Watermarking Capacity for Gaussian Sources

! "#$#% &' ()*"+,-.# / 0214357698:;3=?5A@ BDCFEGEHCFIHJ2KLENMPO...

Author: Cohen | Lapidoth

31 downloads 613 Views 191KB 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

! "#$#% &' ()*"+,-.# / 0214357698:;3=?5A@ BDCFEGEHCFIHJ2KLENMPONOGERQTSLEGOPUWVFX,Y9MZIHJ[S\V^]_VF`Fa bcMZENMZCedHI4JDfCegULXhVFdRij]_MZIkONdGV^S[lmIPE n CFogLd4lmp\`FMFq[Bsrut^vwZxFy z\{^|e}[~e[&Ae~F

/ D3 02>h>32k< 2 lmEHELMZp\MPdHCF]QSLENOPU\VFX,Y9MZIHJLS\V^]mVF`Fa i$YcTMZSONd4KLo n et^yFv2qA>K\dHlmI4JAq 2 l_ONPMPd4]mCFSLp zeF2|}LL2 WT~2~j~^}L¡$H{?}

¢ £9¤Z¥¦?§[¨F¥

©M-IkV^SLEGlmpLMPd$OGJ\M.IkVpLlmS\ÌPCeª[CFIPl_O«aVFXOGJ\M CeONMPdHoCedG¬2lS\`®`^CFo¯M;X°VFd±.CFKLEGEGlCFS'IkV?²FMPdN ONMk³2OGEsCFSLpÉGµ2KLCedGMZp\¶MPdGdGVFdDpLlENONVFdGOGl_V^SLEZU·QS¸ª[CedHOGlmIPKL]mCedZq M¹IkVôºª[CedGM¹QGQT»EGIPCF]CedsCFSLp ²FMZIkONVFd±.CFKLEGEGlCFSDIkV²FMPdHONMk³2O-pLlmEGONdHl_g[K\OGlmV^SLEPU \VFd.g>VFOGJDpLlENONdHl_g[KLOGl_V^SLEPq2OGJ\M®IPCeª¼CFIPl_O«a*p\V2MZE S\VFO.p\MPª>MZSLpV^Ss¬2S\V ]_MZp\`FMVFXWOGJ\MIkV²FMPdGONMk³O½CeO;OGJ\MpLMZIkV2p\MPdUccV MP²FMPdq JLlm]mM½C']_VÊHENa IkVôºªLdGMZEHEGl_V^S+CeONOGCFIG¬lmE®VFªLOGlmo¯CF]$XhVFdºC EGIPCF]mCed±.CFKLEHEGlmCFS¾IkV²FMPdGONMk³OPq%EGKLIHJ+CFS¾CeONOGCFIG¬lmE S\VFO¿VFªLOGlmoCF]AXhVFdRCº²FMZIkONVFd-±-CFKLEGEGlmCFSIkV?²FMPdGONMk³OPU À

Á ÂÃ%Ä¿ÅÇÆÈÉ½Ã%Ê&ÅDÂ

Ë >< =ÍÌ;0k=?1s021ÎLh5>ÏÏ\02D=Ð_ÑÒWÓ^Ô¿Õe025D3[>=&Ö$h×A0°35>Ì½=&14=*3×>14Õ&=4=&ØL×>=&5Õ&=&5>=?=&Ùk3Ú= Õ&32LÛ[14°Ï13k=&Õk=?DÚ=?Ý¶321=->h4k14°Ú>×¼k°325'k3=->×>ÚÖ°°Õ8 Ë ×¼k=&*Þ=?14°35)43'3Í0Þ=?1k014ÛDÌ½hk< 02Õ&Õ?=&4Rk3k=>h4k14°Ú>×¼k=&DÞ=?1h35sÌ½hÖ°Ö Ú=02Ú>Öh= >143¼>×Õ&=0 Ý¶314Ï=&1HÛÇk1Õ?=Í=?ØL×>=&5>Õ?= 05>+Û=¯>3[=&5>3 Õ&325Lk02°5=¯=&ÍÚ=&>=&+D=&4k02Ï2=28 Ë =Ís02°571=?Ø[×°1=?s=?5\kÝ¶31®0sÌ;0k=?1s021ÎLh5>Ï HÛ¼Hk=?ß0214=)àá4â2ãäNåâ2ákæãèçé025>êákë2ì&í\äàãèæPäPä8 î 31cÕ&3\Û[1hÏ1432k=?Õ?kh35,ÒLk1025>>021=?5>Õ?Û s=&025>R°Hk1hÚ>×=& >0P0 43×>Öh)Ú=ï44°DÜ Öð01ñ 3Ík=¯314°Ïh5A02ÖA0k0Ò¼Ì½hÖ°=143Ú>×>Hk5>=?s=&025>;=&¹°5Ý¶321s0kh35s43×>Öh Ú=Í1=?Õ&3FÞ=?1k02Ú>Öh= Ý¶132ò025\Û¹Ý¶31Ï2=&14ÛÇh41°Ú×k=?¾A0k08/532k=?1Ì;0FÛ 32ÝkÎL°5Ï02Ú3×143Ú>×>Hk5>=?¯°®kÖÛ+Ú\Û¾=&413FÛ[°5>Ï)=A0k0)Õ&3×Ö°0)>h1k0= 14=&D3^Þ2= k1=05> Õ&Ö°02°D°5Ï0P04=&ØL×>=&5Õ&=&;021=DïH°D°Ö°021ñhÝA=½>°Hk31Hkh35ÍÚ=?NÌR=&=&5*kÕ?=Õ&025Ú=¯025\ÛsNÛ[=º32Ý A0k0Çù«×ÕP°3s31®Þ[°=&3Lú.kA01=&5Õ?ÛÒ213Ú>×4k5=&?Ò\025>¯k5\j32Ý°5Ý¶314 0k°325kÕ?Õ&=?4Ý¶×Ö°ÖhÛ =& Ú9=?>>=&è8sÿN5 A021Hk°Õ?×>Öð01&ÒèÌR=Dû>5> =*°5Ý¶321s0kh35¼Ü=&321=?°Õçâåâ[ç àé 32Ý.0Ì;0k=?1s021ÎL°5Ï 4Û[Hk=& >=?9=?5>>°5Ï)35+=)ù¶93k=&5\ð02ÖhÖhÛ>þè=?1=?5LPú-1k025A0214=&5>ÕÛ¾05>¾132Ú>×>45>=&4k 025>ÇkÏ HÛ¼Hk=?7Òk='A0P0s>°Hk1hÚ>×=&73s= >×Ú>Ö°hÕ 0257k=& °5Ý¶314 0k°325*Ì½hÖ°ÖAÚ=>þè=?1=?5L&8 Ë h;>þè=?1=&5Õ&=D°Ï23$'GRMS7:9 ;TU !.?V;

_ Z ^ Y

W

Z

5Õ&3[>=&1 [

Y

/.P0ÕPÎ2=&1

b

\ Y

=?Õ&3[>=&1 X

^ ] Y

b Àa

î hÏ×>1=ÇÑdc/ >ð02Ï21k02ß3Ý;Ì;0k=?1s021ÎL°5Ï>8 Ë =sA024=&êÖh°5>=h¯×>=?ê°5Ùk=D>14hÞ0k=Þ2=&14°35èÒ Ú>×5>3½°5k=¯>×Ú>Ö°hÕÞ=&14h35,8 Ú=Õe0×>=?ÚLÛ 0'Þ0214°=?NÛD32Ý,°Ï25A02Ö>13[Õ&=?h5>Ï =&ÕP5>hØL×>=&?Ò>=28 Ï>8Ò>3k3[Õ&3\Û[°5>Ï 31;Õ?13>°5>Ï 025 °s02Ï=D025ü1=?Õ&314>°5Ï+31¯ûAÖh=&14°5>Ï7025¸02×>°3¹Õ&Öh°,8ÿN5>Hk=&02¸32Ý-s02ÎL°5>Ï7025\Ûô024×>Dkh35>35 k02Ö >13[Õ&=?h5>ÏsÖ°02Õ&=ÒÌR=Ís02Î2=¯kh1×ÍkÇÌ½3sÏ2= 32Ý$k=ºÌ;0k=?1s021ÎL°5ÏsHÛ¼Hk=?78 î 31-= g02sÖ°=2Ò0*>°10k=ºD°Ï2°=?3 Ú='32Ýk='°ÏÖh=28¿:;325LÞ2=&14=&ÖÛÒ=-Ì.0k=&14 01ÎL°5>Ï4Û[Hk=& Ì½h=?c3'=?5>×>14=®=& h5Ý¶314 0°35¸Õ&02×>4=& =+A0P0ô3üÚ=+×>5L×>02Ú>Ö°=¹k3ô×>ÕP< 0ü°1k0k=28 óô=+Õ&025 kÍ°5Îü32Ý-Ì;0k=?1s021ÎLh5>ÏÙ020¾Ï\0s=)Ú=?NÌR=&=?5 NÌR3Ù>Öð0eÛ2=&1 c0¾Õ&32sÍ×5>°Õ&0kh35üHÛ¼Hk=? ù«=?5>Õ&3[>=?1½025>Ç>=&Õ?3¼=&1Pú.05>¹025Ç0k02ÕPÎ2=?1&8 Ë =¯>Ö°0FÛ2=&14½021=14Û[°5>ÏÍ31=?=&Õ?hÞ2=&ÖhÛ 0 ghshõ&= 025>¹D°5>hshõ&=½k=02D3×>5\½32Ý$°5Ý¶321s0kh35s=&,8 óü=-Õ?35>4°>=?1NÌ;3Þ2=&14°325>¿32ÝÌ.0k=&14 01ÎL°5>Ï8$ÿN5Í=;åá ih â2àGæ Þ2=&14°35èÒk=.=&Õ&3[>=?1¿Õe025×>4= Ú32k02Ö$>0P0=?ØL×>=&5>Õ?= 14=&Õ&3FÞ2=&1¯k==?ÍÚ9=?>>=?ü°5¼Ý¶31s0kh35,8ÿN5 k(ì j çcÞ=?14°35,Ò¼k=º>=?Õ&3[>=&1½Í×4-14=&Õ&3FÞ2=&1½k=º=?ÍÚ9=?>>=?¹h5Ý¶314 0°35DÝ¶13 k=28 Ë =>1Þ0k=Þ2=&14°357°½32s=k°D=&-Õ&02Ö°Öh=&Ç5>35[ÜN3Ú>ÖhhÞ[°32×>.31½5>35[ÜNÚ>Öh°5>Ì;0k=?1s021ÎL°5Ï 025>ÇkÚ>Ö°hÕÞ=?14°35¹°½32s=k°D=&;Õe02ÖhÖ°=&Ç3ÚÖ°hÞ[h3×>;321-Ú>Ö°h5>Ì.0=&14 0214ÎL°5>Ï>8 óü=D02Öh3Õ?35>h>=&1ºNÌR37°41hÚ>×kh35>325Ù=*31hÏ°5>02Ö A0P04=&ØL×>=&5Õ&=2Ò$Ì½°ÕP=&=&5>=&5\eÒ$°>=?5\k°Õ&02Ö°ÖÛ >h41°Ú×k=?üù¶ÿCÿ X¯ú;4=&ØL×>=&5Õ&= 3ÝD=e025[ÜNõ&=?13Þ021°025>Õ?=VÜ sHut l¯02×>4ð025¹1k05>>3 Þ201ð0Ú>Ö°=?&8 Ó¼w8 v=?Õ?k32" 1 l¯02×>4°02x 5 c Ë =sÕ?3FÞ=&1Hk= g¼ _yn ù _ apBq(qBqp _ rú°¯025Ùÿ4Cÿ X 4=&ØL×>=&5Õ&=)32ÝRD=e025¼Ü õ?=&1z 3 l¯02×>4ð0251025>>32 Þ=?Õ?k321&Ò=&02ÕP4ð025ÇÕ&3FÞ2=&14= g[Zú Ñ r ùD o ú tGS ap 0¼8 ?8 p ùHÑFú

a

a

Ý¶313D= Õ?35>HP025\ aG¡£¢ 025>+Ì½=?1=*08 &84k025>>®Ý¶310Ö°D34×1=&ÖÛÒ9T8 =28ÒÌ½k143ÚA02Ú°Ö°NÛ 35>=8 î 31.0Í4Õe02Ö°021wl¯02×>4ð05Õ&3FÞ=?14k=g[eÒ>kA01=&5Õ?Û)Õ?35>Hk1k0°5\ùHÑFúR°ck=º02s=®=g¼Õ&=? k3D1=?Ý¶=?1=?5>Õ&=?A 3 ¤D3w1 {8 Ë =¯>=&Õ?3¼=&1½Õ&3D>Öh=?k=?.k=¯Õ?3s ×>5>°Õ&0kh35)HÛ¼Hk=? ÚLÛ Õ&32s>×¼k°5Ï*0257=?4°s0k= ^ ] 32Ý$=¯Ì.0=&14 0214Î D=&4k02Ï=8 Ë 5>Õk°325¹3ÝkÚ>Öh°Õ¯Þ=?1h35¾05>02Öh3D32Ý%k14hÞ0k=Þ2=&14°325,8 _ Ë ==&Õ?35>D>Öð0eÛ2=&1°5Í=-Ì.0=&14 0214ÎL°5>Ï¯Ï\02D=.h%k=04P02ÕPÎ=&1&Ò\Ì½3'04k=&D%k3¯>°41×>¼ k3^Ì Ý¶132 =º=&5Õ&3[>=&1.3 k=º=&Õ&3[>=?1&8 Ë 3>33>Ò¼k=º0P0ÕPÎ2=&1-13[>×>Õ&=?½0 Ý¶314Ï=&1HÛ \ 00Ý¶×>5>Õ?°353Ý=-4=&Ï3k= g[ [ 05>h%3FÌ½5D1025>>32 Î=?Û ` t Ò2Ì½hÕP>=&=&5[Ü >=?5L½3Ý$=Þ=?14ÛLkh5>ÏD=&Ö°4=28 Ë 3*=&5Ý¶314Õ&==º13Ú×>45>=&4Õ?35>Hk1k0°5\eÒÌR=¯35>ÖÛ Õ?35>h>=&1.0k02ÕPÎ2=?1 °5Ì½hÕP4°025ÇÕ&3FÞ=?14k= g[Zú Ñ r ù¦ § ú t S p 0¼8 ?8 p ùTÓ2ú

a

t

a

Ý¶31c3D=Õ&35>HP025\ t ¡¨¢ 8 Ë ûA=&DÝ¶31R0'Õ&02Öð01Nl¯02×>4ð05 Õ&3FÞ2=&14=g[ 02.Ì½kDkh35°.02Ú>Ö°=®k3k=04P02ÕPÎ=&1?8 l¯Þ=?5êk=s=&5>Õ?3¼>=?1&Ò¿>=?Õ&3[>=&1*05>¸04P02ÕPÎ=&1&Ò%ÌR=Õe05ê=?Þ02Öh×A0k=Dk= >13Ú>02Ú>°ÖhhNÛ3Ý½=&11431&Ò ©ª,« ù ,ú n~¬ 1^ù ^ 3×>1Õ?35>4=&14Þ0hÞ2=Ç02>>143\02ÕP=?k02°Öh32Ý®=¹=&5>Õ?3¼>=?1 05> =&Õ&3[>=?1¾ù¶Ú>×D5>32*kÍ32Ý _ Ò ^ 05> ×>4°5>Ï+| = ®+ >=?Õ&3[>°5Ï¾1×>Öh=28¹óü= Ík0eÛ kÕ?= 32Ý;02Ö°Öh3FÌ.02ÚÖ°=Í10k=ZÜ =&5>Õ?3¼>=?1025> >=&Õ?3¼>=?1 A02h1c×>ÕPÕ?=º32Ýj02ÖhÖ°3FÌ.0Ú>Ö°=½04P02ÕPÎ=&14&Ò ªK© « Dù Wú=&5>;k3'õ&=&14302§ 7k=?5>>;3 °5¼ûA5>hNÛ28 Ë =)çP(ë ° ã ¹çâåâ[ç àé3Ýk>1=?Í×> 32Ý%02Ö°Ö,02ÕP°=Þ02Ú>Ö°=º10k=?&Ò0257h Ì½1=&² 5 ±w³\ù ap t p s ut úº025´ ±§µ¿ù ap t p u úÝ¶3214Õe02Ö°021025> Þ=&Õk3¶ 1 l0×>4ð05ôÕ?3^Þ2=&1Hk= g[&Ò 1=?=&ÕkhÞ2=&ÖÛ8 ÿN5 khD>029=?1&ÒcÌ;=7>°4Õ&×>4skh5>ÏôÕ&02A02Õ?h°=&?8 óô=Ý¶3¼Õ?×>s325 kh5>ÏºÕe02>02Õ&NÛÍÝ¶31¿hshÖð021 Ì;0k=?1s021ÎL°5ÏºÏ\02D=&¿°5Õ&Ö°×>=Ð Ó[¸ =?5LÍºÐ ¹>=Ò ·Ô 025>¹°>=?5\khû>Õe0°35)Õ&02A02Õ?hNÛ ¼Ð »^Ô8cø®32=kÖÛ¹Ð ·ÔÕ&35>4°>=?1k=Õe0=2Ò02Ì;=®>3'Õ?3¼=&1â2Qã °>=?Õ&3[>=&1RÍ×>4RÚ=143Ú>×>H k3D025\Û)Ý¶=&02hÚ>Ö°=¯0P0ÕPÎ2=&1?8 ½ È

ÍÉÊ?ÃÂ

È ?Ã

¾À¿"Á0¿

Ã~Ä*Å ÇÆ @Å

=Ý¶31=¯4k0kh5>Ïk=¯Õ&02A02Õ?hNÛ)14=&4×>Öh&Ò>ÌR=ûA144ÏÞ=¯3D=º>=ûA5>h°35&8W=?½kÇk°5>Ï Ó

p s t t u

ù Ë"Ô

ú

nÀÊ@Ë

ap

ÏÎ

s0g

Ê

c

p s t t u

ú n

t

a t8Õ

pÍÌ s

Ñ

u

t Ë Ö × Õ

Ñ

Î

t¸Ñ

aÐ

ùË

Ë

Ì s

st u ¹ a s t u

uNÒ

ú

a Ct Ö

q

aÐ

t¸Ñ p

ù·ú ù¹\ú

ëá.ìëài¯ài¯Aæåá ih â2àNæâ2ãQ°;åí>ì(j ç h æáPä ëãä½ëCã;àD¯AæNcâ2àGæáDâ2á ãG\âä æ5å,àD¯Aæ®çPë(° ã çâåâ[ç «à é Gä ih æãì&é

Ø"Ù6ÚÜÛÞÝÚßáà@â

ù

±w³

ap t

ú

p s t u

nÀæ

s0gçQèéJê ë,ìVí ëîí ï î

ðñ t

¢

a

Ö°3Ï6òkÑ

Ó Ò

ù Ë"Ô

ap

ú

ù ap ëàD¯Aæá Fä æ

ã É

p s t ó u t

eëá'â*äFçâäjâ2á¥ö½â2íLäPä â2ã çPë h æ áàNækàiå;â2ãQ° ã

ù

±§µ

ap t

Aú

p ¥÷

n ø

sìV0ùg úûí

ë ì ì üþÿý ì ÿ

ø

sîùh5 úûí

ë î î üþÿý ì ÿ

ù

±w³ a

a

p t

p s t

úp

t

p s t u

ú

õ n ô q

ù»2ú ù ú

eëá'â h æPçàNëáö½â2í\äPä â2ãüçPë h æáZàGækàiå 1¯Aæákæ2s at p(qBqBq(p s t â2ákæài¯Aææ [æã h âäj_íæPä'ëCãF¥÷

ø®32= k0º0DÕ&Ö°32=&[ÜÝ¶314 43Ö°×¼k°3257Ý¶31 ù»ú½Õe025Ú='Ý¶32×>5>¾Ú\Û7=?4kh5>Ï k=&14hÞ0hÞ=¯Ì½h< 1=?=&Õºk3 Ë k3)õ?=&138 Ë °Û[°=&Öh>0sÕ&×>Ú>hÕÍ=&ØL×A0k°325¾°5 Ë k14=&1½5>3k=k0G ±w³ù ap t p s u t ú-h-õ&=?13D35>ÖhÛ hÝ t s u t Ò a Ò Ó s u a 8 ÿN5ê6[=&Õ?°35 ·¼Ò,Ì;=*>hÕ&×ºk=*Õ?35\Þ=&14=DA021Hkº32Ýckhk1h= e>Û+>°4Õ&×>4 3FÌ 1k0=&sÖ°=&4 k025uk143932=& Õ&02A02Õ?h°=&sÕe025uÚ9=+02ÕP°=Þ=? h5 6¼=?Õ?°35>)Ó¼8Ñ025> Ó[8 Ó¼Ò Ý¶31½Õ&02Öð010257Þ=?Õ?3* 1 l¯02×>4ð057Õ?3^Þ2=&1Hk= g[&Ò,14=&=&ÕkÞ=&ÖÛ8 î 310sD314==?P0°Ö°=?+0Õ&Õ&32×>5\=?= Ð_ÑÒ ÔT8Róü=35>ÖÛÇ>°4Õ&×>402ÕPh=?Þ02Ú>hÖ°NÛ Ý¶31-k×>ÚÖ°°ÕÞ2=&14°325,Ò94°5>Õ?=Í025\ÛÇ10k='02ÕPh=?Þ02Ú>Öh=¯Ý¶321 kÚ>Ö°hÕÞ=?1h35¹°½02Öh3D02ÕPh=?Þ02Ú>Öh=Ý¶31.k=¯>14hÞ0=Þ=?1h35,8 ã

"!#$&%')(+*-,/.0"#$ ,213"465758"9;:?,@%@AB%

Ë >< =Õ&32sÍ×5>°Õ&0kh35HÛ¼Hk=? ÕP3¼32=&j0àTâ2á[æàäàNæ[ëàGækàLåAë;æá Ë0C Ý¶13 k=4=? É ù ap t p s u t úÒ 025> >=&4°Ï5>;0'Õ?3¼>h5>Ï Hk10k=?Ï2ÛDÚA02=?35 Ë0C 02Ö°325>ÏÌ½k< =3k=?1;A02102D=?k=?1?8 Ë °RÕ&3[>°5Ï 41k0=&Ï2Û*h¿s3khÞ0=&*Ú\Ûk=½ÌR31Î32Ý lº=&ÖD Ý«025>D025 ¬ °5>4Î2=?1ÐFEÔ$ù0ÖhkÏþè=?1=?5L®h5>Õ&=¯k5>ÎL5>3FÌ½5úZ8 Ë ='=?5>Õ&3[>=?1 G>=?Õ&3[r.> =&ì 1A02°1-×='=&h1½Õ&3Ds325 32×>1Õ?= 32Ý1k025>3s5r. =&½k3DÏ=?5>=&10k='r.I0DH Õ?3¼=&Ú3¼3ÎÕ?35>4°4°5>ÏD32Ý¿Ó ÿ4Cÿ X Õ&3[>=Ì;314>®=&h5L3'Ó Ú>hÎ 5>¿32Ýè°õ?=®Ó =&02ÕP=?ÌR314 °¿×5>hÝ¶314sÖÛ>°Hk1hÚ>×=&*35s025 >s t ù¶3Ú9='ÕP34=&5ÙÖð0=&1kú* 8 lºhÞ2=&5¾= Õ?3^Þ2=&1Hk= g[L K 05>¾k='Ì.0=&1GÜ Ü> J s021N Î M'Òk=e01.Õ&3 Ú>°5A0k°325)32Ý=¯ÕP34=&5+Õ?3¼=?ÌR317025>Çk=¯Õ?3^Þ2=&1Hk= g[&Ò QTS¿Uù K-ú ùHÑ W ú K p ù E2ú V n Ò Ì½=?1= W °*0Õ&325>4k025\D3ÙÚ9=ÇÕP34=&5 Öð0=&1&8 Ë =Ç>°Hk31Hkh35¸Õ&354k102°5\Ì½hÖ°Ö;Ú=Çs=ÍÌ½h< hÏ°ÖhhNÛshÝ ú °½Ö°021Ï=¯=?5>3×>Ï3×>Ï2=Ì;314>h57=&02ÕPh5743 k035>=*°ºÕ?Ö°34=D=?5>3×>Ï2< 3¹rB =*ì Õ&3FÞ=?14k= g[ Ì½h=?Õ&3[>=&1¯ûA5>>¯k=?ÌR314ù«32×c32Ý,0Ö°ÖÓ Õ&3[>=?ÌR314>Pú¿k3ºk=½Ý¶314Ï=&1HÛÒ[025>D=&Hk°s0=&¿k=½Ì;0k=?1s021Î 02=Ú>h5¾32ÝkhÕ?Ö°34=&4¯Õ?3¼=?ÌR31,8¯ÿÝ a °s02Ö°Ö =&53×>Ï132ÚA02Ú>hÖ°hNÛ 32ÝW>=?Õ&3[>°5ÏÍ=&14131cÕe025 Ú9=® 02=0214Ú>h1k0214°ÖÛÍs02ÖhÖAhÝ a °cs02Ö°Ö =&53×>Ï32º3¼3 025\Û+Õ&3[>=?ÌR3143)k×>14°3×35>=hÕ&Öh3=?1 k025ük=)Õ?3¼>=Ì;321,87ÿÝ-k=)Õ?35>HP025\k" sJ t Ò W Ò ú 025 a 0214= ÕP34=&5 02 02>>143>1°0k= Ý¶×>5>Õkh35>.32ÝW=k021Ï=.Hk=&Ï232k= g¼.93FÌR=&1½025>)=32=&1RA02102s=k=?1¯ù¶=&=DÐ_Ñ2@Ò ÔðúÒ¼=&5.Õ&025 Úa =¹43FÌ½5 =Ç0214Ú>h1k0214°ÖÛô4 0Ö°Ö.0DÖh35>ÏÙ02 YX Ö°3ÏùGÑ Ò Ó ù ËZCCÔ a5p t p su t úúZ86¼h5>Õ&=Í=P0214Ï=?ºHk=?Ï32k= g['3FÌ;=?1¯Õe025ÙÚ=ÕP3=?5ê0214Ú>h1k0214°ÖÛ t Ý¶132 kÖ°=8

Ë °s=k3[ÍÕe05Ú9=.14=&Ö°0k=? k3°4314°35'Õ&3D=&5>0k=?Ø[×>025\k°õ&0kh35Íh5>>=g D3¼>×Öð0°35 Ð\[ÔTÒcÌ½°ÕP32=Õ&=s0k14g¥ ÷)Õe05)Ú9=>°02Ï35A0Ö°°õ?=&D×>h5>ÏÍ05 31Hk3Ï235A02ÖAk1025>GÜ Ý¶314 0°35)3[=&5>320þ9=&Õ?½='°4314°35è8 Ë ?ÒÌR= Õ&025¾024×>D=¯kð02Ï235A02 Ö 3)h4º32Ý { Õ?3D935=&5\k?Ò _ n ù _ apBqBq(q(p _ úÒèÌ½=?1==&02ÕP< _ °D0 Ö°=&5Ï2k=?5>Õ&=32Ýs=&025¼Üõ&=&143 Þ0214ð025Õ&=Ü s t l¯02×>4°025 1025>>32!Þ201ð0Ú>Ö°=?&8 W= Dù s at pBq(qBq(p s t ú;>=?5>32k=º=¯=&°Ï2=&5\Þ02Ö°×>=?½32Ý ÷,8 c t n ÿN5ü314>=&1º3¹ =&hÏ5Ùk=D=&5>Õ?3¼=&1'025> >=&Õ?3¼>=?1&Ò k=*Õ?3s ×>5>°Õ&0kh354Û[Hk=& ÕP3[34=& { Ü >hs=?5>h35A02Ö>Þ2=&Õ?31 d a f Ò d e t Ò¼025>h g C ÒLÌ½°ÕP< 1=&1=&4=&5\.Ý¶31=&02ÕP=?5Lck5\c32Ý >h4k3214kh35¾k0ºÌ½°ÖhÖ%°5\k143¼×>Õ&=?,Òj025Ù=&4°s0k=3ÝR=D0k02ÕPÎ2=?i1 D '>h4314°35,Ò 05>k=P01Ï= 4=&Ï3k= g[®93FÌR=&1?8 ë °[æ&ìëF.ë lk;/®°5+6¼=&Õk°325Ó¼8ÑÒAk=Õ&3[>=?Ú93[3ÎÕ&325>h4k®32ÝÓ r. Ú>°5>-Ì½k=Ì;314> j ( °5+=e02ÕPh5, 8 m=&1=Ò,=e0ÕP=?ÌR31Ùh 0 1025>>3 s0k14 gÇÌ½h5>Ý¶31D ×k=? ÖhÛ °41hÚ> 3FÞ=?1;kG= Ü>=?1=Ì½kh×>o n >s J t ÒLÌ½=&14þ= sJ t hRk5>Õ?°35s32JÝ st 7Ò a Ò e t 05> s J t 0Ú93FÞ=8 ËZC 0w ë ° ã qklºhÞ=?5+k=&1û>5>>=ÍÕ&3[>=ZÜ p ãèçP( ÌR31´s0k14 gô°5¸Ú>hs 5 M Ì½34=¹Õ?3D935=&5\k*0214=Ç02>>143>14ð0=&ÖhÛ Õ?Ö°34=)k3¾k3 ×ÕP< Õ?3¼=?ÌR31¸= gh4&Ò=&5´025ü=&11431 h >=? Õ&Ö°021=?,8 k=?5L 0214=Õ&1=&0k=?u0Dh5 ù EúZÒ¿Ì½hk< 0Õ&325>4k025\ W =?9=?5>>D3x 5 st tÒ a Ò e t 025> ËZC 0 W 02Ú3FÞ=28 ë ° ã qk l¯Þ=?570 Ý¶321Ï=?14w Û vRÒ=º>=?Õ&3[>=&1½04k=?s;k3Íû>5>)k=º×5>°ØL×>=ºÕ?3¼>=Ì;321ôù«3×¼ u æPçP( r ê @x= H 32Ýj02ÖhÖ9Ó 1 d ù«3×¼.32Ý 0'93ÖÛ[5>3Dð02Ö>4×>Ú>4=? ñ Õ?3¼=?ÌR31>kú.025>0P0ÕPÎ2=&1;>°Hk31Hk°325DÞ2=&Õ?30 32ÝjÝ¶=e024°Ú>Öh=Þ=?Õ?k321Pú½×>ÕPh4P05>t Õ&=sù¶Ì½ks=3k=?1RA01k02D=?=&1kúÚ=?NÌR=&=?5)k=Ý¶31Ï2=&14ÛD05>s=Õ?3¼>=Ì;321)hR0>>132>1°0k=?ÖhÛ s02ÖhÖ Ô 0 4°D°Öð01k=?ÕP5>°ØL×>=°c×>=?°5+FÐ zÔ8¿ÿÝW5>3 ×>ÕP5°ØL×>=Õ?3¼=?ÌR31)= g¼°4&Ò[k=?5Ç025 =?11321 °R>=&Õ?Öð0214=&,8 =&1HÌ½°=Òk=&Õ?3¼=&1-=&Hkh 0=&ck=Ì;0=&1s0214ÎD3Ú9==Ú>h5)32ÝW=×>5>hØL×>= Õ&3[>=Ì;314,8 {ãä â j_é^ä iä k î 31¯0ÏhÞ2=&| 5 g C 025>| d a ÒjÝ d e t °ºÕP32=&5ü02>>143>14ð0=&ÖhÛ2Ò,k=?5 ÌR=DÕe025 3FÌ k002Ö°Öè1k0=& sîùh5 úûí ù g C p d ap d t p c ú ù [ú }X ü

0214='0ÕP°=Þ20Ú>Ö°=ÒÌ½=&14= ù g ~

p d

a5p d

p t c

ú n

ý ÿ

ø

ëJîq~ ì î ÿ

a

Ñ Öh3Ï Ì Ñ Ó

Ò Ó

ùË

Ô

a

p t

p s t

ú

x

Ð%p

ù zú

20 5> Ó ù Ë"Ô ap t p s u t ú*hs>=ûA5>=? °5·ù¹LúZ8 ÿN5>=&=?,Ò-k=7ÕP3°Õ?=73Ý/d e t kÖ°-k=k0s=02-k¹k=ºP01Ï= 4=&Ï3k= g[®5>314s0 g C 01=¯0½ÕP32=&5+0½k=¯Õ?3s ×>5>°Õ&0kh35)HÛ¼Hk=?PD ®°Õ?1=k°325,ÒA02ÖhÖè1k0k=& sìV0 ùg úûí 0 í g Dîù°5 úûí ù g p d ap d t p c ú ùHÑ ¢ ú X ø ø î î ë,ì ë î ì î ~ ì ì üÿ ý ÿ ù ï ÿ ì ÿ ú ç ÿ ù ï ÿ x ì ÿ ú üÿý ÿ

20 14=Ù02ÕP°=Þ02Ú>Ö°=8 Ë 5A02ÖHk=& °sk3´3FÌ kÞ02Ö°=?5L;k3¹ù úÒ>8 =8hÒ[kk=sh5)3FÞ=&1Zd t Õe025ÇÚ=°5\=&1ÕP025>Ï=?7h5 ùHÑ ¢ ú8 Ë Öh°4°5>ÏÍ=6¼°325¼Ü¯02ÎL×k025>H®7°5>h 0gs=&321=&ßù«4=&=¯=28 Ï>8WÐ_Ñ ¢ Ò $8 ·8 EÔðúÌ½h< 4Ö°°Ï2hûA=? Õ?35>41k02h5\k&8 Ë =?k02°ÖhR3Ý,kh.0214Ï×>D=&5\;025>)k14=?Þ[°3×®01Ï×>D=&5\3Ý$=¯>13[32Ý$Õe025ÇÚ=¯Ý¶32×>5>¹°5ôFÐ ÔT8

0Ä

+Ã¿Ê Á

¿"Æ

Ã$Ã Í ¿ Ét ¿ ÂÆ

Å¥Å ¸

ÅKÂ

¾

Åw Á Ä@Ä*Å1Å%Ê&ÅDÂ

ÿN5'°j=?Õ?°35,ÒÌ;=R>=&4Õ&14°Ú=c=.04P02ÕPÎ=&14$k0Ï2×A021k05L=&=k3®1k0=&jÖð0214Ï=&1WÖh=28 Ë hÍ°'>35>=)Ý¶321 Õ&02Öð01Í025>êÞ2=&Õk3¥ 1 l¯02×>4ð05êÕ&3FÞ=?14= g[k*°5¸6¼=&Õk°325>% ·8Ñ 025> ·¼8 Ó¼Ò91=?=&Õ?hÞ2=&ÖhÛ28ºóü= Ý¶3[Õ&×35¾k1Þ0k=Þ=&14h3532Ýk='Ì.0=&14 0214ÎL°5>Ï)Ï02s=Òè4°5>Õ?= 025\Ûô1k0=sÌ½hÕPÖh=D=&14=°'02Ö°43¾5>32Í02ÕPh=?Þ02Ú>Öh= °5ôk=)×>Ú>Ö°hÕDÞ2=&14°325,8ÿN5 6¼=?Õ?kh3A 5 ·48 ·¼ÒÌR=RÕ?3DA021=%=&4=.04P02ÕPÎ=&14j3®35=Öh=&D=&5\kW3¼k°s02ÖÖ°344ÛºÕ?3s1=&4h35 ÚA024=&Ù35>ÖÛ+325¾k=HP0°4°Õ?32ÝkÕP< 025Ù04P02ÕPÎ=&1ºk×15>º3×¼k3)Ú=3°s02Ö Ý¶310ºÕ&02Öð0 1 l0×>4ð025*Õ&3FÞ2=&14= g[eÒ[Ú>×c5>3¿Ý¶31¿0Þ=?Õ?k321 l¯02×>4°025*Õ&3FÞ=?14k= g[e8ÿN5Ý«02Õ?&ÒLÌR=½3FÌ °56¼=&Õk°32| 5 ·$8 ¹Í5>hÕe0°35s4Û[4=&ÎL5>3FÌ½½°5)02Þ025>Õ?=kÖh=&D=&5\Öh3HÛÕ&3D>14=&4°35èÒ =&5 Í×>ÕP< hÏ=&1º10k=?021=*02ÕP=P02hÖ°ºÕ&025 Ú=¯Ý¶32×>5>Ç°5ôÐ_ÑÒ Ô8

:?,57(*t,2.0"# ,213"465@5@$"9;:|*t?,@%8TAB%

ÿN5 a$3 1=&1+3 >13FÞ=¸=¸Õ&35\Þ2=&14= Ý¶3217k1Þ20k=üÞ2=&14°35èÒÍh+°¾4×@DÕ&°=?5\ 3 3FÌ k1=?h35,Ò02k=sÚ>Ö°3[ÕPÎLÖ°=?5>Ï2k< k=?5>>½k3*°5¼ûA5>hNÛ28 Ë °.Ý¶32Ö°Ö°3FÌ½-Ú\ÛÇ025702>Ö°°Õ&0kh35)32Ý î 025>38D -°5=&ØL×A02ÖhhNÛ8 /50P0ÕPÎ2=&1ckh°35DÕe025sÚ9=>=?Õ&14°Ú=&0RÝ¶3ÖhÖ°3FÌ½a &8 l¯Þ=&5Dk=4=&Ï3Ü k= g[ V Ò°$0k02ÕPÎ2=?1%Õ&32s>×¼k=&jk=¯æ å á çäâ jLäàNæ [ëàGæ kà[åAë ;æá Ë«n V t%025> Ø[×>025\kh°=? h½3s35=3Ý%ûA5>k=&ÖÛ)s025\ÛÞ02Ö°×>=?&ÒAÌ½h025\k°õ?=&¹Þ20Ö°×>=¯>=?5>32k=?¾Ú\Û Ë] « 8 Ë = 0k02ÕPÎ2=?1 k V Ò_¡ Ì½=?1= 2n Ñ ëaç@£ ¤¢î 025> ¡ °;025 ÿ4Cÿ X =?Ø[×=&5>Õ?=º32Ý s=&025¼Üõ&=&143'Þ0214ð025>Õ?=Ü ¥t l¯02×>4ð05 1k05>>3 Þ021°02Ú>Öh=&&8 î 321-025\Û ¥t X t ÒhÝ k=ºØL×A05L°õ&0kh35Ç32Ý Ë« h.ûA5>=¯=&5>32×>Ï°ÖhhNÛ28¯ù«ÿN5¹k='Õe024=&Ì½=?1=Í>3¼=?5>3 k0k°HÝöÛ =-Õ&354k102°5\&Ò[4°D>ÖhÛ4=?- v n V 8 ú î ×14kh5Ý¶31s0°35 32Ý¿h5\k=&14=&HºÕe025+Ú9= 0D34 ta Ö°3% Ï Ì\Ñ Ò Ó ù Ë] «Ô ap ¥t p su t ú Ð$8W3[3=?ÖhÛ¹=e02ÎLh5>Ï>Òè4°5>Õ?= ËG] « Õ&025 P0Î2=¯35Ç35>ÖÛ)0ÍûA5>k=º5L×>ÍÚ=&1-3Ý$Þ02Öh×>=°5=º= É ù ap t p su t ú;02574°5>Õ?= Ó ù Ë"Ô ap t p su t ú °®Õ&35\°5L×>3×>ºh5 t Ò,5>310k=?Öð0214Ï=&1kÖ°=86¼=&=¹Ð_ÑaÒ ÔÝ¶310 D31=º>=P02hÖ°=?Ç025A02ÖÛ¼4°?8 ]

:?,57(*t,^`?a%@*t,313"465758"9Y:|*t?,@%8TAB%

/R°5)6¼=&Õk°325 ·8ÑÒ[ =0Pa 0 ÕPÎ* k02Î2=?;k×=&ck=®=&D>°14°Õe0ÖA4k=?Ï32=g[ 3^ÌR=&-1 g « ÒT8 =28Ò ËG« n V t 8@lºhÞ=?5fg « Ò2=-04P02ÕPÎ=&1¿ÕP3[34=&05>35[ÜN5>=?Ï\0kÞ=.°4314°35 Þ=?Õ?3o 1 d t ×>ÕPÖh=&D=&5\k½k=&5\§ ¤ ÚA024=&735 ËG« 05>¨ t 8có kÌR=¯Õe025¹=?ûA5=& °5 ù zú8 Ë =r0m6ê32ÝÍùHÑ^Ó2ú'Õe025 Ú= Ö°hÕe0°353ÝR=)6[°35¼Ü ¯02ÎL×P05>F®7°5°s0g Ë =?31=?78 Ë &Ò.°×>9=?1sÚ3×>5 35 k=ÇÕ&02A02Õ?hNÛ¸32Ý®k=¹>14hÞ0k= Þ=?14°35 02Ï14=&=?Ì½h5>Ù35¾k02Õ&NÛ+32Ý=Í>×>ÚÖ°°ÕÞ=?1h35,Ò 025¾k= NÌR3 Õe0A02Õ&kh=&-Í×>HÚ9=º=?Ø[×>02ÖT8 &

ù

ø

ï

ÿ

ç

ç

:9®¯%7%@ °

>5 =D932=&5\k°02Ö¿04P02ÕPÎ=&1'°D>Ö°=?s=?5Lº3°s02ÖÖh3HÛÙÕ&3D>1=?4°35 32Ý;=D4k=?Ï32=g[ Ì½kh4314°35 t 8 Ë 0½°?ÒÏÞ=&5)=HP0°4°Õ?.32ÝWkk ù«h57kh4k3214kh35 t 8 óô=¯Ì½hÖ°Ö91=?Ý¶=?1-k3kh02.k=)çPë åákæPäPä ëãÙâ2à«àâ[5ç 8 5>=ºÕe025¹ÕP14=&4°32570P0ÕPÎ Ú\Û)khkh35A02Ö>h4k14°Ú>×¼k°325 ª²±³ ´ 32Ý k ='°D°õ?=& ù [ Ô5\ úc3FÞ=?1.02ÖhÖA>°Hk1hÚ>×°35>¿41k02h5L&8½ù«/®;hR×>4×A02ÖÝ¶31c1k0= >h4k3214kh35,Ò,ÌR=DÌ½hÖ°Ö$×>=s025 = g¼9=?Õ?=&ê>h4314°35Õ?35>41k02h5\3ûA5>ô°ºD°5°Í×> Ô hÝ [ h =&14Ï3[>°ÕÒk=?5¹=¯>hþ9=&14=&5>Õ?=ºÌ½hk41k02h5\½°-s02ÖhÖT8 ú ÿÝk4ð05u=?ØL×>=&5>Õ?=¾Ì½h< Þ0214ð025>Õ?= Ë Ò; kÏ =?5>Õ&3[>h5>ÏsHk10k=?Ï2Û¹Õ&1=&0k=?0D4k=?Ï32= g[ Ì½hÕP>1B3 g¼°s0=&ÖhÛÙÿ4Cÿ Xl0×>4ð025,8 Ë &Ò¿k14=&4°35´0P0ÕPÎôh =&4=?5Lð02ÖhÖhÛÙk4ð025ÇÕ?3^Þ2=&1Hk= g[&8 32Ýj0'Þ=?Õ?321 l¯02×>4ð025 4k=?Ï32= g[ kh 02ÖÖh3HÛsÕ?3D>1=?h35 Ì½kh4314°35 t Ì½h=&5\Þ0214ð025>Õ?=&ÍÏ2hÞ=?5´ÚLs Û g Õe05 Ú=)>=?Õ?1°Ú=& 02ÍÝ¶3ÖhÖ°3FÌ½ Ô =?==28 Ï>8ÒÐ_ÑÑ2ÒR6¼=?Õ?&8 Ñ ·$8 ·$8 ·Ô8 Ë °Hk31Hk°325ê°Í°41hÚ>×k=?ük3¾=Õ&32s35>=?5L×>h5>Ï¾kÏ . Þ=?Õ?36 1 d¶µ¸·Ò>Ì½=&14= ú Ý ¹ ú XË p µ¸· n ùH.Ñ ·\ú æB¹ 32=&1HÌ½°4= Ë 025>´Ì½=?1= ¹ ú °ÕPÕP< k14=&4°35´h =&14Ý¶321D=&Ç35¹=e02ÕP=?5L½×>h5>Ï*>°Hk31Hk°32O 5 µ¸· Ý¶31-Õ?3s35>=?5\ ¤A8 6¼°D°Ö°021ÖÛ.k3.=Õ&3D>14=&4°35¯0P0ÕPÎÒ^k°Hk1hÚ>×°35 Þ2=&Õ?3P 1 d t 025 ×=& =Ù3kh 0Ö04P02ÕPÎ Ý¶31Ç0¸Õ&02Öð021 l¯02×>4ð0257Õ&3FÞ=?14 = g[¯35+Õ?3D935=&5\w ¤ Ì½kÕ&=k='3°s02ÖW04P02ÕPÎÝ¶31®0 Õ&02Öð01 l02×°025Õ&3FÞ2=&14= g[.h%=&4=?5Lð02ÖhÖhÛ 3kh 0ÖÖ°324Û Õ&3D>14=&4°35èÒk=½>þè=?1=&5Õ&=½Ú9=NÌ;=?=&5 k0k02ÕPÎ025>*k×kh3532Ýk=?°1c02ÖhÖ°3FÌ.0Ú>Ö°= >h4k3214kh35,ÒT8 =28Ò9= D°5°D°õ&h5>f Ï d @ °5´ù »ú.Þ=&14×>º d µ¸· 32Ý®ùH.Ñ ·ú8ÿN5>=&=?,ÒèÝ¶310sû g¼=& c Ò d a 025>» g¾Ò>k=¯D°5>hshõ&h5>ÏÞ2=&Õk3135kt L = 0m632Ý. ù [\ú;h-ÏhÞ2=&5¹ÚLÛ

@ t

nÀæ¼ Ë

ù ½

p

¹

ú Ý ¼ ù½ p ¹ ú 32=&1HÌ½°4=

XáË

p

ùHÑB¹Lú

Ì½=?1=

¼

ù ½

½

n

Õ

a

p ¹

ú ùË

n

Ñ

Ò

n

Ól ½ a

¹ds t

Ñ

Ò¿¾$ÀÁ s t

út

p

Ö

a Ë

Ñ p

20 5> ¹ °®ÕP34=&5 ×>ÕP`d|µ¸·021=Íhþ9=&1=?5\ Ý¶310257021Úhk10214Û)ÕP3°Õ?=¯32Ý"gß025Âd a Ú\ÛÇk==&5Õ&3[>=&1?8;ÿN57Ý«0Õ?eÒA=Ý¶32Ö°Ö°3FÌ½h5>ÏÍ=g02D>Öh=¯Ì½hÖ°Ö >=?s354k10k=º»d a 8 óü=*5>3FÌ Õ?35>4°>=?1'05 = g02sÖ°=Ì½=&14 = { n »¼Ò c t n ùHÑ ¢Up [ p p ¹ p ÓúZÒ a n »¹05> n ¹ ¢ 8 î 31;=&4=>021k02D=?=&14&Ò>kÕ?3¼=&1025>¹04P02ÕPÎÙù¶T8 =28Òk=?-k=?)Ý¶31R=Õ&32s>14=&4°3250k02ÕPÎÒkh=ºÖð0214Ï=28 Å

Æ2?58Çt996ÇÈ(+*-,_É:9Ê¯%7%@ °

W=? ×> 5>3FÌôÕ&35°=&1,kÖ°=%10k=?,Ì½=&5¯k0s>Öh=º32Ýjk=¯>14=?Þ[°3×4=&Õk°325,ÒA5>32=¯ Ö°=Ù02Ï02°5>HÇ0¸Õ&3D>14=&4°35 04P02ÕPÎ8 Ë hÇÝ¶32Ö°Ö°3FÌ½)4°5>Õ?=Ù×>ÕP4=& >a h4k3214kh3O 5 µ¸ · h5ÇÕ& 3D 9325>=& 5\§ ¤)025>ÇkÕ&3[>=?1®Õ&025¹®14=&Öhð02Ú>ÖÛ 4=&5>+0½1k0k=&Ö°=?-k025 Ö°3ÏùGÑ Ó ù Ë Ô a p µ¸· p sHt úú®h5¾Õ&32s35>=?5LF ¤¹02Öh35>=28óô=>=?5>32k=Í=s0 g¼°D°õ&h5>ÏÞ=?Õ?k321 °t 5ôùGÑ »ú;Ò Ú\r Û d @Na @ 025¨ g @N@ 8 î 31;k1=Þ¼h3×>½=?Õ?°35 @ Ä ù · q ·zd» p´¢q$¢ ·¸» p ¢q ·\Ó p´¢Uq [d· p ¢q ¹LÓú p d @Na_ g @N@ Ä ùHÑ E q ËÓ [ p z q$¢ z p z q$¢ z p [ q ¹¸E p ¹ q dÓ »ú p

Optimal attack

Compression attack 10

σ2

σ2

10

5

0

1

2

3

4

0

5

4

3

3

1

1

D*1

2

D** 1

4

0

1

2

3

4

0

5

2

3

4

5

1

2

3

4

5

1

2

3

4

5

Dwf and A**

20

15 10 5 0

1

2

20

D*2 and A*

5

1

2

3

4

5

15 10 5 0

î hÏ×>1= ÓUcs:;3DA0214°325ê32Ý.Ì;0k=?1s021ÎLh5>Ï¾4Û[Hk=& A02102s=k=?1 Ý¶31k=s3°s02Öc0k02ÕPÎü05> k1=?h35¹0k02ÕPÎ8 025>Çk°5>ÏÌ;0k=?14ûAÖhÖ°h5>ÏÍÞ=?Õ?k321®h-ÏhÞ2=&5ÇÚ\Û d µ¸· Ä ù z q Ñ ¢Up z q$¢ z p z q4¢ z p [ q ¹¸E p ¹ q Ód»2ú q Ë = 0 ghÍ×> 02ÕPh=?Þ02Ú>Öh=º1k0=º°5kh½Õ&=?5A0214°3>Ò ùGÑ »úZÒ>°½02>13Bg¼°s0k=?ÖhÛ ¢q Ñ ¢ » Ú>hGÞ2=&Õk31?Ò Ì½hÕP< °Ds321=Çk3×>Ú>Öh=¹=+Õ&02A02Õ?hNÛ 32Ý=7Ì;0=&1s0214Î[h5>ÏüÏ02s=8 ÿN5 î °Ï2×>1=7Ó¼Ò.ÌR= Õ&32sA01=k021k02D=?=&14½Ý¶321.k=¯3°s02Ö90k02ÕPÎÇ025>Ý¶31.=¯Õ&3D>1=?4°3570k02ÕPÎ8 óü=®5>3FÌ Õ&325>h>=&1R3D=½32Ý,khþ9=&14=&5>Õ?=&RÚ9=NÌ;=?=&5)k4=&c02Ö°Ö¼32Ýk=-Õ?3s35>=?5\k¿k31k025>4s¿°5¼Ý¶31s0kh35,Ò02Öh3×>Ï2< kÍÚ=&1'32Ý-Ú>k°5ô07Õ&32s35>=?5L °Õ&3141=?Öð0=&ôÌ½k< k=DÞ201ð05>Õ&=D32Ý.=sÕ?3D935=&5\k?8 Ë =Í=?5>Õ&3[>=?1k=?°Ï25>=&+Ý¶321=ÍÕ&32s>14=&4°325¾04P02ÕPÎ¹°½Þ=?14Û7>hþ9=&14=&5\e8 Ë °®=&5>Õ?3¼>=?1 >Ö°02Õ&=?º3D=Í>h4k3214kh35°5+Õ&3D9325>=&5\k'Ó ÌU»¼Ò,Ú>×¯325>ÖhÛ¹k3Ú3¼324=Þ201ð05>Õ&=Í3Ýc=4=&Ï3Ü k= g[02575>3-k3D=&5¾025\ÛÇ°5¼Ý¶31s0kh35,8 Ë =Ì;0=&14û>Ö°Ö°h5>Ï*04P02ÕPÎ=&1°-Ì.04kh5>ÏD>°Hk31Hk°32535 k=?5L)ù¶°5Ý«02ÕeÒj02Ö°D3Ho [ ¢qÍ úÒj025 =&1?8 î ×>1Hk=?1D31=Ò =*=&5>Õ?3¼=&1>=&4°Ï5>=? Ý¶321¯=*Õ&3D>14=&4°35 04P02ÕPÎ 35>ÖÛs1k025Dhk;°5Ý¶314 0k°325 °5)=ºÕ&32s35>=?5L-Ì½kÚ>×½-°-0Ú>Ö°=3 =?5>Ç0;0'14=&Öð0khÞ2=&ÖÛ*°Ï<s1k0=®h5s°cÕ&3D35>=&5\;°5Õ&=;=?Þ3k=&;s3H;3ÝWhR3FÌ;=?1;k3kh Õ&32s35>=?5L'025Ù=D0k02ÕPÎ2=?1'3¼=?532¯ÕP< >°Hk31Hkh35Ö°=ÝöÝ¶31k=&5\&8/ D31=DÕ&Öh=?Þ=?104P02ÕPÎ=&1 ÌR3×>Öhê>×Í02Ö°Ö¿32Ý½k>°Hk31Hkh35ôh5\k3+Õ?3D935=&5\)ÑÒ%025>ê5>3+934hhÞ= 1k0k=&.ÌR3×>ÖhÚ9=¯0ÕP°=Þ20Ú>Ö°=8 Ë °-= g02sÖ°=h;32s=Ì½Õ&3[>=&1>=?°Ï25>=&¹Ý¶31.k=Õ&32s>14=&GÜ h35ê0P0ÕPÎÙÌ½hÖ°Ö¿×>4= 43s=DÕ&32s35>=?5LÍ02>=?Õ&3FÛ[*025>ô= 14=&H Ý¶3102Õk×A02ÖhÖhÛ¾1k025Dh°5>Ï °5¼Ý¶31s0kh35,8

Ä ÄÂÉFÄ*Å

Ã~Ä6ÎdÄ

Ð_Ñ&Ô/ 8 68 :;3=?5ü025 /'8 02>°32k4ð025¾Ì;0k=?1s021ÎLh5>ÏÇÏ\02D=28 ñ¹6¼×>Ú>Dh4k=&¾3 ä Ï Bã ãeëáf Ð ¯AæPëáZéÒÑ025è8Ó ¢d¢ Ñ28 Ïp"p"pÐ á4â2ãi Ð ÓR Ô Ñ8/ 8 D 6¼×>ÖhÖ°Þ205,Ò ¬ N8 ®¹3×>Ö°h5,Ò.05>ÒÑ8§®ô8 W 4kh5>Ï=&1?Ò*ï4ÿN5Ý¶321s0kh35¸k=?31=k°Õ+025A02ÖÛ[° 32Ý¿Hk=&Ï025>3Ï102>,8 E¼ÒAÓ ¢d¢ Ñ8 Ðº¹2Ô'ø'68 ®¹=&10FÞÒï 5ô1025>>3 Õ&3[>°5Ï¹=?11431¯= g¼9325>=&5\k'32ÝcÌ.0=&14 0214ÎL°5>ÏÇ4Û[Hk=&D&Ò ñ Ïp-p"p ä Ï Bã ãeëá f Ð ¯AæPëáZéÒÞ23ÖT8 ¹qÒ>>è=8 ¹LÓ ¢ Ì¹d· ¢ =Ò ®7021&8Ó ¢ä¢d¢ 8 Ð á4â2ãi Ð¼» Ô ×8L6Lk=&h5LÚ9=?1Ï025>Dø'8 ®¹=&10FÞÒ$ï4ÿN>=?5\khû>Õe0°35*°5Í=½>1=?=?5>Õ&=32Ý9°=-°5Ý¶314 0k°325'Ì½h< 02>>Öh°Õ&0kh35Ík3¯Ì.0k=&14 01ÎL°5>ÏÒ ñ Ïp"p"pÐ á4â2ãiä Ï Bã ãeëá f Ð ¯AæPëáZéÒLÞ3Ö¸8 ¹¸E¼ÒL>,8BÑ ¹Ñ ¢ ÌBÑ ¹\ÓÓ¼Ò ®+0eÛ¹Ó ¢d¢ Ñ8 Ð\Ô/ 8698L:;3=?5,Ò Ï Bã ãeëá Dâ2à ëã Ð ¯AæPëákæà -ç {ãäâ j_é^ä äCë ãÙØ7â2àNæá Dâ25á ã oÕé^äàNæ ä8 ¬ Í< X =&4°?Ò ®7ÿ Ë Ò:.02ÍÚ1°Ï=2Þ Ò ®¹/ ÒÓ ¢d¢ Ñ8 ÐFEÔ 698jÿP@8 lº=&Ö D Ý«025>ê025>m ®ô8%698 ¬ °5Î2=?1&Ò'ïk:;3[>°5Ï+Ý¶31 ÕP025>5>=?Ö;Ì½h3 A021k0s=k=&14&Ò ñ ÓRáPë2( ì j°æ ä Cë ã j ëãAàákë jjâ2Qã ° Ï Bã ãeëá Dâ2à ëã Ð ¯AæPëáéÒÞ23ÖT8 zÒ5>3>8WÑÒ>,8WÑ zÚÌU·¼ÑÒ,Ñ z[ ¢ 8 Ð\[Ô È 89:;=&5Ù05>Ç l8Aó 8óô315=&Ö°ÖÒcï Û×A025\khõe0°35¹°5>= g¹D3¼×>Öð0°3H5 cc/ Õ&Ö°0232Ý¿>143FÞ20Ú>ÖhÛ Ï3[3¼¹s=k3[>½Ý¶31>°ÏP02Ö9Ì.0=&14 0214ÎL°5>Ï 025>+h5Ý¶314 0°35Ç=?ÍÚ9=?>>h5>Ï>Ò ñ Ïp"p"pÈÐ á4â2ãiä

ã ãeëá f Ð ¯AæPëáZéÒÞ32ÖT=8 ¹¸E¼Ò>,8WBÑ ¹LÓ ·ÚÌBÑ ¹d¹d·ÍÒ ®70FÛ7Ó ¢d¢ Ñ8 Ï B Ð\zÔ È Ü8 m®×>Ï4°025¾0214Ú>h1k0214°ÖÛ)Þ0214Û[°5Ï ÕP5=&ÖTÒ ñ Ïp"p"pÐ á4â2ãiä Ï Bã ãeëá f Ð ¯AæPëáéÒ>Þ32ÖT=8 ·ä¹Ò>>,8 zzd»ÌÑ ¢d¢ ·ÒA6¼=?e8WÑ z[[8 Ð_Ñ ¢ R Ô Ñ8º6[3¼=?1Ù025> :8ó kõ&Ï02Ö°ÖÒ j ëã h æ k à«é â2Qã °©Ý$åà ßÞ â2à ëã ã â ã àGæ u æãä ëãä Ï 8 6¼>14°5>Ï=?1HÜ v=&1Ö°02Ï>ÒÑ zqE ¢ 8 Ð_ÑÑ&Ô Ë Q8 ®ô89:;3FÞ=&1®025>» Ñ8/ 8 Ë 32 02?Ò p j°æ æãAà¶äCë ã Ï Bã ãeëá Dâ2à ëã Ð ¯AæPëáZé8Rø=?2 Ì ×c321ÞÎ cÑ2325 ó °Öh=?y Û à 6¼35>?ÒWÑ zzÑ28 Ð_Ñ^ÓÔ :8 î =?T,Ò XÍ=8 ¯×5>>×,Ò025>¯ ¯=8 ºÌR35>ÏÒºï Ë =sÕP3hÕ&= 3Ý;Ì;0k=?1s021ÎÙ>3s02h5 h5ô= >14=&GÜ =&5>Õ?=-32ÝÕ&3D>14=&4°35èÒ ñ¯°y 5 ÓRákëFçPæPûæ ° ã äCë ã-Dà ¯Aæ Ï ãAàNæáãâ2à ëãäâ j j ëBã ãeæákæãèçPæ®ëã Ï Bã ãeëá Dâ2à ëã æÐ P5ç ¯[ãèë j°ûë 2lé k j (ë ° ã sâ2Qã ° j ë åí¼à ã 2Ò>è8 ElzÚÌ7[ä¹ÒÓ ¢d¢ Ñ28

The Gaussian Watermarking Game

The Gaussian Watermarking Game

Fingerprinting, Watermarking and Quantization of Gaussian Data

Fingerprinting, Watermarking and Quantization of Gaussian Data

CAPACITY

Resynchronization methods for audio watermarking

Resynchronization methods for audio watermarking

CAPACITY

Capacity

Digital Watermarking for Digital Media

Digital Watermarking for Digital Media

Sources for Ancient History (Sources of History)

Sources for Ancient History (Sources of History)

Gaussian Processes

Gaussian Processes

Gaussian measures

Gaussian measures

Informed Watermarking

Informed Watermarking

Digital Watermarking

Digital Watermarking

Digital Watermarking

Digital Watermarking

Digital Watermarking

Digital Watermarking

Quantization Watermarking

Quantization Watermarking

Informed Watermarking

Informed Watermarking

Gaussian Hilbert spaces

Gaussian Hilbert spaces

Lectures on Gaussian Processes

Lectures on Gaussian Processes

Fair Evaluation Methods for Image Watermarking Systems

Fair Evaluation Methods for Image Watermarking Systems

Sources

Diminished Capacity

Diminished Capacity

Sources

Sums and Gaussian Vectors

Sums and Gaussian Vectors

Lectures on Gaussian Processes

Lectures on Gaussian Processes

Isoperimetry and Gaussian analysis

Isoperimetry and Gaussian analysis

Watermarking Relational Databases

Watermarking Relational Databases

Multimedia Watermarking Techniques

Multimedia Watermarking Techniques

Generalized Gaussian error calculus

Generalized Gaussian error calculus

Leadership Capacity for Lasting School Improvement

Leadership Capacity for Lasting School Improvement

Building Leadership Capacity for School Improvement

Building Leadership Capacity for School Improvement

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