Transcript
Page 1: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

HAL Id: inria-00071999https://hal.inria.fr/inria-00071999

Submitted on 23 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.

Worst Cases and Lattice ReductionDamien Stehlé, Vincent Lefèvre, Paul Zimmermann

To cite this version:Damien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [ResearchReport] RR-4586, INRIA. 2002. �inria-00071999�

Page 2: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--45

86--

FR

+E

NG

ap por t de r ech er ch e

THEME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Worst Cases and Lattice Reduction

Damien Stehle — Vincent Lefevre — Paul Zimmermann

N° 4586

15 octobre 2002

Page 3: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999
Page 4: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

Unite de recherche INRIA LorraineLORIA, Technopole de Nancy-Brabois, Campus scientifique,

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

� ������������ ������������������� ��� ��! ��"�#�$�%�

&('*),+.-0/21�34-05#687-:9!;�<�+=/#>?-0/@3BA�-8C$D-0E�FG-:HI;KJL'*M#6ON"+=)P),-?FQ)R'*/�/RST�U�VWYX!W[ZI\^],_WY`bacWedgfihjacklagWYdWlm�klnidoklpbd�qsrtXIu�fjdcacv*p@Ww�x f?ysWlm%z*{�nik4WYq

| n}{b{�f x m�~@W x WYk U W x k U W�`������i�i�,\ �Y�(f�k4msfiu x W�Z}�i�jZB\ �l�!{�n}hiWYq

�����Y�8�0���i�j� � W!{ x fi{�fjqsW(n�`@Wl��nidghif x agm U X�msf���`b~R��f x qsm�klniqsWYq[�=f x k4f x�x WYk4m x fjpb`b~�ac`@h:fi��ni`2ni`bnidgr*m�ack��pb`bk4m�agfj`�� � W�� x qsm x WY~bp�k4W�m U acq�{ x fiu�dcWYX msf(m U W!¡�¢�£}¤�¥G¦!£}¤=¤�§?£}¤©¨�¢�ª�¡¬«�­4¤c¢4¦®\^a¯�°Wi�@�.f x {�fjdcr*`@fjX:acnidcqL��agm Ux WYnid�k4f�W4±²klacWY`�m�qY� T�U WY`³��W²q U f?�m U n}mBm U acqIqsWYk4fj`b~´{ x fiu�dgWYXµklni`³u�W�q�fjdg¶jWY~·W4±²klacWY`�m�dcri¸�utr¹W4º*msWY`�~bac`@h» fi{b{�W x q�X:agm U�¼ q���f x�½ fj`²m U WB¾=¿�ÀÁ¢ÃÂ�¢l¡�¥G¦!£}¤=¤#§}£}¤Ä¨�¢�ª�¡¬«�­4¤c¢4¦Å\Æ�=f x {�fjdgrt`@fjX!acnidcq���agm U ac`*msWlhiW x k4f�W4±²klagWY`*m�q\Çpbq$ac`@h(dcn}msm�ack4W x WY~�pbk4m�agfj`2È �b¸��t¸��0ÉÃ�Ê f x(Ë fjn}m�ac`@h}ÌÃ{�fjac`*m�`tpbXBu�W x qI��agm U n·X!ni`*m�acq�q$nRdcWYq�q�m U ni`ÎÍ´¸�ni`b~În2{�fjdgrt`@fjX!aonidLn}{b{ x f8º@acX!n}m�acfj`´fi�~@Wlh x WlWBÏb¸�fjp x nidchif x agm U X ��`b~bq�nidcdK��f x qsm�klniqsWYq�n}m%~�acqsm�ni`bk4W!ÐÑÍÓÒ}ÔÖÕÕ ÔÖ×tØ � x fjXÙn:X:nik U ao`@We`tpbXBu�W x ac`�m�acX(WÚ!Û Í�ÔÖ×@ØÕ ÔÖ×@ØGÜ�ÝGÞ � Ê f x ÏÑßàZ*¸�m U acq:aoX({ x f?¶jWYq�fj`ám U W Ú!Û ÍRâ�ã¬ä Ü�Ý Þ k4fjX({�dcW4ºtagmÖrå� x fjXçæKWl� VWl¶ x W ¼ q²nidghif x agm U XÈè�YZt¸��Yé?ÉOmsf Ú!Û ÍRä�ã¬ê Ü�Ý Þ � � W�W4º U acu�agm�qsfjX(W�`@Wl���f x qsmIklniqsWYq[�.fjpb`b~³pbq�ac`@h�fjp x nidghif x acm U X�¸ë�.f x ~bfjp@u�dgW4ÌW4ºtmsWY`b~@WY~!ni`b~:v�pbni~ x p@{�dgW�{ x WYklaoq�agfj`�� Ê f x don x hiW x Ïb¸�fjp x nidghif x agm U X�klni`!u�W�pbq�WY~(msf[k U WYk ½ m U n}mOm U W x W�W4º@acqsm`@f(��f x qsm�klniq�WYq�n}m�~baoqsm�ni`bk4W�Ðìͳí�îIac`ïm�aoX(W Ú!Û Í ØÕ Ü�ð"ñ Øò}ó Þ �ô,õ�ö#÷�ø�ù �?ú��i� û ºtnik4m x fjpb`b~bao`@h@¸Km�n}u�dgW�X!n ½ W x ¼ qI~bacdgWYX:X!nt¸ë��f x qsmIklniqsWi¸�ü ûLûOû ÌÖý}�?�@¸þdon}msm�ack4W x WY~�pbk4m�agfj`�¸» fi{b{�W x q�X:agm U�¼ q�m U Wlf x WYX

ÿ���������� ���������� ������� ����� ��!"�$#$#��%��& �'��(*)+,��-�.��0/1��2436587$9$:<;>=�?@9$ACB$9�D�9$:+=E;GF�HI�JLK�MON>P�Q�N � MRNSPTJVU W,���'-+�"�VXY��.,Z�- U"[Y\U ���]�C���^�$-+.6_4`Gab,�$c U �'���8dCe&�% ���f�C�hg���,�C�'-ia U�j �$-+�lk@�+�"�V���"d"#"monR�'�����6 �>`p�rq���>`p�^��-�.6_

s �0�C�6ti(*)+,�$-+.��&/1�uB"9"F@9"v"H@9�D@B�w�HC7"3L;GF"Hx�JLK�MON>P�Q�N � MRNSPTJVU W,���'-+�"�VXY��.,Z�- U"[Y\U ���]�C���^�$-+.6_4`Gab,�$c U �'���8dCe&�% ���f�C�hg���,�C�'-ia U�j �$-+�lk@�+�"�V���"d"#"monR�'�����6 �>`p�rq���>`p�^��-�.6_

s �0�C�6ti(*)+,�$-+.��&/1�+yC765�5�9$H&5u3"D@B"w$HC7"3L;pF�Haz�'��c��'�C{

Page 5: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

�á�$�� �� ����� þ����� ��! ��"�#�$�%� �( ������ ��� �� �����õ ��� �õ � � fjpbq²{ x fi{�fjqsfj`bq²pb`ì`bfjp@¶jWYd�nidghif x agm U X(W�m x fjpb¶?ni`*m:dgWYq²{�a x WYq!klniq:{�fjp x d ¼ n x�x fj`b~�a�~ ¼ pb`@W�.fj`bk4m�agfj`2X!n}m U _WYX!n}m�acv*p@Wi¸�WY`³klnidcklpbd Vn,¶*a x hjpbdgW Ë fimsm�ni`*msWi�� �ni`bqIpb`·{ x WYX!agW x msWYX({�qY¸"`@fjp�q x _WY~bpbaoqsfj`bq�k4W{ x fiu�d VWYX(W Vn2dcn x WYk U W x k U W�~ ¼ pb`bW�{�Wlm�agmsW,¶?nidgWYp x ~ ¼ pb` {�fjdgr*`��fjX(W Vn2k4f�W4±�klagWY`*m�q x _WlWYdcqY� � fjpbq:X(fj`*m x fj`bqWY`bq�p�agmsW²v*p@Wïk4W�qsWYk4fj`b~å{ x fiu�d VWYX(W:{�WYp@m��Wlm x W x _WYqsfjdcp´W4±²klnik4WYX(WY`*ml¸OWY`_WlmsWY`b~�ni`�m�dgW�m x n0¶?niacd�~@W » fi{b{�W x Ìq�X!acm U \ p@m�aodcacq�ni`*m�dcn x _WY~�pbk4m�agfj`�~@WYq x _WYqsWYniptºï\^q�p x don x WYk U W x k U We~@We{�Wlm�agmsWYq%¶?nidcWYp x q�~ ¼ p�`�{�fjdgr*`��fjX(W Vnk4f�W4±�klagWY`*m�q�WY`*m�agW x q�È �b¸��*¸��}ÉÃ�w fjp x ~@WYq�`bfjXIu x WYq Ë fimsm�ni`*m�q�n0¶jWYk�pb`@WeX!ni`*m�acq$qsW�u�f x `�_WlW�{�n x ͳ¸�Wlm�pb`@W�n}{b{ x f8º@acX!n}m�agfj`�{�fjdgrt`@fjX!acnidgW~@W:~bWlh x _W:Ïb¸�`@fim x W:nidghif x agm U X(W(m x fjp@¶jW:msfjpbq[dgWYq[{�a x WYq�klniq Vn,~�acqsm�ni`bk4W�ÐÍ Ò?Ô¬ÕÕ ÔÖ×@Ø ~ ¼ pb`³`@fjXBu x W(X!nik U ac`@WWY`�msWYX!{�q Ú:Û Í Ô¬×tØÕ Ô¬×tØGÜ�ÝGÞ � w fjp x Ï!ß Z*¸�k4WYdcn:niXå_WYdoagf x W[dcn:k4fjX({�dgW4º@agm�_W[~@WId ¼ nidghif x agm U X(We~@W[æëWl� VWl¶ x W²Èè�YZt¸"�Yé?É\ Ú!Û ÍRâ�ã¬ä Ü�Ý Þ \ n8¶�WYk%pb`bW�k4fjX({�dgW4ºtacm�_W%~@W Ú!Û ÍRä�ã¬ê Ü�Ý Þ � � fjpbq�W4º U agu�fj`bqO~@W�`@fjp@¶�WYniptº!{�a x WYqLklniq�m x fjp@¶�_WYqn0¶jWYk!`@fim x W�nidghif x agm U X(Wi¸ë{�fjp x dcn,~@fjp@u�dgW({ x _WYklacq�agfj` _WlmsWY`�~bp@W!Wlm[don�v�p�ni~ x p@{�dcWB{ x _WYklacq�agfj`K� w fjp x pb`2{�dcpbqh x ni`b~·~bWlh x _W(Ïb¸ë`@fim x W�nidchif x agm U X(WB{�WYp@meX(fj`*m x W x v�p ¼ acd�` ¼ W4ºtacq�msWB{�niq�~@W�{�a x W�klniq Vnï~bacqsm�ni`�k4W�Ð�Í2í�î:WY`msWYX({�q Ú!Û Í ØÕ Ü�ð"ñ Øò}ó Þ ���ù �8� ÷ ��� �õ �P� n x�x fj`b~baKW4ºtnik4ml¸�~bacdcWYX!X(We~bp,�.n}u x acklni`*m�~@W[m�n}u�dgWi¸�{�a x W[klniql¸#`@f x X(WIü ûLûOû ÌÖý}�?�@¸ x _WY~�pbk4m�agfj`~@WYq x _WYqsWYniptº#¸@m U _Wlf x VWYX(W[~bW » fi{b{�W x q�X!agm U

Page 6: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� «}¡Q¥GÀ���£?¥Y¢G¥I£}¿����ë£iÀ¯À�¾��G¢�¢��}¨��YÀ�¾¯«}¿ é

�}�� ����������������������T�U WIü ûLûOû ÌÖý}�?��qsm�ni`�~bn x ~��.f x u�ac`bn x r Ë fjn}m�ao`@h}ÌÃ{�fjao`�m%n x agm U X(Wlm�aok²È��0Éø�n}{b{ x f?¶jWY~�ao`³� �i�i��utrïm U WIü ûOûLûz*m�ni`b~�n x ~bq"!�fjn x ~�ni`b~!m U W�#�X(W x acklni` � n}m�agfj`�nid�z*m�ni`b~�n x ~bq�ü¬`bqsm�agm�pbmsWi¸ x WYv*pba x WYqþm U n}m�nidcdb�=fjp x u�niq�ack�n x agm U ÌX(Wlm�ack!fi{�W x n}m�acfj`bq Û%$ ¸'&e¸�(�¸*) Þ ni`�~·m U W:q�v*pbn x W x f�fimIn x W²k4f x�x WYk4m�dcr x fjp�`b~@WY~�� Ê f x n�hjac¶jWY`¹�.p�`bk4m�agfj`�¸Ë fjn}m�ac`bh}ÌÃ{�fjac`*m�ao`@{�p@m�q��.f x � U ack U agm�acq%~baè±²klp�dgm�msf²hjp�n x ni`*msWlWIk4f x�x WYk4m x fjpb`�~bac`@h@¸bklnidcdcWY~,+O«}¡Q¥GÀ-�Q£?¥l¢4¥Q¸#n x W`tpbXBu�W x qO�.f x � U aok U m U W�W4º@nik4m x WYq�pbdcm�\Çniq�k4fjX({�p@msWY~�ac`�ao`t��`bagmsW%{ x WYklacq$agfj`(\ acq�`@WYn x n(X!nik U ac`@W�`tpbX�Ìu�W x ¸Kf x `@WYn x m U W�X:ac~b~bdgWIfi�LmÖ��f�k4fj`bqsWYklp@m�ag¶�W�X:nik U ao`@W(`*p�XIu�W x ql� T�U acq�acq�m U W���niX(fjpbq/. T n}u�dgW�0Pn ½ W x ¼ q

�acdgWYX!X:n21({ x fiu�dcWYX Û T 0 �.f x q U f x m Þ ��z*Wl¶jW x nid�nip@m U f x ql¸�ac`ì{�n x m�acklpbdon x üÖf x ~bnik U WPni`b~30Rn}m�pbdcnÑÈè�l�}Éøæëni`bhBni`b~40RpbdcdgW x Èè�i�4Éø U n0¶jW�q U f0�%`!m U n}m��.f x m U W�kldcniq$qLfi�"nidghiWlu x niaok���pb`bk4m�agfj`bqY¸�q�pbk U ��f x qsm�klniq�WYq�klni`b`@fimu�W�msf�få`@WYn x � x fjX n·X!nik U ac`@W�`tpbXBu�W x f x m U W,X!ac~�~bdgW�fi�%mÖ��f2k4fj`bqsWYklp@m�ag¶�W�X!nik U ac`@W�`tpbXBu�W x qY�Îztp�k Uu�fjpb`b~bqþWY`bn}u�dgW�fj`@W�msfB~@WYq�aghj`(qsfjX!W�W4±²klagWY`*m�nidghif x acm U X!qþm U n}m�hjp�n x ni`*msWlW�k4f x�x WYk4m x fjpb`b~bao`@h%�.f x nidchiWlu x niack��pb`bk4m�agfj`bqY�5 f0��Wl¶jW x ¸��=f x `@fj`tÌÁnidghiWlu x niack���pb`bk4m�agfj`bqY¸�`tpbXBu�W x m U Wlf x r(u�fjpb`�~bq�n x W�`@fimLq U n x {:WY`@fjpbh U ¸�� U ack U X:n ½ WYqk4f x�x WYk4m x fjpb`�~bac`@h U n x ~bW x msf¹acX({�dgWYX!WY`�ml� T�U aoqeacqe{ x fiu�n}u�dgrRm U W x WYniqsfj`´� U rPm U W²ü ûLûOû ÌÖý}�?�Rqsm�ni`�~bn x ~~@f�WYq�`bfim x WYv*pba x W�k4f x�x WYk4m x fjpb`b~�ac`@h��.f x m U fjq�W%��pb`bk4m�agfj`�ql�60RpbdcdcW x ni`b~²fim U W x nip@m U f x qO{ x fi{�fjqsWY~²ao`RÈè�Y�?É�msfac`*m x f*~bpbk4W�~ba87#W x WY`*m�dgWl¶�WYdcq�fi��v*pbnidcagmÖr��.f x m x ni`bq�k4WY`b~bWY`�m�nidþ��pb`bk4m�acfj`bql� T�U acq�{ x fi{�fjq�nidþ��niq�{ x WYqsWY`*msWY~Rutr

0Rn x�½ q�msWYac`¹n}m�m U W90Rn8rRZ}�i�jZ�X(WlWlm�ac`@h�fi��m U WBü ûOûOû ÌÖý}�?� x Wl¶tacq�agfj`,h x fjp@{K¸�u�p@m�m U W�k4fj`bkldcpbq�acfj`,��niq�m U n}m.:+L¢�; ¡s¢I¿�«iÀ=<�¢YÀ�¡�¢�£>�?<�Àë:¥GÀãi¿��j£}¡@�}¾BA8¢C1G�z*rtqsmsWYX!n}m�ack[��f x�½ fj`Rm U W T 0 ��niq�~@fj`@WButr�æKWl� VWl¶ x WBni`b~D0RpbdcdgW x Èè�Yé?Éø#� U f:{�p@u�dcacq U WY~ï��f x qsm%klniq�WYq�.f x X!ni`*r[WYdgWYX(WY`*m�n x r[�.pb`�k4m�agfj`bqþac`�~@fjp@u�dcWL{ x WYklacq�agfj` Û ÍÓß Z êsä Þ ¸jf?¶jW x m U W��.p�dcd x ni`@hiW��=f x q�fjX(W��.p�`bk4m�agfj`bql�#�dcniql¸*m U WYa x n}{�{ x fjnik U acqLmsf�f�W4ºt{�WY`�q�ag¶jW%msfB~@WYnid���agm U m U W�v�pbni~ x p@{�dgW�{ x WYklacq�agfj`K¸j� U ack U acq�ac`bkldopb~@WY~!ac`:m U Wklp x�x WY`�m x Wl¶tacq�agfj`²fi�"m U W�ü ûLûOû ý}�?�:q�m�ni`b~bn x ~�� T�U pbq�klp x�x WY`�m�dcr(m U W�fj`bdgr!{�fjq�q�acu�dgW�n}{b{ x fjnik U WYq��=f x U agh U W x{ x WYklaoq�agfj`bq�n x W�WYagm U W x msfIhjpbWYq�qOn x WYniqsfj`bn}u�dgW�u�fjpb`b~!fj`²m U W%{ x WYklacq�acfj` x WYv�p�a x WY~(�.f x m U W U n x ~bWYqsm�msf x fjpb`b~klniqsWYq�ni`b~:msf[� x acmsW�nIdoagu x n x rBk4fjX({�p@m�ao`@hep@{:msfBm U n}mL{ x WYklacq�acfj`�¸if x msfB� x acmsW�nehiWY`@W x ack�X�pbdgm�ac{�dgW4ÌÃ{ x WYklaoq�agfj`dcagu x n x ri� Ê f x ac`bqsm�ni`�k4Wi¸FE#ag¶ ¼ q-0Rn}m U æ�aguPdcacu x n x r�~@f�WYq�m U WI�=f x X!W x ¸�� U W x W[m U WIhjp@WYq�qsWY~Pu�fjpb`b~,aoq�ý}�i�:u�agm�q�.f x ~@fjp@u�dcW�{ x WYklacq�agfj`·Èè�8ý0ÉÃ�

5 n8¶tac`@h[ni`²W4±²klagWY`*mOnidchif x agm U Xmsf[��`�~!m U W U n x ~@WYqsmLmsf x fjp�`b~:klniqsWYqY¸*�.f x nehjag¶jWY`:�.p�`bk4m�agfj`:ni`�~:nIhjag¶jWY`Ë fjn}m�ac`bh}ÌÃ{�fjac`*mL�=f x X:n}ml¸*��fjpbdo~ U WYdg{�msf x Wl{�dcnik4W�hjp@WYq�q�WY~ïu�fjpb`�~bq�\Æ� U ack U n x W�pbq�pbnidodgrIf?¶jW x WYq�m�acX!n}msWY~:\utr·q U n x {�W x ni`b~ x aghif x fjpbq[u�fjp�`b~bql��ü mI��fjpbdc~¹m U pbqeWY`�n}u�dgW(fj`@W²msfR~@WYq$aghj`·¶jW x r¹W4±²klacWY`�mIdcagu x n x agWYq���agm Uk4f x�x WYk4m x fjpb`�~bac`@h@� T�U WY`,m U W x WI��fjpbdc~�u�W�`@f:hif�f�~ x WYniqsfj`Pni`*r�X(f x WImsf�W4ºtkldopb~@W[m U fjqsWI��pb`bk4m�acfj`bq�� x fjXm U Wek4f x�x WYk4m x fjpb`�~bac`@h x WYv�p�a x WYX(WY`*m�q�fi�þm U W�ü ûLûOû ÌÖý}�?�:qsm�ni`b~bn x ~��û º U nipbq�m�ag¶jW(qsWYn x k U X(Wlm U f*~bq[k4fj`�q�acqsmeac`·��`b~bac`bh²m U W U n x ~@WYqsm�msf x fjpb`b~·klniq�WYq[fi��m U W!hjac¶jWY`·��pb`bk4m�agfj`ac`�m U Wehjag¶jWY` x ni`@hiWi� T�U Wlrïhjag¶jWem U Weu�WYq�m�{�fjq�q�agu�dcW�u�fjpb`b~K¸@u�p@m�n x WI¶jW x r�m�acX(W4ÌÁk4fj`bq$pbX!ac`@h@�'0,f x Wlf0¶jW x ¸�nqsWYn x k U �.f x n(hjag¶�WY`ï{ x WYklacq�acfj`²hjac¶jWYq�dcagmsm�dgW ½ `@f?��dgWY~@hiW��.f x ni`@fim U W x { x WYklacq�agfj`K�� W�{ x fi{�fjqsW U W x W�n[`@Wl� nidghif x agm U X u�WYdgfj`@hjao`@h�msf�m U n}m�kldcniq�ql�"üÖm�`bn}m�p x nidcdcreW4º*msWY`�~bqþm U W�� x q�m"nidghif x agm U X{ x fi{�fjqsWY~�utr�æKWl� VWl¶ x W:Èè�YZ}Éø�ni`b~�acq�u�niqsWY~�fj` » fi{�{�W x q$X!agm U�¼ q�ac~@WYniqY�8Gw�x Wl¶*agfjp�q x WYdcn}msWY~I��f x�½ ��niq�~bfj`@WLutr û d ½ acWYql¸8� U f%hjag¶�WYq�ac`�È ý0É@n%`@Wl� nidghif x acm U X pbq$ac`@h%don}msm�ack4W x WY~�pbk4m�agfj`msf(��`�~�nidcd x n}m�agfj`bnid#{�fjac`*m�q�fi�þq�X!nidcd U WYach U m�`@WYn x n({�dcni`@W�klp x ¶jW2H@�=f x W4ºtniX({�dgW U acq x WYk4f x ~�I�i�i�iéi�i�i�i�t�Y�jý}�t�YZjZié ä &2�i��ý}�i�?�J�iZi�?��Zi�?�j�jZi�}��Ziéi��ý>�t�Y� â ß �Y�?�b�Y�?��é

k4f x�x WYq�{�fj`b~�q%msf�nï��f x qsm�klniqsW�fi�Om U WB��pb`bk4m�acfj`LK ä�ã¬â �.f x n��ié0ÌÃu�agm�ac`@{�pbm%ni`b~¹n�ý?�0ÌÃu�agm�fjp@ms{�pbmMH U acq%fim U W xW4º@niX({�dgWZiZiZ}�}��ZiZ2�iéiZ ä &´Zi�ié}�j�2�2�i�i� ä &´ZiZt�Y�i�i�i�i�t�8ý ä ß é}�

k4f x�x WYq�{�fj`b~�q�msf�n[��f x qsmLklniqsW�fi��m U W���pb`bk4m�agfj` Û K ä $ON ä Þ G ã¬ä ac`²éiZ0ÌÃu�agmLn x acm U X(Wlm�ack}�P0Pf x W x WYk4WY`�m�dcr!]�fj`b`@WlmÈ°�?Étnidcqsf%p�qsWY~[dcn}msm�aok4W x WY~bp�k4m�agfj`emsf%��`b~���f x q�m�klniqsWYql¸ U f?��Wl¶jW x U acqKn}{b{ x fjnik U qsWlWYX!q�WYv�p�ag¶?nidcWY`�m�msf�æKWl� VWl¶ x W ¼ qnidghif x acm U X��G X U%U �+ c8��� j�Q - U@R ���0�TS"�"� j Z���� ��� j Z+�6U� � j - U -C`G.1W_ [Cj U S�,� [ Z+��.^� [+[ ����.0� j � U - UWV s U$[�[ �6 �W� � j ZE� � RzU Q {

MOM -?X���WY$d

Page 7: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� ������ÀÁ¢���¤��¢�������¢ ���¢4§0¡s¢�������K¾.¦�¦!¢l¡Q¦!£}¿�¿

� ������������� # ��pb`bk4m�agfj`�h x n}{ U ni`b~,m U W[h x ao~�fi��X!nik U ac`@We`tpbXBu�W x ql� � f x qsm�klniqsWYq�k4f x�x W4Ìqs{�fj`b~ïmsf!h x ac~ï{�fjac`*m�q���agm U n:q�X!nidcd�¶jW x m�aoklnid�~bacqsm�ni`bk4W�msf:m U Weklp x ¶jWi�� p x {�n}{�W x acqBf x hjni`ba��lWY~ niq��.fjdcdgf?��qMI�z*WYk4m�acfj` ZPW4ºt{�dcniac`�qIac`åX!n}m U WYX:n}m�acklnid�msW x X!q�m U Wï{ x fiu�dgWYX ��W��ni`*m�msf!q�fjdg¶jWi¸ x WYklnidcdoq�æëWl� VWl¶ x W ¼ q�nidchif x agm U X®ni`b~�ni`�nidgr��lWYq�agm�q�k4fjX({�dgW4º@agmÖri�Oz*WYk4m�agfj`�é(~@WYq$k x agu�WYqLfjp x `@Wl�nidghif x acm U X�¸Ln}�.msW x n2q U f x m!q$p x ¶jWlråfj` dcn}msm�ack4W x WY~bpbk4m�agfj`ìni`b~ » fi{b{�W x q�X:agm U�¼ qB��f x�½ ¸�� U aok U ��W U WYn0¶*aodgrpbqsWi�Îz*WYk4m�agfj` �·{ x WYqsWY`*m�qBq�fjX(W�`@Wl�®��f x qsm�klniqsWYq(�.fjpb`b~å��agm U fjp x nidchif x agm U X �.f x m U W,Z� P��pb`bk4m�acfj`�¸�ac`~@fjp@u�dgW4ÌÃW4º*msWY`�~@WY~Î{ x WYklaoq�agfj`Îni`b~áv�pbni~ x p@{�dgWï{ x WYklacq�agfj`�� z*WYk4m�agfj`��·~bacq�klp�q�qsWYq!qsfjX!WPac~@WYniq:�=f x {�fjq$q�agu�dgWacX({ x f0¶�WYX(WY`*m�qLni`b~�fi{�WY`�v*p@WYqsm�acfj`bql�

Z*�"! ���$# �&%�� �('����)�+*Z*�c�}�-, õ�.0/�1 � 1.ù+/ ��� / ú32 ù �8�b� 1.ù+/ �54�� W!niq�q�pbX(W���W(��f x�½ U W x WB�%agm U Ë fjn}m�ac`@h}ÌÃ{�fjac`*m�`*p�XIu�W x q%��agm U nX!ni`*m�acq�q$n:fi�76åu�agm�ql�IæëWlm�Í ßZ�8�Hë�.f x ac`bqsm�ni`bk4Wi¸KÍ ßZ êsä k4f x�x WYqs{�fj`b~bq�msf�~@fjpbu�dgWB{ x WYklacq�acfj`�¸�Í ß Z:9<;k4f x�x WYq�{�fj`b~�q[msf·~@fjp@u�dgW4ÌÃW4º*msWY`�~@WY~³{ x WYklacq�agfj`K¸"ni`b~åÍÇß�Z G@G ä�k4f x�x WYqs{�fj`b~bqBmsf¹v�p�ni~ x p@{�dcW!{ x WYklacq$agfj`��O#��f x qsm�klniqsW��=f x n���pb`bk4m�agfj`>=³acq�n Ë fjn}m�ac`@h}ÌÃ{�fjac`*m%`tpbXBu�W x K2q�p�k U m U n}m?= Û K Þ U niq"@ ao~@WY`*m�acklnidKu�acm�q%n}�.msW xm U W x fjpb`�~²u�agml�þü ��m U fjqsWA@Óu�agm�q�WYv*pbnid Û x WYqs{K�@~ba 7�W x � x fjX Þ m U W x fjpb`b~�u�acml¸ KPacq�n���f x qsm�klniqsW��=f x ~�a x WYk4msWY~x fjpb`�~bac`@h Û x WYqs{K� x fjpb`b~bac`bh[msf:`@WYn x WYqsm Þ �Ê f x q�n ½ WRfi�Iq$acX({�dcaoklagm ri¸L��WRk4fj`bq$ac~@W x U W x WR~�a x WYk4msWY~ x fjpb`b~bao`@h2fj`bdgr Û msf?��n x ~bqO&"B�¸�msf?��n x ~bq $ B�¸msf?��n x ~bqC�lW x f Þ ¸bq$ac`bk4W%nI��f x qsm�klniqsW��=f x�x fjpb`b~�ac`@hemsf�`@WYn x WYqsm�n}m�{ x WYklacq�acfj`D6Pk4f x$x WYqs{�fj`b~bqLmsf�nI��f x qsm�klniqsWn}m�{ x WYklaoq�agfj`E6 $ ���.f x ~ba x WYk4msWY~ x fjpb`b~bac`bh@� T f(��`b~²��f x qsm�klniqsWYq��.f x ~ba x WYk4msWY~ x fjpb`b~bac`bh@¸���W�m U x f?� n8��n0rm U W:� x qsmA6ìq$aghj`baè��klni`*m�u�agm�qefi��m U W x WYq�p�dgmeX!ni`*m�acq�q�nt� T�U WY`³n���f x qsm[klniq�W:fi��dgWY`@him U @çk4f x�x WYq�{�fj`b~�q�msfF ÍG= Û K Þ X(ft~�� F ÐáZ�í$HI¸j� U W x W'KïX(ft~��/IÄß K�&JI KLK�~@WY`bfimsWYq�m U W .sk4WY`*msW x WY~�1"� x nik4m�agfj`bnidt{�n x m Û q�WlW Ê agh@�b� Þ �� W�nidcqsf(k4fj`bq�ao~@W x m U n}m�u�fim U n x hjpbX(WY`*m'K,ni`b~ x WYq�pbdcm N ßM= Û K Þ n x W�`@f x X:nidca��lWY~�¸*a¯�°Wi� GâDN KPOQ= Û K Þ Ð �}�T�U aoq�aoq�WYniqsrïmsf:nik U agWl¶jW�utr²X�pbdgm�ag{�dgr*ao`@h/K,f x = Û K Þ utr²qsfjX(We�bº*WY~ï{�f?��W x q�fi�þZ*¸�p�`bdgWYq�q�m U W�W4ºt{�fj`bWY`�m�fi�= Û K Þ ¶}n x agWYq�n(dcfim�ac`ïm U Wek4fj`bq�ao~@W x WY~ x ni`@hiWi� T�U acq�W4ºtkldopb~@WYq�m U WeklniqsW[fi�"`tpbX(W x acklnidcdgr(a x�x Wlhjpbdcn x �.p�`bk4m�agfj`bqdca ½ W:q�ac` KÎ�.f x dcn x hiW Kë�³]�ag¶jWY`ånR{�fjdcr*`@fjX:acnidOn}{b{ x f8º@acX!n}m�acfj`SR Û�T Þ msf·ÍG= Û�UV Þ Û �.f x W4ºtniX({�dgW²n T n8rtdgf xW4ºt{�ni`bq�agfj` Þ ¸*m U W T 0 �klni`,u�W x WY~bpbk4WY~�msf!m U W��=fjdcdcf0��ao`@h[{ x fiu�dgWYX��W �$'(#>XY%-'(#Z#\[]'(#J�P�S! ���P^+#Z�L%`_ W �$'(#>XY[7'(#a!"bdc#]�ag¶jWY`ï{�fjq�agm�ac¶jW�ac`*msWlhiW x qfe ni`�~\g[¸bni`b~�n�{�fjdgrt`@f}ÌX!acnid�R���agm U x WYnidKk4f�W4±²klagWY`*m�ql¸@��`b~ïnidcd�ac`*msWlhiW x q F T F Ðhg q�p�k U m U n}mÛ � Þ F R Û�T Þ X(ft~�� F Ð �

ejiW �L%k'���lm��c T�U W�X!ni`*m�acq$q�n¹u�fjp�`b~ÎÍ ~bf�WYq:`bfim:n}{b{�WYn x W4ºt{�dcacklagm�dcr2ac`Îm U W x WYnid�zon�nid w ¸ U f?��Wl¶jW x m U W{�fjdgr*`bfjX!acnidpR Û�T Þ ~@Wl{�WY`b~bq�fj`�Í´¸�ni`b~�qsf²~@f�WYq�m U W�W x�x f x X:ni~@Weac`ïm U W�{�fjdgrt`@fjX!acnid#n}{b{ x f8º@acX!n}m�agfj`K�W �L%k'���lrq+c�üÖ��m U W�� x nik4m�agfj`bnidbu�agm�q�fi�#m U W���pb`bk4m�acfj`(u�W U n8¶jW x ni`b~@fjX!dgri¸���W%klni`!W4ºt{�WYk4m�sutv ��f x qsm�klniq�WYql�T�U W x Wl�=f x W���WeX!n0r²niq$q�pbX(Wwgyxze ac�ë��W���ni`�m�fj`bdcr:�.Wl� ��f x qsm�klniqsWYql� ââ X Z���- U$j � j � U -"{k|~}���� �0k@�����$�$����- j j U {����%(�}+/1{

N � MON>P

Page 8: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� «}¡Q¥GÀ���£?¥Y¢G¥I£}¿����ë£iÀ¯À�¾��G¢�¢��}¨��YÀ�¾¯«}¿ �

Z*� Z*��� õ����õ�� � õ� ��� �� ù � 1 � �� 4 æëWl� VWl¶ x W ¼ q�nidghif x agm U XàÈè�YZt¸þ�Yé?Éþ��f x$½ q%niq%�.fjdcdgf?��ql� � `@WIk4fj`�q�ac~@W x q�n�dcac`@WYn xn}{b{ x f8º@acX!n}m�agfj` msfåm U W���pb`bk4m�acfj`�=Ñfj`�q�X!nidcd�ao`�msW x ¶?nidcqY� T�U fjq�W,n}{b{ x f0º@acX!n}m�agfj`bq(n x WPk4fjX({�pbmsWY~ì� x fjXU agh U W x f x ~@W x {�fjdgrt`@fjX!aonid�n}{b{ x f8º@acX!n}m�acfj`bqOfj`�don x hiW x ac`*msW x ¶?nidoql¸*p�q�ac`@hBni`ïW4±²klagWY`*m�q$k U WYX!W�u�niqsWY~ïfj`ïm U W.¬m�n}u�dgW�fi��~ba 7�W x WY`�k4WYq�1KX(Wlm U f*~�� � `IWYnik U q�X:nidcd�ao`�msW x ¶?nid¯¸?��f x qsmþklniqsWYq�n x W��.fjpb`b~Bpbq�ac`@h�n�X(f�~bag��WY~[¶�W x q�agfj`fi�"m U W û p�kldcac~@WYni`ïnidghif x acm U X�¸t� U ack U hjag¶jWYq�n!dcf0��W x u�fjpb`b~ï�.f x F Í>= ÛPUV Þ X(ft~�� F fj`�m U n}m�ac`*msW x ¶}nid¯�#�q�q�pbX(W\= Û K Þ ß���� $ � G K $ � â K â $ Ú:Û K ä Þ n x fjpb`b~ Káß �t�³ztao`bk4W:��W²`bWlhjdgWYk4mIm U W²msW x X:qIfi��f x ~@W xmÖ��f²f x X(f x W�ac`,ÍG= Û UV Þ ¸#��WIX�pbq�m U n0¶jW F � â t ÕV F x Gv q�f²m U n}m�m U WIW x�x f x k4fjX!ac`bh:� x fjXÙm U WI{�fjdgrt`@fjX!acnidn}{b{ x f8º@acX!n}m�agfj` ~@f�WYq!`@fim(W4º@k4WlWY~ m U W�~�acqsm�ni`bk4W Gv � T fihiWlm U W x ��acm U g x e¸�agm��=fjdodgf0�%q g x Í G ã¬ä �T�U W x Wl�=f x W�m U W�k4fjX({�dgW4º@agmÖr(fi�ëæKWl� VWl¶ x W ¼ q�nidghif x agm U X acq Ú!Û ÍRâ�ã¬ä Ü�Ý Þ ¸tq$ac`bk4W���W U n0¶jW%msf(k4fj`bq$ac~@W x V t sÑÍRâ�ã¬äq�X!nidod�ac`*msW x ¶?nidoqLmsf²k U WYk ½ n:k4fjX({�dgWlmsW[X:ni`�m�aoq�q�n x ni`@hiWi�ü¬`I{ x nik4m�ack4W�æKWl� VWl¶ x W ¼ q�nidghif x acm U X acq"W4º*{�WY`bq�ag¶�W�u�p@m��.WYniq�agu�dgWL�.f x m U W�~@fjpbu�dgW�{ x WYklacq�acfj` Û Í â�ã¬ä s����s�l� G � Þ ¸`@WYn x � x fjX®m U W[dcacX!acm�qLfi�þm U WIklp x�x WY`*m�{ x f*k4WYq�qsf x q��.f x ~bfjp@u�dgW4ÌÃW4ºtmsWY`b~@WY~�{ x WYklaoq�agfj` Û Í â�ã¬ä s ý����l� G â Þ ¸�ni`b~fjp@m�fi� x WYnik U �.f x v�p�ni~ x p@{�dcW�{ x WYklaoq�agfj` Û ÍRâ�ã¬äks ���j�l�iâsâ Þ �

é*������������#o�6���������%�� *M� �����0'��F�����+� W �����6���������ü¬`[m U n}m�qsWYk4m�acfj`�¸}��WL� x q�m�qsm�n}msW�qsfjX(W�u�niq$ack���nik4m�q�n}u�fjpbmþdon}msm�ack4WYqë\ ��W x Wl�.W x msf!Èè�Y�?É@�.f x ni`�ac`*m x f*~bpbk4m�agfj`msf!m U n}m%q�p@utysWYk4mL\Çni`b~���W�W4ºt{�dcniac` » fi{b{�W x q�X:agm U�¼ q�m U Wlf x WYX�¸@fj`�� U ack U fjp x nidghif x acm U X®acq�u�niqsWY~K� T�U WY`��Weac`*m x f�~�pbk4W�m U Wenidghif x agm U X�¸t��W�{ x f0¶�W�acm�q�k4f x�x WYk4m�`@WYq$q�ni`b~���Weni`bnidgro�lWeagm�q�k4fjX!{�dgW4º@agm ri�

é*�c�}�! ù õ#" ��� 1 �%$ë���i�8� 1 / � �@�Y� 1 � õ �²õ ú �ë�i� 1=ùY/'& � õ�ù � ö 4 #�¤g£iÀ¯À�¾��G¢)( acq[n�~bacq�k x WlmsW�q$p@ubh x fjp@{·fi�* 8 ¸�f x WYv*pbag¶}nidgWY`*m�dgr²m U WBqsWlm�fi�Onidcd"ac`*msWlh x nid�k4fjXIu�ac`bn}m�agfj`bq�fi�,+ N 6ådcac`@WYn x dgr�ac`b~@Wl{�WY`b~@WY`*m�¶jWYk4msf x q�f?¶jW x* ¸Km U n}m%acqMI(åß.- /0 1 2 G

61�1F 614365�7

i� W[~@W4��`@W�m U W ��¢YÀÁ¢4¡G¦B¾.¿�£}¿�À�¸�nidcqsf!klnidodgWY~ïm U W!§}«}¤Ä¨t¦:¢4¸@fi�þm U Wedon}msm�ack4W!(ániq I

~bWlm Û ( Þ ß/812 GF�F � 91 F�F O

� U W x W F�F i F�F acq#m U W û pbkldcac~@WYni`�`@f x X ni`b~!È � 9 G O i�i�i O � 9/ É�aoq�m U WL] x niX(ÌÖztk U X!ac~@m�f x m U fihifj`bnidoa��Yn}m�agfj`�fi�KÈ � G O i�i�i O � / ÉÃ�T�U We­�£}¥Q¾=¥�È � G O i�i�i O � / Étfi�9(·acq"`@fim�pb`bacv*p@W�ni`b~Bfj`�ni`�nidghif x agm U X!ack�{�fjac`*mëfi�b¶tagWl�e¸}fj`bdcr�u�niqsWYq"� U ack U k4fj`�q�acqsmfi��q�X!nidcdtdcao`@WYn x dgr[ao`b~@Wl{�WY`b~@WY`*më¶jWYk4msf x q�fi�:(³n x W�fi��ac`*msW x WYq�ml� T�U fjqsW�qsf}ÌÁklnidcdgWY~�¡s¢��}¨��Q¢��!­$£?¥l¢4¥%nidg��n0r*q"W4º@acqsmni`b~�klni`�u�W[k4fjX!{�p@msWY~�ac`�{�fjdgrt`@fjX!acnid#m�acX!W��%agm U m U We��WYdcdèÌ ½ `@f?��`�æëæëæ³nidghif x agm U X Èè�4�jÉÃ�& � õ*ù � õ <;L4!= ¾=§i¢4¿�£B­�£?¥G¾o¥�È?> G O i�i�i O > / Éþ«<��£�¤g£iÀ¯À�¾ �Q¢@(BA

58 ëÀ��t¢'�����´£i¤ Â�«}¡Q¾�À���¦ ª�¡¬«i§8¾ ��¢G¥�¾.¿�ªt«}¤ <}¿�«}¦�¾¯£}¤À�¾.¦!¢[¾.¿)+²£}¿���¾.¿RÀ��t¢(­l¾.ÀDCÖ¤c¢4¿�Â�À��j¥[«<�[À��t¢ F�F >

1F�F ; ¥Q�£ï­$£?¥Q¾=¥E-GF G O i�i�i OHF /

7¥4£iÀ�¾=¥&� <}¾=¿*ÂGIJ � F�F F G F�F N ZLK Õ ~@Wlm Û ( Þ ØK�MN � F�F F â F�F N Z K Õ ~@Wlm Û ( Þ ØK Ò�Ø �Û T�U aoq�aoq�`@fim�m U Weqsm x fj`@hiWYqsm x WYq$pbdgml¸tu�p@m�acq�q$pt±²klagWY`*m��=f x fjp x `@WlWY~bql� Þ» fi{b{�W x q�X:agm U Û qsWlW!È �@¸@�?Éø�f x È°�?É#�.f x nIu�WlmsmsW x ~bWYq�k x ag{bm�acfj` Þ �=fjp�`b~ x WYk4WY`�m�dcr�ni`ïacX!{�f x m�ni`*m�k4fj`bqsWYv*p@WY`bk4Wfi��m U acq"m U Wlf x WYX I"fj`@W�klni`!k4fjX!{�p@msW�m U W�q�X!nidod x f�fim�q�fi��n�X�pbdgm�ag¶}n x acn}msW�{�fjdcr*`@fjX:acnid�X!f�~bpbdcf�ni`(ao`�msWlhiW x Íac`�{�fjdgr*`bfjX!acnid�m�acX(Wi� 5 acq�X(Wlm U f*~�{ x f?¶jWY~²¶jW x r:{�f?��W x �.pbd�msf(�.f x hiWek x r*{bmsfih x n}{ U aok�q$k U WYX!WYq Û qsWlW:È°Zt¸bét¸t�iÉ�.f x W4ºtniX!{�dgW Þ � � p x `bWl� nidghif x agm U X®ac`*msWY`bqsWYdgr:p�qsWYq�m U n}m�msWYk U `bacv*p@Wi�

é*� Z*� & � õPO / � õ õ � ´� � ��Qe� � � õSR � ù � � õ �4 T�U W({ x fiu�dgWYXÓm U n}me��aodcdë{ x f0¶�W!ac`*msW x WYqsm�ac`@hïac`¹fjp x klniqsWacq�m U W��.fjdcdgf?��ac`@h I%hjag¶jWY`¹n�pb`bac¶?n x aon}msWI{�fjdgrt`@fjX!acnid R3#5

È K�Éëfi��~bWlh x WlW!Ïb¸���`b~Rfj`·� U ack U q�X!nidcdþac`*msWlhiW xWY`*m x agWYqIagm U niqBq�X!nidcdO¶?nidop@WYq[X(f*~bpbdgfRn,dcn x hiW�ac`*msWlhiW x ͳ� û v�p�ag¶?nidcWY`�m�dcri¸K��W²n x Wïdgf�f ½ ac`@h,�=f x m U Wïq�X!nidcdac`*msWlhiW x�x f�fim�q�fi�"m U Weu�ac¶?n x aon}msW�{�fjdgrt`@fjX!acnid%IT:Û KPO N Þ ßyR Û K Þ $ N Û X(ft~2Í Þ i

MOM -?X���WY$d

Page 9: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� ������ÀÁ¢���¤��¢�������¢ ���¢4§0¡s¢�������K¾.¦�¦!¢l¡Q¦!£}¿�¿

� W�`@f0� W4º*{�dcniac` U f?� » fi{�{�W x q$X!agm U�¼ q�msWYk U `baov�p@W U WYdg{�q�qsfjdc¶*ac`bh!agml� Ê a x qsm�dcWlm�� u�WIn!{�fjq�agm�ac¶jW[ac`*msWlhiW xÛ m U n}me�%acdcd�h x f0� dcn}msW x msf,ac`t��`bagmÖr Þ ¸#ni`b~·niq�q�p�X(W Û K�:O N � Þ acq�n x f�fimefi� T X(f*~bpbdgf�Í´� � W:k4fj`bq�ao~@W x m U W��niX!acdgr�fi��{�fjdgr*`bfjX!acnidcq T1� � Û K�O N Þ ß K

1T � Û KPO N Þ Í��*í � ��agm U � N � $ Ï�� N Ï�O� T�U WY` Û K�:O N � Þ acq�n x f�fimX(f*~bpbdgf(Í � fi�"WYnik U T

1� � ¸b� U WY`bk4W�fi�þWYnik U dcac`@WYn x k4fjXBu�ao`bn}m�agfj`�fi�þm U WYX��� p x hifjnid"acq�msf:u�pbacdc~�mÖ��f:ac`*msWlhiW x k4fjXBu�ac`bn}m�acfj`bq�fi��m U fjqsWe{�fjdgrt`@fjX!acnidcqY¸� G Û KPO N Þ ni`�~� â Û KPO N Þ ¸�� U aok Um�n ½ Weq�X:nidcd�¶?nidcpbWYq�\ aÃ�°Wi�bdgWYq�q�m U ni`�Í �!\ �.f x q�X!nidod�KPni`b~ N ¸@X(f x W�{ x WYklacqsWYdcr F K F N�� ni`b~ F N F N�� �.f x�bºtWY~Ru�fjpb`�~bq � ni`b~ � � T�U pbqY¸�ag� Û K �:O N � Þ acq�nïq�X!nidod x f�fim�fi�� G ni`�~� â X(f�~�pbdgf²Í´¸ Û K��O N � Þ acq�nidcqsf�nx f�fim�fi�� G ni`b~� â f?¶jW x5¸�ni`b~ Û K �:O N � Þ �%acdcd�u�W��.fjpb`b~�utr²dgf�f ½ ao`@h!n}m�m U Weac`*msWlhiW x�x f�fim�q�fi�"m U W x WYq�pbdcm�ni`�m| WYq�� Û G O� â Þ

365È KbÉÃ�üÖm x WYX!niac`bq�msf·W4ºt{�dcniac` U f?�®msf¹��`b~åm U fjq�W�m ��f¹{�fjdgr*`bfjX!acnidcql� Ê f x m U aoqI��W�k4fj`bq$ac~@W x m U W�dcn}msm�ack4W�fi�

~bacX!WY`bq�agfj` ñ � Ü G ó ñ � � Ü â óâ hiWY`@W x n}msWY~�utr:m U W�¶�WYk4msf x q�niq�qsf*klacn}msWY~��%agm U m U W T1� � Û � KPO � N Þ I�m U W�¶jWYk4msf x niq$qsf}Ìklacn}msWY~ï��acm U nBu�ac¶?n x aon}msW%{�fjdgrt`@fjX!acnid�� 1 � � �

1� � K1N � U niq�agm�q'K

1N ��k4f�f x ~bac`bn}msW�WYv*pbnid�msf �

1� � �1� �i� � Wehjag¶jWU W x W�m U Weq U n}{�Wefi��m U WeX!n}m x aèºï��W�hiWlm�ac`ïm U WeklniqsWBÏBß é!ni`b~��2ß Z*�

��� �! � !" � !# � !$ � !%&� !' � !( � !! � "" � "# � "$ � "! � #

)********************+

,.- ,.-0/ , - / - ,.-0/.1 ,.-0/32 ,.-0/.4 ,.-�/.56 6 6 6 ,876 6 6 6 ,./876 6 6 6 ,./.-�76 6 6 6 ,./.1�76 6 6 6 6 6 6 6 6 6 6 79-

:<;;;;;;;;;;;;;;;;;;;;=

ztac`�k4W���W�hiWlm�n�m x acni`@hjpbdon x X!n}m x aèº#¸tm U Weklnidcklpbdcn}m�acfj`�fi��m U We~@WlmsW x X!ac`�ni`�m�acq�fiut¶tagfjpbqMI~@Wlm Û ( Þ ßÑÍ Ô > � > Ü@?Qñ � > ó � � Ô ÕA.� > Ü@?$ñ � > ó � � Ô AB� > Ü@?Qñ � > ó iT�U W x Wl�=f x Wi¸ëu*r T�U Wlf x WYX �}¸ë� U W x W U W x W(m U W!dcn}msm�aok4W:~bacX(WY`�q�agfj`Rq�n}m�acq¬��WYq�+DC � â � â ¸�m U W:æ�æëæ nidghif x agm U Xhjag¶jWYq�pbq�m ��f(¶jWYk4msf x q � G ni`b~ � â fi�þ`@f x XÙdgWYq�q�m U ni`,Í Õ> � Ü@?Qñ � ó � � Ô > � Ü@?Qñ � ó � � Ø> � Ü@?$ñ � ó � U WY`��åh x f0�%q�msfac`t��`bagmÖri� T�U fjq�W�¶jWYk4msf x q � G ni`b~ � â k4f x�x WYq�{�fj`b~ïmsf:m ��f�{�fjdcr*`@fjX:acnidcqE G Û K�O N Þ ni`b~� â Û K�O N Þ � 0,f x Wlf?¶jW x ac�F K F N�� ni`b~ F N F N�� ¸ F î Û KPO N Þ F N � 1 � � F ñ î ó

1� � FGF F HIF � F JK HMLNJ NPO �lX!n?º F ñ î ó

1� � F NQO � F�F � î F�F �=f x n(k4W x m�niac`�k4fj`bqsm�ni`*mO � T�U p�ql¸�msf:hiWlm F î Û K�O N Þ F ÐìÍ��#¸bacm�acq�q�pt±²klacWY`�m�m U n}mMI

O �8Í Õ> � Ü@?$ñ � ó � � Ô > � Ü@?Qñ � ó � � Ø> � Ü@?Qñ � ó ÐìÍ � O� U aok U niqsrtX({bmsfim�acklnidodgr:hjag¶�WYq�m U Weu�fjpb`b~ � � � xàÍ´�

R q�ac`@h » fi{b{�W x q�X!agm U�¼ q�msWYk U `�acv�pbWi¸�fj`@W�klni`:m U pbqLqsfjdg¶jW%m U W�ac`*msWlhiW x zon�nid w ac`:{�fjdcr*`@fjX:acnidbm�acX!W%niq�dgfj`@hniq � � � ÐìÍ G íTS$�þüÖ`(��nik4ml¸�m U acq�acq�`@fim�k4fjX({�dgWlmsWYdgrBm x p@W�u�WYklnip�qsW���W�pbqsWY~(ni`!n x hjpbX(WY`*m���W�klni`�`@fim�{ x f?¶jW2I��W(niq�q�pbX(WY~Rm U n}m | WYq�� Û G O� â ÞVUß��t� T�U acq U WYp x acqsm�aok U niq�u�WlWY`2X!ni~@W�¶jW x r,fi�=msWY`2ao`Pk x r*{bmsfih x n}{ U r Û qsWlWÈ°Zt¸�ét¸@�}É Þ �é*� é*� & � õ :�XW � � ù � 1 � � �4 ztp@u�q�m�agm�p@m�ac`@h²Í utr³�}¸ � utr g[¸Kni`b~ � utr´�ZY e ac`¹m U W(ac`*msWlhiW x zon�nid w ¸��We��`�~,W4º@nik4m�dgr�m U W x WYnid�zon�nid w � T�U W[fj`�dgr�~baè±²klp�dgm rïacq%m U n}mwR Û K Þ U niq x WYnid"k4f�W4±²klagWY`*m�ql¸�ni`b~Rm U W�æëæ�ænidghif x acm U X ~@f�WYqI`@fim[��f x$½ ��WYdcdþ�%agm U x WYnidLac`@{�p@ml� T�U W!�.fjdcdgf?��ac`@h�nidghif x agm U X f?¶jW x k4fjX(WYq[m U n}mI~�aè±²klpbdgmÖrÛ ��W�{ x WYqsWY`*m U W x W�nIk4fjX({�dgWlmsW�nidchif x agm U XmsfBqsfjdc¶jW�m U W T 0 I¸*u�p@mOm U W�q$p@u@ÌÁnidghif x acm U X�k4fj`bq�aoqsm�ac`@hefi�Kq�msWl{�qé�msf,�i�[X!n8r�u�W�fi�þao`�msW x WYqsm�msf:qsfjdg¶jW�m U W x WYnid�zon�nid w agm�qsWYdg� Þ �[�\T]@^9_Z`.acbMdfefg�hGikjlenmNoqp�jlr�ikh�i�sZtuikevhGtxwZtzyGr.{�of|Xo~}Po~�fof�� ^9_x]@^T_l`3aZ�k���3jZy&r h8gxalr0t�r.a�h��qi�r h&aBefgxt������x}�bMjly�m@�9�,�� bMjZyc� �x�q��|�Z����txh����M� � h��ftu�Na����kjly.t� qpfaBer0i�jZe¡jBb�{¢m@�T�, � d~p£h�j�jZy&�qtzy.�ofaZef�¥¤D¦¨§�©Zª " « §�¬�©Zª #G«#­ �¯®�jZ°�p~dqhGt±a�²jld~ef�¥³´r�dfg&�¥hG�fa�hc� ���M� �@µ {¢m@�N�, � ����³XbMjZyc� �x��� |

N � MON>P

Page 10: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� «}¡Q¥GÀ���£?¥Y¢G¥I£}¿����ë£iÀ¯À�¾��G¢�¢��}¨��YÀ�¾¯«}¿ ý

� ����txhE}��¦�� " � #" ��� ª�� o�� ¦�� �� � � }�� o~aZef� $ ���<��� � ¦ "� ���u���M|�� ����f����txh���� "���������� � ��!�"#��� ��$l� !ubMjZy&%��(') ���* ���v�+ ����txh���, "���������� ,��!�"#�-� © �M|�� � � � ���<��� � /.�10 � � ! bMjly&%´�2'� ���*�� �v�3 �54~jly�° hG�~t±¤76¡¤ °�a�h�yGik 98��8�ftxyGt:8<;�= >Ni�r.hG�~t±g�j�t@? gxiktzelh jZbThG�~t±°�jZe~jl°´i�aB���-;�ikeA,B>C��EDF"G� 6 © �TaBh0h�i�g�t�H8tz�~dfg�tv��8 �I ����txh&J " � J # hG�~tuh �3j�r�°�aB���kt�r h8sltzg�hGjZy&r bMyGjZ°KD�ofaZef�ML " ��� � $ � aZef�1L # ��� � $ �hG�~t±g�jZyGyGtzr�pjlef�qi�e~w�p�jZ����e~jZ°�i�aB��rN �5O�PRQS� � $ ��TVU µ � � �@W��8ikh���� L " ��� � $ � �SX��ujlyc� L # ��� � $ � �YX��lo~_�Z\[�\�y�txh�d~yGeT��4S]�^ � ���%~�_L9� � � "`H8t�r . �aL " �M����| � $ � � L # � ����| � $ ���cb OdP�L9� � � ¦e%´_�Z\[�\�yGt�h�dfy�eT�d4S]&^ � ��Z�Z�fP�gih�tzalg&�¡� ! i�eA^IevhGtxwZtzyjH jvjZhGrz�kL9� � � ��U µ | � |lW �Rm g� ­ � O�P�� {¢m@� ��n, � °�jq�£�v�v� ���x} _�Z\[�\�jZdqhGp~dqh � ! �Û � fimsW!m U n}m�m U W!X!n}m x agº ( U niq�ac`*msWlhiW x WY`*m x agWYq�q�ac`�k4W O Rpo Û K Þ U niq�ac`*msWlhiW x k4f�W4±²klacWY`�m�qY¸ëni`b~Geqoþ~bag¶tac~@WYq

O � Þé*�°�@�sr ù �?� õ �i� /�õ �8� ù9� � � õ � � ù � 1 � � �4& � õ*ù � õ utp4�v ¿ �$£}¥l¢�£}¤°Â�«}¡G¾�À���¦ � � �,�j«8¢4¥�¿�«iÀ"¡s¢YÀ�¨t¡Q¿xw�y v ��K¾�Àþ­Q¢ �*£}§i¢G¥-�$«i¡Q¡�¢��8À�¤ <��¾ ��¢ ��¾�À@ª�¡Q¾.¿�À.¥�¢�z�£>�8À�¤ <£}¤.¤K¾.¿�ÀÁ¢ÃÂ*¢4¡Q¥ T

3È &fg-O<g�ÉL¥G¨�����À��*£iÀ F Í>= Û UV Þ X(ft~�� F Ð �ZY e �

��¡s«Y«<� � !�WYklnipbqsW:fi�Lm U WB��`�nid�k U WYk ½ ao`Rq�msWl{ì�YZ*¸ë��W�fj`bdgr U n8¶�WBmsfPk U WYk ½ m U n}me`@f���f x qsm�klniqsW:acq�X!aoq�qsWY~��ztp@{�{�fjqsW�m U W x Weacq T �3È &fgkO<g�ÉK��acm U F Í>= Û U�{V Þ X(ft~�� F Ð �ZY e � T�U WY` F R Û�T � Þ X!f*~�� F Ð �ZY e $�| N Gâ v~} ¸ni`b~ F Rpo Û K Þ & R Û g�K Þ F N � Ü Gâ@� �.f x F K F N �}¸Lm U pbq F Rpo Û�T �BY g Þ X!f*~�� F Ð Gâ v~} $ � Ü Gâ@� N �ZY e�o¯� � U WY`bk4W

Rpo Û�T Y g Þ $F� ß��eX(ft~�� U niq�n x f�fim Û�T �:O � � Þ ��acm U F � � F Ð �ZY e�o��Bztac`�k4WM� G Û�T O N Þ ni`�~x� â Û�T O N Þ n x W(dcac`@WYn xk4fjXBu�ac`bn}m�acfj`bq#fi��R o Û�T Y g Þ $ �v~} ni`b~eagm�qK{�f?��W x ql¸8m U WY` Û�T �:O e o � � Þ acqën�k4fjX!X(fj` x f�fim�fi��� G Û�T O N Þ ni`b~�� â Û�T O N ÞX(f*~bpbdgf:�}¸tni`b~:Wl¶jWY`!f?¶jW x m U W x WYnidcqLq�ac`bk4W F � G F O F � â F Ð �}� T�U pbq T ��aoqOni`:ao`�msWlhiW xLx f�fim�fi� | WYq�� Û � G O�� â Þ ¸tni`b~��acdod�u�We�.fjpb`b~�n}m�dcao`@WB�i�}� �é*� �*�sr � ùY1 � õPù� R ���0� õ � õ �?�(� / ú r ù ���� õ��P1 � ö � / � � ö � 1 �54é*� �*�c�}� ��«Gªjª@¢4¡Q¥Q¦�¾�À��6; ¥s��«}¨t¿�� � !�WYklnipbqsWIfi�þm U W[pbqsW�fi�þm U W » fi{b{�W x q�X!agm UK¼ q�msWYk U `�acv�pbW�ao`�fjp x nidghif x agm U X�¸msf:ac`�q�p x W�m U W[nidghif x agm U X®~@f�WYq�`@fim x Wlm�p x ` Ê #�ü¬æ³n}m�q�msWl{O�*¸bm U Weu�fjpb`b~4. � � � x ÍO1 U niq�msf:u�We¶�W x aè��WY~��ü¬`�fjp x klniqsWi¸ � k4f x�x WYq�{�fj`b~�q�msf g[¸ � msf,�ZY e�oKni`b~�Í msf,�}¸�qsf(��W�hiWlmMI

gMxue ØÔ ié*� �*� Z*� ���t«}¾��G¢ï«<�!À��t¢ ��¢ÃÂj¡s¢Q¢[Ï � ¾�À��L�¢G¥Áª@¢��8À%Àë g��%æKWlm Û �

1Þ1m U W T n0r*dcf x k4f�W4±²klagWY`*m�q�fi�f="�²ztac`�k4WI��W`@WlhjdgWYk4m T n0r*dgf x k4f�W4±²klagWY`*m�q(fi��~@Wlh x WlW�Ï $ ��ni`�~ h x WYn}msW x ¸�m U W�W x�x f x X!ni~bW�ac`åm U W�n}{b{ x f0º@acX!n}m�agfj`´msfÍG= Û UV Þ u*r R Û�T Þ aoq7s � � Ü G g

� Ü G ͳí � �Oztac`bk4W���W�n x W%dgf�f ½ ac`bhe�=f x ��f x qsmOklniq�WYqO��acm U F R Û�T Þ X(ft~�� F Ð �ZY e ¸��W���ni`�m]g � Ü G ͳí � x �ZY e ¸�aÃ�°Wi�$e~g � Ü G x Í � �é*� �*� é*� ��«i¦�ª�¤g¢�zi¾�À�<�y�¿�£}¤ <0¥G¾o¥ � T�U pbq���W U n0¶jWemÖ��f:u�fjpb`�~bq��.f x g I�m U W�� x qsm�fj`@W g~x e G ã � k4fjX(WYq�� x fjXm U W » fi{b{�W x q�X!agm U�¼ q�X(Wlm U f�~�¸�m U WeqsWYk4fj`b~,fj`@WAg � Ü G x Í � Y e k4fjX!WYq�� x fjX m U Weniklklp x nik4r�fi�þm U W T n8rtdgf xW4ºt{�ni`bq�agfj`K� T�U W x Wl�.f x W%�.f x e xàÍ ÔÖÕÕ Ô¬×tØ ¸ » fi{b{�W x q�X:agm U�¼ qLu�fjpb`�~²�%ac`bq�ni`b~�acX({�doagWYq0gyx e G ã � ¸t� U W x WYniq�.f x e � Í ÔÖÕÕ Ô¬×tØ ¸�æ�n}h x ni`@hiW ¼ q�u�fjpb`b~�hjag¶�WYqfg � Ü G x Í � Y e � T�U W[dcn x hiWYqsm�u�fjpb`b~��.f x g acq�fiu�m�niac`@WY~��.f xe CÑÍ ÔÖÕÕ Ô¬×tØ ¸*��agm U gyxàÍ ÔÕ ÔÖ×tØ � Ê f x Ï�ß �}¸t��W���`b~:m U W�k4fj`bqsm x niac`*m0gMxàÍ G ã¬ä � x fjX æëWl� VWl¶ x W ¼ q�X(Wlm U f*~�H�.f x Ï�ß�Z*¸�m U aoq�hjag¶�WYq�g xÆÍ â�ã¬ê ��agm U e C Í ; ã¬ê H��.f x Ïïß�é*¸�m U acq%hjag¶jWYq-g x Í ä�ãc� �%agm U e C Í(� ãc� �� agm U e CÑÍPî�¸b��W�hiWlm�n(u�WYqsm�{�fjq�q�agu�dcW�ao`�msW x ¶?nid#dgWY`@him U gQCÑÍ ØÕ í Ø� ò Ü@?Qñ Øò?ó �$ X Z+�^- U�j � j � U -7�����%(��P{�/���� �0��-�� j Z�� j�R �^ U ��-�� j U]j Z+�^-��0�& ��� j ��- j � S"�6O�0��.1Z . U �j� .�����- jOUWV ���%(��P{�/1{

MOM -?X���WY$d

Page 11: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� ������ÀÁ¢���¤��¢�������¢ ���¢4§0¡s¢�������K¾.¦�¦!¢l¡Q¦!£}¿�¿

{ } | � � t�r h��qh�i�°�t�qjld~²~��t ­ 'G$ ­ #�� ­ "0' � � +3 %´�~a��qr­ 'G$ ­ 'G$ ­ #�! ­ ­ � ­ %´�~a��qrp~yGtzgxi�r�i�jZe ­ 'G$ ­ "I!&( ­ #G' � ­ � + �~a��qr�qjld~²~��t ­ ( % ­ $G# ­ "�� � � ��� %��ZtzaZyGrtx �h�txe�qtz� ­ ( % ­ ( % ­ # % ­ ­ � � �ltzaZyGrp~yGtzgxi�r�i�jZe ­ ( % ­ " #�� ­ $�! � ­ N �ltzaBy&r� dfaZ�qyGd~p~��t ­ "G" $ ­�� ! ­ $G' � � � 3 % %�� �ltzaBy&r­ "G" $ ­ "�"0$ ­ % $ ­ ­ N �� �Zt�aBy&rp~yGtzgxi�r�i�jZe ­ "G" $ ­ #�#G( ­ 'G$ � ­ �l� 3 � �ltzaBy&r

� ���������Sq+��!LWYq�m�W4º*{�W x acX(WY`*m�nid�{�n x niX(WlmsW x q��=f x ~@fjpbu�dgWi¸t~@fjp@u�dgW4ÌÃW4ºtmsWY`b~@WY~�ni`b~�v*pbni~ x p@Ì{�dgW�{ x WYklaoq�agfj`�¸tni`b~�WYqsm�aoX!n}msWY~�m�acX(W��.f x ni`�W4ºt{�fj`@WY`*m x ni`@hiWefi�þÍ¢Y}Z(¶}nidcp@WYql�é*� �*�°�@� � «}¡�?¾=¿*ÂD��¡�¢��l¾=¥Q¾¯«}¿+�%ü¬`�qsmsWl{å�}¸b��W[klni`,pbq�W Ë fjn}m�ac`@h}ÌÃ{�fjac`*m�k4f�W4±²klacWY`�m�q%ac`�m U W T n0r*dgf x W4º*{�ni`�q�agfj`R Û�T Þ ac`bq�msWYni~·fi��qsr*XBu�fjdcack�k4f�W4±�klagWY`*m�ql¸�niqIdgfj`@h,niqeacm[ac`*m x f�~�pbk4WYq�`@f�W x�x f x ac`·qsmsWl{´é�� U acdcWBk4fjX({�p@m�ac`@hR o Û K Þ � � acm U ÏiÌÃu�agm Ë fjn}m�ao`@h}ÌÃ{�fjao`�m%k4f�W4±²klagWY`*m�ql¸#n�`@WYk4WYq�q�n x r�k4fj`�~bagm�agfj`Pacq�m U n}m�Z � � O Íàmsf�WY`bq�p x Wem U Wk4fj`bqsm�ni`*m�k4f�W4±²klagWY`*m�� x fjX`Rpo Û K Þ aoq�k4f x�x WYk4ml�W �L%k'���l��pc � U WY`2qsWYn x k U ac`@hï��f x qsmeklniq�WYqe��agm U e xÇͳ¸"~@Wlh x WlW²Z�acq�WY`@fjp@h U �(üÖ`b~bWlWY~�¸ëÍ G í � g � xÍ G í � e q�ac`bk4W g � x e Ûû fi{b{�W x q�X:agm U�¼ q�u�fjpb`b~ Þ ¸�ni`b~P�=f x Ï�� é*¸�Í G í � g � x ÍRâ í � xÇ�ZYGÍuxÇ�ZY e �T�U pbq�nidcd T n0r*dcf x msW x X!q�fi��~@Wlh x WlW���é[hjac¶jW�n�`@Wlhjdoaghjagu�dgW%k4fj`�m x agu�p@m�agfj`:msf!Í>= Û UV Þ ¸bni`b~�m U W�dcn x hiWYqsm�¶}nidcp@Wfi�0g�acq�Í â�ã¬ê ¸#hjag¶tac`@h:n�k4fjX({�dgW4º@agmÖrïfi��Í ä�ã¬ê msf�q�WYn x k U n²� U fjdgW x ni`@hiW[fi�LÍ¢Y}Z:¶}nidcp@WYqY�-0,f x WIhiWY`@W x nidcdgri¸�.f x e x ÍPî�¸�~@Wlh x WlW[Z���acq�WY`@fjp@h U ¸@hjag¶*ao`@h(n(k4fjX({�dgW4º@agmÖr:fi�þÍ Õ ò� ò ×tØ �

�@�������$������%k� ���Z'P# W �+* �(#T��*� W U n8¶�W:acX({�dcWYX(WY`*msWY~·nidghif x agm U Xàztæ6E ac`·m U W w n x a��i] w qsrtqsmsWYX Û ¶jW x q�agfj`åZ*� Z*�°�}ÌÁnidg{ U n Þ Èè�4É�ni`b~2W4º*Ì{�W x acX(WY`*msWY~ agmBfj` n #�m U dgfj`�� w �Y�}�i� $ p�`b~@W x æëac`tptº#� � W U n0¶jW�k U fjqsWY` m U W�Z� ¹�.pb`�k4m�agfj` q�ac`bk4Wïagm(acqm U W�WYniq�acWYqsm!fj`@Wi¸���agm U fj`bdcr³fj`@W�W4ºt{�fj`bWY`�m x ni`@hiW�msf³qsm�p�~@ri� Ê ach@�LZ¹q U f?��q��.f x WYnik U m�n x hiWlm!{ x WYklacq�agfj`Û ~@fjp@u�dgWi¸�~@fjp@u�dgW4ÌÃW4ºtmsWY`b~@WY~K¸#v*pbni~ x p@{�dgW Þ ¸�ni`b~R�.f x e s�Íçni`b~ e sÍ â ¸Km U WBu�WYqsm�{�n x niX(WlmsW x q Û g[¸�Ïb¸ni`b~ � Þ �.f x fjp x X(Wlm U f*~�¸#msfihiWlm U W x ��agm U m U WBWYqsm�acX!n}msWY~¹m�aoX(W[msf�k U WYk ½ m U W�� U fjdgWIW4º*{�fj`@WY`*m x ni`@hiWi¸Ka¯�°Wi�Í¢Y}Z Ë fjn}m�ac`bh}ÌÃ{�fjac`*m�`tpbXBu�W x qY� Ê f x WYnik U { x WYklaoq�agfj`�¸þm U W�� x q�m x f?� hjag¶jWYqBm U Wïu�WYqsm�{�n x niX(WlmsW x qB�.f x m U WÏ�ߨ�áß �(klniqsWi¸K� U ack U acq�� U n}m[]�fj`�`@Wlm�k4fj`bq�ac~@W x q�ao`åÈ°�?É%H�k4fjX({�n x ac`bh�m U n}m%� x qsm x f0��msf�m U WB�.fjdcdgf?��ac`bhfj`@WYq(q U f0��qBm U W�qs{�WlWY~bp@{Îfiu�m�niac`@WY~�� Ê f x e s ͳ¸Lm U W�qs{�WlWY~�p@{Îac`bk x WYniqsWYqB� x fjX éRmsf �8ý�¸O� U WY`bk4W�acq`@fim�~ x niX!n}m�ack}� 5 f?��Wl¶jW x �.f x e s Í â ¸���W[hiWlm�n:qs{�WlWY~bp@{Pfi�On}u�fjp@mB�l�i�i�ïac`Pv�p�ni~ x p@{�dcW�{ x WYklacq�agfj`���agm Ux WYqs{�WYk4m�msf�m U W(`bniac¶jW(X(Wlm U f�~ Û Ï,ß �áßÙ� Þ ¸ë�%agm U Û Ï$O�� Þ ß Û ��O$Z Þ � Ê agh@�Kéïq U f?��q�nï�=Wl����f x q�m�klniq�WYq�.fjpb`b~�pbq$ac`@h!nidghif x acm U X�ztæ*ER�.f x ~@fjpbu�dgW4ÌÃW4ºtmsWY`b~@WY~�ni`b~Rv�p�ni~ x p@{�dcW%{ x WYklacq�agfj`�� T�U WYq�WeW4º*{�W x acX!WY`�m�q�msWY`b~msf(q U f?�Ñm U n}m���agm U nBkln x Wl�.pbdodgr(m�pb`@WY~�acX({�dcWYX(WY`*m�n}m�agfj`�¸�ni`b~ïqsWl¶�W x nid#k4fjX({�p@msW x q x p�`b`bac`@hInI�.Wl� X(fj`*m U ql¸qsfjdg¶tac`@h�m U W T 0 �.f x m U We~@fjpbu�dgW4ÌÃW4ºtmsWY`b~@WY~ï{ x WYklaoq�agfj`ïacq�`@f?��ni~bn8rtq��=WYniq$agu�dgWi�

�*�"!-��* * �&^+#Z� Q%������! P�L%-������* '����#" �$���%$��P�+* ��������*� W U n0¶jW�{ x WYqsWY`*msWY~ånR`bWl� nidghif x agm U X�¸�u�niqsWY~åfj`Îdcn}msm�ack4W x WY~bpbk4m�acfj`�¸�msf2qsWYn x k U �=f x ��f x qsm�klniqsWYq��.f xk4f x�x WYk4m x fjpb`b~bao`@h,fi��ni`bnidgr*m�ack��.p�`bk4m�agfj`bql� T�U Wï� x qsmIW4ºt{�W x acX(WY`*m�nid x WYq�pbdcm�q(q U f?�Åm U n}m:nidghif x agm U X z@æ6Eacqev*pbagmsW(W4±�klagWY`*ml¸ëWYqs{�WYklacnidcdgrPmsfP~@WlmsWYk4mI��f x qsmeklniqsWYqBn}mI~baoqsm�ni`bk4W!X�pbk U dgWYq�q�m U ni`´Z�í$8�¸�� U W x WD6 aoq�m U Wm�n x hiWlm�{ x WYklacq�agfj`K� 5 f?��Wl¶jW x m U W�W4±²klagWY`�k4rIdcn x hiWYdcr[~@Wl{�WY`b~bq"fj`�m U W���pb`bk4m�agfj`Bk4fj`bq�ac~bW x WY~�¸jdca ½ WLac`�æKWl� VWl¶ x W ¼ qnidghif x acm U X��z*Wl¶jW x nid�fi{�WY`�v*p@WYqsm�agfj`bq x WYX!niac`K�� �f�WYqïm U acq!n}{b{ x fjnik U W4º*msWY`�~ádca ½ W�ac` m U WPX(f*~bpbdcn x klniq�W Û È �iÉ Þ msf��pb`bk4m�agfj`bq�fi�þmÖ��f(¶}n x acn}u�dgWYq�dca ½ W�K � f x n x k4m�ni` �'&� p x nidghif x agm U XÓacq�k4fjX({�dgWYX!WY`�m�n x r,msf�m U n}m�fi� û d ½ agWYq�È ý0ÉøK� U ack U ��f x�½ q%��WYdcd"� U WY`>e xÆÍ Û ac`¹fjp x`@fim�n}m�agfj` Þ ¸�a¯�°Wi��� U WY`Î��WïW4º*{�WYk4m²X!ni`*r´��f x qsm!klniqsWYql¸�� U W x WYniq!fjp x nidghif x acm U X acq(X(f x W,W4±²klagWY`*m�� U WY`

N � MON>P

Page 12: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

� «}¡Q¥GÀ���£?¥Y¢G¥I£}¿����ë£iÀ¯À�¾��G¢�¢��}¨��YÀ�¾¯«}¿ �

{ � ! { ­ 6 " � # ª ��n � , °�jq�£�­ ( % +I3 %�C��-CCq� C 3 3N 3B� % � � � % � % %~�l�Z�Z� � � �­ ( % +N ��% 3I � N % +BI I + C � % � %E% % � ��%% %~��% � � �­ ( % I N � +BI3 � Iv­ ���SC � IBI % � � � '0! % %~��%%% � � �­ ( % N %~��� �I � II Nl­ % ­ ����C % � %8� '�$ %f��% %~�Z� � � �­ ( % N 3 % ­I3 3 % ­�I�+Z­B3 � � % � %E% ' % �l�Z��%%~� � � �­ "G"0$ �Z�l� N� C � Nl­Z­ %�C ­BIB3�+ � N+ % � %8� (�$ % %% %%% � � �­ "G"0$ INv­B�N 3 % � C ­� % 33+ %B% 3 � % � %E% ( % ��%~��%~�Z� � � �­ "G"0$ � �3 � 3 �� + � %f�Z� ­I + CB% ­Z­ � % � %8� (�' %f�Z�l�Z��% � � �­ "G"0$ +B�3 % I % �I 3 % % NN 3I %�% �3 % � %8� ( � % %% %%~� � � �� ��������� �p��z*fjX(W·��f x qsm�klniqsWYq,�=fjp�`b~��=f x m U W³Z� ��pb`bk4m�agfj`Ñac` ~bfjp@u�dgW4ÌÃW4ºtmsWY`b~@WY~Ñni`b~v�p�ni~ x p@{�dcW�{ x WYklaoq�agfj`��

e � Í´¸#a¯�°Wi��� U WY`���W[W4ºt{�WYk4m�fj`�dgrï�=Wl� ��f x qsm�klniq�WYql¸#f x `bfj`@Wi� 5 f0��Wl¶jW x ¸�ac`,m U WBklniq�WIfi�7= Û K Þ ß K�ä�ã¬â}¸x WYdcn}msWY~Pmsf 5 nidcd ¼ q%k4fj`?ysWYk4m�p x Wi¸ û d ½ agWYq�{ x fi{�fjqsWYq�n²qs{�WYklacnidèÌÃ{�p x {�fjqsW[nidghif x acm U X msf²��`b~PnidcdK��f x qsm�klniqsWYq�n}m~bacq�m�ni`bk4W(Ð �ZYGÍàac` Ú!Û Í G ã¬â Ü�Ý Þ � %f�WYq�m U acq�nidghif x agm U X hiWY`@W x nidca��lW�msf:fim U W x nidghiWlu x niack��.p�`bk4m�agfj`bq &� U n}m�aoq�m U WBu�WYq�m%k4fjX({�dcW4ºtagmÖrïfj`@W�X!n8r�fiu�m�niac`,�.f x m U W T 0 p�q�ac`@h » fi{b{�W x q�X!agm U�¼ q�X(Wlm U f�~ & » fi{@Ì{�W x q�X!agm U hjag¶�WYq�ao` È°�}É�q�fjX(W!n x hjpbX(WY`*m�q�hjag¶tac`@h�Wl¶*ao~@WY`bk4W�m U n}m"g x e G ã � X!agh U m%u�W(m U W�u�WYqsm�{�fjq$q�agu�dgWu�fjpb`b~(�.f x ��`�~bac`@h�ac`({�fjdgrt`@fjX!acnidtm�aoX(W�m U W x f�fim�q�fi�#hiWY`@W x nid�X!f�~bpbdon x pb`bag¶}n x acn}msW�{�fjdgrt`@fjX!acnidoqþfi�#~bWlh x WlWÏb� » fjpbdc~Pfj`@W(acX({ x f0¶jWIm U W�msWYk U `bacv*p@W[utr�k4fj`bq�ao~@W x ac`@h²m U W(q U n}{�WIfi�Lfjp x {�fjdcr*`@fjX:acnidcql¸#dca ½ WBacq�~@fj`@WButr!�fj`@W U ni`b~ �p x �=WlW:È°Z}ÉK�=f x m U W[q�X!nidcd#ac`*¶jW x qsW%{ x fiu�dgWYX ��acm U F � F ÐìÍ � � â � â &

W ���a���������+�+*� e������� �����������������������! "��������#�$��#�%�&'�)(*�+���,�-.��$/�!01���!�$.%324��&65�&+��#���78�:9�;=<�>@? ;BA)C�DFEG<IHKJ"L�MN�O'PQA/L �<m$#$#"#4{F"?�R/SUTVT65�9�W�H@9$y�; 5u3$?$A<;KX�YGZCw$H�2@9�3[XG\�;GF"H�T[RVXVZ�T[RC3$HC7�T�5u3�:GXC3"B@=�T[X+="9$HC=Y;KR@2@F+{

� m]���^,�$.��-���(I���^�$.%_(4 �#�`G�����ba*� s W_ [Cj �$-��$� _C�W�'� U�V�M � P R � j Z [ ���$� j � Q �6_dc �����W� j Zu��-8e {=f ÕUgÃÕ { N -hL)>iJ]jk<l<iE�D�m�n�;oJUpq C�>iJ]j=>kr�stHu? vVv (>e]w�w�w�/1��� U �r{Ve&�[w�m UWV�x <ij=HyC�>i<bzbJ�H'<�;bD�m|{/J�}^s~C�H'<�>b��j=DK<�m~j�<1�8� [ �'-CS"�6W`pn<�6 �'�WSC� [+[ {�e���e"e${� � ���^,�$.��-���(*�+�b(� .#�`G�����4a*�+���$�%�0�,t�4��#~�5����ia�#~�-.����b�1� )u�$. j U ��- Sde �����=� V U �'��%S$���4{ N -�L)>iJ]jk<l<iE�D�mGn�;�JUp

q C�>iJ]j=>kr�stHu? vVv (>e]w�w�w�/1��� U �r{Ve&�[w�m UWV�x <ij=HyC�>i<bzbJ�H'<�;bD�m|{/J�}^s~C�H'<�>b��j=DK<�m~j�<1�8� [ �'-CS"�6W`pn<�6 �'�WSC� [+[ { � m�du� ��� !@{� �[���,����@��#�����&��~-/��(*� )L��-u�C��- S � �W�o�����R U@U$j UWV � c����"�& �'� j �h��- j � S"�6��0k@�u� j � U -�� V ��. j U ��- S R � j Z Z+� S$Z c�� j � Q - U@R -E{ N -

L)>iJ]jk<l<iE�D�mGn�;�J�p q C�>iJ]j=>kr�s~Hu? vG� (>e�w�w$d�/1�8�f{�� ���+ �60�u�b��{'��� U �r{Ee0#�!�# UWV�x <ij=HyC�>l<�zbJ�H'<�;bD�md{/J�}^s~C�H'<�>4�~j=DK<�m�jk<1�V� [ ��- S$�6W`n<�6 �'�WSC� [+[ {�e&!�Yu��e@Y�w4{

� �]���,����@��#�����&��~-/��(I�@)L�'-��+��- S��^�W�o�����+ U@U$jbUWV �^�+-����"�& �'� j � � U �+�+�'��<�0k@�u� j � U -E{ N -1L)>iJ]jk<l<iE�D�m�n�;�JUp q C�>UJ]j=>�rks~Hu? vG�h(>e]w�w"d�/1��f{t� �$�+ �10�u�<�V{��d� U �r{Ve0#�!�# UWV/x <lj=HyC�>i<�z�J�H'<�;�D�m|{/J�}�stC�H'<�>��~j=DK<�m�jk<1�u� [ ��-CS"�6W`pn<�6 �'�WS+� [�[ {�e&�"���ue�d��4{

� du���,����@��#�����&��~-/�^(*�z)L�'-��+��- S �W�o�$���R� U ��� j � U -+� j U �W�o�$�����C� S$ �6� [8U � _C- U � �'���'��{ N -8L)>iJ]jk<l<lE[D�m�n�;�JUp�{QM x {�? ���T(pm$#$#Ce&/1�� U �r{�m4e��"d UWV�x <ij=HyC�>l<�z�J�H'<�;bD+m|{/J�}�s~C�H'<�>b�~j=DK<�m�jk<1��� [ ��- S$�6W`pnb�1 �l��S+� [�[ {�m$#u� � e${� !]�:�)����&��@�����1� M � j � U -��$� [8U ��- j � -��0�&o.��+ �"��� �$-�� �W�o�����O- U -@���6 Uh  {q>�¡E} Õ   �4�'�i�'� jWj ��.�� ���+��. j � U -E{ N -hL)>iJ]jk<l<iE�D�m�n�;oJUpMz:¢£�~¤�O�¥ (pm�#"#$#�/1��¦ {ua U �W�o�C���<�V{���� U �r{Ee Y � Y UWV/x <ij=HyC�>l<�zbJ�H'<�;�D+m|{/J�}�s~C�H'<�>���j=DF<�m�jk<1�8� [ ��- S$�6W`pnb�1 �l��S+� [�[ { ��� �@d � {� Yu��a4,�$.$.���£��aI� P - U�j � U -�Uu-u�C��- Si�+� � .��+� j �$�$������� j U �<�"�����u� j � -4��� �6 ��.0���'� _@{EA@?�?�R/S�TVT[§V§V§�;W7&:@F�;G9"?"A@y�;U¨&A�T[R49$HC="w�:C3"BVT

W@w�:�:49"?GT�©�ªG«�¨�¨uX@H43G¨u¬�T[­C3@=&?V¬�®C3�BuX494=Y;KRC=4�Y�C� [Cj {um�#"#"m4{ � [ �WS$����{� wu� N �b�<� � j ��-u���&,� V U c+�'-���W_4¯ U � j ��- S�` [8U ��- j �& � j Z+� � j ��."{"XY��.,ZE{ M � [ { P ��� N ` N �<�b� � j ��-u���&,� !"�&�$` e]wWY��@�+�R� R±°zU Q �Ce]wWY��@{PO[�[ U �"�0�8� �& .1Z m@e"��e�w Y��G² N �<�<� � j ��-u�+��,�+��a U ��,���O� [�[ U ���0� g"�+� _ m�dC��e]wWY���² P � �6 ��.0��- ��� j � U -��$��� j �$-�����,�C�N -�� j � j � j �$�8e@Y [ ��S"����{

� e0#u��³i,�#�%~�´~-.���!����µ£���/�$.%�7���~ ��[/��(I�/¶·� N -CUu-+� j ��� _ [ ��.����W�% U ��-u�C��- S V U ��C� �4���W� U -V�L� k@�u�& �] U@U$j �L�$-u�h� k@�u�& �] U@U$j �6.�� [ U .0���*{ N -¸L!>UJ]j�<l<iE�D�mGn�;4JUp4�=¹@H+ºIO q�q/q �@r�}^s�J�;lD�C�}»J[m±{/J[}�s~C�H'<�>�M>kD�H+º�}1<�HyDyj%(>e]w�w�w�/1� [�[ {�m ��� �4m��"#C{

� e"e���¼��$.�½�¾����!�$.%¿7À �������#��Á�� �U78�<a U �+-u�+� U - �+-�� U�V �6�6 U � ��-u� U -���� V U �$� S"��cC,�$��. V ��-�. j � U -���{ N -±L!>UJ]jk<k<iE�D+m�n�;"JUpMN�O�¢�Âo? ��à (rnb�$���r� s U � U ,�"� U �8m$#"#4e&/1����{�az�+%S$���W�O�$-u� J { s ��� �'-+���6,�C���b�C��{�� N �<�<� s U � [ � j �1R� U .���� j _@� [+[ {�e � �Cm�#C{

� e&m]��¼���`.Ä��5�#�����Å1��ÆÀJ�rV<�mt;�Ç�>kD�H+º�}hÈ<�HyDFÉ=Ct<�;!s�J�C�>bC�mÊjlÇ�Ë6j=C�Ë�̽ÇVÍ�ËÎ<1{��<Z@Ï X Z��6�W�'����Ð�<. U ����� U �o�����f�C� [ Ð�6 �����+ �^�+� J _ U -V��g"�$-E{m�#"#$#C{

� e � ��¼���`.Ä��5�#�����Å1�����$.%Ñ7Ê .�����@#���Á.� �U78��¦ U � j .0�$�W��� V U . U W ��. j U ��-��+��- S UWV^j Z+� ������� ��- j ��W_ V ��-+. j � U -��o��- � U �+c����[ ��.����W� U -V{ N -IL)>iJ]jk<l<lE[D�m�n�;�JUp^H+ºt<���Ã�H+º1O q/q�q �@r�}^s�J�;lD�C�}ÒJ�m�{/J[}�s~C�H'<�>M>kD�H+º�}�<�HyDFj:Ó�M^N�O�¢�¸? ��Ã=Ô�(pm$#$#Ce0/1�+�f{4az�C%S"���W���-u� J { s ��� �'-+���6,�C�u�b�C��{'� N �<�<� s U � [ � j �6 � U .���� j _@� [+[ {�e"e$e��ue$e@Y4{

MOM -?X���WY$d

Page 13: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

�l� ������ÀÁ¢���¤��¢�������¢ ���¢4§0¡s¢�������K¾.¦�¦!¢l¡Q¦!£}¿�¿

� e��[��¼���$.�=��#~/������ "����¼��@$.�=�~#~���01�£¶·�����$.%±¼�,t5������@��¼�E)u�$. j U ��- S [8U � _C- U � �l����� R � j Zh,� j � U -u����. U �j� .�����- j ��{£ÆÀÇ�H+º@<=¤}:Ç�HyD�;�jkºt<�Mmtm�Ç�ËÎ<�m����V� (>e]wWY�m"/1�u�4e0�]�4� � �C{

� e&�]��¼�,t5!Ä����t��¼�� P - �$� S U � j Z�� ��. j Z�� U W_ UWV -4����c8�6 ��� S$,� [ Z+�R��-u��. U - ���6tC� j _@{���OiM^Æ ËÎ<ij=HyC�>l<�;�<�>kDK<�;4ÃG�i(>e]w Y$d�/1{� e0du�:7À �������#��)Á.� �i7 ��� U$[8U � �$��� V U ]� � [ ��.6� U�.0� j � U - UWV j Z��]������� ��- j ��W_ V �+-�. j � U -���{ N -±M�Í=;lHy>iÇ�j=Hy;"J�p���{QM^z¸? ������� (pm$#$#�m"/1�

gC{ ` J { J �$� U$jWj �^�$-u� )z{ M ��. U ���<�+��{�� J �$c U ,� j U W_ JEN �bd4�u��& �����8)+,��-�.��"� [+[ {����u�C�"�@{� e&!]�Q&65������$)u�$� j �<�$�$����� j � U - UWV �6�'�6� ��- j ��W_��o� j Z+���o� j ��.0��� V �+-�. j � U -�� R � j Z]. U W ��. j � _� U �+-u�+�����'��� j c�� j {VM4{QÆ ¢@>iÇ�m@;���ÆÀÇ[H+º �

�~JUp�H��������&� � (>e]w�wCe0/1���Ce0#u���@m � {

N � MON>P

Page 14: Worst Cases and Lattice Reduction - InriaDamien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. inria-00071999

Unite de recherche INRIA LorraineLORIA, Technopole de Nancy-Brabois - Campus scientifique

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

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

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

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

ISSN 0249-6399


Top Related