>.;&�3-(�_�wsy�² !Ig¶¯°y|lm¸$c � nCª�aª�N8os��cMwst�_wi J s�t q�u���;q�� � t t¶qrxacMy�yµwskKos���Rc$��cM�2nmluy�v0�¤�3os¢Xn � ��xanmcMy�¬�_aory cI�a��lmk8t4lm�8w�n�v cM� by_�wX�sc�² !Ig¶¯¹y lm¸$c ��� � n � ª

�8ª�N8os�-cMwrt�_\iKJ s"t q�u����q��tet�_acMt�ºv�_8wAv��¤�Cws�a�anmlpcT�Kv�o s B���. � zPT i�tK� s B<~�t�qrc$kacM��w�v cMyxY��R.GRª ��_8cMt�ºv�_8wAv� ��� � w��a�anmlmcM��v o s B���. � zwTB�I� t tK� s B<~�t�qrc$kacM��w�v cTy{B �2�� J�Iz�ªEaªaR¹¢�cMwrt�_�t�_acTt�ºKluy�y|x8tMt4cTy y|¢¥xan v�_ac$k·w�k�y|¬c$�´6<8`�ª

^�oI�a��oA�sc�v�_acµt$os��� cTt�v lmosk�os¢Gv�_aluy3�a��oRt4cT�Rxa��cC¬�cµy _8wsnpnGy|_aoA¬,v _�wAv�¬cµtMw�ke�axRv�w�~Eosxak8��ork�v _ac�npcMkaq�v�_eo�¢akaor� bwsnswAv v�wst�ºRyMª� cI¬�lmnpn8¦8��y�vµqrlp�rc��a��os�Ec$� v lmcMy�ws~�orxRvµ�Rc$��lm�Xw�v lmosk8yMª � cI¬�lmnmn�w�nuy oeqslm�sc��Eosnm{Gk8osb2lmwsna~Eosxak8�8y�osk2v�_ac\y xa~8y|v lpv xRv�lporkP.�v _8w�vluy�x8y cM��lmk)w�kaor� bwsnzwAv|v�wst�º�ª

- . - ��4 258��><�� 8�*;P?R°k�v _aluy�y cMt�v�lporkz»}¬�cj¬�lmnpn�qslm�scjy|orb2c¶x8y c4¢¥x8nz�Rc4¦8k8l£v�lpork8y-w�k8���a� or��cM�|v�lpcTyµork��RcM� lm�AwAv�lpork8y$ª � c�y _8wsnpn�lmkrv�� oR�Rx�t4c¶w�kaosv lmoskos¢�kaor� bwsn��ac$��lp�AwAv�lporkz»z�RcMkao�v�cM�,~G{�-��� �i G # &�HI(�ª ! � cMnmw�v cM� kao�v�lpork,os¢�kaor� bwsn��ac$��lp�AwAv�lpork _8wryj~�cMc$k�y�v�x8�RlmcM� lpkBA ��C¹ª�\wAv _8c$�-v _�w�k·w�k8wAv�xa��wsnz�RcM�ax8t�v�lpork��a��cMy c$kGv�wAv lmosk5lmkMA ��C3¬c�x8y ce_ac$��cjv c$��b���c$¬���l£v�lpk8q8ª

85½ � m��¥��� Á m�� ������,� Z ,��,���� Z ���2��� - �AH�~wsk� � H � sk� + ��� s5� , H Z ������) zK�4� Z *�� Z ���/.�- � . z�J H Z �<,z �J2H ��� @

N8os�2lmk8y�v�w�k8t$c�lp¢6zLJ 3({ I� �}&RH�(9v _acM� c)c4¡Rluy�v�y2w,�RcM� lm�AwAv�lpork ¬�lpv _¨qsoGw�n8z���¬c5v�w�ºrc5w �RcM� lm�AwAv lmosk�- � Hdsk� �����"sk� , H� ¬�lpv _~zKJ2H� 3w�k8��v _acMk)¬�cjv�wsºscjv _8c�y|bw�nmnmcMy|v-�8� c$¦a¡�o�¢ - t4oskGv�w�lmkalpk8q�z�ª^-_8ckac4¡}v9nmc$b2bw·y|_8oA¬Iyjv _8w�v9¬�cKtMw�k�c$nmlmb�lmk8w�v c�y osb2c��cM�Rxak��aw�k8t$lpcTy�lpk w·�ac$��lp�AwAv�lpork��e¢¥or��lpk�y�v�w�k8t$cl£¢Iw5v cM� b

z9lmyt$osb2��oGy|cT� wAvy osb2c5y|v cM� os¢\v _8c5�RcM� lm�AwAv�lpork v _8c$��c�luy�kao,k8c$cM� v o³�RcTt4osb2�Eory c�lpvKwAvw«¢¥xa� v _8c$�y�v�c$� y|lmk8t4c�v _8ct$osb2�EoskacMkrv�y-o�¢�z�ws� cew�nm��cMws�a{�a� cTy|cMkGv-lmk5v _ac�t$xa��� cMkrv�y|v c$�3ª

�������3�J�9��

Page 11: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

� �6� ����1���������������3����� Z �$�

\µ½ � � �3� .azyJ 3({ I� �}&�HI(O������,�M�����,����� ���4���2� Z , ������ Z ���2��� - .����) Hy��������* � Z �$zO Z �-, ���,��� .�����}*���ork8�Rlpv lmosk � �.,��� Z ����)+���� Z * �,�W�$q�EQ� @�� .0!$#�& s s �$t ' t ' �"��(xJ.- �����,�/! $ & s s ��t ' t ' �"��(��J/-� @�� .0!$"& s �$t ' (xJB- � �3��� ! u & s s ��t ' t ' �"��(��J/-� @�� .0! u & s�s �$t ' t ' �"��({J.- � �3��� ! " & s s �$t ' t ' �"��(��JB-� @�� .0! $ & s s �$t ' t ' �"� (xJ/- � �3��� ! # & s �$t ' ( �J.-

� � ÁzÁ�] � h3c4v- ~�ceorkac�o�¢�v _aceb2lmkalpbwsn3�Rc$��lm�Xw�v lmosk8ylmk)npcMkaq�v�_�¢¥��osbjH ¬�lpv _)qsoGw�n1z�ª�^-_acMk5nmc4v�x8yI~axalmnm�.-P zwskao�v�_ac$��RcM� lm�AwAv lmosk)¬�lpv _)v�_ac9y�w�b2c�lmkal£v�lmwsn�wsk8��¦8k8w�n�y|c$v�v _8wsk -·»E�sc$��lp¢¥{Glmkaq2b2os��c$oA�rc$��� ���-,����[�2��� � »�w�k8�)b2lmkalpbwsn3lmk·npcMkaq�v�_w�b2orkaq�v _ac9�ac$��lp�AwAv�lpork8y�¬�lpv _)v�_acMy c�lmkal£v�lmwsn3w�k��5¦8k8w�n�y|c$v�y\wsk8�5�scM� lp¢¥{}lpkaq�� ���<,������2��� � ª � c���cMwsy osk5~}{�lmk8�Rx8t4v lmosk·orkv _8c�npcMkaq�v�_)o�¢*-·ª#R¹¢*- ����¬c�v�w�ºrc�-P ��aª � v�_ac$��¬�lmy c�- �=&�H�~hs� � �u�>s� , H (�w�k8�5~}{�lmk8�Rx�t�v lmosk·_}{}�Eo�v _8cMy lmyv _8c$��cec4¡Rlmy|v�y0-w � �=&�H� ~ s � � ���}s ��, �"� H� � � (¬�l£v�_2H ~ � H� ~ »eHy � � � H� � � »8�sc$��lp¢¥{Glmkaq�� ���<,������2��� � »�wsk8��b�lmkalmbw�nlmk5nmc$k8q�v _3ª � c�y _aoA¬[_aoA¬Uv oc$¡Gv�c$k8��v _8lmy-v�oKw2�Rc$��lp�Aw�v lmosk.-P zwstMt4or���Rlmkaq�v o�v _8cenmwry�vIy|v c$�)os¢ - ��rª�l£¢! � ! # & s�s�� t��1t t � �"��(�wsk8� v _8c$��cKc4¡Rluy�v�y�i T=n±y x8t�_�v _8w�v !K � � ! $ & s s�� t��1t � �"��(gJ -P � »�v�_ac$k nmc4v ! �!$"& s�� t��9(4ª.!K tMw�k�~Ec�w��a�8nplmcM�³y lpk�t4c � J�H� � � � w�k8��� J�Hy� ��� »Cwsk8� ¬�c_8wX�rc ! u & s s�� t��1t � �"��(�?J -P � y|lmk8t4co�v _8c$��¬�lmy c -w � ¬osxanu��kao�v�~Eceb�lmkalmbw�nzlmk)npcMkaq�v�_zª

daª�l£¢ ! � ! " & s�� t � («wsk8�1v _acM� c c4¡Rlmy|v�y�i T[n6y x8t�_Wv _8w�v ! � � ! u & s s�� t � t � �"��( J - � »�v�_ac$k�nmc4v�! �! # & s s�� t��1t � �"��(4ª !K tMw�k�~�c�w��8�anplmcM�2y lpk�t4c s s�� t��$t � �"�WJ�H� � � � w�k8���CJLH� � ��� »rwsk8��¬�c-_8wX�rc�! $ & s�s�� t��1t � �"��(�?J-P � ~}{Kb2lmkalpbwsnplpv�{�os¢ -P � ª

�8ª�l£¢ ! K� ! u & s s�� t � t � �"��(-wsk8��v _acM� c�c4¡Rluy�vyiKTkn�y|x�t�_�v�_8wAv�!K � � ! " & s s�� t � t � �"��(xJ.-P � »Rv _8c$k��� R¹¢ ! $ & s�� t��e(�?JB-P � »R¬c�v�w�ºrc !K � ! # & s�� t��e(-y|lmk8t$c s�� t��5wsk8���³ws� c�lmkLHW� ��� ª� R¹¢ ! $ & s�� t��e(xJ.-P � »Rv _8c$k s�� t��"� s s�� 2t � �"� t���wsk8��¬�cjv�wsºsc ! � !�"�& s�� t � �"� (-y lpk�t4c � 3w�k8����� �¶ws� cjlpkH ��� ª

��ª�l£¢ ! � ! $ & s�s�� t��1t � �"��(-w�k8��v _acM� cec$¡Rlmy|v�y{iKTkn�y|x8t�_�v _�wAv�! � � ! # & s�� t��9({J/-w � »Rv _acMk��� R¹¢ ! u & s�s s�� t � t � �"� t � (�?J.-P � »a¬cjv�wsºsc�! � ! " & s s�� t � t � �"� (4»8y|lmk8t$c s�� t � wsk8���³ws� c�lmkLH � ��� ª� R¹¢ ! u & s�s s�� t � t � �"� t � (xJ.-P � »a¬cjv�wsºsc�! � !�#�& s s�s�� t � t � �"� t � (�ªGaª�o�v _8c$��¬�lmy c¶lmk·w�nmn � cMbw�lmkalpk8q2tMwsy cMy¬c¶v�w�ºrcbYb $� Y�ª^-_acMk«¬�c�t$wsk«kao�v�lmt$c�v�_8wAvhH� � H ¢¥or��H� � � s� �, H� w�k8��-P �� &�H�~ s���I�u�<s� , H� (\�scM� lp¢¥{�� ���-,������2��� � ª

XIcMk8t4cr»�v _ac�y c4v�os¢I�Rc$��lp�Aw�v lmosk8ye¬�lpv _�v�_ac�y�w�b2clmkalpv luw�n�wsk8�,¦8k8wsn�y c4v�y�v _8wsk -&w�k8���rc$��l£¢¥{}lmkaq�� ���-,����[�2��� � lmy9kao�vc$b2�Rv�{r»8w�k���¬cet$w�k·t�_ao}oGy|cjoskac�os¢�lpv�y-cMnpcMb�cMkGv�y�b2lmkalpbwsn lpk)nmc$kaqsv _zª

XIcMk8t4c�¢¥��osb w��ac$��lp�AwAv�lpork«�a��oA�}lpkaqKv _8w�v'zbJ 3({ I� �}&�HI(\¬c�tMw�k,~axalmnm�,w�kaosv _acM�¶orkac��rc$��l£¢¥{}lmkaqKb2os��c$oA�rc$��� ���-,���7���2��� � ªe^-_ac�b2lpk8lpbw�n��a��c4¦8¡)os¢�v _8lmy��RcM� lm�AwAv lmosk«v _8w�v�t$oskGv�wslpk8y6z�luyjw��Rc$��lm�Xw�v lmosk«os¢µqsoGw�n�z\y w�v luy�¢¥{}lmkaq�� ���-,������2��� � »w�k��Kv�_almyI�a� oA�rcMy�v _8cenpcMb�bw8ª �^-_aluy¶wsnpnmoA¬IyIx�y\v�o��Rc$¦8kac2y|orb2c9kaor� bw�n��RcM� lm�AwAv lmosk3»ElOª csªe�Rc$��lm�Xw�v lmosk«b2lmkalmb2wsn�lmk nmc$k8q�v _ w�b2osk8qv _aoGy|c��scM� lp¢¥{}lpk8q

� ���<,������2��� � �85½ ��m��¥�� Á m �"! ��, ���������-"� i G # &�H�( Z , ������ Z �[�2���M�9.6)�������) Z � �;�,�}*�� � Z )+���}*+� �3��,��,���� Z ���2������.����)=H ����� �g* � Z �3zZ �<,K� Z ���4� .�����"*���osk8�Rlpv lmosk � /��3�J����� Z ��*,����� Z ������� Z )+���}*�� �3�a:��J����2*,���K�����,� � @

R°k or���ac$�jv o5~�orxak8� v _ac2nmc$kaqsv _ o�¢�y x8t�_ �Rc$��lp�Aw�v lmosk8yM» ¬�c2t$wsk,k8oA¬ x8y c9v�_ac�Rc4¦8k8l£v�lpork,os¢�� ���<,������2��� � v�o��8� oA�rcv _8c�v�¬�o�¢¥osnmnmoA¬�lpkaq�nmc$b2bwsy� !InmnzlpkGv�c$��b�cT�RluwAv cjv�c$��b2y-lmkB-��� i G # &RH�(-luy�w�y xa~Rv�c$��b¤os¢�H os�az�ª\µ½ � � � � . � �3����� � ���4���2��z0 ������~� � Z � !$#"&2z0 �({J.-��� i G # &RH�(����! u &2z)({J.-��� �i G # &�HI(-� �3����z0 g�4� Z ����*,������)��/.�H� � ÁzÁ�] � hzc4v�-y� -��� i G # &�H�(4ªI�{«b2lmkalpbwsnplpv�{·o�¢ -'wsk8�«~}{�� ���<,������2��� � »3¬c�_�wX�sc ! " &2z (g?J - w�k8��! $ &2z ( ?J -·ª^-_acMk#cMl£v�_ac$�+z0 IJ H'wsk8� ¬c)_8wX�sc�v _8c)t$osk8t$npx�y|lmosk o�¢¶v _ac)nmc$b2bwaª � v _acM� ¬�luy c�v�_ac$��c)c4¡Rlmy|v�y2w ��xanpcB! # &2z � 1 z0 8�(�os�! u &2z � 1 z0 8¤(µlpk/-6qscMkac$��wAv�lpk8q-z0 OªD�xRv�w�k}{2��xanpc¶lpk/-6qscMkac$��wAv�lpk8q-z � 1 z0 u8�b�x8y|v�~Ec¶lmk ! # � ! u § lp¢3kao�vT»Gv _ac��RcMt$osb2�Eory l£v�lpork¬osxanu�·~Ec�x�y|cMnpcTy yjw�k8�·v _ac�RcM� lm�AwAv lmosk ¬�orxanm�«kao�v�~�c�b2lpk8lpbw�n ©��¶¬�c2t$wsk«lpv c$��wAv�c9v�_aluy¶��cMwry|orkalmkaqKoskFz � 1 z0 8¹»zw�k��)v _aluyc$k��ayCv�_ac\�a��o}o�¢/� z0 almk8t4��cMwry|cTy�y|v ��lmt4v nm{�w�vµcMwrt�_l£v�c$��wAv lmoskwsk8��v _acj�Rc$��lm�Xw�v lmoskorkanp{2t4orkGv�w�lmk8y�we¦8kalpv c\k}xab�~�cM�µos¢Ev�c$��by$ª�

¡�¢ �X¡¤£

Page 12: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� L

\�½ � � � � .������,��� � ���4��[��z0 ������K� � Z � ! " & z0 ¤(xJ/-"� i G # &�HI(6��� ! $ &2z0 ¤(xJ.-"� i G # &�H�(6� �3����z0 �4� Z ���*����,��) �/. s zt<�bH� � ÁzÁ�] � hzc4v�- � -"� i G # &�HI(�ª��{)b�lmkalmbw�nml£v�{·os¢ - w�k��«~}{·�Rc4¦8k8l£v�lpork o�¢������-,����[�2��� � » ¬c9_8wX�rc�!$#"&2z0 �( ?J?- wsk8�! u & z0 ¤(�?J.-·ª XIcMk8t4c�c$lpv _8c$�az0 FJ s zt �\H wsk8�Kv�_ac�npcMb2b2w�luy-�a��oA�scM�zª � v�_ac$��¬�lmy c¶v _acM� c�luy�wAv�nmwry�v-osk8cj��xanmc�x8y|lmkaq�z0 ���lp¢k8o�vM» ! " & z (¬�orxanu��~Ecex8y c$nmcMy�ywsk8�.-��� i G # &RH�(-kaosv-b2lmkalmb2wsn?ª�^-_8c$k)oskac�os¢�v�_ac�¢¥osnmnpoA¬�lmkaq2t$wry|cTy�t$w�k5~Ec�w��a�anmlmcM� �

� ^-_acM� c¶c4¡Rlmy|v�yx�y x8t�_Kv�_8wAv0! # & s ��t # � �"��(�J/-·»}_acMk8t4c s �$t # � �"�-luy-w9y xa~Rv�c$��b6o�¢cH ~G{�v�_ac'h3c$b2bw��8»Rw�k8�Ky o�lmygz0 Oª� � �ev�_ac$��c�c$¡}luy|v�yIE2y x8t�_³v�_8wAv�!$"& s z0 t ' (PJ -·»Cos� !$"& s EJt # � (PJ -·»�os� ! $ & s�s z0 2t ' t ' �"��(4ªFR°k�v _8lmy�t$wry|cr»�¬c�tMw�klpv c$��wAv�c�v _8lmy��cMwsy osk8lpkaqeork+z � � s z0 t ' » z � � s EIt # � os�gz � � s�s z0 �t ' t ' �"�Tª�^-_aluyµcMk8�ayµv _ac¶�a� o}os¢�»r~EcMtMw�x8y c5z0 Ey�v�� lut�v�np{lmk8t4��cMwry|cTyw�v�cMwst�_5lpv c$��wAv�lpork5wsk8��v _ac��RcM� lm�AwAv�lpork�orkanp{�t4oskGv�w�lmk5w�¦8kalpv cek}xab�~�cM��o�¢�v cM� byMª

�� c�y _aoA¬Ulmk5v _ac�k8c4¡}vI�a��os�Eory l£v�lporkKv _�wAvIv�_ac$��cew�nm¬wX{Ryc4¡Rluy�v�yI�Rc$��lm�Xw�v lmosk8yos¢�w�v�c$��b zµ¢¥��osb w2y|c$vWH ¬�lpv _«w�k}xab�~�cM�os¢���xanmcMy-~Eosxak8�acM�5~}{v _ac�²"!Ig¶¯¹y lp¸Mc�o�¢�lmkalpv luw�n3wsk8��¦8k8w�n�v�c$��by�z�q`Hª�^-_aluy�¬�lmnpn ~Ece�scM� {x�y|c$¢¥xan v o2~Eosxak���v�_acenmc$kaqsv _os¢�v�_ac��RcM� lm�AwAv lmosk�y�lmk}�sornp�rcM�Klmk5v _ace��cMy cMws��t�_�o�¢Cw�k)wAv|v�wst�º�ª

� � Á$_µÁ �� �� Á m 9 !���� Z �$� ���,�"�9.D���,��) �\H�~ Z �-, .,��� Z �$�O���,��) zO�� ./-��� �i G # &�HI(M� H�~ s� � �u��s� , H � �3���n � ���[q`H"� � S�� Z �-,�.,��� Z ��� � � i � n>g� Hy��� � S�� � ���[q`H"� � S�� @� � ÁzÁ�] � hzc4v�x8y��a� oA�rcv _�wAv9v�_ac5kGx8b9~Ec$��o�¢\y|v c$��y9lmk -��� i G # &�H�(9lmy�wAv2b2ory|v~� z�q`H"� � S�� ~}{�c4¡awsb�lmkalmkaq·v�_ac�v�c$��byt$osb2�Eory cM��os�I�RcTt4osb2�Eory cM��¢¥os�IwskG{��xanmc�Y�v _�wAvI_8wry-~�cMc$k·w��a�8nplmcM��lpkB-��� i G # &�H�(,�� Na��osb h3c$b2bw¶�¶c$�rc$��{¶v�c$��b �RcMt$osb2�Eory cM�eor��y lmb��8nplp¦8cM� § ¬�lpv _�! u or� ! # ©zluy��Rc$��lp�rcM��¢¥� orb�H1~}{��RcMt$osb2�Eory l£v�lpork8yc4¡at$npx8y lm�sc$nm{ § ! u or� ! # ©4ª+XIc$k�t4c2c$�rc$��{)v c$��b ¬�_alut�_ lmye�RcTt4orb��Eory cM� ¬wry�w5y xa~Rv�c$��b o�¢{H.wsk8�,lmyet$osxakGv cT�«lmk

� H"� � S�� ª� Na��osb h3c$b2bw��)cM�sc$��{5v cM� b t$osb2�Eory cM� § ~}{ !$"jor� ! $ ©jluy¶x�y|cT�,wryjw5y xa~Rv cM� b o�¢w�ºscM{·os��o�¢8z�ª�X\c$k8t$c2l£veluyt4orxakGv cT��lmk � z�q�H"� � S�� ª� <��rc$��{e� xanmc�Y[cMl£v�_ac$��t$osb2��oGy|cos���acMt4orb2��oGy|c-w¶v c$��b)»s~axav Y±kac$�rc$�Ct4orb2��oGy|c § ��cMy �zª��RcMt$osb2�Eory cT©�wIv�c$��b ¬�_alut�__8wryIw�nm� cTws�R{�~�cMc$k t$osb2�Eory cM� § ��cMy �zª��RcTt4osb2�Eory cM�8©4ª�XIcMk8t4c9v oKcTwst�_·y|x8~Rv cM� b o�¢ H or�5z�t4or� ��cMy ��ork8�ayIw�v\b2ory|vosk8cj��xanmcew��a�anmlut$wAv�lpork�lmkB-��� i G # &RH�(¢¥os�It$osb2�Eory lpkaq�os�I�acMt4orb2��oGy|lmkaq�l£vTª § b2c$��qslmkaq�lm�ac$kGv lut$wsn3y|x8~Rv cM� by�©

X\c$k8t$c5v _ac·k}xab�~�cM��os¢\v�c$��b2yt$osb2�Eory cM�#os��RcMt$osb2�Eory cM� lmk -��� �i G # &RH�(�lmy2~Eosx8k8�RcM�#~}{³v _8c5k}xab�~�cM�o�¢��Rlmy|v lmk8t4vy xa~Rv�c$��by�os¢cH+q�zµw�k8�Kv _acj¦8��y|v�8ws�|v-o�¢3v _ac���cMy xan£v�¢¥osnmnpoA¬IyMª `}lmk8t4cjcMwrt�_�lmkrv�c$��b2cM�RluwAv�cIv cM� b6luy�w9y xa~Rv�c$��b�o�¢�H+q9z�»Gv _8cy cMt$osk8���8ws�|v�o�¢�v _8ce�a� or��oGy|lpv lmosk�¢¥osnmnpoA¬IyMª �

!\k osv _acM�¶º}lmk8�«os¢µx8y c4¢¥xanC�RcM� lm�AwAv lmosk�y\luy¶y _aoA¬�k«lmk«v�_ac�¢¥osnmnpoA¬�lmkaq��µ� or��oGy|lpv lmosk,daª R¹vj¬�lmnpnCwsnpnmoA¬�x8y¶v o��a� oA�rcev _8ch3c$b2bw+GRª� � Á$_µÁ �� �� Á m�� >����az J 3({ I� �}&�HI( Z �-, ��J 3({ I� �}&RH�(T*� * ����,� ����� ��-��� �i G�� &�H�(~���-,����"*M����� � Z � Z : :1���2� Z ���2����9. Z �����;�K���B! " ��� ! $ @��1�3���f� �3�����K�4� Z ,��,���� Z ���2��� - ����� �L*�� Z �1z ��� Z ���[���}* .����) H Z �-, ���,��� .�����}*���osk8�Rlpv lmosk�d5�! # & �F(��J/- Z �-,�! u & �X( �JB-� � ÁzÁ�] � h3c4v�zKJ.3({ I� �}&RH�(w�k8� ��J.3({ I���}&�H�(�~Ec¶¬�lpv _/-��� �i G�� &�H�(�cMk8�Rlmkaq�¬�lpv _5w�k5w��a�anmlut$wAv�lporkKo�¢�w���xanpcjlmk ! " or�! $ ª hzc4v - ~Ec -��� i G�� &RH�(�¬�lpv _8osxRv�l£v�� y¶nuwsy|v¶��xanpcr»�lOª csª -��� �i G�� &�H�(Ilmy - ¢¥ornpnmoA¬cM�)~}{/! " os� ! $ » w�k8�«npc$v -P �~Ec�v _8c�ac$��lp�AwAv�lpork�or~Rv�wslpk8cM��¢¥� orb -��� i G # &RH�(~}{���c$�anuwst$lpk8q�cM�sc$��{K�RcTt4osb2�Eory l£v�lpork/!�#�os�! u os¢��«~}{ -·ª�^-_8c$k��� -P 3lmy¶wt4os����cMt�v\�ac$��lp�AwAv�lpork��µy lpk�t4c�!¨qrc$kacM��w�v cMy � w�k�� ��»av�_acev�¬o��Rluy�v�lpk8t4v¶�Rlm� cTt�v¶y xa~Rv�c$��by�o�¢ ��ª § y|lmk8t4c��,luy

os~av�w�lmkacT��~}{�t$osb2��oGy|lpv lmkaq2¬�lpv _B! " or�! $ ©� !³�Ro}cMy�kao�vµt$oskGv�wslpkwe�RcTt4osb2�Eory l£v�lpork !$#-or� ! u o�¢ �K¢¥��osb v _ac�¢ wrt�vCv�_8wAv$-"� i G�� &�H�(�_�wsy ��wryCqsoGw�nOª � v�_ac$��¬�lmy cv _8cenmwry�vIt4orb2��oGy|lpv lmosk5¬�orxanu��~Ecex8y c$nmcMy�y$ª� XIcMk8t4c�-P 3y w�v luy�¦8cTy�� ���-,����[�2��� � wsk8��v _acenmc$b2bw�¢¥osnmnpoA¬IyMª

�������3�J�9��

Page 13: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�M� �6� ����1���������������3����� Z �$�

-�.�� 1 �*6��=; �*9 8K< 6�� ���P;�P? �*;� � 259:< 6 � � � <���� ?� cIy|_8wsnpn}�a��oA�sc�v _�wAvC¬�_ac$k�v�_ac$��c-c4¡Rlmy|v�yCwsk�w�v|v�wst�º�v _8c$kw\k8os��b2wsnawAv v�wst�º�tMw�k2w�nm¬wX{Ry�~�c�t$osk8y|v ��x8t�v�cM�9¢¥� orb y xa~Rv cM� byv _�wAv9w���c2w�nm� cTws�R{·oRt$t$xa��lpkaq)lmk v�_ac�a��os~anmc$b'y|�EcMt$l£¦�tMwAv�lporkzª^-_almy�¬�lpnmn�w�nmnpoA¬ v o)qslm�sc2~Eosxak��ay�osk,v _acb2cTy y�w�qrcMy�y|lm¸$cw�k���ork�v�_acek}xab9~Ec$��os¢���c$¬���l£v�lpk8qry-lmkG�rosnm�scT�Klmk)y x8t�_)w�k)wAv v�wst�º�ª

XIcMk8t4cjnmc4v�x8y-wsy�y|x8b�c�w��a� osv oRt4ornF] � s Yb a Z B; a �'^ J fatr»aw�y|cTt4��c4v-b�cTy y�w�qrc�B �2�� J�Iz¨w�k8�5w�y|c$vos¢�b2cMy�y wsqscTyKB<~wsyIv _ac�lmkal£v�lmwsn�lmkGv ��x8�RcM�\º}kaoA¬�nmcM�Rqrcsª � c�wry y xab2cev _�wAv\v�_ac$��c9c$¡Rlmy|v�y¶w�k w�v|v�wrt�º5�acMy�t4��lp~EcM�)~}{·wqs��osx8k8�·y|x8~8y�v�l£v�xRv lmosk. wsk8�«w�t$os��� cTt�v\c$¡RcMt4xav lmosk or���ac$� r ��f �-s t q�����,q� § ¬�_acM� cK�·luyIv�_ac2t$w����Rlmk8w�n�o�¢cf�©�ª � c2�Rc$¦8kacgY��#�+Yb � �"� w�k��B � � B; � �"� ¢¥os��i�� t q�����,q��ª� cIw�nuy|oe�Rc$¦8kac ��� M wsy�v�_ac\y c4v�os¢ y xa~Rv�c$��byCo�¢Ev _acIv c$��byClmk�v�_ac\y c4v;MW� s Y � � z � t q������q���tF� s B � � z�����q�����,q��tr»

w�k���� M�� � v _8c�y|c$v-os¢�y xa~Rv�c$��by-o�¢ s Y � � z � t q����,q)i�t � s B � � z����3q�����,q`i�trª� c�wry y xab2c�¬�l£v�_aosxavInpoGy y-os¢�qscMkac$��w�nmlpv�{2v _8w�v ���9�" &� �\J2B ~ lOª csª�v _8celpkGv ��x8�RcM��lpk8l£v�lmwsnpnm{�º}kaoA¬Iylpv�y�k8wsb2c��

85½ ��m��¥�� Á m�� >��,� z Z �<,Kz *�K���#� �%����)�� Z �<,�� Z *��������-,�����*,����������[�2��� @��1���,�Dz��4� Z �37�) Z �%��U�/.'z � .'z��4� �1��� Z� Z ��� Z *���� Z �-,(z����@z0 @ �1� �4�-��������*�I,��,�1������,�*,�(z���8z0 ^-_ac\¢¥osnmnpoA¬�lmkaq9nmc$b2bw�lmyosk8c\o�¢3v _acjºscM{��8� or��cM�|v�{2o�¢3v _aluy�8w��Ec$�Tª�R¹v-wsnpnmoA¬Iyµx�yµv�o9�a��oA�scIv _8w�v-cM�scM� {2y|x8~8y�v�l£v�xRv lmosk

.)lmkKwekaos��bw�n8w�v|v�wst�º9luy�orkanm{�~ax8lpnu�2¬�l£v�_K�8ws�|v�yCo�¢�v _8c\�a��o�v�o}t$osnay ��cTt4lp¦�t$w�v lmoskzª�R°k2v�_almyµ¬wX{r»s¬cI¬�lmnpn�~�cjw�~anmc-v o��8� oA�rcv _�wAv\w�nmnzy xa~8y|v lpv xRv�lporkL.,lmk)w�kaor� bwsnzwAv|v�wst�º_8wry�w² !Ig¶¯°y|lm¸$c�~Eosxak8�acM�5~}{�w���ornp{}kaorb�lpk5v�_ace�a��o�v oRt$osn ² !Ig¶¯°y|lm¸$crª

\µ½ � � � � �8������ Z �1����) Z � Z ��� Z ��b�.(�.,��� Z ��� � Z ��� Z *�����37#�����,��� � ���4��[��z��! I.;&�3-(-�������T��� Z � zKJ"� M @� � ÁzÁ�] � hzc$vI.¨~�c5w·kaor� bw�n�wAv|v�wst�º�»�w�k8��nmc4v�x8ye¦8��y|v�wsy�y|x8b�cKv _8w�v9v _8c$��cKc4¡Rluy�v�y�3#y x8t�_�v�_8wAv�¢¥or��wsnpngz�y x8t�_�v�_8wAvz�� .;&�3<(�¬c�_8wX�sc'zW?J#� MK»awsk8�5npc$vIx8y��RcM� lm�scew2t4orkrv���wr�Rlut�v lmosk3ª h3c4vIx8yI�ac4¦8kac��

� $�� |�i|n s z �G.;&�3-(KJ#� M � ��t��$9lmy�v�_ac\¦���y|v-y|v cM��os¢zv _acj�a��o�v�o}t$osnE¬�_aory c\b2cTy y�w�qrc\t4orkrv�w�lmk8yx.;&�3-(-wry�w�y|x8~Rv cM� b)ª�XIoA¬c$�rc$��y lmk8t4c¶¢¥os�-w�nmn1zµy|x�t�_

v _�wAv-z%� .;&�3-($» zjlmy�k8o�v�lmk&� MK»zv _8c$��c�c$¡Rlmy|v�yew5�Aw���lmws~anmc�� ¬�_almt�_�luyey|x8~Rv cM� b'o�¢xY('*)�or��B+',)5y x8t�_,v _8w�v�.;&�3-(elmywKy xa~Rv�c$��b os¢ .;& �>(4ª § o�v�_ac$��¬�luy|cev�_ac$��c�¬osxanu�·c$¡}luy|v¶wP.E¯¹b2w�v�t�_«os¢ .;&�3-(¶¬�lpv _ y osb2c9y xa~Rv�c$��b.os¢ Y-'*)�©4ª\^-_ac$k npc$v¶x8yy _aoA¬UkaoA¬Uv _ac�¢¥ornpnmoA¬�lmkaq�t$nmwslpb �� & �>� � .;&�3-({J.3({ I� �}&RBF~�.1q�u���u�;q�B.',) � � .X(�ª_ � Á3Á$] � hzc4v -��� �i G �./ ) &_B<~.1q�u�;q�B.',) � � .X(�~�c�Hy~bs� � H � �u�'s� , H ª�`}lmk8t4c�.;&�3-(�luyIwy xa~Rv�c$��b¤o�¢ Y(',)�wsk8�)y|lmk8t4cY ' )G.�JB3({ I� �}&RB ~ .1q������u�;q�B ' ) � � .X(4»a¬cj_�wX�sc �� l£¢�v _acM� c�c$¡}luy|v{i � n y|x�t�_�v�_8wAvW.;&�3-(�JLH � »Rv _8c$k)os~}�}lporx8y np{P.;&�3-({JB3({ I� �}&_B ~ .1q�u���u�;q�B ' ) � � .X(�ª� � v _acM� ¬�luy|cr»8¬c�¬�lmnpn��a��oA�sce~}{�lmk8�Rx�t�v lmosk«v _8w�v�.;&�3<(\oRtMt4xa��yIwsy¶wKy xa~Rv cM� b lmk·c$�rc$��{�lmkGv c$��b2cM�Rluw���{�y c4v�Hy�|ª � c

_8wX�scb.;&�3<(�y|x8~Rv cM� b.o�¢�HW�y lpk�t4c�Y ' ) ./JLHy »aw�k�� �0 R¹¢x.;&�3-(�y xa~Rv�c$��b o�¢KHy�-wsk8�«lp¢µv�_ac$��c�c$¡Rlmy|v�y�z � iIy x8t�_·v�_8wAv�}2t$� cTwAv cT�«~}{ ! � � !$"&[} (¶os� ! ��� ! $ &�} (�»v _8c$kD}g?� .;&�3<(�y|lmk8t$cb.;&�3-(h?J�H �Gª�X\c$k8t$cs»9.;&�3-(-luyIw2y xa~Rv cM� b¤o�¢;H � � � ª

0 R¹¢ .;&�3<(\y xa~Rv�c$��b o�¢ Hy�w�k8�)lp¢Cv _acM� c9c4¡Rlmy|v�yxz � i�y$ª vMª5}�t$� cTwAv cT�)~}{/! �(�:! # &[} (Ios��! �(� ! u &�} (�»Ev _acMkP}w�k���.;&�3-(�ws� c�y xa~Rv cM� by-os¢DH � ��� ª

0 R¹¢Cy|x8t�_)w�v cM� b�}��Ro}cMy-k8o�v�c4¡Rluy�vT»Rv _acMk2.;&�3-(�lmyIw2y xa~Rv�c$��b�o�¢;H ~ w�k��Kv�_acel£v�c$��wAv�lpork�luy-¦8kaluy _acM� ª^-_almy�lpv cM��w�v lmosk,lpb2�anmlpcTy�v _�wAv�.;&�3<(�lmyew)y xa~Rv�c$��b2y�os¢xH ~ � B ~ .1q�����q�B ' ) � � .�ªP�xRvel£v�� yelmb2��oGy y lm~anpc�Rx8c�v�o5v _act�_aoslut4cjo�¢;� $ ª

½ m�ÂXIcMk8t4c9v _acM� c�c4¡Rluy�v�y¶wK�ac$��lp�AwAv�lpork -��� i G � $ � &RB ~ .1q����u����q�B ' ) � � .X(\wsk8�·¬c�tMw�k«kaosv lut4c9v _8w�vjlpv�y¶nmwry�v�y�v�c$�«x�y|cTy\kacTt�¯cMy�y�w���lpnm{�wt4orb2��oGy|lpv lmosk)��xanpc�y|lmk8t4ceosv _acM� ¬�luy c-hzc$b2bw�¬osxanu�5lpb2�anm{�v _8w�v�.;&�3<(�luy\wy xa~Rv cM� b o�¢ B ~ .1q����u���;q�B ' ) ��� .�»w�k��Kv�_ac$��c4¢¥or� cew2t4orkrv���wr�Rlut�v lmosk3ª

hzc$vIx8yI�Rc$¦8kac�v�_ac9y xa~8y|v lpv xRv�lporkL.F zv o~Ec�cM�Gx8wsn�v oP. osk·wsnpnz�Aws� luw�~anmcMy-c$¡at4c$�av�¢¥os��3«¬�_acM� ch.F |&�3<(g�:���e�� &�2�Eª � c¬�lmnpn�a��oA�sc�v _8w�v .F �Rc4¦�kacMy�w�k w�v|v�wrt�º ¬�lpv _³v�_ac�y�w�b2cc$¡}cTt4xRv�lpork³os���Rc$��v _8wskk.�ªP`}lmk8t4c\.¨lmy�w�k wAv v�wrt�ºE»�¢¥or�9w�nmnDz�»Y �.�J.3({ I� �}&_B<~�.1q�u�;q�B�� � � .X(�ª

¡�¢ �X¡¤£

Page 14: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� �s�

R¹¢ zwTl� $ v _8c$k«y lpk8t$c�.;&�3<(�_8wry�kaooRt$t$xa� ��c$k�t4celmkLY � .1q�B ~ .1q�u�;q�B � ��� .�»8¬ce_8wX�schY � .F� Y � .F O»-B ~ .F� B ~ .F O»3ªMª$ªM»B � � � .~�!B � � � .F Oª�^-_ac$��c4¢¥or� cj¬c�wsnmy o2_8wX�scbY � .F XJ.3({ ���}&RB ~ .F [q�����q�B � � � .F ¤(R¹¢Xz��@� $ v _8c$k)v _8c$��cjc$¡Rlmy|v�y�w2�RcM� lm�AwAv lmosk5¬�lpv _)qsoGw�nFY � .

H�~�s� � H � s� + �u�����"s� ,�� H � ¬�_ac$��chH�~'� B<~.1q�u�;q�B�� ��� .�{)����os�Eory l£v�lpork dR»F.;&�3<(¶lmy¶kac$�rc$�jy|lmb2�anplp¦8cT�«os�j�RcMt$osb2�Eory cM�·lmk·v _8lmy��RcM� lm�AwAv lmosk ���<i � n���» ! ��?� ! # &�.;&�3-(�(\wsk8�

!K�{?� ! # &�.;&�3<(�(�ª hzc$vIx8y�~ax8lpnu��¢¥� orb�v _8lmyI�ac$��lp�AwAv�lpork5w�kacM¬1orkac�¬�_ac$��c�cMwrt�_�.;&�3-(�luy-��c$�anuwst$cM��~G{.���9�" &� ��ª � c�y|_�w�nmn�ac$kaosv c�~}{"z���v�_acev cM� b os~av�w�lmkacT��¢¥��osb�z-~}{���c$�anuwst$lpkaqcM�scM� {KoRtMt4xa��� cMk8t4ceos¢D.;&�3-(\~}{/���e�� &�2�EªaN8os�\t$osk}�scMkalpcMk8t4c�¬cy _8wsnpnzt$osk8y lm�ac$�v _8w�vyH+s�H luy�w2�Rc$��lm�Xw�v lmosk)y�v�c$����x8y|v lp¦8cM�5~}{v _8cjlu�RcMkrv�l£v�{�� x8npc �RªC^-_acMk�¬cey|_�w�nmn �a� oA�rcIv�_8wAv�v�_ac$��cc$¡Rlmy|v�y�w��Aw�nmlm�)�RcM� lm�AwAv�lpork��

H ~ �hs � � H � �hs �+ ���u���"s �,�� Hy � �¬�_8c$��c9c$�rc$��{)� xanmc�! � lmyjc$lpv _8c$� ! ��os� �Rª-XIcMk8t4c�¬c�orkanp{)_8wX�rc�v�o�v�w�ºsc9v _ac2y�w�b2c9��xanmcMy�wsy¶lpk v _ac�lmkal£v�lmwsnC�Rc$��lp�Aw�v lmosk~8xRv¶�Eory�y|lm~anm{·y|º}lm�«y|orb�c�y�v�c$�8yMª5S·or� c9�a� cTt4luy|cMnp{5nmc4v¶x�y¶y _aoA¬Wv _8w�v\¢¥or�Wig� t �����)n��e¬�_8c$k�Hy� � � s��KHy��v�_ac$k c$lpv _acM�H � ��� �hs �cH � ��os�yH � � � �hs�yH � �}ª�sª-lp¢ !K� � !�"�&�o � q ��r�(-w�k8���

§ wG©Klp¢;.;&�3-(h?� o � q ��r$»}v _acMk5&�H� � � � q � q �c(���s �x&�H� � q � q � qo � q ��r�(���lmyIw��Aw�nmlu��y|v cM�)y lmk8t4c+o � ��q � ��r �Wo � q ��r��}ª§ ~E©�cMnmy c�.;&�3<(#�Wo � q ��r�wsk8��¬�cet$wsk�v�w�ºrc !K � � ��y lpk�t4c����e�� &�2�PJ2Hy�|»a¢¥os�Iw�nmnFi�ª

dRª�y�w�b2c�� cTwsy oskalmkaq�¢¥or�! � � !$"& s�� t��e(-w�k8�/! � � ! $ & s s�� t���t � �"��(�aª-lp¢�! � � !$#�&9o � q ��r)(�v _acMkw.;&�3<(�?�Wo � q ��r$»Rw�k8�2&RH� � ��� qIo � q � r`(��bs �x&�H� � qo � q ��r q � q �c(���luy��Xwsnplu�Ky lpk�t4c�o � ��q � ��r �

o � q � r�}ª�8ª�y�w�b2c�� cTwsy oskalmkaq�¢¥or�! � � !$#�& s�� t � (-w�k8�/! � � ! u & s s�� t � t � �"��(�ª

N�lmk8w�nmnm{,¬c�qrc4v�¢¥or��w�nmn�z�» Y �.F WJ 3({ I���}&RB<~ .F 2q�u�;q�B�� � � .F (4ªFXIcMk8t4c�l£v�¢¥osnmnpoA¬Iyev _8w�v .F �lmy�w�k w�v|v�wrt�º«¢¥or��v _8c�y�w�b2c�8� osv oRt4orn�or���RcM��v�_8w�k�.�ª�`Rlpk8t$c�.F �luy�os~Rv�w�lmkacM�K¢¥� orb .«~G{Ky lmb��8np{2��c$�anuwst$lpkaq9v _acj�Aw�nmxac\os¢ 35~G{Kw�y�v�� lut�v�np{y bw�nmnpcM��orkac§ ¬eª �Tª vMª+��p-� ©I¬c�_8wX�rc"o�� Y � .F ��q�������qJ� Y � .F �� r¶y|v ��lmt4v nm{5y bw�nmnpcM�Iv _�w�k o�� Y � . �,q����u�;qI� Y � . � rIw�k��)v _aluyjlmy¶t4orkrv���wr�Rlut�v or� {�¬�l£v�_v�_ac�wsy�y xab2�Rv lmosk5o�¢�kaos��bw�n wAv v�wrt�º¢¥os�y.�ª�� c\tMw�kkaoA¬ x8y c�v _aluy�nmc$b2bw�v�o�~�orxak8��v�_ac¶² !Ig¶¯°y lp¸Mc�o�¢ c$�rc$��{I.;&�3<(�ª�^-_aluy�luyµy _aoA¬�k2lmk2v _ac\¢¥osnmnpoA¬�lmkaq�^-_acMos��c$b~�

� ^ ½}Á � ½ � 9 2.�.i�4�(�����(����*,����������[�2���T��� Z �1����) Z � Z ��� Z ��b������,�F�#�'� Z ���g.,��� Z ���e32JL���" � .;&�3<(I� � S�� � � M�� � S��� � ÁzÁ�] � gjlm�scMk�w�y|c$vos¢3�Aw���lmws~anmc���»}¬cjy _8wsnpn�¬���l£v�c � � s .;&�3<((��3�J�� tGª�hzc4v-x8y-~ax8lpnu�K~}{lmk8�Rx8t4v lmosk)w�y cM�GxacMk8t4cjo�¢y c4v�y�H w N � M6w�k8�5w2y|cT�Gxac$k8t$c�o�¢Cy|c$v�yy� w o�¢��Aw���lmws~anpcTy-y|x8t�_�v _�wAv � .;&�3-(� � S�� ���� H w q � w �� � S�� �

� hzc$v�&�H�~ q��9~ (~EcP& �3q s 3 t (4ª � ce_8wX�rcK� .;&�3<(�� � S�� ���� H�~ q �9~ �� � S�� w�k8�LH�~hN � MKª� !\y�y xab2c-v _8w�v�¬�c-_8wX�rc~8xalpnpv�&RH w q�� w (Cy x8t�_�v _8w�v'� .;&�3-(� � S�� ���� H w q � w �� � S�� w�k8�gH w N � M�»snmc4v�x8y��ac4¦8kacyH w � �w�k���� w � � �

R¹¢-� w ?���nmc4vCx8y�t�_ao}ory c 3- XJ2� w ª�^-_ac$k9v _acM� cc4¡Rluy�v�y�z�� .;&�3e ¤(�y|x�t�_ev _8w�v�zKJ"� MKª � c-�Rc4¦�kac�H w � � � H w � s ztw�k��C� w � � �����" '& z)(;�/� w � s 3- tGª `}lmk8t4c+z J � MK»�¬c_8wX�rcgH w � � N � MKª hzc$v9x8y�y _aoA¬ v�_8wAv~� .;&�3-(�� � S�� ��� H w � � q � w � � �� � S�� ª#hzc$v��(� s 1 �g4 .;& �>(|8 �J�wJ2���" '& z)(�tGªK��{�w��a�anm{}lpk8q2v _ac ��or� ornpnuw���{)�eoskLH w � s ztK� � w �/3 ¢¥os�-v�_ac�y xa~8y|v lpv xRv�lpork��}ª § �5��) Z ��b ��z�� � .;&�3- ¤(|© � ceos~Rv�w�lmk��

�� H w ��q � w �� � S�� � ��� H w q � w �/3 q�z�q ���" '& z)( ��� � S��w�k��Kv�_ac$k

�� H w q � w �� � S�� ���� H w ��q � w �� � S�� ���� H w � � q � w � � �� � S��

�������3�J�9��

Page 15: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�Td �6� ����1���������������3����� Z �$�

N�lpk�w�nmnp{r»Cv _aluyt4ork8y|v ��x8t�v�lpork³v cM� b2lmk8wAv�cMy2y|lmk8t$c�� ������� � .;& �>(���y|v ��lut�v nm{³�RcMt$� cTwsy cMyMª�!v�v�_ac5c$k8� ¬�c5qsc$v � w � �,w�k��� .;&�3-(� � S�� � � H w � � S�� ¬�lpv _2H w N � M���y lmk8t4c�� H w � � S�� � � M�� � S�� v _aluy��a��oA�scTy�v�_ac�v _8c$os��c$b)ª �^-_acKt$osk8y cM�GxacMk8t4cos¢-^-_acMos��c$b­�lmy�v�_8wAv9v _ac�²"!Ig¶¯¹y lp¸Mco�¢-v _8cb�cTy y�w�qrcMyjv�_8wAv�ws� cKy c$kGv�or��� cTt4cMlp�rcM� �Rxa��lpkaq«w

kaor� bw�n�wAv v�wst�º�lmy¶~Eosxak8�acM�·~}{)w��ornp{}kaorb2lmwsn3lmk·v _ac�² !Ig y lp¸Mc9o�¢�v�_ac9�a��o�v�oRt4osnOªI^-_aluy\��cMy xanpv\_8wry\t4��x8t$lmwsn3�a��wst4v lut$w�nlmb��8nplut$w�v lmosk8y�y|lmk8t$c�lpv�b2cMwsk8y�v�_8wAv�¬�_ac$k#y|cTw���t�_alpk8q)¢¥os�2w�k w�v|v�wrt�º,¬c�tMw�k qrlp�rc�w y lpb2�anmc Z :1���2�����µ~Eosx8k8�³osk v _ac�awsq�¯°y|lm¸$cjo�¢�v _aceb2cTy y�w�qrcMy�kac$cT�RcM��v o2~Ec�¢¥os��qscT��~}{v�_acelpkGv�� x8�ac$�I�� Á � Á & &?�R� ¾ � .�.V�4�K� �3������*,����������[�2���M��� Z �1����) Z � Z ��� Z �b �����,��.,��� Z ���Fi�� t q�����,q�� �+� Y��_. � � S�� � � M�� �� S�� Z �-,� B � . � � S�� � � M�� �� S�� @� � ÁzÁ�] � � B � . � � S�� � � B � q s .;&�3-(�� 3 J ���" '&RB � (�t � � S�� ~}{���os��osnmnuw���{³�sª~Na��osbyh3c$b2bwFG5¬cKw�nuy|o·_8wX�rc�¢¥os�9wsnpnK3lJ���" }»5� .;&�3-(� � � M�� � S�� ªMhzc4vT� ���" ��3~EcKv _8c�k}xab�~�cM�9o�¢\�almy|v lmk8t�v��Aw���lmws~anpcTy�lmk³v _8c��a��o�v�o}t$osnµy �EcMt4lp¦�tMwAv lmosk3ªFXIcMk8t4c� B � . � � S�� � � B � � � S�� �+&�� M�� � S�� (�@� ���" ��µw�k8� y lpk8t$c � B � � � S�� � � M�� � S�� w�k8� � ���" 3� � tk� � M�� � S�� ¬�c)_8wX�rclmb�b2cT�Rlmw�v cMnp{U� BF�R. � � S�� � � M�� �� S�� ª�^-_acey wsb�c��a��o}o�¢�_aosnu�ay¢¥os��Y��R.�ª �

-�.� 132 � � � � � 6� ; ? 4 �+�=2 8�� �� �8����� 8 ; 8�� 4���47? ? 8�*;P?!8 ? 80;� 1� ce� cTt$w�nmn _acM� cjv _8c�®I�¨�a��o}t$cM�Rx8� c¶¢¥os�It�_acTt�º}lpkaq�v�_acec4¡Rluy�v�c$k8t$cjos¢�wsk)w�v|v�wst�ºKw�k��5y _aoA¬Iylpv�yIt$os��� cTt�v k8cMy�y$ª� c�wry y xab2c9qrlp�rc$k w��a� osv oRt4orn�y �EcMt4lp¦�tMwAv lmosk s &_^9q�Y� a Z[B; a ( �F^yJ�fatrª-hzc$v�] � s Yb a q�B; a �<^yJLf�tr»zw�y|cTt4��c4v¶b2cTy y�w�qrcB����� ��Iz w�k8�ew-¦�kal£v�c�y c4v�o�¢Rb2cMy�y wsqscMycB<~�¢¥or�zlpk8l£v�lmwsnrlmkGv ��x8�Rc$��º}kaoA¬�npcT�Rqscrª�R¹¢>]�q�B<~luyzkaosv3qslm�scMkelpk9²"!Ig¶¯O��c$�a��cMy c$kGv�wAv lmosk3»v _8c${)w���c�¦8��y|v\t4ork}�sc$� v cT�Kv�ov _8lmyI¢¥os��b2w�v § lmk«�Eosnm{Gk8osb2lmwsn v lmb2cT©4ª � c9wry y xab2c�v _8w�vIv _8c9² !Ig¶¯°y|lm¸$ceos¢;]�q�B<~9luy�nC»8v _ac¦8kalpv c�y c4vIos¢��Aw���luw�~anmcMylmkL] luyy�»8wsk8�IfO�V��ª

�sªIgjxacMy�yIw2t4or� ��cMt4vc$¡RcMt4xav lmosk5os���Rc$� r ��f �-s s"t q�������q��trªhzc$vWY���� Yb � �"� � � � w�k8�LBF���!B; � �"� � � � ¢¥os��iKJ s"t q�u����q��t

dRªIgjxacMy�yIw�qs��osx8k8��y xa~8y|v lpv xRv�lpork·y|x8t�_5v�_8wAv-¢¥os�¶w�nmnF32J2�K»>.;&�3<(-_8wsy�²"!Ig¶¯¹y lp¸Mc � nCª�aªaNaor��cTwst�_Pi J s"t q�������q��� t tjqsxacTy y�w�kKor���ac$��cM�2nmlmy|v �¤��o�¢Xn � ��xanpcTyµ¬�_8ory cI�a��lpk8t$lp��w�n�v c$��by_8wX�sce² !Ig¶¯°y lp¸Mc ��� � n � ª

�8ªaNaor��cMwst�_�i J s�t q����u�;q��1t�t�_acTt�ºv _�wAv�� � w��8�anplmcM��v o s B � . � zwTki�tK� s B ~ teqsc$k8c$��wAv cTy{Y � .GRª"��_acMt�ºKv _8w�v�� ��� � ws�a�anmlpcT�Kv�o s B � . � z T � � t tK� s B ~ teqscMkac$��wAv�cMy�B����� ��Iz�ªEaª�R¹¢�cTwst�_)t�_acMt�º�lmy�y x8t$t$cMy�y�¢¥x8n�v _acMk«w�k8y ¬�cM�´6<g`�ª

hzc$vIx8y¦8��y�vI��c$bws� º2v�_8wAv-v _8lmy��8� oRt4cT�Rxa��c�lmy�®\���� !Wt$os��� cTt�v�c4¡RcTt4xRv�lpork�or���RcM�-lmy�w2�Ec$��b9xRv�wAv�lpork�os¢-f�»8wsk8�)t$wsk�~EceqsxacTy y cM��lmk��Eosnm{}kaosb2luw�n�v lmb2csª� `}lpk�t4c�.;&�3<(�_8wsy�² !Ig¶¯°y lp¸Mc � nC»�orkac�t$w�k�t�_ao}ory cw�² !Ig[� cM�a��cMy c$kGv�w�v lmosk9os¢-.;&�3-(�lmk9v�lpb2c��g&�n (�w�k8� .�lmk��g&�n � (�ª� `}lpk�t4cjcMwst�_�� x8npcjlpk.�¤��_8wsy² !Ig¶¯°y|lm¸$c � n � wsk8�v�_ac$��cjlmyxn � � x8npcTy$»}osk8cjtMw�k�t�_8oGoGy|c¶cMwrt�_ �¤��lpk�v lmb2c��g&�n��G(�»aw�k��w�nmn �¤��lpk5v lmb2c�g&�n�� (�ª �6�,) Z ��b���cTwst�_�v c$��b�lmk5v _ace��xanmcMy-luy�lpk·² !Ig � cM�a��cMy c$kGv�w�v lmoskzª� ��osb2�axRv�lpkaqKv _ac�� cTy|xanpv�H� �o�¢�v�_ac�ws�a�anmlmtMwAv lmosk·os¢�w� x8npc !�$>&2z)(�osk�HK»�¬�lpv _�H¤w�k8�~z�lmk ² !Ig ��c$�a��cMy c$kGv�w�v lmoskz»t$w�k)~Ec9�Rorkac�lmk·��ornp{}kaorb�luw�nzv lmb�c9lpk)v�_ac�² !Ig¶¯°y lp¸Mceo�¢ H ~}{���cMb2ws� º)dRª��xRv�� H� %� � S�� � � H"� � S��3��� § y|lmk8t4cw�k}{«��xanpcws�a�anmlmtMwAv lmosk�lmkGv ��o}�ax8t4cTyewAv�b2ory|v�d5y|{}b9~Eosnuyelmk:H9©�w�k8�,v�_ac$��c4¢¥or� c2cMwrt�_,lmkGv cM� b2cM�almw�v cKy c4v�o�¢v cM� by_8wsy�² !Ig¶¯°y lp¸Mc ��� � n � »�w�k���cTwst�_)w��a�8nplut$w�v lmosk5� xanmc�v�wsºscTy-��ornp{}kaorb�luw�nzv lmb�c9lpkLnCª8`Ro8»8t�_acTt�º}lpkaq�v�_8wAv\wsnpn��¤�µw���ct4os����cMt�v�np{ews�a�anmlpcT��v�w�ºscTy���ornp{}kaorb2lmwsnrv�lpb2c�os¢enCª�^3o��scM� lp¢¥{�v _�wAv Y��R.�lmy�lmk�v _ac�nuwsy|v�y|c$v�os¢8v�c$��b2y�v�w�ºrcMy�os~}�}lporx8y np{��ornp{}kaorb�luw�nEv�lpb2c�v o}o�ª

� c�t$w�k5kaoA¬Wy|cMc¶v�_8wAv�v�_aluy��a��o}t$cM�Rx8� cjlmy\t4os����cMt�vI��lpv\w�k�y|¬c$��y�´6<g`5lp¢�wsk8��oskanm{Klp¢�v�_ace�a��o�v oRt$osn _8wry�w�k)wAv|v�wst�º1�� R¹¢Cwsk !v|v�wst�º�c$¡Rlmy|v�yIv _acMk w®Ior� bw�n�!v|v�wrt�º�c$¡Rlmy|v�y�8`}lmk8t$c - �:-��� i G � � & s B���. � z�Tli�t{� s B<~�tG(-_8wry\wAv¶b�oGy�vn � � xanmcMy § ����os�Eory l£v�lporkz��w�k8�^-_8c$os��c$b)�X©�»�cTwst�_�lpkGv cM� b2cT�Rlmw�v cIy c4vµos¢�v�c$��byC_8wsyµ² !Ig¶¯¹y lm¸$c ��� � n � ª�XIc$k�t4cs»rcMwrt�_v c$��b lpk}�rosnm�scM�Klpk5w���xanpc¶lpk.-6_8wry-² !Ig¶¯¹y lm¸$c ��� � n � ª�^-_acM� c$¢¥os��c�-�¬�lmnmnE~Ec�w9�Eory�y|lm~anmc\qrxacMy�y�¢¥or���¤� § wry¬�cMnpn�wsyv _acec$¡}cTt4xRv�lpork�or���ac$��wsy�y|oRt$lmw�v cM�Kv o2v�_almy�w�v|v�wst�ºa©�ª

¡�¢ �X¡¤£

Page 16: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� �M�

� R¹¢Rv _8cµ�a��oRt4cM�axa� c�wsk8y ¬�cM��y�´'<g`�»Mv _act�_acMt�º}lmkaq��Ec$� ¢¥os��b�cT��oskev�_acµqrxacMy�y cM���Rc$��lm�Xw�v lmosk8y3�a��oA�scMyEv�_8w�k�v _ac��a� osv oRt4orn_8wry�w�k)wAv|v�wst�º�ª

% ��& �� _ & ½ � ½ �T��� Á m��r� � _acMk1t4ork8y lm�RcM� lmkaq y c$�rc$��w�nI�8� osv oRt4orn¶y cMy�y|lmosk8y�y|lmb9xanpv�wskac$orx8y np{ ¬�c wsy�y|x8b�c«v _8w�v�v _8c${[w���clmk8�RcM��cMk8�awskrv § lmk·v _8c�w�~�y|cMk8t4c�os¢ClpkGv ��x8�RcM��©Iw�k8�)v�_ac$��c4¢¥or� ceorkac9tMw�k«wsy�y xab2cev _8w�v\v�_ac$lm�¶y ��cTt4lp¦�t$w�v lmosk8y¶�RoKkaosvjy _8w���c�Aws� luw�~anmcMyMª

R°kor���ac$��v�o���cM�Rx8t$c�v _8c¶y cMt4x8� lpv�{��8� or~anpcMb ¢¥or��y c$�rc$��w�n8y cMy�y|lmosk�y�v o9v _acjy|cTt4xa��l£v�{��a��os~anmc$b ¢¥os�oskac¶y|cTy y lporkz»sorkac\tMw�ky lmb��8np{)t4orb��Eory cev�_ac9y cMy�y|lmosk8yIlpkGv o�wxakalu�Gxac9y cMy�y|lmosk)o�¢�wb2or� c�t$osb2�anmc4¡)�a� osv oRt4orn?ª-^-_aluy\t$osk8y|v ��x8t�v�lpork)lmyjy|ºrc4v�t�_acM�~Ec$nmoA¬eª

gjlm�scMkv�¬o9�Rluy¥��orlpkGv���w�� v luw�nmnp{2os���RcM��y�T[wsk8�2Ty ��Rc$¦8kacM�Kosk��Rluy ��oslmkrv�y c4v�y{Q w�k��wQ� E¬�cj�Rc4¦8k8cIv _8c¶�8ws�|v�lmwsn8or���RcM�T��-Ty v�o~�cev�_ac���c$nuwAv�lpork/T �@Ty <� s�� T � �� � J�QUq � �J�Q� 2t (4ª-gjlm�scMk�v�¬o�a��o�v�o}t$osnzy ��cTt4lp¦�t$w�v lmosk8yW] � »e] � »�npc$vM � »>M � v _8c$lm��� cTy|�EcMt4v lm�scey|c$v�y-os¢��a��lpk�t4lm�8w�nuy$ª � c�t$wsk�wsy�y xab2cjx8��v�oy|orb�cj��c$k8wsb�lmkaq�v _�wAv-¢¥os��cM�sc$��{�a��lpk�t4lm�8w�n�x1v _8cwry y oRt4luwAv�cM�«��w�� v luw�nmnp{·or���RcM� cT� y c4v�y�Q �S w�k8�CQ �S lmk,�a��o�v�oRt4osn;] � w�k8�C] � � cTy|�3ª�ws� c�Rluy¥��orlpkGvTª2^-_8c$k�¬�c2qrc4v�w5kac$¬�8� osv oRt4orn�¢¥or�-v _ac�y cM�GxacMkGv luw�nzt4orb2��oGy|lpv lmoskL] � � ] � ~}{v�w�º}lpk8q

s YbacZ B a��9^DJ f�t¬�_8c$��cDfO� s &[x-q)i0(a��x�J\M � �WM � wsk8�wiKJLQ �S ��Q �S t�Naor�CcMwst�_��a��lpk�t4lm�8w�n3x¨¬c�wsy�y|oRt4luwAv�c�k8oA¬ v _8c��8w�� v luw�nmnp{9os���Rc$��cM�y c4v�&_Q �S �LQ �S qT U �V �9T U +V (� c�qrc4v�y lmb�lmnuw���np{�w«�a��o�v�oRt4osnK] � ��� ] � ¢¥or�9�8ws��wsnpnmc$nt4osb2�Eory l£v�lpork³~}{ v�w�º}lpk8q«¢¥os��cMwrt�_³�a� lmk8t$lp�8wsngx¤v _ac��8w�� v luw�nmnm{

or���ac$��cM��y c4v�&_Q �S �LQ �S qT U �V �kT U +V (4ª

� %'�178 ��e� ���5���3�

� c¶y _aoA¬¨kaoA¬ v _�wAv�v�_ac\c$¡}luy|v c$k�t4c\os¢zw�kKwAv v�wrt�º�¬�_acMkv _8c\lmka�axRv-w���cIw��a��o�v�o}t$osn8y �EcMt4lp¦�tMwAv lmoskKwsk8�2lmkal£v�lmwsn�º}kaoA¬�nmcM�aqscos¢�v _ac�lpkGv ��x8�RcM�9luy�®\�C¯¹_8w�����~}{,��cM�Rx�t�v lmosk�¢¥��osb+�A¯/` !C^�ª�^-_acK�a��o}o�¢�luy�y lpb2lmnmws�ev o v _ac�oskac�qslm�sc$k�~}{iAp�,C�¢¥or��v _8c$lm�b2oR�RcMn?ª

� ����os�Eory l£v�lpork8w�n���ws� luw�~anmcMy�� 3 � q�����q`3 ª� R°k8y|v�wsk8t4c�o�¢��A¯/` !C^ ���D& �s 3I( ����&�3�� �� �� % ��� 3���� +� % � � 3��

�� �� % � (¬�_8c$��c�� � % � J s ��q t t�wsk8�w3 ~ § ��cMy �zª 31�$©b�cTw�k8y{3 § � cTy|�3ª ��3�©�ªh3c4vIx8y��ac4¦8kac

� �F&�� � % ��q`3e� % � ( �l3-� % �elp¢�� � % �5� �2w�k8� s 3-� % � t � l£¢�� � % �6� t� �-i;J2m$q��G�)& �s 3�( � o �<&��G� % � q`3e� % � (,qo �<&��G� % � q`3e� % � (,q/�F&�� � % ��q)3-� % � (9r�rh3c4v-x8ylpkGv�� oR�Rx8t$c¶k8oA¬¨v _ac¶¢¥osnmnpoA¬�lmkaq��a� osv oRt4orn § ¬�_ac$��c\�Aws� luw�~anmcMy�oRt$t$xa� ��lmkaq�lmkKv _8c��acMy�t4��lp�Rv�lporkKo�¢�y�v�c$��& ��q|z"(�y _aosxanu�~Ec�t4ork8y lm�RcM� cT��wrylpk8�ac4¡RcM�5~}{�& �aq_z}(��av _acelmk8�Rc$¡�luy-osb2l£v v cT��¢¥or��� cTws�aws~almnplpv�{�©��

� ����lpk�t4lm�8w�n�!��0 &[x-q t (���o�3 � % � q�u�;q)3 % �r Z s o�� � & �s 3�(�qIo�� � & �s 3�(�qIo0���u�;qo�� & �s 3�(,q���nXo r�r9r�rt��� ����lpk�t4lm�8w�nX�'�0 &R��q`i%(a� s o/o! qo�3 q���r�r1q 6 r�t��CZ s 63t��0 ¢¥or� th� i � � �D& �s 3�(��

� ����lpk�t4lm�8w�nX� �;�0 &R�� �q)i0(8� s o9o s �" �t � qo�3 q���r�r1q 6 r�t��CZ s 6�t#�0 ¢¥or� th� i � � �D& �s 3�(��

� ����lpk�t4lm�8w�n��6�0 & �'q)i0(�� s o9o�3 qo! q9�3r/r�q 6 r�t#�:Z s 63t��

�������3�J�9��

Page 17: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�$� �6� ����1���������������3����� Z �$�

0 ¢¥os� th� i � � �D& �s 3I(�� ��� lmk8t$lp�8wsn�� ���

0 &7� q`i%(8� s o9o�3 qIo s �" �t � q9�3r9r1q 6 rt � Z s 6�t �0 ¢¥os� th� i � � �D& �s 3I(�

� ��� lmk8t$lp�8wsnz²��0 & -Dq)i0(8� s o/o�3�qo2��q� r9r1q 6 rt#�:Z s 6�t#�0 ¢¥os� th� i � � �D& �s 3I(�

� ��� lmk8t$lp�8wsnz² �;�0 & -w [q`i0(8� s o�o�3 qIo2�1q s �" �t � r9r1q 6 r�t#�:Z s 6�t#�0 ¢¥os� th� i � � �D& �s 3I(�

� ��� lmk8t$lp�8wsn <��0 &�H�q t (8� s ��nXo3t � ZdB����� ��z

� c9v�w�ºscgB<~ � s � ��� q q �" �t9¢¥os�¶v _8c�lmkalpv luw�n�lpkGv ��x8�RcM�¶º}kaoA¬�nmcM�aqscsª XIcMk8t4c9v _acM� c�luy¶wsk wAv v�wst�º)ork·v _aluyj�a��o�v oRt$osn3l��v _8c�b2cMy�y wsqsc�y c$kGv\~}{��a��lmk8t4lm�8w�n ! t$wsk)~�c�� cT�Rx8t4cT��v o s �nXo�t#�#lOª crª�¢¥or��wsnpn i�»8v _acM� c9c4¡Rlmy|v�yxzKy x8t�_)v _�wAv��F&�� � % � q)3e� % � (yJs q s �" �t � tr»ClOª csª«v�_ac5lpkGv ��x8�RcM�9_8wry9qrlp�rc$k�v�oPx w)v cM� b+��c$�a��cMy c$kGv lmkaq w,y|ornpxav lmosk³o�¢\��¯�` !C^ §-s �" �t � � cM�a� cTy|cMkGv �©�ª XIcMk8t4c�v _8c9�a��o�v�o}t$osn�ws�Rb2lpv�y�w�k w�v|v�wst�º5l ��v _8c�t$os��� cTy|�Eosk8�alpkaqK��¯�` !C^��a� or~anmc$b _8wsy¶w�y osnmxRv lmosk3ª'S·or� cMoA�scM�-v _ac��cM�Rx8t4v lmosk5lmyIos~}�}lporx8y|nm{�Eosnm{Gk8osb2lmwsn?»R_8c$k8t$c¶v�_ace�a��os~anmc$b¤o�¢�¦8k8�Rlmkaqw�k·wAv v�wrt�º¬�l£v�_5~Eosx8k8�RcM�)y cMy�y|lmosk�y�luyI®\�C¯O_�w���� ª

Na��osb�v _ace��cMy xanpv�y�w�~EoA�scj¬c¶¦�k8w�nmnp{�t4osk�t4nmx8�Rce¬�lpv _5v _acebwslpk5� cTy|x8n£vI�� ^ ½}Á � ½ � � !#���-,����"* Z � Z �[� Z �b5.,��� Z :������%�I���������� � Z � � ��,K����)+*�,�-�9. ��������2����� �4� Z � &6��79����)�:1�����%�a:�����*,�;�,) @

� ���5���$�����4���

� c\_8wX�rcI�a��oA�scM��v�_8wAv-¬�_ac$kKv _ac¶kGx8b9~Ec$�o�¢3y cMy�y|lmosk8y�lmyµ¦a¡RcM� »}lmkKos���RcM��v o�¦8k8��wskKwAv v�wst�º2w�k�lpkGv ��x8�RcM�µkacMcM�ay�osk8np{�v o¢¥os��qscµb�cTy y�w�qrcMy3¬�l£v�_9�Eosnm{Gk8osb2lmwsnry lp¸Mcs»A¬�_acMk�x8y lpk8q¶w�v�_ac-�aw�qs¯O��c$�a��cMy c$kGv�wAv lmosk3ª � c�_8wX�scµwsnmy o\qrlp�rc$k�wsk9®I�C¯¹�a��oRt4cM�axa� c¢¥os��¦8k8�Rlmkaq«w�k³w�v|v�wrt�º ¬�lpv _ w5¦a¡RcT�,k}xab�~�cM��o�¢Iy cMy�y|lmosk�y$ª � x8��¢¥os��b2wsn�b2oR�Rc$n�os¢��a� osv oRt4ornmyewsk8��wAv v�wst�ºRyey xa�a�Eos� v�ykaorkR¯¹w�v orb�lutjy|{}b2b2c4v ��lut�ºsc${RyMª `}c$�rc$��w�n�lpkGv�c$��cMy|v lmkaq��Aws� luw�kGv�y-os¢3v�_almyIb�oR�RcMn3t$wsk�~EcecMwry|lmnm{K� cT�Rx8t$cM��v o2l£vTª

R¹¢Ev _ac�lmkGv ��x8�Rc$�Cluy�wsnpnmoA¬�cT�ev�oeqsc$k8c$��wAv c-w�k}{�k}xab�~�cM��o�¢Ekac$¬¨�aw�v�wsyM»Av�_ac$k2¬cIt$w�k��a��oA�sc¬�l£v�_2v _acIy�w�b2cv cTt�_akalu�GxacMyv _�wAvIw�k}{wAv v�wst�º_8cjtMw�k�nuw�x8k8t�_�tMw�k�~Ecew�nuy|o�os~av�w�lmkacT�K~}{� cM�anuwst4lmkaq�lpk5c$�rc$��{�b2cTy y�w�qrc\_aluy�¢¥� cTy|_anm{qscMkac$��wAv�cM�~}{K_8lmyk8wsb�c § ��_8w���nm{R©4ª�^-_8cIkaos��bw�nR�a��lmk8t4lm�8w�nuyC¬osk�� vµy c$cIw�k}{��Rl ��c$��c$k�t4csª�XIcMk8t4cIv _ac�lmkGv ��x8�Rc$���Ro}cMyCkaosvµqrwslpk2w�k}{�y�v�� cMkaq�v�_¬�_acMk5~Ec$lmkaqw�~8npc�v�o2t$� cTwAv�cjk8osk8t$cMyM»Rlpk�v _ac�¦�kal£v�c9y cMy�y|lmosk5t$wsy csª

Naor�9lmk8y|v�w�k�t4c�¬c�_�wX�sc�t$osk8y lm�ac$��cM� v _�wAvw«�a��lpk�t4lm�8w�nlmy�xak8ws~anmcKv o,� cTt4orqskalm¸$cv�_8wAv2w«b2cMy�y�w�qsc�y|xa�8��oGy|cT��v o,~�cc$k�t4��{G�av cM��~}{�y osb2c-ºscM{��1_8wsyC��cMwsnpnm{�~Ec$cMk2t$osk8y|v ��x8t4v cM��~}{�wsk�cMk8t4��{}�Rv lmosk2¬�lpv _��2» § y|cMc�c4¡aw�b2�anmc�lpk !\�a��cMk8�Rlp¡���ªp�X©�ª^�o9os~av�w�lmk�we�8� osv oRt4orn8b2oR�Rc$nE¬�_ac$��c\�8� lmk8t4lm�8wsnmyµb2wX{2��cMt$osqsk8lp¸Mc�¬�_ac4v�_ac$�-w���cMwsn8c$k�t4��{G�av lmosk_8wryµ~Ec$cMkK��cM�|¢¥or� b2cT�oskacy lpb2�anm{�c$¡Gv�c$k8�5w�k}{�t${}�a_ac$��¬�lpv _·w2y ��cTt4luw�n ¦a¡RcT�K¦8cMnm�zª� c9_8wX�rc�t4ork8y lm�RcM� cT�5v _8w�v¶v�_ac�lpkGv ��x8�RcM�jtMw�k«cTwX�scTy�¯°�R��os�z»8�alp�rc$� v¶b2cMy�y wsqscTy$»�w�k8�«lpb2�Ec$��y|ork8wAv�c�o�v�_ac$�j�a��lpk8t$lp��w�nuy$ª

XIoA¬c$�rc$�C¬c\t$wskb2o}�ac$n�web2os��c��8wsy�y|lm�sc�lmkGv ��x8�RcM�M»}wsy��acMy�t4��lp~EcM�2lmk�!I�8��cMk8�Rlp¡2��ª da»s~}{2c$k8y xa��lpk8q�v _8w�vy osb2c�b2cMy�y wsqscMyt$wskakaosv-~Eceb2oR�Rl£¦�cM� § ¢¥or��lpk8y|v�wsk8t4ce¬�_8c$k�v _acM{�w���cet4orkG�rc${rcM�~}{�w2y�wA¢¥cet�_8wskakac$n ©�ª� c_�wX�sc-t4osk�y|lu�Rc$��cM��y cMt4��cMt${e�a��os�Ec$� v lmcMyMª `}lmk8t4c�t$os��� cTy|�Eosk8�8w�k8t$cw�v|v�wrt�ºRy�tMw�k2w�nuy|oj~Ec�c4¡R�a��cMy�y|cT��~}{9w�k�c4¡RcTt4xRv�lpork

os���RcM�-wsk8�)w���ornp{}kaorb�luw�n k}xab�~�cM��o�¢ 3({ ���9t$osk8y|v ��w�lmkGv�y�v _8c${�tMw�k5~�c��ac4v cTt�v�cM�5lpk·®\�#v�o}o8ªN�lpk�w�nmnp{�osxa�C�a��oRt4cT�Rxa��c-tMw�kw�nuy o¶~Ec\ws�8w��Rv�cM��v oe�a��o�v�oRt4osnuy�ws�Rb2lpv|v�lpkaq�t�_8oslut4c��Eoslmkrv�y$»r¬�_ac$��c�we�Rl��EcM� cMkrvµy xa~a�8� osv o�¯

t4ornEtMw�k~Ec¶c$¡RcMt4xav cM�K~G{wex�y|cM�wrt$t4or���alpkaq�v�o�y osb2cI��cMt$c$lm�scM�2b2cTy y�w�qrcsª���� osv oRt4ornmy�y|x�t�_Kwsy�`3`�h wr�Rb2l£v-t�_aoslut4c¶��orlpkGv�y$ª^-_ac�b2o}�al£¦�tMwAv�lpork«o�¢�orxa�¶b2oR�RcMn�luyj�RcMy�t4��lp~EcM� lpk !I�a�Ec$k8�al£¡ ��ª �8ªj^-_8c��Rc$v cMt4v lmosk,o�¢µw�k wAv v�wst�º5��c$bw�lmk8y¶lpk ®I��ª � ct$wsk5y xab2bw���lp¸McM��v _aceº}kaoA¬�k5��cMy xan£v�y-lpk5v�_ac�¢¥osnmnpoA¬�lmkaq�v�ws~anpc��

¡�¢ �X¡¤£

Page 18: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� �IG

� l£v�_aosxRv\®\osk8t$cMy � l£v�_·®Iosk�t4cMy®Io2~Eosxak8�8y(A L�C k\k8�RcMt$lm�8w�~anmc k\k8�RcTt4lu�aw�~anmc

��orxak8�RcT��b2cTy y�w�qrcMy'A L�C ²5< Q¶�µ^�R/ST< k\k8�RcTt4lu�aw�~anmcN�lmkalpv c9y cMy�y|lmosk�ywsk8�5t�_aoslut4c��Eoslmkrv�y ®\�C¯(��osb2�anmc4v c ®I�C¯(��osb2�anmc4v�c

N�lpk8l£v�c�y|cTy y lpork8y ®\�C¯(��osb2�anmc4v c ®I�C¯(��osb2�anmc4v�c²¶lm� cTt�v lmosk�yµ¢¥or�µ¢¥xav xa��c�¬�or� ºRy�lpk8t$npx��Rc¶~8� oGws�RcMkalpk8q�v _8c�y�t4or��cjo�¢�osxa��w��8�a� oGwst�_2v�o�y osb2c�t$wsy cMy�¬�_acM� c¶v _ac�k}xab�~�cM�

os¢�y cMy�y lpork8ylmy-xak}~Eosxak��RcM�5os�-v�ot4osb2b�xRv�w�v lm�Glpv�{Kos¢�cMk8t4��{}�Rv lmosk5os�Ec$��wAv�os��y$ª

� % :�: ��� �5� $ .0/ �!; ���=4�;=4�� 47? ? 8� � ������� 2 �P6K47?R¹v�¬-wsy¶kacMt$cMy�y ws� {�v o�lpk8t$npx��Rc9v�_ac�! $ ��xanmcMy¶lpk,os���Rc$�\v o5_8w�k��Rnpc��a��o�v oRt$osnuyIo�¢µv _ac�¢¥osnmnpoA¬�lmkaq�v�{G�Ec § ¬�c�_8wX�rc�osb2lpv|v cT�i|nFi%z�wsk8� �nXoKb�cTy y�w�qrcMy�©��

R°kGv ��x8�RcM��lpkalpv luw�n º}kaoA¬�nmcM�aqsc�� s B����� ��Izt �&)&[x-q t (�» s 3�t � �"� Z 3-(&)&[x-q���(�» s s ��t�� t � Z �>(

^-_8ce�a� osv oRt4orn ws�Rb2lpv�y-v�_ac�¢¥osnmnpoA¬�lmkaq2wAv v�wst�º�¬�_ac$k�v _acelmkalpv luw�nzº}kaoA¬�nmcM�aqscjo�¢�lpkGv ��x8�RcM��lmy s ���9�" &� �1q s B����� ��Izt � t��s B����� ��Izt#�:s s s�s B����� ��Izt#�#t � t � �"�{s s�s B����� ��Izt#�gt � sdB����� ��Iz

`Rx8t�_·w�k)wAv|v�wst�º�tMw�kak8o�v�~�c�¢¥orxak8��l£¢�v�_ac�! $ � x8npcTy�w���cjk8o�vIlmk8t4nmx8�RcT��lmk�v _acelmkGv ��x8�Rc$����c$¬���lpv cey|{Ry|v cMb5ª . - �Q809 8�� 8 ; ���=4�80; � 2 �-�=4 2R°k#v _8lmyy cMt4v lmosk ¬c·y|_8oA¬6_aoA¬6v o���cM�Rx�t4c�v o�osxa�2b2oR�Rc$n�osv _acM�2b�oR�RcMnmy�¬�_acM� c�v _8c·lpkGv ��x8�RcM�2lmy�xak8ws~anmc § ¢¥os�y osb2cb2cTy y�w�qrcMy�©Cv�ocMwX�scTy�¯°�R��os�z»G�Rlm�sc$� v\w�k���b2oR�Rlp¢¥{s»aor��lpb2�Ec$��y|ork8wAv�csª^�o��a��c$�rc$kGvCv _8c\lmkrv�� x��Rc$�µ¢¥� orb cTwX�scMy|¯°�R� or�a�almkaq�~�c$v�¬�cMc$k2v�¬o9y|v c$��yW&2x�q)i0(�� ���u�}Zd� � wsk8�2&2x�q_z}(��I�5�WZ��u���m»r¬c

lmkGv ��oR�Rx8t4c�wKkacM¬ y|{}b2c4v�� luteºscM{�] orkanm{�º}kaoA¬�k·~}{Tx wsk8���¤wsk8�)kacM�sc$�¶�axa~anmluy|_acT� � � lpv _«v�_ac�y|v cM�8y &2x-q`i0(5�:�����eZs o2x�q�� � rtJu� w�k8�5&[x-q|z"((� s o2x-q��:��r�tJu� Z ���u�m»�v _ac�lmkrv�� x��Rc$�j¬�lpnmn�kac$�rc$�j~�c2w�~8npc�v o�lpkGv cM��t$c$�Rvjv _8c�b2cMy�y�w�qsc � � wsk8�y c$k��5y osb2c$v _almkaq2c$nuy|cjv�og� lmk8y|v cTws� »acM�scMk�¬�lpv _5v _8c�! $ � x8npcTy$ª^�oI�a��c$�rc$kGv v _8cµlmkrv�� x��Rc$�3¢¥� orbW�Rlm�scM�|v�lpk8qIw�k8��b2oR�Rlp¢¥{Glmkaq\v _ac�b2cTy y�w�qrcC~�c$v�¬�cMc$kev�_acCv�¬o\y�v�c$�8y�w�~EoA�scr»M¬cµt$wskex8y cµw

k8c$¬ �a� lm�AwAv�c�ºsc${��PS % ��lmk8y�v�cMwr��os¢<]2»}w�k��9v�_acIy|v c$��yW&2x�q)i0(8�g�u���}Z s o2x�q�� � rt w � V � w�k8�L&[x-q_z}(�� s o2x-q�� � r�t w � V � Z �u���pª^-_8lmy¬-wX{s»}l£¢�v _ac�lmkGv ��x8�Rc$�º}kaoA¬V� � �S % � v _8c$k5_ac�ºGk8oA¬Iyx� � »R~axRv�_ac�t$wskakao�v-b2oR�Rl£¢¥{�l£vT»Rc$�scMk�¬�lpv _�v _ac�! $ ��xanpcTy$ª�!\k8�y lmk8t4c(� S % � luy-oskanm{�x�y|cT�K¢¥or�-v _aluyIy|v c$�3»Rv _acelmkGv ��x8�RcM�It$w�k�� v�x8y cew�k}{Ko�v�_ac$�Iy|v or� cT��b2cTy y�w�qrcsª^�o5�a� cM�scMkrv¶v _ac2lmkGv ��x8�Rc$�¶¢¥��osb lmb��Ec$��y osk8w�v lmkaq�w�b2cMy�y wsqsc � y|cMkrv�~}{)v _ac2�a��lmk8t4lm�8w�n�x v oL�K»z¬c2wsy�y|xab2c9v _8w�v

v�_ac$��c·c4¡Rluy�v�yKw��a��lp�Aw�v c)ºscM{ � osk8np{ º}kaoA¬�k¨~}{ v�_ac·�a��lpk�t4lm�8w�nuyKw�k��#k8c$�scM��axa~anmluy|_acT� § l£v�kac$�rc$�Kw��8��cTw���y2lpk v _8ct$oskGv cMkGvos¢�w�b2cMy�y�w�qscX©�»Rw�k��¬�c�wsy�y xab2cIv _�wAv��f���-luy�º}kaoA¬�k�~}{�cM�scM� {}~Eo}�a{sª�XIcMk8t4c¶v _aluy-w�b2osx8krv�y�v�o���c$�8nmwrt4c���~G{s o2x�q�� rt w � lpkMx wsk8����� y�y|v cM�8y��j^-_ac�lpkGv ��x8�RcM�jwsk8�·v�_ac��a� lmk8t$lp�8wsnmyjt$wsk ��cMwr��� » ~axRvjv _ac2lmkGv ��x8�Rc$�j¬�lmnpn�kacM�sc$�j~Ecws~anmcIv o�~axalmnm��w�b2cMy�y�w�qsc\lpk�v _acjk8w�b2c¶o�¢ x�»RcM�scMkK¬�l£v�_v _8c ! $ ��xanpcTy$ª § ��xav_acjt$wskKx8y c\osnu�"x � y�b2cMy�y wsqscTy�lp¢3¬�c¶x8y cM�v�_ac�y�w�b2c�ºsc${��,©

^-_8cjk8c$¬W�a� osv oRt4orn�_8wsy�w�nmlpkacTw��Iy lp¸Mc�lpk5v _8celpkalpv luw�n3oskacr»Rc$�scMk�lmk·w2�aw�q2��c$�a��cMy c$kGv�w�v lmoskzª

.�� � ��P80; � � �*8 � 4�� �*80; � ?� c\c$¡}v c$k���v _ac¶�a� osv oRt4ornab2oR�Rc$n�lpkor���RcM�Cv o�w�nmnpoA¬#¢¥os�t�_aoslut4c��EoslmkGv�yMª�^�{}�alut$w�nmnm{�v _8cI¦8cMnm�2os¢ w�b�cTy y�w�qrc�bwX{�t4orkGv�w�lmklmkR¢¥or� bwAv�lpork ws~�orxRvv _8c·v�{}��c·os¢�t$� {}�Rv�osqs��w��8_G{³k8c$qsosv luwAv�cM� ¢¥os�v�_ac·� cTy�vKos¢¶v�_ac«y cMy�y|lmoskzª@X\c$k8t$c·v _8c«y|xa~�y|cT�rx8c$kGvb2cTy y�w�qrc¶c$¡Rt�_�w�kaqrcMybwX{��Rc$�Ec$k���¢¥��osb�v _ac�t$oskGv cMkrv�os¢�v�_almy�¦8c$nu� ª� c�y _8w�nmnzt4ork8y|lu�RcM��8� osv oRt4ornz�RcMy�t4��lp�av lmosk8y¬�_acM� cey osb2cey�v�c$�8y�&[x-q`i%(bwX{~Ec�t4orb��Eory cM��~}{��a��lmos��l£v�{~anmoRt�ºRy �&[x-q�&�iq �"()(�� Y ~� Z[B ~�&[x-q�&�iq t ()(�� Y-�� Z[B#��ªmªmª

�������3�J�9��

Page 19: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�E �6� ����1���������������3����� Z �$�

&2x�q&�i�q�>()(,� Y �� Z[B ��^�¬�o5y�v�c$�8y¶lpk,v _8c�y�w�b2c9�8� lmos��l£v�{5~anmoRt�º«t$w�k8kao�vj~�c2w��a�8nplmcM� lpk v _acy�w�b2c9c$¡}cTt4xRv�lporkz» w�k8���'z�q_z" » &2x-q�&�iq_z}(`(hT��<V

&2x�q&�i � t q_z ()(4ª�^-_8lmyj~anmoRt�º)t$osk8y|v ��x8t4v lmosk«luy¶y lmb�lmnuw��¶v o)w�t$wry|c�y�v�� x�t�v x8� c�lpk«�8� orqs��w�b2b2lpkaq�nmwskaqsx8wsqscTy$ª ! �a��o�v�oRt4osnc4¡RcTt4xRv�lpork,¬�lpv _�y|x8~8y�v�l£v�xRv lmoskC.#b9x8y|vekaoA¬ w�nuy o5y�wAv�lmy|¢¥{1�'N8os�ec$�rc$��{)�a� lmos��lpv�{·~anpoRt�º § lpk8�ac4¡RcM� ~}{2i�©�»z¬cw��a�anm{ y|v cM�&2x�q&�i�q|z}()(lp¢ �>z�» � �}»>Y � � .5?� Y ~� ��w�k8�,ªpªmª�w�k��\Y � � . ?�lY � � �� �}ª

Naor�v cMt�_8kalmtMw�n���cMwsy osk�y$»G¬�c�lmkGv ��o}�ax8t4c¶¢¥os��cTwst�_��Aws� luw�~anmc�33»aw�v cM� b ���e�" &� � $9v _8w�v\�Ro}cMykaosv�ws�a�EcMw���wskG{}¬�_acM� c¶lpkv _8c9�a��o�v�o}t$osn3�acMy�t4��lp�Rv�lporkz»av�_8wAvjlmy\lpk8l£v�lmwsnpnm{�º}kaoA¬�k·orkanm{�~}{�v _ac�lmkGv ��x8�RcM�¶w�k��·y|x8t�_)v�_8wAv+� ���e�" &� � $�� � �8ªI^-_ac$k3»EwskwAv v�wrt�º9luy�kaoA¬ qslm�scMk�~}{2w�k2os���RcM��¬�_alut�_lmy�t4osb2�8w�v lm~anmc�¬�l£v�_v _ac\qslm�sc$k2�8ws�|v�lmwsnaos���RcM�µw�k8�2w�y xa~8y|v lpv xav lmoskP.)�scM� lp¢¥{}lpk8qv _8cIw�~EoA�sc�y�v�c$�8yµt4osk��Rl£v�lpork8yM»Av _ac\x8y|x�w�n9Y���./J.3({ ���}&RB<~ .1q����;q�BF� � � .X(Cw�k��gB �2�� J�IzKJ.3({ I���}&RB<~�.1q�����q�B .X(Ct$osk8�Rlpv lmosk�y$»w�k��·¢¥os��w�nmn��Xws� luw�~8npcTyW3�»����9�" &� � $�bwX{5orkanm{«w��a�EcMws�\lmk�.;&�3<(�ª9^-_8lmyjnuwsy|v�t$osk8�Rlpv lmosk,�Ro}cMy¶kao�v�� cTy�v�� lut�v¶v _ac�lmkrv�� x��Rc$�y lpk8t$c�_acelmy-k8o�v�¢¥os��t4cT�Kv�o2x8y|c����e�" &� � $8ª

N�lp��y|vj¬c�tMw�k,� cMb2ws� º�v _�wAv�w�nmn��a� or��cM�|v�lpcTy$»�nmc$b2bwsyM»zw�k8� �a��oGos¢ y�w�~EosxRve�Rc$��lm�Xw�v lmosk8y¶��c$bw�lmk �Aw�nmlu� y lpk8t$c�¬c2�Rlm�kaosv�t�_8w�k8qsc�v�_ac\lmkGv ��x8�Rc$�µ� xanmcMyMªC^-_ac$��c4¢¥or� cr»�v�_acIosk8np{2�a��o}o�¢Ev o9~Ec¶wr�aw��av cM�2luyCv�_ac\orkacIos¢ hzc$b2bw�G3� � cI~axalmnu�Kw�kacM¬y xa~8y|v lpv xRv�lpork�.F �¢¥� orb .¨»�wsk8�·¬c9b�x8y�vj�a��oA�scev�_8wAvjl£vey�v�lpnmn�y w�v luy�¦8cTyIv�_ac9��cM�Gxalm��c$b2c$kGv�yI¢¥os��w�k w�v|v�wst�º�ª¶^�o��Ro�v _8w�vM»¬c�wry y xab2cj¦8��y�v-v _�wAv ���e�� &�2� $ lmy-x�y|cT��lmk8y|v cMwr�5o�¢ ���e�" &� �K¢¥or�y.;&�3<(-lpk5v�_ace�a��oGos¢�os¢3v _8cenpcMb�bw8ª� c«_8wX�sc�Y � .F J 3({ I���}&RB ~ .F [q�u�;q�B � ��� .F ¤(Kwsk8� B����� ��Iz�J 3({ I���}&RB ~ .F �q�����q�BF'.F ¤(�ª !Ik�� ¢¥os��cTwst�_±wAv v�wrt�º y|v cM�

r � � &[�>(~��&2x�q&�i�q|z}()(�»�¬c)_8wX�sc2Y � � .O?� Y ~� � w�k��1ªmªmªUw�k8��Y � � .O?� Y � � �� � ~}{ �Rc4¦8k8l£v�lpork os¢b.�ª hzc$vKx8yKy _aoA¬¤~}{t4orkGv ��ws�Rlut�v�lpork2v _8w�vµv _8c$��cIlmy�kao �ey|x8t�_Kv _8w�vxY � � .F �� Y � �� �j¢¥os�y|orb�c{z" FTCz�ªC^�¬�o9tMwsy cMy�w���cI��oGy y lm~anpc���R¹¢X3)w��8��cTw���yClpkY � � »�v�_ac$k ���e�" ��2� $ew��8��cTw���yClpkPY � � .F Ew�k8��lmkPY � �� �}ª�^-_8c$��c4¢¥os��cs»r¬�c�_�wX�sc�Y � � .~� Y � �� �G R¢¥or���G acT�Gx8w�nRv�o �j¬�l£v�_ ���9�" &� � $��c$�anuwst$cM��~}{�.;&�3-(�ª;�xRv�v _8lmy�luy�lpb2�Eory�y|lm~anmc�~}{��Rc$¦8kalpv lmosk�o�¢e.�ª�R¹¢93�Ro}cMy�k8o�vCw��a�EcMws��lmk Y � � »Av�_ac$kgY � � .T� Y � � .F ��lY � �� �w�k��¬�cjw�nuy|o9_8wX�sc\w�t4orkrv���wr�Rlut�v lmosk3ª�^-_aluyµ¬-wX{s»G¬c\_8wX�rcI�a��oA�scM��v�_8wAv�.F ��Rc4¦8k8cMyw�k�w�v|v�wrt�º�¬�lpv _�v _acjy wsb2c\c4¡RcTt4xRv�lporkos���RcM�-wsk8�5¬�_alut�_�luyIy b2wsnpnmc$�v�_8w�k2. § y lpk8t$c"� ���e�" &� � $������8ª�^-_8cenpcMb�bw�¢¥ornpnmoA¬Iy$ª

`}lmk8t4c�w�nmn3~Eosxak��ay\ork �RcM� lm�AwAv lmosk�y\w�k��·wAv|v�wst�ºRyI��c$bwslpk·�Aw�nmlu�·¬�c�oskanm{5kac$cT�)v o5ws�a�)v�oKosx8�\lmk8y|cTt4xa��lpv�{)�ac4v cTt�v�lpork�a��oRt4cM�axa� cr»µw�k c4¡}v���w,qsxacTy y lmkaq,y|v cM�#¢¥or��v _ac)~a��w�k8t�_8cMy�v o�~Ec�v�w�ºrc$k#w�vKt�_aoslut4c5��orlpkGv�y2lpk os���Rc$��v o³�RcM� lm�sc)w�k¨®I��a��oRt4cM�axa� cj¢¥or�-v _ac�b2os��c�qsc$k8c$��w�n t$wry|c�o�¢��a��o�v�o}t$osnuy¬�l£v�_)t�_8oslut4csª

=.� ��� &<+��

� 4 47?!132�� � � � �*6��������� ������������� ��� ��������� ��� ��!�" � �$#����%'& ���)(*��+-,.����!���/ 0����2143

56%*�7*�8#6/9#6��"9�:<;>=?;A@CB>D ���"<�( � � ; (�+8� ; ( ��+ B �6�6���8����"9#�����& ���<�E ;GF � ;HF + ;HIJB 7 D ��+ ��"<�

(�7���K0���%�L*�:MB>=<;N@O; ( � �$�=PBQ@R; (�+8�$�@CBS:T;G=?; ( � � ; (�+8���

E ��������L ���U 3 :WVXY=PB E ;H:<;G=?;NZ�F � ; E ;H:<;G=K[ ( � �\ 3 =]VX)@CB E ;H:<;G=?;NZ�F � ; E ;H:<;G=K[ ( � � ;NZ�F + ; E ;G:T;G=K[ (�+9�^ 3 @_VXY=PB E ;`Z�F � ; ( ��+ [ (*��� ;`Z�F + ; (*�+ [ (�+8�a 3 =]VX�:MB E ;`Z�F � ; ( ��+ [ (*���b<3 :WVXY=PBAZI�[ ( ��+

hzc$vx�yµ¬���l£v�cIv�_ac � v�¬-wX{1pI��c$cTyC�a��o�v�o}t$osn�y|�EcMt$l£¦Et$wAv�lporkK¬�lpv _�orxa�µk8o�v�w�v lmoskzª�N8os�-y|lmb2�anplut4lpv�{2¬�cj¬���l£v�c�� q��� [q��� 8¢¥os�o9o|� q��� ;r1q��� 4r� &`&2x-q t (4» i|nFi%z Z oR� q�x-q`��r1q s ��S8qo_� q�x�q`� r�t � V�c ©

&`&���q t (4» o�3e��q)3 � q`��r1q)3 � Z 3e� q)3 � q`�"q)3 � q s � � q)3e��q)3 � q��+t � � c ©&`&RB�q t (�» 39d q�x-q`�"q s 39e q`3Kd�q�x�q`�+t � V�c q s 3Kf�q)39d q�x-q��+t � � c Z 3Kd q s 39e q���g ' t � V�c q s 39f q���g ' t � � c ©&`&���q���(4» 3 � q`3 � q s �I��q)39h�t � � c Z 3 � q)3 � ©&`&2x-q���(4» � q s � S q)3 � t � V�c Z s B����� ��Izt $ � ©&`&���q � (4» s B����� ��zt $ji Z ��nXo ©  

¡�¢ �X¡¤£

Page 20: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�������������� ������������������������ �"!#���$���%�'&(��)+*�,�-�/.(01�,�����2���3�(�4�5&6�879���)5:��;�,��� �Jj

!\kc4¡RcMt$xRv lmosktMw�k�~�cIor~Rv�wslpk8cM��~}{�v�wsº}lpkaqjv�_acI�a��o�v�o}t$osnay|v cM�8yClmk�v�_acIqrlp�rc$k�os���Rc$��wsk8��~G{9v�wsºGlmkaq�v�_acIy|x8~8y�v�l£v�xRv lmoskluy�3 � � � g ' »93-�'� � »>3 � � x�»e3 � � s o�� S qIoR� qo[x-q�� r�r9r,t � V�c »93 � � s 3 e q�� g ' t � V�c »93 h � � g ' »>3 d �!� »e3 e � � S »3 f �l� � ª�!Ik�wAv|v�wst�º2t$wsk~�c¶��cM�|¢¥or� b2cM�2orkv _8lmy��a��o�v oRt$osna¬�lpv _Klmkalpv luw�nElpkGv�� x8�ac$�µº}kaoA¬�nmcM�aqsc �DB ~ � s ���e�� &�2�$tG»sx8y lmkaqv�_ac�y xa~8y|v lpv xRv�lporkC1 3 � 4 oR� qIo2x-q�� r�r|8�wsk8��v _ac�y|v cM�8y�&2x�q t (�» &[x-q ��(-y|lmk8t$c��

o_� q�x-q`��r1q s ��S8q�� q�x-q`��t � V�c s���Gs���Gs� � � q s ��S8qo_� q�x-q�� r�t � V�cwsk8� s B����� ��zt ��� % S % ��� qo_� q�x-q�� r1q s �IS#qIo_� q�x-q`��rt � V�c s���Gs���2B �2�� J�Iz

.� 1 2�� ��� ����� 479 9:< /gjlm�scMk5wy c4v�os¢�v�c$��b2y�HK»8w��Xws� luw�~8npc�3 w�k���w�b2cMy�y wsqsc'z�»a¬cj¬-w�kGv-v oy _aoA¬(��� Hw1 3�4 z|8�� � S�� � � H+q�z�� � S�� �

h3c4v-x8yµ¦8��y|v-� cMb2ws� º�v _�wAv-¬�cj_8wX�sc�� z�q�z�� � S�� � � z�� � S�� ª � c¶��cMtMw�nmn8v�_8wAvyBD6FEG&�H� �(��Rc$k8o�v cTy�v�_acey|c$vos¢3y|x8~Rv cM� byo�¢H� ª � c¶lmkGv ��oR�Rx8t4c�w9¢¥xak8t�v�lpork �F� BD6FEG&�HP1 3w4 z|8�(��-s BD6FEG&RH+q9z)(µw�k8��¬�c�y|_8oA¬¨v _8w�v �)luyoskac$¯?v�o�¯¹oskacrª�h3c4v�x8y-�Rc4¦�kac�·¢¥os� � JLBD6FEG&RHP1 3\4 z|8¤(~}{D�

� �D& � ( � � l£¢ � J2BD6FEG& z)(¶ª� �D& � ( � � lp¢ � � � |1 3L4 z|8z¢¥os�Iy osb2cey xa~Rv cM� b � os¢DH� _8c$k±y|cM�scM��wsn � \ws� c5�Eory�y|lm~anmc·lpk v _8c«�Rc4¦8k8l£v�lpork¨ws~�oA�rc�v�_ac$k¨¬c)v�wsºsc)oskac·ws� ~alpv ��w���lmnp{rªBh3c4vKx8yKy _aoA¬6v�_8wAv �Uluyorkac4¯Ov os¯Oorkacsª ��osk�y|lu�Rc$� � q �CJ�BD6FEG&�Hw1 3\4 z|8�(¬�lpv _ � ?� ��ª

� R¹¢ � q �/J�BD6FEG&2z)(�v�_ac$k �D& � (#� � » �D& �c(#� ��»8wsk8� �D& � (�?� �D& �c(�ª� R¹¢ � JlBD6FEG& z)(ewsk8��� � �c _1 3@4 z|8¹»�w�k8��� WJ BD6<E &�H�(4»3v�_ac$k � 1 3�4 z|8�� � ?� � |1 3�4 z|8O» � ?� � w�k8��y o�D& � (�?� �D& �c(4ª� R¹¢ � � � _1 3�4 z|8�w�k8���P� �c _1 3\4 z|8¹»R¬�lpv _ � �q � XJ2BD6FEG&�H�(4»}v _acMk � ?��� 3w�k8� �D& � (�?� �D& �c(4ª

^-_8lmyj�a��oA�scTyIv�_ac��8� or��cM�|v�{r»�y lmk8t4c�v _ac² !Ig¶¯°y lp¸Mc�os¢�w�y c4v�o�¢�v cM� by¶lmy�cT�rx�w�n�v o)k}xab9~Ec$�jo�¢-�Rlmy|v lmk8t4vey xa~Rv�c$��by\v�_ac${t$oskGv�wslpk3ª

��/A�-����������

Ap��C��eª !\bws�Rlmo�wsk8�«²�ª�h3xaqslmc$¸rª � k·v�_ac9��cMwrt�_8w�~almnml£v�{��a� or~anmc$b lpk,t4��{G�av osqr��ws�a_alut��a��o�v oRt$osnuy$»�R°kU�8���I�C@ ����&��� <�������� »1`}�a��lpkaqrc$��h3cMt�v�xa��c�®\o�v�cMy-lmk ��orb2�axRv cM�5`Rt4lmc$k�t4c�M� j jRª

A d�C�²9ª �ornplmqsk8wskao8ª¨^�oA¬ws���ay9v _8c)¢¥os��b2wsn��scM� lp¦�t$w�v lmosk#o�¢�c$nmcMt�v�� orkalut5t$osb2b2c$��t4c5�a��o�v oRt$osnuy$ªiR°k ������ ����)�:1���%���01��,����������! �����-, Z ���2���3� ! ����b��9����:�»8�8wsqscMy��T�s���E�$� Eaª�R�< < < ��osb2�axav c$�5`Ro}t$lpc$v�{s» �IL L jRª

A ��C(R�ª ��c$���scTy w�v o8» ®9ª3²\x8� qrlpkz»���ª�hzlmk8t$osnmkz»���ª S·lpv�t�_acMnpnO»�!9ª�`Rt4cT�R��oA� ! b�c$v�w�¯Okaosv�w�v lmosk«¢¥or���a��o�v�o}t$osnCw�k�w�nm{}y luy$ªKR°k��ª3`}{}�rc$��y|orkz»rcT�Rlpv os� � � ���� ������ � ��)�:1���%���50 ��,���������(!������<, Z �[�2���3� ! ����b����3��:UR�< < < ��osb2�axRv�c$��`}oRt4lmc4v�{2����cMy�yM»�L L�Laª

A � C(<Iª S ª ��nmws� ºrcs»�w�k8� `�ª��s_8w8» � ª S«w���� cM� o�ª"k\y lmkaq y|v�wAv�c�y �8wst$cc4¡R�anmos��wAv lmosk�w�k��³w)k8wAv�xa��wsnµ�RcT�Rx8t�v�lpork y�v�{}nmcb�cTy y�w�qrc��RcM� lm�AwAv�lpork c$k8qslmkac�v o)�scM� lp¢¥{«y|cTt4xa��lpv�{·�a� osv oRt4ornmyMª-R°k ����o}t$c$cT�Rlpk8qryjo�¢�v _acKR/N R°� � or� º}lmkaq ��orkR¢¥c$��c$k�t4cosk5��� orqs��w�b2b2lpk8q ��ork8t4cM�Rv�y�wsk8��S«c4v _8o}�8y § �µ� � � � ST<�^I©�»z�IL Ls�8ª

A G�C9g�ª3²\cMkaºscM�M»��8ª S«cMy c$qsx8c$�T»zw�k8� �¶ªz^�w�nut4o�v vMª����o�v�oRt4osn�y �EcMt4lp¦�tMwAv lmosk�w�k��,wsk8w�nm{Ry|luy�lpkfS«wsx8�Rcrª�R°kfX\c$lmkrv�¸$cr»�®�ªw�k8� � lpk8q8»��8ªm»�cT�Rlpv os��yM»µ����oRt�ªCo�¢ � or� ºRy _aos� oskiNaor� bwsn�S«c4v _8o}�8yKw�k8�i`}cTt4xa��l£v�{ ��� osv oRt4ornmyM»d�G��rxakac³�IL Lr�a»R°k8�Rluw�k8ws��ornpluy$»�R°k��Rlmwsk8wa»

A E�C9g�ª}²¶c$k8ºsc$�-w�k�����ª S·lmnmnpcMkzª � !\�g`�hMR°kGv c$qr��w�v cT�����o�v oRt$osn�<µkG�}lm� orkab2c$kGvMª�R°k�² !¶����! R°kR¢¥os��bwAv lmosk�w�k8� `}xa���Glm�G¯w�~almnplpv�{���orkR¢¥c$��c$k8t$c�w�k8� <C¡R�Eory l£v�lpork § ²6R9` �g< Q � �s�G©�»�XIlmn£v�oskTXIcTws� »$`RosxRv�_ �ws� ornplmk8w8»��rwskGx�w���{�d GX¯�d�jR»ad��s�r�a»}�a�zªd�� jX¯°drdR�r» R�< < < ��orb2�axRv cM�5`}oRt4lmc4v�{���� cTy y

AHjJC9g�ª3²\cMkaºscM�M»��8ª�S«lpnmnpcMkz»�N�ª �! 8y�v�c$�-N�lmnplm��c�w�k8��!�ª�gj��wsxzª � �Rv lmb2lp¸Mlpk8q·��� osv oRt4orn���c$¬���lpv c��xanmcMyeos¢ �gR/h@`R��cTt4lp¯¦�t$w�v lmosk8yMª$R°k �M�Av�_DR�< < < ��osb2�axav c$�(`}cTt4xa��l£v�{TN8osxak8�8wAv lmosk�y � os��ºRy|_aor� § �T�Av _PR�< < < ��`�N � ©�»"�rxanp{«��¯�Ga»�ds�s�r�a»�w�b9~8� lu�Rqscr» <�kaqsnuw�k��

�������3�J�9��

Page 21: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

�M� �6� ����1���������������3����� Z �$�

A ��C�²�ªs²¶osnmc$�9wsk8� !�ªA´µwso8ª � k�v _8cIy|cTt4xa��l£v�{9o�¢��8xa~anmlmt-ºrc${��a��o�v�oRt4osnuyMª$ ���� �+�3� Z ��� Z �,���2�����6���K ��.,����) Z �[�2���:�1���������A»R�^¯�d�L��p�ILs� �ad��r�a»a�ILs�s�8ª�!\nmy o"`R^ !\®�¯(��`G¯°�a�$¯¹� GA�8» S«wX{·�ILs�a�r»3`Gv�wskR¢¥os����k9ª

A L�C�®9ª�²¶xa� qrlpk3»���ª1hzlmk8t4ornpkz» �8ª�S·lpv�t�_acMnpnO» !9ª�`Rt$cM�R��oA�@kIk��RcMt$lm�aws~almnplpv�{�os¢ ��orxak8�RcT�D`}cMt$xa��l£v�{���� osv oRt4ornmy � os��ºRy|_aor�ork N8os��b2wsn�S·c4v�_aoR�ay�w�k��~`RcMt4x8� lpv�{K����o�v�o}t$osnuy �rxanm{ Ga» �L L�La»}^3��c$kGv�o8»�R¹v�wsnp{ § �8ws�|v�os¢ N�h � � � L LG©

Am�M��C ��ª8gjosxa~�w�xanpvMª ! b�c$v _aoR��¢¥os�IwsxRv orbwAv lutet4��{}�Rv orqs��w��a_8lmt¶�a��o�v oRt$osn��scM� lp¦�t$w�v lmoskzª�R°kF�8����2@ !���� ��� + 0 :������"* ����7�$����� Z *�»�ds�s�r�aª

Am�s�,C ��ª�XIcTwAv _8c$�µwsk8��` ª�`Rt�_akacMlm�ac$�Tª�^�oA¬ws���ay�w�xav osbw�v lut-�sc$��lp¦�t$w�v lmosk9os¢�wsxRv _8c$kGv lut$w�v lmosk��a� osv oRt4ornmy�ork2wsk�x8kG~Eosx8k8�RcM�kac$v�¬�or� º � � � ��� ��)5:������,�-01��,����������! �����-, Z ���2���3� ! ����b��9����:�» �rxanp{5d��s�r�aª

Am�TdJC-Nª��Gwst$�GxacMbw���� » S ªa��x8y lpkaoA¬�lpv�t�_5w�k8� h�ª �¶lpqrkac$��osk3ª ��osb2�almnmlpkaqwsk8� �CcM� lp¢¥{}lpkaq�`}cTt4xa��lpv�{��� osv oRt4ornmyMª�h3osqrlmt\¢¥os�����osqr��wsb2b�lmkaq)w�k8��!\xRv orb2w�v cT� �IcMwsy osk8lpkaq�ª `Gv�gjlmnpnmcMyM»��Ic$xakalmoskUR�y nmwsk8� ª�`R�a� lmkaqrc$� �Cc$��nuw�q8»�d��r�s�aª h�® ��`�ªz�rosn�IL�G GaªG�r���zª�S³ªa�Cws� lmqsosv-wsk8� !9ª �Cos��osk8ºsoA�2cM�Rlpv or��yMª

Am�M��C9g�ª�hzoA¬csª �wry|�Ec$�I� w t$osb2�almnpcM��¢¥or��v�_ac,wsk8w�nm{Ry|luyKos¢9y|cTt4xa��l£v�{¨�a��o�v�oRt4osnuyMª���������� Z �'�9.�� ��)5:������,�F01����������[�A»E § �T©,� Gs� �R���8»��L�Ls�aª

Am�$��C9g�ª h3oA¬�crª8^�oA¬ws���ay�wet$osb2�anmc4v cMkacMy�y���cMy xan£vC¢¥or��b2oR�Rc$n8t�_8cMt�º}lpk8qjos¢�y cMt$xa��l£v�{��8� osv oRt4ornmyMª3R°k � � ���� �� ��� � ��)5:��������0 �������������-!������<, Z �[�2���3� ! ����bJ���3�:E»��8w�qrcMyaL E��E�T��GRª�R�< < < ��orb��8xRv cM�6`}oRt4lmc4v�{r» �L Lr�aª

Am�IGJC �¶ªIS·cMwr�RoA¬Iy$ª !\�a�anm{}lpkaqI¢¥os��b2wsn�b2c4v�_aoR�ayzv�o�v _8cµw�k8wsnp{Ry lmyzo�¢aw�ºscM{\bw�k�w�qscMb2c$kGv3�a��o�v�o}t$osnOª�� ������� Z � �9. � ��)5:��������0 �������������A» � § �X©��HG �R� Ea»E�L�Lrdaª

Am�E�C �¶ª S·cTws�RoA¬IyMª�^-_ac2®\��h �a��o�v�o}t$osn�w�k8wsnp{}¸Mc$�I�\w�k,oA�sc$���}lpcM¬eª�� ������� Z �#�/.�> ��* �2� �8����* � Z )�) ���}*�»�d�E § ds©,�p�r�M� ���M�8�s»�IL L E8ª

Am�JjIC ��ª#S·lmnpnmc$k3ª � !I�8`�h � ��osb2b2osk !IxRv�_ac$kGv�lmtMwAv lmosk±�µ� osv oRt4orn6`}�EcMt4lp¦�tMwAv lmosk h�w�kaqrx8w�qrcsª[^�cMt�_akalut$wsn\��cM��or�|v"S·�L jG��r�8»G^-_ac-SDR�^��a< ��os����or��w�v lmoskz»��L�L jRª

Am�M��C ��ª S·l£v�t�_ac$nmnO» S³ª S·lpv�t�_acMnpnO»3w�k��Fk�ª `}v c$��kzª !\xRv orbwAv cT�,wsk8w�nm{Ry|luy¶o�¢t$� {}�Rv�osqr��ws�a_alute�a��o�v oRt$osnuy\x8y lmkaq S«xa���ª-R°k ����� 0���)�:��J������) ���D0 ��,��������� Z �-,K�8���� Z ���A»8�8wsqscTy��$�8� �E�JGA�8ª�R�< < < ��orb��8xRv cM�5`}oRt4lmc4v�{s» �IL L j}ª

Am�L�C-hµªR��wsxanuy|orkzª�^-_8celpk8�ax8t�v�lp�rc�ws�a�a��orwrt�_v o2�scM� lp¢¥{}lpk8q2t$� {}�Rv�osqr��ws�a_alutI�a��o�v�o}t$osnuy$ª��������� Z ���/.�� ��)�:1���%��� 01����������[�A»E § �T©,� � G ���Td��8»8�IL Ls�8ª

A d���C!9ª � ª���oGy t$oGcrª S·oR�RcMnpnmlmkaq#wsk8�±�scM� lp¢¥{}lpk8q ºscM{G¯Oc$¡Rt�_�w�kaqrc5�8� osv oRt4ornmy�x8y|lmkaq ��`}�.wsk8�VN�²j�eª R°k���� �B ������� ��)5:�������� 01��,����������! �����-, Z ���2���3� ! ����b��9����:�»8�8wsqscMyaLr� ���M� jRª R�< <#< ��osb2�axRv�c$�6`}oRt$lpc$v�{s» �L�L�Gaª

A dR�,C�`M`Rt�_akacMlm�ac$� �CcM� lp¢¥{}lpkaq !\xRv _acMkGv lut$wAv�lpork,�µ� osv oRt4ornmyj¬�lpv _ ��`}��ª��T�Av _PR�< < < ��orb��8xRv cM�(`RcMt4x8� lpv�{DNaorxak8�aw�v lmosk8y� os��ºRy|_aor�z»��L�L j}ª

¡�¢ �X¡¤£

Page 22: Protocol Insecurity with Finite Number of Sessions is NP ... · LORIA, Technopôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy

Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399


Recommended