On Watermarking Numeric Sets

"! #%$'&)(+*,.-%& /10203547680:9 ; =@?ACBED>F+GIHBKJCLMBKN+O5PQmog...

Author: Sion | Atallah | Prabhakar

6 downloads 795 Views 236KB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

"! #%$'&)(+*,.-%& /10203547680:9 ; <>=@?ACBED>F+GIHBKJCLM<>BKN+O5PQL+GIAC?MFSBKNUT VQJXadcc pxtu^xZoa cf5Y"vwZ\jxpx[hi^x\gy]_admcb^>`bkn`bhzhZ\`adXcfj{Xe>mogdjChiadadp}kj_`5|fgQaQIadk knmaytul mmycbadg]Cadr `nm`bcnhza\`bj{x[~~adj>jRtu>`Zcbmo[8ojC\mp \`bRhzZ\@j fexknkn ^xcmj_gQao ©ªª ¤w\«¬\+¬Cw£I¢®£¥¤x¦o§\oR X¨7¡R£¯¢¨\I§\£¥>¤¬ X© ¦Q§o_R¨R¨7\£¬R¨\\§o\ 8°ÅÌy±7Ç ²Q³QÅÌy´ËµCÇ"¶ogQ³\mo·}j¸ Î@¹iºoa1¹u»¥kn¼\^x½[¾{[8¼m»¯cb¿ÁhzÀÁÏQÂ1aypÐ¼oÀb` Ãr ¹EasÄRº `b aQg Å Ær ÇÈj_hiÅ ÑX^_ÇÈaÅ ÉZoÇÈtÅ ÊaQÇ[Å Î@yÇÈayp_Å Ëp>ÇÈhzj_Å Ò8ÇÈ^_Åj>Ì\ÓÌdp>ÇadÅ`bÌyaQÍgdÇÈ`mÅÎ_ÌyÊÔza Ç RmÕ ^xjChipxjxp{aQÓ¥Z>]@p>®adhikcb`b`ngQadcbaQÛRhzhzÎx`ÁoÖ^xmo`bÎx[8hzÔiZ\aymoj2Ö hzjxrÎRÔzhi}p_p>aQ`bj_aQZÐjmo]xÎxhzjxcbÔihzZtujxZ`bÒ1aQcbgd[8]x`cbm`Z`brohzZ\moa}jÎ_pxÔzhza jRm``bcbmZfhzÒ p_tEr cbm`bZo`k [Üm×ZZo^_Î>adjCØcaQm` gd^xr `b` ka r ÕgQZohÚZ\cbÙ a\hzjXÏQÙ`bayhiaQ[8pjX`ymp>Ù Ò\^_aQ]xkQÔiohzgymm^C`bpxhzZ\hzZ>j ~¯knjÝhzÔzhzaQ` jXr `"a1hz]>jxcbtEZoaQcbkn[8aQjXm`s`bhi]CZomj ]@r adchip>Þ hijxa}Òx\m\hipxjsp>` cbr aQa knkÈtKc` mor [aadhzÞ"knknZo^xcba}ßsZZot×tSjRp_^_m`[madkncbaQhzggQ^>pxcbmhz``¯mRÙo`Ýr cba\Z\à cb^xaÒ Ôir ZXZocbßXadÓÓ hzmj_jCÒpÐhi]xjX`bcbZZo]@` Z\r knaatucb^xaQjCknhzpxÔzhzmoaQ[jX`aQjXmo`ÔzÒomoZoÔ7cb]>h`cbrZ\Îx[ÔiadkQ[áÙ q ZoZ8t ` Þ r asm`bÎ@adcbaQ[8k`ÈmZcbt"ß>hzZojx^xÒcÈjRßR^_j_[ZyÞ adcbÔzhzayg1p>Ò\gdaZ\ÔzÔz` aQr gdad`bcbhzasZ\jxhikk jxZtwZp_Þ mZ`mRcbßÙ q kn]@r aQagdÞ hzâ_higyp>moaÔzÔmcbaymom1p_pRZocbtUaQmknkn]_hz]_j_ÔzÒ{hzgy`mr Î_ahzÔzhz]>`¯{cbZ\ZÎxt7Ôiad` [ár aÈZo]xt cbZoÞ Î_mÔz`baQad[ãcb[8cmmcbj_ß>Ò\hzhzjxjxÒÒ8` tKrcbZ\hzkÈ[ä`>jR]@^xa Ó [hzjX`nadcbcbhzhzÒ\g^_pxhzjxm`Ò2mÎCÞ mor knada"jgQZ\gQjXZo`bj_aQknjXhipx`åaÁ`bcbZfhijxkÒ{`bZXm8gÁßÒ\[8aQjxmadcbcbß\hzg1ad`7knmZ\jCÔz^>mo`bÔhi>Zoknjhzkwmp_jCmp`mR]C\m[8cn`bmhzßgQaQ^xkIÔimhz`+cbhad`bhzknaQ]@kÈaQgQZohit"moÔzhzÔ`b k ommknkncb^_hzZ\[^x]xk+`bmohz]xZo]_j_ÔzkfhzgyÞ"ma`bhzZ\]>jxcbkQZÙXRæ5hipxhzaadmjknmZ\Ôzc^xmo`bjxhzZoÒ\jÝa Zmt@jCmopknknmoZXj_gdmohKmÔ>`bayÏQpa1jRm^_knkn[ZXgQaÁhicbmhigw`baygQpZoj_mk`n`n`cmomoghzßRjXkQ`bÙIk+ç5moj_^xp c knknZo^xÔiÎ_^>kn`badhz`"Z\jknaQhiÔzk×aQgdcb`baQhzknZ\hzj ÔzhiadÕ jR^_`5]1`b`bZ8Zsm8Æ\\[èé^_Ôpx`bmh`b`^CmÈp>ÔzaÈZ\knZok"t"`bmZo`nÔz`admocmogßRjxkQgQ@aÖÁhzj_RgQÔzhzÔzj_^_aypxmhzj_c Ò}p_mp_`mm`m1g r cbmadj_Ó¥knÒoZoaQcnk"`bhzadj_`bÒ>goÙ êmÔihzÒojCZomcbÔzhÔz` r Þ"[{a5Ù ]>cbaQknaQjX`mjCp}pxhzkngQ^_knk ms]>cbZXZotEÓ¥ZtEÓ¥gQZ\jxgQaQ]>`hz[]_ÔzaQ[aQjX`m`bhzZojZotåZo^xc ë ì_í×î@ïSðÈñòsó®îSôbð×í õöCPÁ÷oFMøBEù>÷VÁ÷\ø÷\ø2WI÷o÷FüúD>FM=S?MúyPd÷o=ýDxFýPÁLS÷þSVdD>WSNE÷ÿ,D <XPd÷VdÿÐ÷oøoGåùCBE=@÷oD÷oN NEBiPÁPÁNE÷ fDxVÁJ D>F5GCPÁLM÷ D@ú?Mø D <XPd÷Vdÿ{<>VÁJCBEF D>VÈ=@B >BKPdL_Pdø5þSVdDRPd÷oúyPdBKDxFBuøÈøLMB ÚPdBKF {Pd D 5<>Vd==S<XPQ<8P Cþ®÷oøÈøÁ?MúQLÝ<xøfPÁ÷öCP\GMø D ÚP5<>VÁ÷xG D>VdBKPÁLSÿÐø ®ACBEFMú÷2PÁLM÷oøÁ÷=SVÁ÷=@÷oøÁB xFS÷o= D>V1ÿÐ<>úQLSBEFS÷}BEF x÷oøPÁBED>F <>FM= LM<ù>÷ 5÷NEN =@÷ MFM÷o= øÁ÷ÿÐ<>FxPdBEúoø ¯<>øúD>ÿ{þM<>VÁ÷\=PÁDPÁLSD_ø÷2D BEÿÐ< >÷\øG®ùCBE=S÷DMGID>Vÿ}?MøÁBuú yG PÁLS÷Bu=@÷oFxPdB ®ú<XPdBKDxF{D ®PÁLS÷s<ùX<>BKNuFM= ÈBu=CPÁ!L " D>#V <XPd÷Vdÿ{<>VÁJCBEF BEøf<>ø BEÿþ®D>VÁPd<>F_P $ Zocn`bhzZ\jxk Zotå` r hik Þ"ZocbßÞ aÁcbaÈkn^_]_]@Zcn`bayp8ÎR2æfcmjX`bkfv+~ Ó¯\\oÆÉXÉR_~n~neXÓo\Ë\Éo>ÌQ1moj_p2~n~eRÓ \ç&\%\ÍogQËoa×Ë\ZoÆ tItE cbZomy[ omo`Ô_r |×aUaQ knaymm`bcbhzgZor jC>momoÔ_j_e>p1gdhiadÎRj_gQkna ]@ê Z\Zojx^_knj_Zocbp_k m`bZhztIZo`jIr \a5Y"Y"Z\jXadjR`nc`bmoaÁgÁc"`+tuZ c \vwo>pxÌd^_ogQÓ¯m`bÍyhzZ\ÓÌÁj}Ó¯\moÆ\j_ÊpÈ|ftEcbaQZ\kn[ aQm`cbrg r a hzj{~j>tuZocb[8m`bhzZ\j{×knkn^xcmj_gQamjCp{eRaQgQ^>cbhz`¯XÙ

~n~ <úQLI÷<>ø5PdLS÷8DxVÁBKPÁLMÿ{ø& DxVÈBKFMøÁ÷VÁPÁBEFPÁLM÷5÷\ø'Sõ öCBuøPÁBEF VÁ÷\ø÷\øs<x=S=@Vd÷oødø÷\=ÝPÁLS÷2þSVdD>WSNE÷ÿÐø D ×ø D ÚP5<>VÁ÷ <XPd÷VdÿÐFM= FMF_?M<XPdÿ?S÷oVdVÁ<>BuNÈú1NEú<>DxF NK>NE÷o?Iú< PÁ>BED>1÷ FM5øÈPÁFM÷o=VÁÿÐþMVÁ*35DxVÁ4BKPÁ÷oLMVÁÿ{÷ ø&5÷DxVÈøPÁVÁ?I÷\=øBEýNKBE÷PÁF_LMP÷ BEød<XøPÁ?M÷o÷ÝVÁÿÐ D <>6VÁJCBE<XFPd÷ Vdÿ{<>VÁJCBEF 7 L_?IøGXBEFPÁLSBuøþM<>þI÷o8 V f÷ ÷ö@þSNED>Vd÷ LS D üOsNKBuú÷ ÿÐFÐøÁ< ÷N }=@BuøbPdVÁBEWS?@Pd÷ <ÈùXNENE÷oúyPdBKDxFPÁ9D 7 BEÿ'H ÷ÈNu<XPd÷V5D>FÐPdD <_ ø×þ®DxødøÁ÷oødøBED>?F ¥þID_øÁøÁBKWMN ÿ{<>NKBuúBED>?IøN ÿ{D@=@B I÷o= 5BuøÈLS÷oVúVd÷o<XPdBKDxF<>FM!= @DxVÈ<>ødø÷oVPøÁD>ÿ{÷1DRPÁLM÷VÈVdB xLxPQø5DXù>÷V BKP 7 LS÷D>VdBKPÁLSÿáBEF_PÁVdDC=S?Mú÷\=þSVdDXù>÷\ø PdD{WI÷1Vd÷oøÁBKNEBK÷oF_P5PdD{ù<>VÁBED>?IøfBEÿ{þIDxVPQødø÷\ø D VÁPÁBEF MGwÿ{<xøÁøÁBEù>÷øÁ?SWMøÁ÷PÐø÷oNK÷\úyPdBKDxF+GUNEBEFS÷o<>V =S<XPQ<úQLIÿÜBKPÁ÷oÿ ¥Bø C ø DxNKNE D ø MAC÷oúPÁBED>?F )þSVd÷oøÁ÷F_Pdø5PdLS÷}úDxFMøÁBE=@÷oVÁ÷\D= VQødøD@úBE÷\'ø AC÷oúPÁBED>FF E þSVd÷oøÁ÷F_PdøD>?MVøÁD>NE?@PÁBED>GF A=@C÷oBEødúúPÁ?IBED>øÁIFøÁ÷oøH úBKD>F_FMPÁúVdNKD@?M=@øÁ?MBED>úFM÷oøø<>< FMþS={VQ=@<>÷ úMPÁBuFMú÷oøF PÁxVQD><XBEPdF BKF 1 ÷ KåD>D>?MVÁVPdø<>N<>xø D>þMVdBi<>JF =@÷o0 V Vd<>ÿ÷1 D L ?@Pd?SVd÷÷FCùCBEøÁBKDxFS÷o=VÁ÷\ø÷\VÁJCBEF I[ W ÿ{BEFSD>VFMøPÁDüBi.P > øúDxF_PÁ÷F_.P CnMD>VÝFS D o5÷û<xøÁøÁ?Sÿ{÷ kR^p_q b cdddcUq e frhsi Buøs<>NEøÁDÐD øÁB to ÷ \8!Zw÷P<ÐøPÁVdBEF D WSBKPdøoG u D øB tm÷ vxwwy\ W®÷ PÁLS÷1=@÷oøÁBEVÁ÷\l= 5~F u PÁDÐ=@÷oFSDRPd÷1PÁLS÷ PÁLÝWSBKPD u

~ ~B

nF2PdLS÷C D>NENED ÈBKF"5÷ =S÷MFS÷<1ú÷VÁPdNK?M÷]¯NENE÷oúyPdBKDxFPÁLI<XPÈBKNENC÷FM<>WSNK÷5?MøUPÁD=@÷Pd÷Vdÿ{BKFS÷fPÁLM÷& <XPd÷Vdÿ{<>VÁJCBEFVd÷oøÁ?SNKP"<>øUW®÷BEFùX<>NK? VÁFMDx=F ùXPÁ<>LSNK÷ Bu=7JCG FSÈD BKÈPÁLSNE÷oBEF=>þ®÷÷VdDÿ{1PBiÁPÁLMPÁ÷\÷?=!@+>x?M?M<>VQ×V55?SFMD>Dx?MVÁVdBKPÁúLSBKFÿ PÁBEÿ+÷ 5VÁ÷\øbPdVÁBuúyPdBKDxFM|ø @þSVdD>þ®÷VÁPÁBE÷oøfVÁ.÷ _?SBKVd÷o=D UPÁLS÷8<xúyPd?M V5÷o<xúQLVd÷NE÷ùX?M<>Vd<>F_PÁ÷÷\=ýVd÷oøÁ?SNiPÐ÷oVÁVdD>VWIDx?SFM=SBø ?MødNE6ø ©]^p_:«ª b U¬/ e c.«ª b ¬ |®|c' dddc.U«ª ¬C e cª ¬| ®{|fxGåø?IúQLPÁLMNENK D ÈBEF ?MødFM=SBiPdBKDxF LSD>Nu=Sø ¯ :«ª ¬C e c«ª ¬|®° ±¯U, c²g| nFDRPdLS÷C V 5D>VQ=S&ø f÷1=@÷ IFS÷PÁLS÷1?SFI=S&ø D>V5PÁLS]÷ xBKùx÷FBEFSþS?@P =S<XPQ1< [LwBKFPÁ÷VdÿÐø5D 9Á?MøÁ<>WSBENKBKP 8ÿ{÷PdVÁBuú]ø ¥øÁ÷~÷ ?IøÁ<>WSBKNEBKP }BKF (,'H*8ÈLMBEúQLúyPd?MÿÐ
´9µ ¶I·¸!

~n~n~

FS÷øÁBEÿþMNK÷1WS?@PÈ÷ö_PdVÁ÷oÿ{÷NÐVÁ÷oNK÷où<>F_P5÷ö@<>ÿ{þSNK÷Buø5PÁLS÷l¹º'»{¼«¹½ ¹Yº ¾«¾¿À5º Á'¾Âr¹lÂBºÃ ú<>øÁ÷>GIBKF¨ÈLSBEúQL PÁLS÷?MødVÁ÷2=@÷MFM÷o=BEF Pd÷Vdÿ{øsD ÿ{÷o<>F ø _?M<>VÁ÷\=÷VdVdD>V5PÁDxNK÷oVd<>FMú÷\ø5<xø/ D>NENKD ø ; Ä Å ½º ÆÂÇÂ'Æ|Æ:¿ Æ :

a.È qUÉw³¥Ê°g^, c'dddc:\Ë,. Ì

aWÈÍq3É]wJ¥¬|®Î)

ÈLS÷Vd ÷ ÏÐ^Ñ_¥:b c'dddcU¥egf1hRi VÁ÷{=@÷IFSBKFPdLS÷l>?M<>Vd<>FxPd÷÷\=÷VdVÁDxV WIDx?SFM=Sø8VQ=SøÏã=@÷IFS÷oø1PdLS÷ÐBEFM=@BEù_Bu=@?M<>N ÷NE÷ ÿ÷oF_Pdø5VÁPÁBED>FIø BEFÐPÁ÷oVÁÿÐøfD+ÿ{÷o<>Fø:_?M<>VÁ÷\={÷VdVÁDxV9¥H A@õ&f÷VQFxPdBEú1<>FM=FC?Sÿ{÷oVÁBuúúD>FMøPÁVQFW®÷BEÿ{ < >BEFS÷o=1ÈBKPÁLÝVÁ÷\øþ®÷oúP PÁD2m< >BEù>÷oFV úD>ÿ{þSNE÷öúDxFMøbPdVd<>BKF_Pú<xø÷WM?@PsVQ<XPÁLM÷VDxFPdLS÷8DXùx÷VQFMú÷þ@.P IOsFMFMøPÁVQW Òb÷oúyPÈ D <{=@BuøbPdBKFIúyPoG ?@PÁ?SVd÷Vd÷oøÁ÷oVdB>BEFMVÁJCBEF&7 LM<XPBuøGfLSD =@DI5÷yVd÷oúD{>FSBt÷. <>F BiPd÷ÿ:< ÚPd÷VBKPLM<xøÐW®÷÷F úQLIV2BiPd÷ÿ NE<>WI÷oNKBEF PÁLMLPÁDÐW®÷8ø?MBiPQVDx?SVC5
ür'.î.çG

4s<ùCBKFD D>Vdÿ}?MNEWSNE÷ÿ)<>ø1<>WIDXùx÷>GMW®÷ DxVÁ÷<XPÁPÁ÷ÿ{þ@PdBKF<>F øÁD>NE?@PdBKDxF+GINE÷P.> ø MVdøPÈD>?@PdNKBEFS÷8úNE<xøÁøÁ÷oø5D ÷FCùCBuøBED>FS÷\=úQJ@ø' 7 LS÷<XPÁPd<>úQJx÷V5úFM=@D>ÿ{NøÁ÷NE÷oúP5<>FM=Ð?MøÁ÷<2ø?SWIø÷P D PÁü?LS÷gD>LVdýGBxíLBKFM<>.¸N r=S<XýWPQ¸< ¸ øÁî÷PoG"â!øÖ&?MWMø÷P8PÁLI<XP}ÿ{B>L_P2øPÁBENKNfþSVÁDXùCBu=@÷{ùXV}BiP.> ø2BKF_Pd÷FM=@÷\= þS?SVdþID_ø÷ ¥øÁ?SW@PdVd<xúyPdBKùx÷1úQJ | 7 LM÷1úQJ>÷oV5<x=S=Sø <{ø÷P5

D UFC?Sÿ2WI÷oVdøfPÁDPdLS÷1D>VdB xBKFM<>N ø÷P {ü~7 GLMBEø ýW<xíW=S=@BKPÁ¸BErD>FÐü=BuøÔÔLFSD>P âgPÁD8Ö&øÁ B xFSB Iú<>FxPdN ÷oV > ø þI÷oVdøÁþI÷\úyPdBKùx.÷ 5þSVdD>þ®÷VÁPÁBE÷oø5D "PÁLM÷BKFSBKPÁBuVdB xBKFM<>N =S<XPQ<}ü1øÁþW÷rP5øÁýG?MíLúQLÐPd¸LM³<XP5ü1PÁLM÷¸Vdæ ÷s Bu.ø5âgøPÁÖ5BENE NIùXødøD@úBEøÁ÷ÈFS÷o÷o=Sø PdDWI÷sD>?@PdNKBEFS÷o={LM÷Vd÷>GRFM<>ÿ{÷N 1¥6O E {< <1NEBEFS÷o<>VPÁVQVdÿÐ<XPdBKDxF}þ®÷:V D>Vdÿ÷\= ÿ ~jZo` r adc Þ ZcpxkQMh moÔzknZ r ZoÔip2tuZc×` r a tcbaQéhkn^_iÔ`bkÈhzj_ÒoÒ1hzaQp_jm`momjCp Ù r ZoÔipxk5tEZocÈ` r ahzj_h`bhimoÔwhijx]_^>`Èp_m`m> Ð` r aQj k r Z\^xÔip

~ S? FSB D>VdÿN{PÁDVPdBEú?SNuø5ø?IúQL<PdVd<>FMøU D>Vdÿ{< PÁBED>FûþSVÁ÷\ø÷oVÁùx÷oøsÿ{<> F =S<XPQ < ÿBEFSBEFÝVÁ÷oNEþ®÷VÁPÁBE÷oøDfPÁLS÷Ð=SFMøÁBE=S÷VQGCÿÐVQ=@÷oV}ú<>FýWI÷BKÿ{þ®DxøÁ÷o=ýDxF PÁLS÷Ý=S<XPQ< PÁLS÷o1F ü L<X"PdýG÷VdíLÿ{<>VÁ¸J{ Vd1÷Pd¸{VÁBEõ÷.ùXâ?SNu=W®÷Vd÷oøÁBKNEBK÷oF_PfPdDVd÷ nøÁD>VÁPÁBEF }<XPÁPd<xúQJCø <>FM= øLSDx?SNu=FSD>Ps=@÷þ®÷FI=DxFÝPÁLSBuø þSVd÷o=@'÷ MFS÷o=ÝDxVd=@÷oVÁBEF !

BKùx÷F PdLS÷{<XPPQ<>úQJ@ø÷>GåøÁ÷ù>÷oVd<>NwþSVdD>þ®÷VÁPÁBE÷oøD <øÁ?Múú÷oødø ?MN øÁD>NE?@PdBKDxF øÁ?SUV ¥<xú{÷ nSD>V}BKÿ{ÿ}?MFSBiP PÁD ü¨ GwPÁLS ÷ <XPd÷Vdÿ{<>VÁJ LM<>øPdD WI÷Ð÷oÿ}W®÷o=S=S÷o= BEFûDXùx÷VQF þSVÁDxþI÷oVPdBK÷\ø8PdLM<XP{ø?MVÁùCBEù>÷øÁ?SWMøÁ÷P{ø÷oNK÷\úyPdBKDxÐF {÷ !UFC?Sÿ{÷VdBuúúD>F I=S÷FMú÷BEFxPd÷VdùXødø?Sÿ{þ@PdBKDxFüBuø{ÿÐ<>=@÷ÝPdLM<XPÐPdLS÷ <XPÁPd<>úQJ <>NiPd÷VQ<XPdBKDxFMø{WSNE÷ÿ =@BEøPÁDxVPdBKDxFW®D>?SFI=SøG@=@'÷ MFS÷\= W {PÁLM÷?MødFM]ø ¥DRPÁLM÷V ÈBEøÁ÷sPdLS÷1=S<XPQ< ÿB >L_PÈNED_D_ø÷BKP > øÈ<>ødøD@úBu<XPd÷o=ù<>NK?M.÷ yGCPdLS÷F øÁLSDx?SNE=W®÷1=S÷ ÷o÷1ÿÐNED>WIFW®÷=@'÷ ÷oVdÿÐFøPÁ÷oþ BKDF ÈLMBEúQL<2úDxÿÿ{DxF=@BKü~ùCBuþG=@÷ VfPÁDFM=Ð=@BEù_Bu=@÷\= W nSDxV < xBKùx÷FBKPÁ÷oÿ G D>V FSD>Pd<XPdBKDxFþS?MVÁþ®DxøÁ÷o&ø f÷1D>BEF }PÁD{=@÷FSD>PÁ÷PÁLSBulø FMD>Vdÿ{<>NKB t÷\= m ù>÷VQøÁBKDxFD BKPW êÍ f÷\ú<>?Mø÷ÝBKPÐ<x=S=SøFS÷ =SøÁJ®G W®÷oúoÿÐVPdLS÷ úQJ>÷oV /7 LM<XPÝBEøo/G f÷ LI<ù>÷PdD W®÷ VÈPdLS÷8<XPÁPd<xúQJ>÷|V # D>V PÁLM÷ >BEù>÷F=SBK÷oNE?@F_PÁP×BED>ÿ{lF ÷PÁDxLMVDCPÁ=DLM÷ D>VfÄ ¼«Bi¹Pd÷£ÿá¾ ¼ &ÂNuBNKFMBEFøÁBuøbPQPdøLSD ÷wøVdÁ÷÷_ùx?S÷BEVQVdVÁJCBEF ý=@÷Pd÷oúPÁBED>F PdBKÿ{2÷ ¥B ÷ < ÚPd÷1V 5FM!= @DxV <XPPQ<>úQJ@Bø |nF PdLS÷FS÷öCPÐøbPd÷Jþ f÷Ý÷FMøÁ?SVd÷WSBENKBKP W d?÷ <XPd÷Vdÿ{<>VÁJCBEF ûÿ÷PÁLSD@= /7 LS÷ F ÷ Kå÷oúPBEø <>úQLSBE÷ùx÷o= W PdLS÷2=S÷þSNE D Cÿ÷oF_PsD ×øÁ÷oúVÁ÷PdøBEF øÁ÷NE÷oúPÁBEF úDxNKNE÷oúPÁBED>F øÁ?SWMøÁ÷PQø ÈLSBuúQ?L ÈBENKN WI÷\úD>ÿ{÷ BEFSþS?SDP D>VPÁLS÷ IFM÷x#G ÈLS÷oVÁ÷ <ýþMVÁBEÿ{BiPdBKùx÷ 5÷\=DxFÝøÁ÷NE÷oúPÁ÷\=øÁ÷oúVÁ÷PÈøÁ?SWMøÁ÷PQ'ø þG

"ð!¸æ+ðG¸å

Osø<>FDXùx÷Vdù_BE÷'G_PdLS÷8øDxNK?SPÁBED>F1 D>V PÁLM÷8øBEÿ{þSNKBM÷\=þMVÁDxWSNK÷oÿÜVd÷o<>=Mø5<xø/ D>NENKD ø ´ Öî âGÔÖè

´ AC÷oNK÷\úyP <8ÿÐ<Xö@BEÿ{<>NMFC?Sÿ2WI÷oV×D +?MFSB _?S÷xGxFMD>F BKF_Pd÷VQø÷\úyPÁBEF D¯ø÷o÷sW®÷NE D 9 ø ?SWIøý÷.Pd¸{ø · D"PÁLM!÷ D>VdBxBKFM<>N7ø÷PoGM<xøÈ=@÷oødúVdBKW®÷o=BEFA@÷oúyPdBKDxF~E E ´ nMD>V ÷o<xúQL2úDxFMøÁBE=@÷oVÁ÷\=}øÁ?SWMøÁ÷P\G ¯õû) ,.w÷ÿ2WI÷\=2<65F2BKFA@÷oúyPdBKDxFE H8V×=S

WIDx?SFM=Sø37?Mød?MFM=Sø÷ö@ú÷÷\=@÷o=7G!¥õû) E{"Vd÷PdV:2=@BKå÷Vd÷F_P×÷oFMúD@=@BEF1þM<>Vd<>ÿ÷PÁ÷oV ù<>VÁBu<XPdBKDxFMøDxVoG®B føbPdBKNEN"FSDøÁ?Múú÷oødøG ¥9õ ) Ex{< 5Pd:V ÝPdDÿÐNKBu³= ¥B ÷ øú÷o?S÷8VdVÁ÷÷oFMF_úPÈD@øÁ=@÷BKPF É úDxFCù>÷F_PdBKDxFÝBEF AC÷\úyPdBKDx¨F E HGMD>VB ×øPÁBENENUFSDø?Múoú÷\øÁø ¥õ û) HÈB xFSD>Vd÷PÁLM÷ } ÷VÁ÷oþI÷\<XP2øPÁ÷oþ õ û)?SF_PÁBEN FSD ÿ{D>Vd÷øÁ?SWMøÁ÷PQø2<>VÁ÷<ùX<>BKNuV2÷FMúD@=@BKF 7 LSBEø VÁ÷\ø?SNKPdøwBKF2ÿ}?SNKPÁBEþSNE5÷ <XPd÷Vdÿ{<>VÁJCBEF ÈþMNENK÷\úyPÁBED>WF H PÁLS÷oV+BKÿ{þSNE÷ÿ{÷oFxPQ<XPdBKDxFMø7D CPdLSBEøwøDxNK?@PdBKDxFMø+<>VÁ÷ þ®DxødøBEWSNE ÷ 'nMD>V+÷ö@<>ÿ{þSNK÷ úQLM÷oúQJ BKF m D>V5=S<XPQ<8?MødFM÷<XP ÷oFÐÿDxVÁ÷<XPÁDxÿ{BEúsNK÷où>÷NGCø?MúQL<xø×BEFMøÁBE=S÷ PÁLS÷WSBi:P ÷oFMúD@=@BEF þMVÁD@ú÷\=@?SVd÷>G"ø÷oNK÷\úyPÁBED>Fý D BKF_Pd÷VQø÷\úyPÁBEF øÁ?SWMøÁ÷PQø}úD>?SNu= WI÷Ý
øÁBKF ÐPÁLS÷}J>'÷ @ø<>FM=øÁ÷oúVd÷PQCø VdD>ÿ øPÁ÷oþ õ ,>GSVd÷oúDXù>÷oV <ÿ{<ÒbD>VdBiPD PÁLS÷8ýøÁ?S.¸{WM·øÁ÷PQ¨ø BK!F$õ# ,>GS<>NKN7D UPdLS÷ÿ B FSDÐ<XPÁPd<xúQJCøC5÷Vd÷1þI÷oVU DxVÁÿ{÷\=DxFPdLS÷8=S<XPQ< nSD>V{÷\<>úQL úD>FMøÁBE=S÷Vd÷o= øÁ?SWMøÁ÷P\G?MøBEF PÁLS÷÷FIúD@=@BEFûúD>FCù>÷oF_PÁBED>F BEF AC÷oúPÁýBED>.¸{F · E ÍHMGC VdG÷o úDXùx÷V5PÁLM÷÷ÿ}W®÷o=M=@÷o=ÝWSBKPùXFIøbPdVÁ?MúP/ <XPd÷VdÿÐ
D &%û){BEøs<Ðø÷PD úDxþSBK÷\ø D PÁLM÷8øÁ<>ÿ÷ <XPd÷VdÿÐVÁBEýD>.?I¸{ø×·sþ®DRÍPd÷F_þWPÁ BuVQ'ø 7 LSBuøfNu<>øPÈøbPd÷þ?Iø÷\ø5<2ÿÐ < ÒbDxVÁBKP {ù>DRPdBKF 2ødúQLS÷ÿ{÷PÁDVÁ÷\úDXù>÷oV PÁLS÷LSB >LM÷oøP NEBEJ>÷NEBELSDCDC=ÝBEFSBKPÁBu ÷ þG«

'|.¸ ¶Ë

&¸ÖèÍä3â!æ=.¸{l¸ ¸ ¶I¸ Ö£

ûF Vd÷_?SBEVd÷oøsPÁLS÷{FSBt÷.¹l¿ Ä ù D ×PÁLM÷ úD>NENE÷oúyPdBKDxF BiPd÷ÿÐø}W®÷ DxVÁ÷<>FM=ý< ÚPd÷V5NKN BEÿ{þIDxVPQø<Vd÷oøÁþ®D>FMøÁ÷PÁD <XPÁPd<xúQJCø1D fPCþ®÷ ü GnF DRPdLS÷VmfDxVd=SøB j ÿ BKF Nu<>)N úo(ú÷\GSøÁPdøÁLS÷o!=÷F@Xÿ{LMDCD>=Sþ®B÷ M?S÷\=NEN W®÷<X6P D>Vd5l÷ FåPÁBEWIÿ{÷oBK÷8F ÝPdLSBu÷}=@÷oødFxWSNE"÷ ÈBKPÁLÝPdLS÷8øÁ<>ÿ{÷NuBEF PÁLSBuø<ÝNKBKPPdNK÷ÐWSBK P ?SVÁPÁLS÷oVoG7ÿÐ<>J_BEF øDxÿ{÷NE÷oødø1Vd÷oøPÁVdBEúPÁBEù>÷{<>ødø?Sÿ{þ@PdBKDxFMøoG f ÷ 5D>?MNE=ÐNEBEJ>÷ÈPdD}W®÷
< ÚPÁ÷oC V 5FM!= @XD>VfúQJ@'ø COs/ø 59÷ ÈBENKN+øÁ÷÷xLG Áÿ{BEødøBEF }<}øÁÿÐVÁJCBEF {ÿ÷PÁLSD@=ÝBuø VÁ÷\øBENKBE÷F_P PÁD{PdLM<X.P } LSBKNE÷VÁ÷\ø÷\VÁPdø5 D +PdLS÷VQøfBKFMúNK?I=@]÷ fDxVÁJ{BEFPÁLMBEøÈ<>VÁ÷\D< ¥øÁ÷÷ bPdD>NE÷VQFSBuú<>N NE<>WI÷oNKBEF BEsF (,E *]5÷DxBKF ÝPdD þSVd÷oøÁ÷F_P}<øBEÿ{þSNEB M÷\=ýøDxNK?SPÁBED>F LS÷Vd÷>G PdVd÷o=PÁDPÁLS÷þIWSNE÷ÿ ;?SV×øÁD>NE?@PÁBED>FÐBuø WM<xø÷\={D>FÐNE÷ö@Buú

D xVd<>þSLSBuú<>NKN }øÁD>VÁPÁBEF 1PÁLS÷BiPd÷ÿÐø BEFÐPdLS÷úD>NENK÷\úyPÁBED>FwG øDxVPdBKF D@úú?SVÁVdBEF 2WI<>øÁ÷o=D>F<}DxFS÷ 35+< xG@ø÷\úVd÷PÁN ÐJ>÷ >÷\=7G@úV _þSPÁ D xVd<>þSLSBuúsLI<>øÁLD "PÁLM÷ ø÷P D Uÿ{DxøPÈøB >FSB IúoVdÿ{<>NKB t÷\¨= ¯ø÷o÷A@÷oúyPdBKDx~F ) ){×ùx÷VQøBED>FD 2 q r hzk5Ôzaymyadk mojÐhzjRomoÔzhipÞm`baÁcb[8mcbßÎxh`5aQjxgQZRpxaQp{hij{` r asp_m`m` r m`×Þ hzÔiÔ7Î@agQZcncbaQgd`baypÐÎX [8mØZocbh`2Z`bhijxÒ1m`faÁÛ>`ncmgd`bhzZ\j`bhz[a\Ù }

f~ PÁLS÷1BKPÁ÷ÿÐø7 LS÷ø÷\úVd÷P D>FS÷' <+{LM<>øÁLSBEF÷FMøÁ?SVd÷oøfPÁLMúQJ>÷oV5úoVQ=@÷VdBEF2Buø7 LS÷FwG@BKFÝPÁLM÷FS÷öCPsøPÁ÷þñ¯ø÷o÷A@÷oúyPdBKDxF E E{yGøÁ?SWMøÁ÷PÁúQLC?SFSJ@ø D PÁLS÷BKPÁ÷oÿ{øÐ<>VÁ÷Ýø÷oNK÷\úyPd÷o= WI<>øÁ÷o= DxF PdLSBEøÐøÁ÷oúVÁ÷PDxVd=S÷VdBKF! 7 LSBEø1=@÷' ÷oø65÷NEN <xø<>F <XPPd÷ÿ{þ@PQøsPdDÝ<xúyPd?MVdC÷ D>VdÿÐBEù>÷oF<8úDxNKNE÷oúPÁBED>FÐ D åBKPÁ÷oÿÐøf<xø×÷x G [D^R_ a b c'dddca e fhi5G®<>FM= <2øÁ÷oúVÁ÷rP døDxVPdBKF }J>'÷ 4365 G 5÷BEFM=@?Iú÷<2øÁ÷oúVÁ÷PfDxVd=@÷oVÁBEF 2D>FBKPÈ W øDxVPdBKF <>úoúD>VQ=@BEF PÁD <ýúV Cþ@PÁ{D >VQ÷ >÷o=üLM<xøL D 1PÁLM÷ÿD_øbPøÁB >FSB Iú<>F_P{WMBiPQøD PÁLM÷FSD>VdÿÐ1?êa U|c@3 5 | 7 LS÷H 1 A øÁþM<xú÷LS÷oVÁ÷sBEø5<>ødø?Mÿ÷\=PÁD2WI÷1<8=@Dxÿ{<>BK=F ÈLS÷Vd÷ÿ{BKFMD>VfúQLINENE÷oúyPdBKDxF BKPÁ÷ÿÐlø ¥úQLIBEù>÷oF Vd÷ _?SBEVÁ÷\=?Mød÷<ÿ{BEFSBEÿ{<>NIBEÿ{þM<>úP×D>F{PdLS÷H AA NuøÁ÷o6ø ¥<>ø ?Mø?IWSBKNEBKP 8ÿ{÷PÁVdBuúøVÁPd<>F_P þMVdB xBKFM<>NU=SWSNE÷>Gå<=@B K®÷oVÁ÷oF_PNuú÷2úFM÷ ÈLS÷Vd÷>GS<xøÈ÷>GCÿ{BEFSD>VúQLM<> F >÷\ø D>FPÁLS÷8úD>NENK÷\úyPdBKDxFBKPÁ÷oÿ{øLM<ù>÷1<ÿBEFSBEÿÐúP

þG þ

ü1ègâæ .×L¶

?SVÁVd÷F_WP <XPd÷VdÿÐN xD>VdBiPdLSÿÐøw=@VQ<+ ÿD_øbP"D@PdLS÷BEV"úD>?SVÁPUþ®÷VQø?M<xøBED>F1þ®D f÷oV&(,'H * ÁV Dxÿ:< ø÷\úVd÷PPdLM<XPúD>F_PdVÁDxNKNE÷o= <XPd÷Vdÿ{<>VÁJü÷ÿ2WI÷\=S=@BEF¥B ÷ /5úQJ BKÿ{ÿ2?SFSBKP <>ødøD@úBu<XPd÷o=JÈBKPÁL <>Fs <XPÁ÷oVÁÿÐ<>VÁJCBEFýD>VdBKPÁLSÿBEø WM<>øÁ÷o=D>FPÁLSBuøÈJx÷'<>FM=ÝBiP.> ø NE÷ùx÷N+D "øÁ÷oúVd÷oú'

BKùx÷F ñ < 5÷o<>JüþM<>VPdBE<>NsÿÐVÁJCBEF ÿ{÷PÁLSD@= ú<>F W®÷{=S÷VdBKùx÷o= W Vd÷þ®÷oþSþSN_BEFPÁLS÷l5÷o<>JPÁ÷oúQLMFSB_?S÷{BEF <J>÷ >÷\~= ¥<>øÁLSBKDxF D>F =@B Kå÷Vd÷F_lP ¥øÁ÷oúVd÷PdN øÁ÷NE÷oúyPd÷o= sþM<>VPQøD PdLS÷D>W Òb÷\úyPsPdDW®m÷ <XPd÷:V ÿ{<>VÁJx÷o= nF PdLSBuø8ø÷\úyPdBKDxÍ F 5÷{þSVÁ÷\ø÷oF_PPÁLS÷{P 5Dÿ{<>BKFýøPÁ÷oþMø1D D>?SV <XPd÷VdÿÐÿ{÷ÝBKF < ÚPd÷VÐNuNENK÷\úyPÁBED>FüBiPd÷ÿÐø ÚPdLM<XPÐBuø8G 5÷<>ødø?Mÿ÷PÁLM÷ ÷ Cø D L\ûWSBKPdøÈ÷\<>úQGL E ùúÂFHGCÃ£ÂJI8K¿À5ÂÆL6¹£¾ ¼ NMBº ù¼¿ Ã FMVd<>ÿÐøÁ÷oø=÷D>öSF<>øÁú÷PÁPwN D SøPÁLS÷oO÷ NK÷\úy¿ PÁÀ5BED>Â'FÆ=øÁ÷oº ú¹!VÁg÷Pd¾ ¼ø|PM@BJ>º ÷ù@¼ø¿ IÃ ø?MVdD>WMNEø÷>÷G7Pd<>øwø1DCPDxÁ?@LSPÁ÷×NEBEBKFSFS÷oBK=ûPÁBuPQ÷x< G D>¨F ÈLMBEúQLPdDÝ<>þSþSN <øÁBKÿ{þSNEr÷ 5FwGå<xúQLSBE÷ùCBKF DXù>÷oVd<>NKNU< LSB xL=@'÷ xVÁ÷o÷ D "Vd÷oøÁBKNEBE÷FMú÷<>FM=ÝþI D 5÷.V Q } ÷Ý=@÷ IFS ÷ G]^x_ a ¯Ð[/z3{TSU ^x,f{c:m^x, c'dddc:vÍnF DRPÁLM÷]V ¸fÖDx¸Vdæ =MøîJ÷\ýW<>âgúQÍL íLG9âghïÖNR [ BEø=@'÷ MFS÷\= W øÁ÷NE÷oúPÁBEF Ý<ÝøÁ?SWMøÁ÷PD C[? ?SNEN =@÷Pd÷Vdÿ{BKFS÷\= W Bi.P > øúD>VdVÁ÷\øþ®D>FI=@BKF 2Jx'÷ 3{8¯VCô¥øÁ÷÷ n"B >?SVdl÷ ,{¯{< U 7 LS÷ÿÐ
D ÈPÁLSBuø{øbPd÷þüBEø}PÁD N 5úQL PÁD{W®÷ÿ{<>VÁJx÷o=ø÷oþMF <ýøBEÿ{þSNE÷ 5øP< v=WSBK1 P B {÷ W"^j, cdddcUvDXù>÷VQVÁJWM<>FM= ÈBu=CPÁsL ¥?SFSNE÷oødø8ÿ2?SNiPdBKþMNK÷÷oÿ}W®÷o = =@BKF xø1VdBKP ùxDRPÁBEF ÝFMøÁBE=S÷Vd÷o=7Gåø÷o÷AC÷\úyPdBKDxÍF Eû0 sBE¨F ÈLSBuúQL ÷\<>úQL WMBiP1BEø ÷ÿ}W®÷o=M=@÷o!= @XLSBu=S=@÷FBEF÷o<>úQLÝD UPÁLS÷ÿÐ<>VÁJx÷o= B

5~n~

s1 s2

collection subset

s3 collection subset

weak mark

si

collection

sort H(ks,MSB(NORM(s i)),ks)

weak mark

sn collection

collection subset

sorted collection

collection subset

nk[8WY^xX[m[Z cb·DßRaQ\Ck5hijx· ` Ò8r amocbj_hz[mop`Îxhz_hi`bZÔzhzhc`am`n]{`b`Z{mmgÁcb^xßRß j_kQhiÙ ÑXÕ^_ZyÎCÞ aQÖ5ÔaÁcseR^_mfjCÎ_[pknaÁ]_`5gQÔzZ\hknâCjxaQkngQÔzaQhimkgÁ`b`b`bhzadhiZ\ZojRjMjÝ`bÙ ÔmÕ mtEhi`bNÖpxadaQc^ÈjXknaÁ`bZoXhtEcnay`bpÐhzj_h`bknÒ}aQaQ[ÔzZ\aQjgdkÈ`bhzÎ@ßZ\adadjtEZoayZocbp t"a8rknm^_mjCÎ_k pÝr knaÁmZ`btEkQtU`bÙ ad` q cr ar Þ hz[mk`bZ\moadkcnk`ÓÓ knmhz`nÒo`j_mohgâCßRgykQm_jXhzj_`UgdÎxÔih^_`bpxk hzÕj_]ÐÒ6e cbaadÓ¥Ö®k r Zc^ tCb` r hza j_dÒ jx_ZokncbZo[8cn`bmhzÔijxhzÏQÒ8aQpmojCp_p}m`Ôzhimfjxhay`bmaQcf[g kQr Ù moq jxr Ò\hzaQk+kQaQÙ j_moÎ_ÔzaQk7cbaQgdZ\adcnmtE`baÁcUomcbhzZo^_k e nF WS?SBENu=@BK F PdLS÷øÁ?SWMøÁ÷PQ"ø 5÷2=@DFSDRPúDxFMøÁBE=@÷oV÷oNK÷oÿ{÷F_Pd]ø aÈLSBuúQL ÷o:V VÁ÷\øbPdVÁBuúyPdBKùx÷?IøÁ<>WSBKNEBKP ÿ{ ÷PdVÁBuúø ÷ ÈBiPdL úD>VdVd÷oøÁþIDxFM=@BEF ¥ ^-MGSB ÷ FSD<ù<>BKNuF }D ®PÁLS÷úD>FMøÁBE=S÷Vd÷o=8Jx'÷ @øFM÷ D"PÁLSD_ø÷1?SFM<>NiPd÷VQ÷oFS÷VQ<XPd÷<>FSDRPdLS÷VÈJx'÷ BEFMøPÁ÷o<x= } ÷þMVÁ÷\ø÷oFxPd÷o= PÁLS= ÷ >÷FM÷VdBEúÐøÁD>NE?@PÁBED>Fý<>WIDXùx÷ DxV8BENKNE?MøPÁVQ<XPÁBEù>÷ÐþM?SVÁþ®DxøÁ÷oø GmP 5D>VdJ@ø f÷oNKNg D>VÈúo<>øÁ÷oø/ÈLS÷F÷öS<>úP5BKPÁ÷oÿÜNuBKNuFMú÷VdFMøfD <XPPQ<>úQJ@ø D PÁLS÷}P _þ®÷oø F+GåøÁ÷NE÷oúyPdBKDx!F |g7 LS÷ DxNKNE D ÈBEF BE=@÷\<2PdNEøÁD2BEF_PÁD<xü~úú Dx?SF_P PÁLSü?÷\ø÷úDxFMú÷oVÁFI'ø B D>FMøÁBu=@÷VdBKF 1PÁLS÷s<>WSBENKBKP }PdD8BEFM=@?Mú÷<8øÁ÷oúVÁ÷P D>VQ=@÷VdBKF D>F{PÁLM÷ úD>NENE÷oü=úyîPdBK.DxíLF{Bi5Pd÷ýGÿÐâgø íL<>ø â!þSÖNVÁ÷\R ø÷oF_PÁ÷o=BEFAC÷oúPÁBED>F Eû)@G 5÷=@'÷ MFS÷NiPd÷VdFM<XPd÷Èÿ÷PÁLSD@=2PÁD2ø'÷ NK÷\úyPÈø?SWIø÷Pdø5<>øfVÁ.÷ x?MBKVd÷o=Ð W {þ® D f÷oVfGF Z+÷gP 3h55PÁLM÷øÁD>VÁPÁBEF }Jx'÷ {<>ø5{÷ O ÚPÁ÷oV}BEFM=@?MúBKF PÁLS÷ø÷\úVd÷P8D>VQ=@÷oVÁBEF W ûøÁD>VÁPÁBEF ÝPÁLS÷úD>NENE÷oúyPdBKDxIF [I^ _ a b c'dddca e fh i D> F 3\87{¤:9La.^i;I.365+c@>=êa.:|c@365G 5÷ WS?MBKNu= PdLS÷ýø?MWMø÷Pdø Ð<xø ÁúQLC?SFSJ@ø Ð D BKPÁ÷oÿ{6ø VdD>ÿPÁLM÷úD>NENK÷\úyPÁBED>FwG®2< ÁúQLC?SFSJ ÐW®÷BEF Ý<øÁ÷PD f<x+= Ò<>ú÷F_PBKPÁ÷oÿ{ø BKFÝPÁLM÷8øDxVPd÷o=ù>÷VQøÁBKDxFD UPÁLS÷8úD>NENK÷\úyPdBKDx F ¯ø÷o÷ nUB >?MVÁl÷ ,{¥!W U 7 LSBEø5BEFMúVd÷o<xø÷\øPdLS÷úQJ@øfBKFMúNK?I=@BK F dú?@BP "V Á<x=S= <XPPQ<>úQJ@Cø {÷ ! ü? G ü~ yGX W ?d=@Buøþ®÷VQøBEF ÈPÁLM÷BEV÷ Kå÷oúyP PdLSVÁDx? >LMD>?@P PÁLM÷ =S<XPQ<SG<>øw<VÁ÷\ø?SNKPU D @PÁLS÷5ø÷\úVd÷P"D>VQ=@÷VdBKF '7 LC?MøoGoB MúQJVÁ÷oÿ{DXù>÷oLø 06jéD CPÁLM÷fBKPÁ÷oÿ{ø PÁLSBuCø ÈBKNEN7Vd÷oøÁ?SNiPÈBEF÷\<>úQLø?MWMø÷P G WI÷oBKF D0hj øÿÐ<>NKNE÷.V 3 BuøÈøÿÐ<>NKNå÷FMD>? xLÝ<>FM!= @DxV B ÈPdLS÷þSVdBEÿBKPÁBEù>= ÷ 5VPQø8 D 5PdLS=÷ <XPd÷VdÿÐ
÷ ,1WSBKBP 5BE?F Buø ÿ{<x=@÷1VÁ÷\øBENEBK÷oFxP PdD2PdLS÷oøÁ÷1JCBKFM=Ý D / ÿ{BEFSD>|V ×PÁVQVdÿÐ<XPdBKDxFM]ø ¯AC÷o÷

f~n~n~ nUB >?MVÁ÷H fPÁLS÷oFPÁLS÷8þMVÁDxWMFIVQ=@÷oV PÁD{þSVdDXù_Bu=@÷1Vd÷oøÁBKNEBK÷oFMú÷PdDÿÐ<xøÁøÁBKùx÷ Áú?SP {<XPÁPd<>úQJ@øo G f÷ ÈBENKN ø÷oNK÷\úyPPdLS÷2øÁ?SWMøÁ÷=P dúQLC?SFSJ@:ø 2PdDWI÷2D øÁB to÷oø.÷ _?MPÈD WMöC÷\=<>WMøÁD>NE?@PÁ÷øÁB to÷oBø |7 LSBuCø >?M<>Vd<>F_PÁ÷÷\øf<ú÷oVPQ?SVø?SWIø÷Psø÷oNK÷\úyPÁBED>F ødúQLS÷oÿ÷xGM<>ødøÁ?SVÁBEF {Nu<XPÁ÷oVD>FÝPdLS÷}Vd÷PdVÁBE÷ùX÷o=F VdD>ÿGSød+< >G@LM<>N DUPÁLS÷DxVÁB >BEFM<>Nå=M<XPd< nSD>VÈ<>FSDRPdLS÷VÈ=@BuøÁú?MødøBED>FD>FPÁLS÷8øÁB to÷

D wPdLS÷8ø?MWMø÷Pdø øÁ÷÷8AC÷\úyPÁBED>F Eû0 nMD>VÈFS D Bi8P Òb?MøPøÁA? ú÷\ø5PÁDød+< {PÁLMFIøBu=@÷Vd÷o=Ý<þMF PdLS÷þMøPdLS÷Ð=S<XPQ< > ø ?MøÁ<>WSBENKBKP }=@Dxÿ{<>BKFwG>=@B K®÷oVÁ÷oFxP Jx'÷ x÷o=ø?SWIø÷PøÁ÷NE÷oúPÁBED>F{ÿ{÷PdLSD@=SøÿÐ+< 8W®÷=@÷ùCBuø÷\=7GRÿ{D>Vd÷ þMVÁBu<XPd÷sPdD2PdLS÷÷FCùCBuøBED>FS÷\=P Cþ®÷oøÈ D "
D sPÁLM÷PÁD W®÷ 35÷\= ø÷\úVd÷P Jx'÷ >÷\= ø÷PdDø 1Buø=@÷PÁ÷oVÁÿ{BEFS÷o=7G PÁLS÷ þSVÁDxWSNE÷ÿVÁ÷\=@?Mú÷\øBKPdøÁ÷N PÁñD IFM=@BEF ü<ýVd÷o<xøDxFMÑF ÁúD>BEF oIBKþ MG VQD>VdBKPÁLSÿ D>1V 5WSNE÷ÿ)LM÷Vd÷2BuøPÁ1D MFM=û<=SBKVd÷oú"P <XPd÷Vdÿ{<>VÁJCBEF ÿ÷PÁLSD@={PdLM<XPÈNE÷WSBKP5ùX
þG[

´ ÖLî{âGÔÖèW)p]æ+¶I.ðg¸ Ó

¸æ ¶I æ+çG{

7 LS÷þSVd÷ùCBKDx?Ml ø Á<>ÿ{þSNKB Iúo<XPÁBED>!F 2þMVÁDXùCBu=@÷oø ÿ{DxøPD"PÁLM÷LSBE=SBKFþID 5÷V DDx?SVF¨¯<>øfLMVÁJCBEF1PÁ÷\úQLSFSB x?M÷oø&ÈLS÷Vd÷søÁ÷oúVd÷PQø×<>VÁ÷s<>F BKÿ{þ®D>VÁPdV LMBE=@BEF 8<x8ø f÷oNKNI<>ø þSVdDRPd÷oúyPdBKF1PÁLS÷ <XPd÷Vdÿ{<>VÁJ7 LS÷ÈFS÷öCPføPÁ÷oþ <>úyPd?MNENK÷\úyPdBKDxFMø' ;FS÷=@÷\øBEVd÷o=ýþSVdD>þ®÷VÁP D <>F ÷FIúD@=@BEF ÿ{÷PÁLSD@= Buø2PÁLS÷VdÿÐ<XPÁBED>F ÈBKPÁLSDx?@P}LM<ùCBEF ÝPÁLS÷D>VdB xBKFM<>N =S<XPQ< W7 LMBEø2úVÁPd<>F_P ÷oøÁþI÷\úBuøÁ÷ D Uù>÷o:V ÐNu÷1= CFM<>ÿBuú1=S<XPQVÁPÁBED>FM1ø 5÷Vd÷ D>?SPdøÁD>?SVQú÷\= <XPùXBEF_PdøÝBEF PÁBEÿ{ ÷ PBEø ?SFSVd÷o<>øÁD>FIúQLýDx?@PdøÁD>?SVQú÷\= úD> þ D ÈPÁLM÷ D>VdB xBKFM<>N7=S<XPQ< !;?SVÈÿ{÷PdLSD@=øÁ
q

âgÖLÕÔ¸ ÖLî{¸Vr1âgâ!æ+{ D# g

l "¢ =GU7 6 Z c q s.u tv5xw Dq UyTzDw ){¯ï-c',. Wqcs+@tv5|wwpqDUy=zDw D { !){=^~} -`j q UyTzDw ^ ,.h - j q s+@ tv5xw ^ 6jr|} { M¿ÃJ/Ç Â'ÃMBÂøºcM'ùú¿ Æ q s+@tv5xw cUq Uy=zDw g MB¿ JÃ /Ç Â' Ã MB Â ¼¿¾º ùú¿ Æ Ä /`- Ä ù3Â'Æ:Â Ä ¼ Ä 7 6 8

} '{ J® u @ ?F J® °9 ¯IU @Ù ]&Ú xxUß! ^ ^ H Ú Z q{ª£6 x G9~ Dx P{"¯ñ-c',+

' "c q .3 rÃ½ ¹ÁÂ'Æ"¿Uø

÷s<>VÁ÷ xBKùx÷F ¥B ÷ xD>FS÷sD ®PÁLS÷søÁ?SWMøÁ÷PQø×øÁ÷oúVÁ÷PÁN 2øÁ÷NE÷oúyPd÷o={BEF{PÁLS÷sþSVÁ÷oùCBKDx?Mø øPÁ÷þ >< ø 5÷NENw<>ø5PdLS÷8ùXVÁJÐWMBiP sPdLM<XPsBuø PdDÐWI÷8÷oFMúD@=@÷\=ÝBKF_PÁD LM÷WSBiP ÷FMúDC=SBKF {þSVdD@ú÷\=@?SVd÷ DxNKNED ø +÷P GG G Gû÷ úNG LS÷oøÁ÷V <XPd÷Vdÿ{<>VÁJ=@÷PÁ÷\úyPÁBED>F2PÁBEÿ{÷ û÷ú<>F2úD>FMøÁBE=S÷V PÁLM÷ÿ <xø þM÷ +÷P G BEù>÷oF U<>FM=ÐPdLS÷sVd÷o<>NIFC?Sÿ2WI÷oV

<>ø5÷xG 5÷=@÷ MFS÷ PdD}W®÷sPdLS÷ }

.~ ¼«ù3Â'¹ ¿øù0/º ù/ºÆ:Ân Æ:ÂBº ùúÂÆ]ù/ºÃ q ª {g¡ G}û÷ú<>NKN£qc.3×PÁLS÷FC?Sÿ}W®÷V D þID_Ä øBKPÁBEù>÷ùCBED>Nu<XPdD>VQø:D UPdLS÷{1úDxF I=@÷oFMú÷1DXù>÷oVbGMøÁ÷÷n"B>?SVd÷ ) Uì{ì!qcs+@tv5|w ÖÔ qDUyTzDw ] 5â ð!¸ ì m Ü!Û6£Ù{ÚY¤+ßh¥fúÚWØ ¤.ß!Ú&¦Ù{Ú&|Uß!ÚY Q ð!¸ Ö å ¸ v h§3 ¯ _.,¥ §.¡£¤=c ¨©`ªa.¤cU3\q``ª,7f ÔL¸ÕÖL¸ V â 5¸ ¥,§+¡£¤ ä q ¬«q Uy=zDw ¡z z ì ¨©hªa.¤ ä q /wJq s+@ tv5xw ¡1z z ÖLÔ \`q `ª, 7 ä q /¯q s+@tv5xw ¡1z zcUq Uy=zDw ¡1z z

¢

¯distribution(S °) i

±c ²stdev(S ®) x

i

avg(S ®) i

Si Vc(Si)

`WYr X[a Z ·Jknhz³MÏQ·al Zotåhzk` `nr cbhza·Î_^x¶]@`bhzZoZoknjÝhz`bhzZot aÈh`bRaQhz[áZ\Ôimkn`baÁZo` cbk,´¸8µnknÙIadvw`y6jx¹JgQºZRÕpx´hzj_µ¥ÖÁÒÙ Zot` r aÞm`baÁcb[8mcbßÎ_h`ÈcbaQÔzhzaQksZ\jmÔz`baÁcbhijxÒ nF2D>PÁLS÷o V 5D>VQ=SøoG\PÁLS÷ <XPd÷Vdÿ{<>VÁJ1Buø ÿ{DC=S÷NE÷o=2W8PÁLS÷Èþ®÷VQú÷oFxPQ÷ldúD>F I=@÷FIú÷ùCBED>Nu<XPdD>VQ:ø ÝþSVd÷oøÁ÷F_P8BE2 F DxV2< xBKùx÷FýúDxF I=@÷oFMúD÷ ¥<>úyPdD>4V {ÐF I=S÷FMú÷ ù_BED>Nu<XPdD>VQø5L @øPÁ÷oVÁ÷\øBu]ø qcs+@tv5xwcUqDUy=zDw|| OsúyPd?MøÁ÷ VÁ÷\ø?Sÿ{÷\ø7PÁDsÿD@=@B CBKF PÁLM÷×úDxÿþ®D>FM÷F_PU÷oNK÷oÿ÷oF_Pdø+ D MøÁ?MúQL8<>ø+PÁD CBE÷Nu=1PÁLM÷×FS÷\ú÷oødødNK?M]÷ D>V qc.y!G ÈLSBKNE÷øÁBKF_PQø ò PBEø PdDW®÷FSDRPd÷o=ÝPÁLI<XPÿÐVPÁ÷oúQLMFSBEúo÷VÁ<>JúVÁPÁ'÷?M<>÷NåVd÷÷FMFMúúl÷ DC=S¥BBKF ÷ {SPÁÿ{LS÷÷Pdÿ{LSD@÷o=<>FBKPd DøÁ÷ <N >NKN7nSùXDx<>VÈNK?S÷öS÷\øGC1G BE{Fq{DxªVd=S ÷VfUPÁGSD{BiPdÿÐ÷ÿÐPÁúDÐPÁ?MW®<>÷ N JCø5LM<x=PdDÐWI÷1PQ
÷PPdBKF WM<>úQJPÁDPÁLS÷2÷FMúDC=SBKF þSVÁD@ú÷o=@?SVd÷>GSPÁLSr ÷ x?M÷oøPÁBED>F FLS D 'PdDþI÷oUV D>VdÿPÁLM÷ Vd÷ _?SBEVÁ÷\= BKPÁ÷oÿ<>NiPd÷VQ<XPÁBED>FIøøD <xø{PÁD <>NEøÁD ød<XPÁBuUø PÁLM¨÷ >BEù>÷opF Á?MødV PdLS÷8=SøÁ ÷ 7 LS÷oVÁ÷þSþSVdDx<xúQLS÷oø 7 LS~ ÷ MVdøPúQLüøÁBEÿþMN þ®÷:V D>VdÿÐøPÁLS÷þSVdBEÿBKPÁBEù>÷ <XPd÷VdÿÐrV G8<>FM= PÁLS÷oF úQLS÷\úQJCø D>V2=M<XPd<?IøÁ<>WSBKNEBKP ÍÈBKPÁLýVd÷oøÁþI÷\úyP8PdD ò &3 <XPd÷Vdÿ{<>VÁJCBEF BKFý?MødFMøoGwøÁBKÿ{þSN B >FMD>Vd=÷ s<>FM=ýúDxFMøBu=@÷oVPÁLS÷FS÷öCP{ø'÷ úVd÷PÁN øÁ÷NE÷oúPÁ÷\=ýø?SWIø÷P8PÁD ÷oFMúD@=@÷{PdLS÷Vd÷oøP}D 5PÁLM=÷ <XPÁ÷oVÁÿÐ<>VÁ£J 7 LSBuø ÈBENKNfVÁ÷\ø?SNKP}BEF ÷VdVÁDxVdø×BKFPÁLS÷1÷oFMúD@=@÷\=ÿÐVdBKP ùxDRPdBKF ¥øÁ÷÷AC÷\úyPÁBED>F Eû0 ×DXù>÷oV

<NE<>:V x÷ÈFC?Sÿ}W®÷VfD +÷FIúD@=@÷o={ÿÐ<>VÁJúD>þMBK÷\ø BEFÐPÁLS÷=S<XPQ<SG>PÁLS÷÷oVÁVdD>VQø8ÈBENKN LSDxþI÷' ?SN ×W®÷ ÷NEBKÿ{BEFM<XPd÷o=ÝBKFÝPdLS÷VÁ÷\ø?MNiP. 7 LSBEøûúQLãBuø ùx÷V'úPÁBEù>÷ DxVû= CFM<>ÿ{BEúo÷oFS÷VQ<XPd÷o=!@RøLM<>þI÷\=ã=S<XPQ< ÈLS÷Vd÷PdLS÷ø÷oNK÷\úyPÁ÷\= øÁ?SWMøÁ÷Pdø6ÈBENKN LM<ùx÷øÁÿ{<>NKNÈøÁBto÷oøFMú ÈBKNEN+W®÷LSB xL÷où>÷oFBEFøÿÐÿDx?SF_Pdø5D "PÁLM÷D>VdB xBKFM<>N7=S<XPQ < 7 LS÷þSVd÷ùCBED>?MøfþSPÁBEÿ{<>NM?MøÁ÷s

D åPÁLS÷÷ö@BEøPÁBEF 8WI
< ÚPÁ÷o~ V ÂBºcM@/ BiPd÷ÿ F += Òb?MøP}PdLS÷÷oFMúD@=@BEF WI÷oLM<ùCBKDxV}<>úoúD>VQ=@BEF >N L7 LSBEø úFSDRPdLS÷V8BKPÁ÷oÿ ¥B s<ùX<>BKNuFIøBu=@÷Vd÷o= DxFS÷ =@DC÷oøFSD>PþSVd÷oøÁ÷Vdù>÷ =SVd÷ D>þ@PdBKÿÐþSþSVdDx<xúQL þSVÁ÷\ø÷oF_PdøLMB xLS÷VÝúD>ÿ{þS?SPÁBEF DXùx÷VdLS÷o<x= VÁJCBEF PÁBEÿ{÷ 7 LS÷Bu=@÷o<>N5<>þSþSVdDx<xúQLýBEø2þSVdD>WM<>WSN < úD>ÿ2WSBKFI<XPÁBED>Fý

D PÁLM÷þSVd÷ùCBED>?MøÁN þSVd÷oøÁ÷F_Pd÷o= BE=@÷\<>ø ;?SV BEÿ{þSNE÷ÿ{÷F_PdFÐBuøWM<xø÷\={D>FPÁLS÷ MVdøP×úQL{L{ÿ{BEFSD>VP 5÷o<>J@ø úD>?MNE=WSVdBKF {BKPsúNEDxøÁ÷V PdD{<{úD>ÿ}WMBKFMFF 2PdLS÷1P 5D þG0»

1¸.¸ Öî ¸gríLÖL¸æ

B

O =@÷oúBEøÁBKDxFFS÷÷\=Sø PÁDÐW®÷8ÿÐ<>=S÷8 =@÷Pd÷VdÿBEFSBEF{PÁLM÷8øBt÷D UPdLS÷8ø?MWMø÷PdøøÁ÷NE÷oúPÁ÷\=BEFPÁLM÷ ?MVfÿ{÷PdLSD@=÷oÿ}W®÷o=Sø6,WMBiP þI÷oVÈø?SWIø÷P&5÷ LM<ù>÷}<PÁD>Pd<>NUÿÐPdVÁJ÷oFMúD@=@BEF{WM<>FM= ÈBu=CPÁLG 7 LSBEøUú<>F1<>FM=øÁLSD>?SNu=1W®÷fúDxFMøÁBE=@÷oVÁ÷\=1<>ø+V+=S<XPQ< ?MøÁ<>WSBENKBKP ÿ{÷PÁVdBuúøþSVdDXù_Bu=@÷\= £3×PÁLMDxøÁ÷2ÿ{÷PdVÁBuúø1<>VÁ÷2PÁDCDÝVÁ÷\øbPdVÁBuúyPdBKùx÷>G®ÿDxVÁ÷BKPÁ÷ÿÐ]ø ÈBENKN WI÷2FS÷÷\=@÷o=BEFMøÁBE=S÷ PdDW®÷2<>WSNK÷8PÁD÷oFMúD@=@÷2D>FS÷8WMBi"P ÈLSBKNE÷}øPÁBENKN"þSVd÷oøÁ÷VdùCBKF ÐVd÷ _?SBEVd÷o= ?MøÁ<>WSBENKBKP 5;F PdLS÷ÝDRPÁLM÷VLMVÁ÷ÿDxVÁ÷D>F PÁLS÷ÝVÁ÷oNEWG G5úo÷o:V øÁÿÐÿ{÷PdBKÿ{÷\ø÷où>÷o2F ,.- B,.0{BKPÁ÷oÿ{ø G7 LSBuø1÷FM<>WSNE÷o6ø D>V1ÿ{D>Vd÷ ÷FMúDC=SBKF {WM<>FM= ÈBu=CPÁLDXùx÷VQNKN BuøÈNu÷1÷FSDx ? >LPÁDÐþMVÁDXùCBu=@"÷ D>VÈÿ}?MNiPdBKþSNE÷8÷oÿ}W®÷o=S=@BEF _øÈD "PÁLS÷}D>VdB xBKFM<>N ÿ{<>VÁJ ÈLSBuúQL Buø{< Vd÷ _?SBEVÁ÷oÿ{÷F_P2BEFüDxVd=S÷V8PdDûøÁ?SVdùCBKùx÷øÁ?SWMøÁ÷P Áú?SP úQJ@1ø ¥B ÷ "PdD þSVÁ÷\ø÷oVÁùx÷sPdLS÷ÿÐG < ÚPd÷VVÁ÷\úDXùx÷VdBKF <>NKNfPÁLS1÷ 5þMBK÷\ø VdD>ÿ PÁLS1÷ >BEù>÷oF =S<XPQ< DXùx÷Vs<>NKN7PdLS÷8Vd÷oúDXù>÷Vd÷o1= <XPÁ÷oVÁÿÐ<>VÁJWMBiPQøBuøs=@÷oþSNK D x÷o=BKF D>VQ=@÷V PdD =@á÷PdÜd÷½ Vdß!ÿÛcxBEFS|¾¿÷1Pd¦LSß÷xúÿ{ÚWDxØ øP NEBEJ>÷N BEFSBKPÁBuVÁJÐWMBiPQøGMøÁ÷÷ n"B >?SVd÷ E OFSD>PÁLS÷oVBKF_PÁ÷oVÁ÷\øbPdBKF þIDxBKF_PPÁDW®÷ÿÐ<>=S÷ÈLS÷Vd÷ BEø úD>FMøÁBu=@÷VdBKF PdLS÷ÈBEFSLS÷Vd÷F_P×WSBENKBKP D 5PdLSD÷ <XPd÷Vdÿ{<>VÁJCBEF øÁúQLS÷oÿ{ ÷ UOø}øÁLS D ÈFýVdúQL (,H*GUWSVdBE F >BEF PdLS~÷ 5÷\=û=M<XPd< <>ø{úNEDxøÁ÷<xø}þ®DxødøBEWSNE÷PdD PdLS÷Ý<>NKNE D VÁPÁBED>FûW®D>?SFM=Mø ?IøÁ<>WSBKNEBKP ù_BuúBEFSBKP NEBEÿBKPdBø Buø1D 5=S÷ MFSBKPÁ÷ÐW®÷FM÷ SP8BEF ÿÐ<>J_BEF ÝPÁLM÷ VÁ÷\ø?SNKP > øÈ=SBENK÷<>ø þ®DxødøÁBKWSNE÷PÁD<>F úQ£J OF <XPÁPd<>úQJx÷V ÈBENKNsWI÷ ¥<xú÷\2= ÈBKPÁL PdLS÷ þSVdD>WMNK÷oÿ D 1Vd÷ÿ{DXùCBKF @X<>NiPd÷VdBKF PÁLS¨÷ 5< PÁ÷VdÿÐFM= FS D á÷øÁ÷o=

5~ bits

5

4

3

2

1

0

Àww12

1 1 1

0 0 0

1 1 0

1 0 0

1 1 1

0 0 1

À wresult

1

0

1

0

1

0

À w0

ÞWYmX[Z `bad·JcbÁM[8· m]{cbß@mØÙ Zocbh`8Z`bhzj_ÒÈZadcw` r cbaQacbaQgdZ\adcbaypÞm`badcb[8mcbßsgQZo]_hzaQkUtEZoc mÈÊÎxh`"knhzÏQayp1ZcbhzÒ\hzjCmÔ NKBEJ>÷oNKBELSDCD@= D 5ÿÐ<>J_BEFPdLS÷=S<XPQ<BKFCùX?M<>Vd<>FxPd÷÷\= ?MødG@PÁLC?Mø Vd÷ÿ{DXùCBEF2DxV<XPÈNE÷o<>øPs=@BEÿ{BKFSBuøÁLSBKFBKPdøù<>NK?M÷ nF÷ Kå÷oúyP6f÷8BEF_PÁ÷>VQ<XPÁ÷\=PÁLSBuøBu=@÷o<?SVþSVdBKÿ{BKPÁBEù>÷8÷oFMúD@=@BEFÐBEÿþMNK÷oÿ÷oF_Pd< PÁBED>GF ÃsDRPDxFSN =@lD f÷1÷ÿ2WI÷\=<>ø ÿÐVÁJúD>þSBE÷oøÈ<xø þID_øÁøÁBEWSNK÷xG@WS?@Ps<>NEøÁDMGS<xø 5Vd÷oødøBEù>÷oFS÷oødø ¥<xúyPÁDxV BKF úVd÷o<>øÁ÷oøoG>=@÷PÁ÷oVÁÿ{BEFSBKF 2<>úPÁ?M<>N®úQLIVdÿ{÷o=Ðÿ{D>Vd÷s<>FM=Ðÿ{D>Vd÷ ?SþÝPÁDPÁLS÷þ®÷Vdÿ{BiPÁPÁ÷\=NKBEÿ{BiPs<>FM=ÝFSDRPÈD>FMN <xø VÁ.÷ _?SBKVd÷oG= e 7 LS1 ÷ ¼«ÃM'Æ:Â'¹lÂ'Ãùúº ¾@BEFMúVd÷o<xø÷ D UPdLS÷2 < xVÁ÷\øÁøÁBKùx÷FS÷\øÁ5ø ¥<>úPÁDxVBEøsFS÷÷\=@÷o=øÁD<xø PÁWID1÷' ÿÐD>VdúQ.÷ LM¸føÁ÷fBEPÁFLI1<XPÁP LS<X÷P=SNE÷o<X<>PQø<1P PÁøDC÷oD}ù>÷o< Vd <>>N_Vdú÷oD>ødþSøÁBKBEùx÷o÷ø"NDIPÿÁLSB>÷ L_ÿÐP×NK÷NEVdD ÷f2÷oÿ}D>W®V×÷o÷o=SF_=@PÁBE÷\VÁ=2÷ÈøÁÿÐ?M?SþMNKBKN ÷\>ø G PÁDÐW®÷÷ÿ}W®÷o=M=@÷o= B ÷ ÈLSBENE÷ÿÐ<>BKF_Pd<>BKFMBKF Ð=SúQ J VÁ÷oNEFüùX÷o:V þSVÁDxÿ{BEøÁBKF 1 ÷\<XPd?SVÁ÷\øD D>?MV8÷ÿ2WI÷\=S=@BEF ÿ{÷PdLSD@= L7 LSñ÷ dúDxF I=@÷oFMú'÷ ù_BED>Nu<XPdD>VQ:ø þSVdBEÿBKPÁBEù>÷ø÷Pf÷oFMúD@=@BEF 8þSVdDXù>÷\ø PÁD}WI÷VÁ÷\øBENKBE÷F_P PdD2<8úD>FMøÁBu=@÷VQVdÿ{N 1=@BuøbPdVÁBEWS?@Pd÷o=}øÁ?SV >÷oVÁBE÷o#ø ¥B ÷ <XPÁPd<xúQJ>÷LV ÈBiPdL8FSD÷öCPÁVQ< J_FM D ÈNK÷\= >÷xGXBKFPÁLMBEøfú<>øÁ÷ BiPd÷ÿ%VÁ÷oÿ{DXù<>NEBø UWI'÷ D>VdC÷ 5FMø"D@úú?S.V >õ ùx÷F PÁLS÷oF+GMPdLS÷Vd÷}÷ö@BEøPdøPÁLS÷ú÷ D>VVdB >BEFMVd÷÷ ¥B ÷ R W }PÁV _BEF PÁDBEF ÷oVPÁLS÷ÈDxVÁB >BEFM<>N@ÿ{<>VÁJ}ù<>NK?MC÷ VÁDxÿ NKBu= ø÷BP |®HDxøPÈD PÁLM÷8úD>FIøBu=@÷Vd÷o=Ý=SVd÷ÿ{<>VÁJ NEøÁD{BKF n"B x?SVÁ]÷ H! 7 L_?IøGCD>?M/ V <XPd÷VdÿÐV<ø÷PÈD "úQJ{P CþI÷\øG BKFMúNK?I=@BKF ýVÁ'÷ øÁD>VÁPÁBEF 2 ü yG×øÁ?SWMøÁ÷PøÁ÷NE÷oúPÁBED>RF ÷ ×?Sþ Pd³D ) 0 3E{-hj =S<XPQ<ûVÁ÷oÿDXùX<>N G ÿ{<xøÁøÁBEù>÷×øÁ?SWMøÁ÷P ø÷oNK÷\úyPÁBED>DF ¥ ÷ .5N SD @PÁLS÷ =S<XPQ<ÈB M=S<XPQ< LM÷\= VQÿ{N G®NEBEFS÷o<>V1úQLI=S=@BEF @ÿ}?MNiPdBKþSNEBuú<XPdBKDx!F yG7VQÿ)BKPÁ÷oÿ ¯ø úQJ@ø5PdLM<XPþSVd÷oøÁ÷Vdù>÷=S<XPQ<?MøÁ<>WSBENKBKP Ä a"aQgym^_kna"` r awÞ m`badcb[8mcbßRhijxÒf]>cbZXgQaQknk+mÔzcbaQm\p>smÔ`badcbaypÈ` r aUp_m`m ^_]È`bZ×h`bkå^_kbmÎ_hzÔzhz`¯[ad`ncbhzgQk Ôzhi[h`bkQÙ

f~n~

1.5 exact watermark recovery approximated watermark recovery watermark alteration

1

Å

0.5

0

-0.5 0

2

4

6

8

10

12

14

attack surgeries

n`bkWYaÁZ8X[`1Z jxkn·ÇZohzÆ®cbÏQ[8a}· v+mogQÔzZ\Û>Ôjx]@knadp>hKcbp>hihzkad[`ncbcbayaQhzpjXÎx`b^xÞk8`baymZ\pÐk1j Æ\pxcbÉRmaQ7`knm>hzadÔzÙ1Û>hzaQU]@j_admgdcbcbahzhz[Zo`b^_aQZk5jX`bknZkÈ^x` rcbÞ ÒoadadaÁc5cbcbhz]Ca2aQmk ]@cmoadÕ cn[pxtEmZoad`cb`bm[adcbayÔzk@Z\pÊ`kn¹dkQZoË@ jýÌu¶¯ÍÉÎFgQÌyÏ¬^xx`Ð ¸pxhÉÉ\ÈSmèÒad`n`cbÑ,moaQ¹JjXgÓÕß>`ÔxkÖ knÖÁÏ adÙ `bÐ q ksr ZoatfègQÑxh`bÔz×gZ\aQkn[Ð a ÎXËoË}è}[Ù Z\q kr`a ` r mya5ad`bcaQmkÒ\`ba aypÎ@ap_r mm`RmRhzÙ Zocwhzkw]_ÔzZo`n`bayp r adcba\Ùo5]s`bZsÍ\Éoè p_m`m ÔzZoknkwÞ mok+`bZ\Ôzadcm`baQp1Òocmogdadtu^xÔz Ø ì VÚÙ"TPWV PwífAî Û"îSôbð×mí XÜÞÝ¬ÙPwïMUô V Pwíf1î ßgX 7 LSBEøø÷\úyPdBKDxFÝþMVÁ÷\ø÷oFxPQø<>øÁþ®÷oúyPQø

D Dx?SVþSVdDCD «D «núDxFMú÷oþ@PÈBKÿ{þSNE÷ÿ{÷F_PQ<XPÁBED>F <>FM=øÁD>ÿ{÷ D "PÁLS÷Vd÷oøÁ?SNKPdø D>WSPd?MøÈ=S
à×¸?ÖLæ µ á·îçè!¸

µ Buø5D>?SV5PÁ÷\øb:P W®÷o=BKÿ{þSNE÷ÿ{÷oFxPQ<XPdBKDxF D +PÁLS÷1DxVÁBKPÁLMÿ{øfþSVd÷oøÁ÷F_PÁ÷\=BEFPdLSBEø5þM<>þI÷oV ÖWP æ BEøÈá VdBiPÁPÁ÷oF?IøBEFPÁLS÷<_<ùX<sNu?M< >÷ 7 LS÷ þM<xúQJX<>÷5Vd÷oú÷oBKùx÷oø <>ø BEFSþS?SP×<6 <XPd÷VdÿÐ PN5dMLSNE÷÷1þSD>NE?Vdÿ{BKD@ø=@f?SNE÷o÷oNKøsNÈBE<>ø ø2PdD< N9FS÷\½ =Ä º ?IÁ'øÁ¼«<>¾ ¼«WSù.- BKNEBK£P¾ ½cÿ{ncâú÷¼«ÃJPdVÁBu¹úø ¿.PdÇ DÐ½ ¾WIÂ Ä÷}?M7 ø÷\LM= ÷ <XPVd?SF PÁBEÿ{¨÷ ÈBKPÁLSDx?@PVd÷oúD>ÿ{þSBENEF <>FM!= @DxVøÁD ÚP øÁ÷2 D ×PÁLM÷Ðÿ{WSNK÷ÿ{÷o<>Fûø _?MVÁ÷PÁLS÷oF?MøÁ÷oø5PÁLMDxøÁ÷ÿ÷PÁVdBEúoø PÁDÐVd÷ ÷ùX<>NK?MúQL<XPdD>ÿ{Buú 5ø ÷ö@þSNE<>BKFM÷o=BKF AC÷\úyPÁBED>FIø E E<>FM= E, ;FMú÷?MødVÁ÷ =S÷ MFS÷\= G PÁLS÷ 5VPQø PÁLS÷2þSVdDCú÷oødøÈD # <XPd÷VdÿÐúQJNE D BuøwJx÷þ@LP D>V"÷o<xúQL}<XPdD>ÿ{BEúføbPd÷þ8þ®÷:V D>Vdÿ{÷oD= ¥B ÷ ,'WSBKPU÷oFMúD@=@BEF 7?SF_PÁBENC=S<XPQ< ?MøÁ<>WSBENKBKP BuøÈ<>ødø÷\øÁøÁ÷o=NENKWI<>úQJ@:ø 8BKFú<xø÷?Mød÷o= W ÐPÁLS÷8ú?SVdVÁ÷oFxPÿ{BuúD>þ®÷VQ<XPÁBED>WF }ûVÁJúD>þSBE÷oø LM<ù>÷WI÷o÷F ÷oÿ}W®÷o=S=S÷o= BEF_PÁD PÁLS÷=S?I
5~n~n~ usability metric plugin A

usability metric plugin B

evaluate

usability metrics plugin handler

mark

user key

WM

alteration rollback log

usability metric plugin C

data

· M· q r a¬å ´Çæw·0ç ]_mogß\mÒ\a\ÙCç5adcbRhzadÞÈÙ }û÷oV: Pd<>J>÷\ø2<xø2BEFSþS?SPPdLS÷J>÷ý?Mø÷\= BEF ÷ÿ}W®÷o=M=@BKFMG"PÁLS÷BEFSþS?@P =S<XPQ<2JCFSD ÈFPÁDÐúD>F_Pd<>BKFPÁLS÷ <XPd÷VdÿÐFM=ÝVÁ÷\úDXùx÷VQøPÁLM÷8ø÷PDL <XPd÷VdÿÐþMBK÷\ø BKFSBKPÁBuVdBKP ýù>DRPdBKF ýDXù>÷VPdLS÷ Vd÷oúDXùx÷Vd÷o=üúD>þMBK÷\ø úD>ÿ{þSNE÷Pd÷oø5PdLS÷VÁ÷\úDXùx÷V þMVÁD@ú÷\øÁø WXvZ äã

W«

´6µ ·5¸æ ¶I¸ Ö£

SD>V÷ö@÷ÿ{þSNEBIúo<XPdBKDxFþM?SVÁþ®DxøÁ÷oø5÷>÷oFS÷VQ<XPd÷o=øUCF_PÁLM÷PÁBuú BEFSþS?@P×=Sø~,.- - -{-SGUPÁLS÷øÁ÷PÐBKPÁ÷ÿ ù<>NK?M÷oø{W®÷BEF ?SFSB D>VdÿN =@Buø PÁVdBKWS?SPÁ÷o= BKFPdLS÷8BEFxPd÷VdùXFIøBu=@÷Vd÷o=PdLS÷}H A@õ ?MøÁ<>WSBENKBKPÿ{÷PÁVdBEú8úo<>øÁ÷ ?MNK DFI =@Biõ ?MV úQxLM÷VdD÷ ÷÷oF_VdPd<><>NKNUN_BEFMVd==@BEùCFIøBuøbPd÷o=}D I<m, 4t B T NEBKFC?@ö8W®Dö ÈBiPdLAC?MrF D VÁBKPÁLSÿ%þM<>Vd<>ÿ÷PÁ÷oVdø 5÷Vd÷È<># += Òb?MøPÁ÷\={VÁ÷oþI÷\<XPd÷o=@N }BEF÷oøPd<>WSNEBEøÁLSBEF PÁLM÷ÿá?SFI4= {^íëcë`jDc:q s+@tv5xw ^Ð0hj=cUq Uy=zdw ^ }hj= øÁBKF sPdLS÷ÈúD>F I=@÷FIú÷ ùCBKDxNEVQø+÷oFMúD@=@BEF 1ÿ{÷PdLSD@m = ÈBKPÁL2PÁLS÷È<>WIDXùx÷5øÁþI÷\úB M÷o=2þM< Vd<>ÿ÷# PÁ÷oVdLø fC÷ 5÷Vd÷5<>WSNE÷×PdDVd÷þ®÷oV =@B Kå÷Vd÷F_8P >÷FM÷VQ<XPÁ÷\=BEFSþS?@P×=S÷8V <XPd÷VdÿÐFIøBuøbPd÷o=D )H}WSBKPdø }û"÷ 5÷Vd÷<>WSNE÷sPdDVd÷oøÁBENKBE÷F_PÁN Ð÷ÿ2WI÷\=þMBK÷\ø D PÁLSm÷ 5Nw÷oF D>VQúBEF ÐøbPd÷þ D ÿÐ< ÒbD>VdBiP ùxDR:P BKF IG_<xø×=@Buødú?Mødø÷\={BKFÝAC÷oúPÁBED>FM&ø E,súQJ2÷ö@þI÷oVÁBEÿ{÷F_Pdø D>F þMNKN <xøÁøÁ÷oødø1÷FIúD@=@BEF VÁ÷\øBENEBK÷oFMú÷<>FM= D>W@PQ?MVd < >BEF Vd÷oøÁ?SNKPdøoG+øDxÿ{÷ÐþSVÁ÷\ø÷oF_PÁ÷o=ûBEñF nUB >?MVÁl÷ HUT Vd÷ þSVÁD@ú÷oødø÷\= þM<>VPQø D ×PdLS÷{D>VdB xBKFIøPÁLM÷BE"V <XPd÷VdÿÐ÷o= ù>÷VQøÁBKDxF F PdDWIr÷ D>?MFM= <XP ×£'·NR0îfîålålå~î {·íWæ ÔíL¸g¸Ôíî×â!¶³¸Dîâ!Öî å¶ïîÖWæ+å¶ ð ñ ðfísó T ò ßCôbð×í ß!Xòòî@òsï PôóJP*ß{PõÛ"ïIó OX nFPÁLS÷þSVd÷oøÁ÷F_PþINE?@PdBKDxFPÁDPÁLS÷2þSVÁDxWSNE÷ÿ D # <XPd÷Vdÿ{<>VÁJCBEF <F_?Mÿ÷oVÁBuúÐúDxNKNE÷oúPÁBED>F W 2¥B =S÷ MFSBEF <FS÷ áø?SBKPd<>WSNE÷Ðÿ{<>VÁJ ÷FMúD@=@BKF ÿ÷PÁLSD@ = DxV n

f~ F_?Mÿ÷oVÁBuú1ø÷Pdø<>FM= BEB È=S÷oøÁBxFSBKF{<>FÝ<>NxD>VdBiPdLSÿ{BEú1øÁ÷oú?SVÁBEFÿÐVÁJ<>ÿr þSNKB Iúo<XPÁBED>gF VdD>ÿ<ûøBEÿ{þSNE÷÷oFMúD@=@BEF ûÿ÷PÁLSD@= PÁD < ÿ{D>Vd÷úD>ÿ{þSNE÷ñö <XPd÷Vdÿ{<>VÁJCBEF VÁLSD>ÿÐVdBKøÈPÁLS?Mÿ FM=@÷} V ÷PdLSVd=@ÿÜ÷ùxD ÷ lþ®<_÷o<= ù<< øÁþSDVdÚPD_ D sDúQFMJXú<÷>þ@÷xP{G BEÖWÿæ þMµ NK÷oá ÿ ÷oF_Pd<XPdBKDxF D sD>?SV{ D nS?SVÁPÁLS÷oV Vd÷oøÁ÷o÷\øbPdB _<XPÁ÷È<1ÿ{D@=@÷NID 7<XPÁPd<>úQJ@ø BEF{PÁLSBuø FM'÷ =@D>ÿÐWSBKNEBKP <>FM÷\=úDxrÿ ÿ÷oVdúBE<>N 5F úD>?MNE= W®÷=S÷VdBKùx÷o³= VÁDxÿD>?SVÐþMVÁDCD «D «núDxFMú÷oþ@P ø D ÚP5<>VÁ{÷ nUBEFM<>NKN >G=SB Kå÷Vd÷F_PU<>þSþSNEBEúo<XPÁBED>FIgø D>V+Dx?SV+FC?Sÿ{÷VdBEúúD>NENK÷\úyPdBKDxFÿÐ?MNE=WI÷÷FCùCBEøÁBED>FS÷\=øfD>?SV5D>F >D>BEF 5D>VdJBK1F 5FM<>N+=S<XPQ< óJPõöPwï Pwís£ó P*ß Ì\!Ù r]moÙ [÷xÙay pS`wmmoÔiÔij_m pûr u e@sÙUÙy5|5momhzknßxßRÖÁhiÙûj 5Õ Þ mh`b` ^>r c]moÔ ÙÔiYUmoj_cbZ\ÒoÒ\^CmomjMÒ\a{YfÞ@Ù mø5`baQad[cb[8]@maQcbÔz[8ßRhzj_mj_ÒcjIÊw l ê aQuÙ kn^ÈhzÒ\adjMcb kng mr jCÎCmomÔ^_>[{knhzkQQ lmäÙ jC]p Zom Ó ]> cb» ZXZotKÄxÓ¯Z»¯tE¿ÁÓ¥ÀÁgQÄCZo¼oj_»ÚgQ¹ aQþo]>ÄC`}¼\½ hi[Ä ]x@ÔzþoaQÀ[ÂaQ¼ojX»Ú`¹ þmÄ`bhzZ\ jM¹ Ùû¹u~¯ÄR$j ºûù 4¾ x¿ údþo0» ÀbûRäÃ Ànÿý¿ 6þügþo»¯ T¿ ÿ{¹u»Ú0» ¹uÿÄ =ûRõþÀ º o Â `fûR¿Á»¯¿ÁÄxÀ hÄ ÿ`oúd½ ¼¹¥¿ÁoAÄ Ä_ú¹ ¿!¼ "#,ÀCþuÀú¹Ú ½ ÍRÙs×Y$%&r^_%&cbÒo'hz^_kÙC`bke>hK`Èm]>jÌycbohiY"jx\ËRÒ\Z\adÔzÙ ÔzcÎ@adwcbadÒ5cbÔimmojCÒ>p@ÍY"\Ôi>mcbÌoß Ù q r Z\[Î@ZocbknZojIÙç5j5` r a+Ôzhz[hz`bkMZot>knZtE`Þ mcbawÞm`baÁcb[8mcbßRhzj_Òx Æ>ÙÈ~¯¾j_Òo¼oaQ»¯[8¿ÁÀmÂc7¼oY"ÀnÃZ¹uÛSÄXJº ÷\1Ù a=]È@cbZoaÁcb!Ò\moa"·j ÔzZX^Z\[{m^xotumo[8j_mp j_]{jMCmÍ`n`\r RadÌ\Þ Ù ]hzÔzÔzadcyÙ>l hzÒ\h`moÔXÞm`badcb[8mcbßRhzj_Ò>ÙR~j ¸È¹zºo¹u»¥¼\½ xÙÈ`b~¯j_ZÒo`aQmo[8[m]@Yc ad÷>cbhzÙ@j_Y"Ò>ZyÙ Û2~¯j moj_4pÄ_»¯÷\¿ÁÀayÄ@mjx¼o»ÚÓ ¹0 þomoÄ@^x¼\©Ô ½(]õþÙ_Ä)æy¿ÁÙ_ÀnI¿ÁhzÄAj_új_¿ mþocn`bÄÏ\ÙÂ ^x¼yºXÎ_¿ ÔzhzgÞÀFþumú`b¿TaÁÿ|cbÿb[8¹uÄXmºcbßR*+!k ,moj_.p}- /cb0aQ1 knChzkex`momjxjXgQ`ma ÉRÙÈ~¯gQa"j_mmÒo`bcbhzaQÎCZ\[8jxmckmm>)c Þ @÷xhzY `ÙxrmoY"ÔzhZykntuÛShiZopxcb6 a1jx]{hihzm>j>mStu`nZo` cb+[8Ù e@6Ù mÙ ]`bhzÙzZ\hiCÔzjIÔzÍadÙ Ê cy2Rx ÍomoÀ,}j_þup8úç5¿bs¿3gdo`bÙRZ\¹uÄR+Î@ºJ6ÙaÁÿcs]þÌQ@g\^È\» _aQRÔz¿4ÙCÔzhi~]xvU5kQvwÙ®56v57Ù m*!`b89adcb:[8"(m1 cbIß>Ëhzjx ÒÕ omoÖTÊzk Ì\gQÌZoÍo[)2_[Ì\ÌQ^x>j_Ì\hÓ Ê>Ù÷ovw ^_aÁp>Ô`nÞ cmmÌyc\ìp oZ>÷xrÙ Ù7Ôz[8l5aQmoÔzg ]Ir Ùadcy mm`bad`ncbcbhz[8gÁß4mcbø5ßRhzZoj_cbcÒ k`bÊ®aÁ cy@r moZj_p{gQm|5cbaQmk Ôt ;{eXp>`bZXaQaQhzj_k[h`ad`bÞ"ÏoCZocbay<ß p>; hz`bZ~cbìj kQ>÷= mjCû>mÐ½ »Ú¹ul Âh¿`n`b[8¹ ¼2mo¼jxÄ?jI @,¿ ú=û>À¹u+» @ü4¾ þoÀbäÃ ÿ 6þ ¼o» "4 =A= û>½ »Ú¹uÂ3¿ o¹ ,¼ - /CB 7ZoÔi^x[a8xÌ2ZoEt D6= G Ì`nycbÍo^x)Æ [ã2CÌyÆ\~j>Rtu Zawcb[8cbhikm`b`bZohzZ\ÔÚjx5k`bjxaQhg `br ayj_p hzß2^Èhzæ5j_[Ò\pxÎ6Zoø[{RlÈeRmaQ]xcb[`badk[`m\Î@p>ad`yc _ÌQæ5\adocbË>[8ÙX×mjXY X]Ù gæ ]Il 2 ¸ ê ZF×cbH¿kncg rþoÀ^x» j_wÒ\]CknmÏdÒ\aQjxaQkÓ RÙs`beXad`bg adr tÚmj_Þj hiÑX^^_aQmk`bÏQtEaQZojRc8Î@kaQ`bhzaQknÒ\knadmoc jxZ\Õ Òoayp>cmhz`b] Zr cÁûÖ momoj_jCp p ê pxmhzÒoÎ_hzhz`aQmj Ô× Þ adm`b`bhad`bgQcb[8Z\ÔimomkcbßRÕ hiayjxp>ÒxhzÙ `bZcÁ~jÖÁÙ1~¯)Ä jx@tuþZÀÁcbÂ1[8¼om»Ú`b¹ hzþoZ\JÄ j 5r hi¹ pxohz¹EjxÄRÒº Ë>Ù aUDK Z\¿xÀnúÎ¼ X<Ä_v+ X¹¹ ÔzL=ú=Ôiû_hzÿ kQ@¿TÙ ÿÆ\Æ@þoÕ ^_ÌÀÎxTÖ _ÊÔzÆ\hi»¯g) ¿ 2Rº]@Æo¼oZo>ÄAÔzhiþbgÁºoê ÀaQ¼Ê®Î><5cb ^Cad{mÞ cn¼oÄ?Z\j>ÌQÓ¥\¸ÈÔiohzjx>¹zºoaÙ ¹u»¥kn¼\^x½ cb¾{ad>¼o»¯@k ¿ÁÊIÀl Âhz¼Ò\Àbh`Ãmo¹uÄXÔwº ÞÙ@m `bcnad`bcbaQ[8g rmcbø5ßRhzZoj_^_Ò>knÙ a\CõþÍooo Â >Ì\`Ù ûR»¯¿ÁÀ >Ù!|]q ÙYR^ÈÌy\ZoRÎCCmQR.~ amk]r hÚÙ'÷mo]_l5mohijIÒoh `q moÔÈZoßRÞmZx`bh ad÷cb[8mo]_mmocbjIßRChzj_fÒc]>Êcbø5hiÔ7hzÌQk`b\ZocbXhzgyÙ mÔÈcbZXZo`bkQÙ ~.a] |×aQknaymcbg r |fad]@Zocn` ÌQ>Ù ]@÷>ÙaÁcb hiadmj_ÔzkngQÎ@aadÞ cbÒxh ` r eS8Ù knZo^ tE`¯cbÞhzk mr cbj_aÝmokÞÞ mmo`b[adcbX[8* m]cbßRõÙ hzj_^ Ò>Þ"Ù Z\jI~¯j +l&Ù ÀFþu]{ú¿mR¿3# o¹EMÄRºJÙUÿe r þQmZx"4URmoj_:Op"4RN6Ù '&P S r »mo TjxÒx"×ÙûÄxÄhv+ûxÛR¼\Ó ½ ÌoÌ\Ù gdê õþZ\mo]XÎ_Â RhzaQcb`jhiûRÒ »¯r s¿Á`®UÀ Ù [8S m,¿ Ù cbú=ß>ûR hzÀjxad¹u`b+»ÒhV`bkgQx"(Zok&ÔK`bmCadkQ[½ ¹ úbkQ|f¼oÙ\»ÚZo~¯¹ jÈknþoÒk 6Ä lÈÿ.÷xmyÙõþ>hi×opÈ)Ä jCy×px¿Á^_ÀnaÁ¿ÁcbgQAÄknknZ\[újI¿ hC"` ]_r momodj_Òoay p pxaQk5h]{`bZÆocym\cb)Ë ßR2RÄ)^xÆ>@k{ÌQþÊ>ÀÁæMÂ1Í)Ù ¼o\^È»Úo¹>þo^ ÄWÙr jM Ù ¹ ¹uÄR`nºY`X:mo@gß>¿,ú|k8þZoÄ?j ÍXÄxÌy»¯)Ë ¿Á2>ÀÁÍoÄCÆo¼oË>»Ú¹ þo ÄCZ¼\cn½"`bÔi4¾ mojCþopSÀnäÃSÿ3çf ccbþaQ ÒoZ\\jMZ\_Ôz^_s[Ù e@a5Ù sÌyÉ\ÙzIÍoÉ5ÌQ\Zoot Ë>ù ÙS,¿ eRúd]x0» ûRcbÀbhzj_Y¿ Òoügadcnþo?Ó »¯UT¿ ÿaÁcb¹uÔKZÄ mÒxõþh oa" Â adcb`ÔzûRhzjI»¯¿Á@[À æ5Aadúdcb¹¥[8¿Á`Ä moúÁjX¿ X\Ù ]CmÒ\aQk .

P

ÌyÍRÙ rê himopxÎ_hzhzj_aQÒ}j Ó×sm2Ù kn ^xÙ cb\ adadX`bÙ h`bgQ ZoÀ,ÔKþumúkQ¿b ¿3o|×¹uZ\ÄXknºJkÿÒ÷>þÙ ×» _jC¿VpxaÁ5cbkn5Z\5jIwMmË\jC Õp \ÖT]{ÊzÌQm\cbÊßRÍ^_2Ck}ÌQæoËRÙõ*^È÷\^ ^_r Ô jIÙ Ìyo~¯\jx>tEZoÙ2cb[8eR]@maQ`bgQhzZohimoj Ô ÌQÆ>ÙÈ`bhz|fkn^>knm\cb^xp>aQaÈkQ^Ù+Z\eRj{~jhiZo]>jI*cbÿ|Zoû<`b]daQÂgdhi`bß ¹ÕhzrÿxZoÿmoj{¹0hzþoÔ ZoÄVt7 =[`þomoÀ^_ÔzÔiÔmÀb`b¿rhz[y¹7ay¿mpx\>jChi1)pm]gQeR>5Z\^_#jXj_`bF[hzaQÔ 3jX" `yÙc mÎ K r F moßo$:m%^cy%&Ù{'C_Hç5`j MÞÍ\m`badÍXcbÙ [8mcbßRhzj_ÒÐknad[hik`ncb^xgdÓ ÌdxÙÈÞ |fmm\`bp>ad^cb[8e>hzmZ\cbjMßRfhijx]Òhzmoß Ôzr Ò\moZhzcbÔ+h` r `[mokQÔzÔiÙMm ~r j M mojC,À pþuú¿beR3¿ ^_oj_¹uÄRhzÔ Jº ÿ¬cþmÎ r 5m5ß\5JmcyÙ K ZyÞ ad%uc$ Êf#]>56aÁ`nF[cbhi3gd"ak5 tuZKcÈF aQo$:mo%&Ôz^C%&m'!`b_bhi`&jx` Ò Ù ÌyÉRÙÈÞ ~|vUmÙ vw`bwadv aQcb[8j_Y"ßomZomcb[`bßRaQhi]_kbjxm^>ÒxjI`bÙSadD ~¯cfsj e>Ù ZX"gQFÀ mþuhzadÏQú`¯h¿c 3¿ moojx¹EÄRh cbxJºaQmknÿgkQjCþb@p2×ÍoeS»o _Ùx¿ ÍRe>Ùchzj þär û>m>ÀÙå» .ýÄ_Òo»¯c¿ÁmoÀ] Ä@r ¼»Ú` ¹ r þoaQÄ@Zo¼ocb<½ ad`bhzg5¹ mo¹uÄR]_º]>cb4¾ Zmþog Ànr äÃ ÿ`b 6ZþknZ,tE`3Þ m%&cb' a ÌQÊ>ÙsÍæe@\Ù Ù Rq UÌ\r Ù ZyaQRZRmp>`bZoÏQcbhzkQhipx®hzsk Ùåad`×fmohiß\ÔÚÙzZ\xÔiaymopxhipxh`bhzkQZo+cbkQm jC_pÝ¹zºo~Ä@Ù ¼\ ½ hC`,ÀmoþukQúÙÐT¿ ÿxl ÿb¹uhzÄRÒoVº hz`mdUÔ"Þ7»m _`b,¿adþocb[8À¹¥T¿mÿcbßR¼hz?Äj_8cÒ Ê7¼ m&j@½ 0¹Zú¼ad»Úcb¹ RþohÄhzadÿeÞÈXÙÐC,À ~þ j _ ú¿b¿3o¹uÄXºJÿ þZ5&ûÿ¹ cú|þ _ /^BCü5¹uÄ_» T5&ûRÀ,þx¿n¼oÄf_¹zºoÄ@¼\½ À,þuú¿Tÿxÿ¹EÄRºgõþÄ)y¿ÁÀn¿ÁÄAú¿VF( 6þ)X¿Tÿ D Àb¿b,¿ úÁ¿ B&@ '&' SH¿ _»¯¿Á.Â ¿ÁÀ ' /&/^B ®]CmoÒoaQk1) 2CÌyÍR m`ncmokQ+æfcbadaQgQa\ ÌQ\oË>Ù q x]@Zcmo[8m{vwpxhÓ b ` ÌyRÙ÷o¼hzhK?Ä Zomj_jkQD ÙP Ànr ¼ mo:Z8 R¹mú=jCÿ p{SÍov+Í gÁÕ ßXrTÖ Ê mÆocpO) 2^ÈZXoÆ>g r IÙ f ^xÒ\Ò\^_adkj_`ÈadÌQcb\hzgsoË>pxÙ hzÒ\h`mÔ®Þm`badcb[8mcbßRhzj_Ò1[ZRp>aQÔÚÙ õþÂ`ûR»¯¿ÁÀ|ÿ ÌQË>Ù [8÷ohKmmEjcbßRP hijxr Ò6moÊXZ>Þ v+r mgÁß` r Þ mhzcÔzÔIp h^È`×ZXp>gZsr tE: Zo÷\c×ZX[afaç;5à |fmjC^_pmojCÞ mr hip mr ` \h`moj_Þ"p Z\jI]à `)hzj_h7aÁ~¯cbj}\m f]Y]!Ù N">aday^_p>j_h`bÒ>ZoÙ_cyl5 :hiÒo hDa`moD Ô_F[Þ "[m`b#adcn Ó /^/Y i ,À þuú¿b¿ ¹uÄRJº ÿþ» _¿ ' /^/&/Q: DaD F]"[6 ¼oÄxhÄ ûx¼\8½ úxþo)Ä Q¿ÁÀb¿ÁAÄ ú)¿ X õþoÄ y¿ÁÀb¿Á`Ä ú¿}&¼ =ÿb»ÚÀd¼ úd0» ÿ ÌQ¼\?Ä o>¼Ù@&×CY½ ¹ ]úb¼o»Ú¹ þocb6Ä aQÿ kn_kQÙY"Zo[]_^x`baÁcæfcm] r hzgQkQ>]_moÒ\adk5ÌÉÆ 2CÌyÉ\ÉXC5aÁIÞ NUZocbß@X N ÌQ\\ÆoÊ>xÈe>s

On Watermarking Semi-Structures

On Watermarking Semi-Structures

Attacks on Watermarking Systems

Attacks on Watermarking Systems

Lectures on Convex Sets

Lectures on Convex Sets

Informed Watermarking

Informed Watermarking

Selected problems on exceptional sets

Selected problems on exceptional sets

Digital Watermarking

Digital Watermarking

Digital Watermarking

Digital Watermarking

Transactions on Rough Sets XIII

Transactions on Rough Sets XIII

Function Algebras on Finite Sets

Function Algebras on Finite Sets

Transactions on Rough Sets XIV

Transactions on Rough Sets XIV

Transactions on Rough Sets I

Transactions on Rough Sets I

Transactions on Rough Sets V

Transactions on Rough Sets V

Transactions On Rough Sets I

Transactions On Rough Sets I

Digital Watermarking

Digital Watermarking

Quantization Watermarking

Quantization Watermarking

Informed Watermarking

Informed Watermarking

Watermarking Relational Databases

Watermarking Relational Databases

Multimedia Watermarking Techniques

Multimedia Watermarking Techniques

Robust Digital Image Watermarking

Robust Digital Image Watermarking

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Harmonic Analysis on Totally Disconnected Sets

Harmonic Analysis on Totally Disconnected Sets

Transactions on Rough Sets 9 conf

Transactions on Rough Sets 9 conf

Transactions on Rough Sets 6 conf

Transactions on Rough Sets 6 conf

Transactions on Rough Sets XI (Lecture Notes in Computer Science Transactions on Rough Sets)

Transactions on Rough Sets XI (Lecture Notes in Computer Science Transactions on Rough Sets)

Transactions on Rough Sets X (Lecture Notes in Computer Science Transactions on Rough Sets)

Transactions on Rough Sets X (Lecture Notes in Computer Science Transactions on Rough Sets)

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Putnam on Reference and Constructible Sets

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close