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ˆðÿB w$›H¡Z§r¡Š•¢wŽZÆ`҃À¡š³¹r³Z­B›0ƒH24@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`€#"ñ ¿`¿ðqâ +Ž ðv ð<,  ð2B+CºDEÁ°FÁŒ€‚ƒ„‰Š‹ŒdŽ•—Á˜™š›œ@¿ÿ ˆ,,ðÿ–Z* 6HZl ~*œZŽ–ºÑŽœ~+l+Z+H6Ñ–––Ž`®0œ~ lZ H60` –Ë û6%H%Z%l~ûœË®–Ž–Ž[\@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`€#"ñ ¿`¿ðáò ¬ ðÆðb ð è‘  ð=,# ð ˆ"ñ¿ð è‘ ðŒ2 ð>, à ðH¿?€ƒ¿ÿ?¿ÿ #"ñ ¿`¿††ð è‘ ð’2 ð?, Ó ðN¿?€ƒŒd¿ÿ?¿ÿ #"ñ ¿`¿††ð с ðŒ2 ð@, à ðH¿?€ƒ¿ÿ?¿ÿ #"ñ ¿`¿††ð7% Žw ð’2 ðA, Ó ðN¿?€ƒŒd¿ÿ?¿ÿ #"ñ ¿`¿††ðR5 ›d ðæ ðB, £ ð<€D˜«‡¿ƒ¿Àÿ "ñ¿ð*°Ðpðà « ðTŸš Click to edit Master title style¢!ª !ðÝ ðC, ƒ ð0€ u«ƒ¿Àÿ "ñ¿ð ` à ðà « ðWŸš#Click to edit Master subtitle style¢$ª $ðì ðD, “ ð6€Ì²K ‡ƒ¿Àÿ #"ñ ¿`¿ð` `€ðà « ðZŸ *¡øªðî ðE, “ ð6€·K ‡ƒ¿Àÿ #"ñ ¿`¿ð`°Ð€ðà  K  ð\Ÿ *¡úªðî ðF, “ ð6€À»K ‡ƒ¿Àÿ #"ñ ¿`¿ð` `€ðà K  ð\Ÿ *¡ØªðB ð, s ð*“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆ‘Љº___PPT10‹ië.™ À œä+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îƒï€ šð’ ðð*ð( ð ððx ð c ð$€ÇK ¿ÿB,ð*°Ðpðà K  ð žðr ð S ð€äÇK ¿ÿC,ðp ` Àðà K  ð žðH ð ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ‘Љº___PPT10‹ië.Jgà @ÀJ+Dñ='ñ ¹ÿÿÿÿ=ñ @Bñ +îôï € #ðÐð 4ð³ ð( ð ð4ð¢ ð4 ƒ ð0€ÈÁ¿ƒ¿Àÿð °Ð 𞟚 H(X) · Shannon-Ent(X)¡d 2  ý]þG ý]þ ý]þ ý]þçÿªðr ð4 S ð€À Á¿ÿC(ð ÿ `nðà  Á ð žðœ¢ ð4 ƒ ð0€ Á¿ƒ¿Àÿð ð  ð<Ÿš*Definition: H(X)žk iff maxx Pr[ X=x ]<2-k¡°+ 2  ÿÿþ  ý]þG ý]þ ý]þ  ý]þ ý]þçÿ  ý]þ ý]þª> ð4¢ ð4 ƒ ð0€Á¿ƒ¿Àÿð à ðg ðԟš( X r.v. over {0,1}n )¡€ 2ý]þ ý]þý]þªðµ¢ ð4 ƒ ð0€ØÁ¿ƒ¿Àÿð,ðàÀ ðUŸš Properties:¡. 2 $ÿÿþ$ð¢ ð4 ƒ ð0€ $Á¿ƒ¿ÀÿðC °Ð°  ð Ÿš H(X)=n iff X~Un¡@ 2  ý]þ ý]þçÿª,ð ¢ ð4 ƒ ð0€ä*Á¿ƒ¿Àÿð³ °Ð  ð­Ÿš H(X,Y) ž H(X) (concatenation) ¡r  2  ý]þG ý]þ ý]þÿÿþ ý]þðŠ¢ ð 4 ƒ ð0€Ì5Á¿ƒ¿Àÿð#°ðv ð*Ÿ † If H(X)žk then 9 (efficient) f s.t. f(X)~eðUk/2 (extraction)¡TD 2   ý]þG ý]þ ý]þ G ý]þG    ý]þ    ý]þ ý]þçÿ ý]þ  ý]þçÿ ÿÿþÿÿþçÿª,!ð•ðl ð€P€  ð4ð€PPPð,$ñD 0ðn" ð 4 ƒ ð0…‡¿ÀÿÿËԔÿ"ñ¿`ð€P€ ðÁ¢ ð 4 ƒ ð0€ÐEÁ¿ƒ¿Àÿð@ü@   ðYŸšOur Objectives:¡. 2$ÿÿþ$ðÚ¢ ð 4 ƒ ð0€˜JÁ¿ƒ¿Àÿð° €À ðrŸšŒ1. Investigate possible defs for computational Min-Entropy. 2. Check whether computational defs satisfy analogs of statistical properties.¡€ 2!    ÿÿþ    N   ª,@,ŠøØø8XðH ð4 ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆyŠqº___PPT10‹Që.Jgà\Øœ+UöñDñ%'ñ ¹ÿÿÿÿ=ñ @Bñ Dñà'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñg'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*4ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*4ÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*4ÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*4ÿÿÿÿÿÿÿÿ+îž ï € \ðT°ðlðìð( ð ðlðr ðl S ð€˜bÁ¿ÿC(ð¯ `}ðà  Á ð žðr ðl S ð€ÄgÁ¿ÿD(ðÀ ðð ðà Á ð žðÙ¢ ðl ƒ ð0€ iÁ¿ƒ¿Àÿð À`Ž ð,$ñ 0 ðEŸš In this talk:¡ 2$ÿÿþðß ðl ƒ ð0€ŒmÁƒ¿Àÿ ð ð`ð,$ñ 0 ðKŸš7Present the 3 variants. Show 2 results + proof sketchesðH ðl ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆŠüº___PPT10‹Üë.âhÃ`"ޚ+~eE™Dñ@'ñ ¹ÿÿÿÿ=ñ @Bñ Dñû'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ2'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*lÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*lÿÿÿÿÿÿÿÿ%ñ(ñ+p+0+lK  ++0+lK  +îï € æ ðÞ àð 8ðv ð( ð ð8ðr ð8 S ð€XvÁ¿ÿC(ð¯ `}ðà  Á ð žð¢ ð8 ƒ ð0€ÈyÁ¿ƒ¿Àÿð`ðÍ 𚟚Def: X is pseudorandom if ¡r 2 ÿÿþ  ý]þ  ÿÿþ   ðŒ¢ ð8 ƒ ð0€ ‚Á¿ƒ¿ÀÿðàÐÐM ð,Ÿ ,maxD2C biasD(X,Un) < eð¡Â 2 ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þª ð–¢ ð8 ƒ ð0€(Á¿ƒ¿Àÿð` ÀÍ ð,$ñ 0 ðŸ rC  class of efficient algorithms (e.g. s-sized circuits)¡t: 2G ý]þ ý]þ  ý]þƒðТ ð8 ƒ ð0€ –Á¿ƒ¿Àÿð@ À°ú ð,$ñ 0 ð<Ÿš$biasD(X,Y) = | EX[D(X)] - EY[D(Y)] |¡â% 2 ý]þ ý]þçÿ ý]þ  ý]þ(ý]þ ý]þ ý]þçÿ  ý]þ ý]þçÿ ý]þ(ý]þ‡(ý]þª ðT¢ ð8 ƒ ð0€4£Á¿ƒ¿Àÿðð°‡ð,$ñ 0 ðÀŸ ` eð  parameter (in this talk: some constant > 0)¡D1 2 ‡ ý]þ  "ðN¢ ð 8 ƒ ð0€Ô©Á¿ƒ¿Àÿðð@]ð,$ñ 0 𺟚4i.e., X is computationally indistinguishable from Un¡j5 2 ý]þ+ý]þý]þçÿŽý]þçÿðH ð8 ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆÞ ŠÖ º___PPT10‹¶ ë._gàA@¢+q•q›Dñª 'ñ ¹ÿÿÿÿ=ñ @Bñ Dñe 'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ)'ñ =ñ%ñ(ñDñÑ'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* 8ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ)'ñ =ñ%ñ(ñDñÑ'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*8ÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*8ÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*8ÿÿÿÿÿÿÿÿ%ñ(ñ+à+0+8Á ++0+8Á ++0+8Á ++0+ 8Á +î Fï € àðØðð<ñð<ð( ð ð<ðx ð< c ð$€°ŸÁ¿ÿC(ˆðÒÿ ` ðà  Á ð žðô¢ ð< ƒ ð0€ ÀÁ¿ƒ¿Àÿðà € 𔟚*X is pseudorandom if ¡b 2 ÿÿþ ý]þ  ÿÿþ   ðŒ¢ ð< ƒ ð0€xÈÁ¿ƒ¿Àÿð° Pm ð,Ÿ ,maxD2C biasD(X,Un) < eð¡Â 2 ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þª ðRB ð< s ð*D¿ÀÿÿËԔÿð€ð—¢ ð < ƒ ð0€HÓÁ¿ƒ¿ÀÿðpÀ Ýð,$ñ 0 ðŸšDef 1 [HILL]: HHILL(X)žk if¡ª 2 ÿÿþ ÿÿþ ý]þ ý]þ ý]þG ý]þ ý]þ   ª ðF¢ ð < ƒ ð0€šàÁ¿ƒ¿ÀÿðÓ@À@ð,$ñ 0 𲟠R9Y s.t. H(Y)ž k and maxD2C biasD(X,Y) < eð¡* 2G ý]þ ý]þ  ý]þG ý]þ ý]þ  ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þª, ð¢ ð < ƒ ð0€íÁ¿ƒ¿Àÿðó@À`ð,$ñ 0 ðtŸ @minH(Y)ž K maxD2C biasD(X,Y) < eð¡ì! 2 ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þª$ ð†B ð < s ð*D¿ÀÿËԔÿð€à€ð,$ñD  0ðe¢ ð < ƒ ð0€ ¿ƒ¿Àÿð£À ð,$ñ  0 ðџšDef 2: HMet(X)žk if¡€ 2 ÿÿþ ý]þ ý]þ ý]þG ý]þ ý]þ ª ð¢ ð< ƒ ð0€€ùÁ¿ƒ¿Àÿðó@À` ð,$ñ  0 ð|Ÿ @maxD2C minH(Y)ž K biasD(X,Y) < eð¡ì! 2 ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þª, ð¢ ð< ƒ ð0€Ø ¿ƒ¿Àÿð@ Àðà ð,$ñ 0 ðmŸšIDef 3 [Yao]: HYao(X)žk if X cannot be efficiently compressed to k-1 bits.¡ÔJ 2 ÿÿþ ÿÿþ ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ$  ý]þ ª, 4ð†B ð< s ð*D¿ÀÿÿËԔÿðã€ãð,$ñD  0ð†B ð< s ð*D¿ÀÿÿËԔÿð€ €€ ð,$ñD 0ðëðl ðÀÀ À  ð<ðÀÀ À ð,$ñD 0ðÄâ ð< ³ ðB€X  ¿G&HáÏÿÿ¿ÀÿÿË|ÿ "ñ¿`ðÀÀ À  ð<Ÿ¡ª 𣢠ð< ƒ ð0€`" ¿ƒ¿Àÿð ð  ð;ŸšOi.e., X is computationally indist. from some Y with žk statistical min-entropy.¡®P 2BF ý]þ!BFÿÿþBF ý]þBG ý]þF ý]þBª/ðrðl ð€ € ð<ð €€ð,$ñD 0ðÊâ ð< à ðH€”/ ¿G)HáÏÿÿ¿ÀÿÿË|ÿ ˆ"ñ¿`ð€ ð ð<Ÿ¡ª ð$¢ ð< “ ð6€Ð4 ¿ƒ¿Àÿˆð€@ €à  ð¶Ÿšhi.e., 8 efficient D, X is computationally indist. by D from some Y=Y(D) with žk statistical min-entropy.¡i 2BG ý]þ BF ý]þBF ý]þBF ý]þBFÿÿþBF ý]þBG ý]þF ý]þBª*9ðH ð< ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?/ð ð<ð<ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆè-Šà-º___PPT10‹À-ë._gàA@¢+ÏÚj—DñÔ+'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ+'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ2'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñg'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*<ÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*<ÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñÙ'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ)'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñçDñØ'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ7'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(left)*ñ<3ñ<ñû* <ÿÿÿÿÿÿÿÿDñé'ñ =ñ%ñ(ñôDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñDñ¹'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñ6'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû* <ÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñÏDñ)'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû* <ÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñÏDñš'ñ =ñ%ñ(ñÐDñB'ñ =ñA@BñBñ@ Bñ°Bñ0Bñ%ñ(ñ)ñ?)ñ?Dñ‰'ñ µè=ñ.ñQ7ñ ØÈBÈBÈBÈBBñcM -3.33333E-6 4.16185E-6 L -3.33333E-6 -0.07538 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€BñBñ–^œ<ñû* <ÿÿÿÿÿÿÿÿDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* <ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñg'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*<ÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*<ÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñ®'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ)'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñçDñÜ'ñ =ñ%ñ(ñèDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñèDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñ+À+0+ <  ++0+ <  ++0+ <  ++0+ <  ++0+ <  ++0+ <  ++0+<  ++0+<  +îžï € Žð†ð @ðð( ð ð@ðx ð@ c ð$€P[ ¿ÿC(ˆðÒÿ ` ðà    ð žð‡¢ ð@ ƒ ð0€0] ¿ƒ¿ÀÿðÐ Àpð,$ñ 0 ðóŸšAClaim 2: For k=n all 3 defs equivalent to pseudorandomness.¡bB 2  ÿÿþ  ÿÿþ ý]þ1   ª,ðRB ð@ s ð*D¿ÀÿÿËԔÿð € ðz¢ ð@ ƒ ð0€|i ¿ƒ¿Àÿð ÀÀ  ðŸ bHHILL(X)žk if minH(Y)ž K maxD2C biasD(X,Y) < eð¡^2 2 ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þ  ª6  ðv¢ ð @ ƒ ð0€@{ ¿ƒ¿Àÿð“ð° ðŸ `HMet(X)žk if maxD2C minH(Y)ž K biasD(X,Y) < eð¡\1 2 ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þ ª6  ðÌ¢ ð @ ƒ ð0€(Š ¿ƒ¿ÀÿðÀý ðlŸ vHYao(X)žk if X can t be efficiently compressed to k-1 bits.¡À< 2 ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ"  ý]þ ª 3ðRB ð @ s ð*D¿ÀÿÿËԔÿð€€€ðRB ð@ s ð*D¿ÀÿÿËԔÿð`€`ðRB ð@ s ð*D¿ÀÿÿËԔÿðR€Rð¢ ð@ ƒ ð0€ðš ¿ƒ¿Àÿð Àý ð,$ñ 0 ð€Ÿš,Claim 1: H(X) · HHILL(X) · HMet(X) · HYao(X)¡- 2  ÿÿþ ý]þG ý]þ ý]þ ý]þ ý]þG ý]þ ý]þ ý]þ ý]þG ý]þ ý]þ ý]þ ý]þª,ð¢ ð@ ƒ ð0€<Š ¿ƒ¿ÀÿðÐÀp=ð,$ñ 0 ð럚5Claim 3: All 3 defs satisfy extraction property.[Tre]¡f6 2  ÿÿþ  ÿÿþ  ý]þý]þª, ðH ð@ ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆÒ ŠÊ º___PPT10‹ª ë._gàA@¢+a3 lDñÖ 'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ‘ 'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñè'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ8'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*@ÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*@ÿÿÿÿÿÿÿÿDñè'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ8'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*@ÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*@ÿÿÿÿÿÿÿÿDñè'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ8'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*@ÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*@ÿÿÿÿÿÿÿÿ+š+0+@  ++0+@  ++0+@  +îÎXï € Àðžð&D ñð(ð( ð ðDðr ðD S ð€š¹ ¿ÿC(ðÒÿ ` ðà    ð žðv¢ ðD ƒ ð0€ÄŸ ¿ƒ¿ÀÿðP ÀÀœ ðŸ ^HHILL(X)žk if minH(Y)žK maxD2C biasD(X,Y) < eð¡^0 2 ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þ  ª6   ðr¢ ðD ƒ ð0€Ñ ¿ƒ¿ÀÿðCð°° ðŸ \HMet(X)žk if maxD2C minH(Y)žK biasD(X,Y) < eð¡\/ 2 ý]þ ý]þ ý]þG ý]þ ý]þ  ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿO ý]þçÿN ý]þçÿ ý]þ ý]þçÿ ý]þ‡ ý]þ ª6   ðRB ðD s ð*D¿ÀÿÿËԔÿð0€0ðRB ðD s ð*D¿ÀÿÿËԔÿð € ðu¢ ð D ƒ ð0€à ¿ƒ¿Àÿð 0  ð,$ñ 0 ð៚Thm 1: HHILL(X) = HMet(X)¡€ 2 ÿÿþ ý]þ ý]þ ý]þ ý]þ ý]þª$ð\¢ ð D ƒ ð0€ìé ¿ƒ¿Àÿðp` €Äð,$ñÜ 0 ðȟ J(For C = poly-sized circuits, any eð)¡b& 2ý]þ‡ý]þðF¢ ð D ƒ ð0€äñ ¿ƒ¿Àÿðc0ÀÐð,$ñ 0 𲟚Proof: Suppose HHILL(X)ñ+Bñ#style.visibility<ñû* Dÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñÜDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Dÿÿÿÿÿÿÿÿ%ñ(ñDñ4'ñ ·=ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Dÿÿÿÿÿÿÿÿ%ñ(ñDñ  'ñ ·=ñ%ñ(ñÿÿÿÿDñH 'ñ =ñ%ñ(ñDñt'ñ =ñA@BñBñ7 Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*DÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*DÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñt'ñ =ñA@BñBñ7 Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*DÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*DÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñ2 'ñ ·=ñ%ñ(ñÿÿÿÿDñÌ'ñ =ñ%ñ(ñDñt'ñ =ñA@BñBñ7 Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*DÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*DÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñ'ñ =ñ%ñ(ñèDñ7'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(left)*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñ‹'ñ =ñ%ñ(ñÜDñ3'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ§'ñ ô=ñ-ño6ñBñwipe(up)*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñÜ'ñ =ñ%ñ(ñÐDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ› 'ñ ·=ñ%ñ(ñÿÿÿÿDñ‹'ñ =ñ%ñ(ñDñ3'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ§'ñ ô=ñ-ño6ñBñwipe(up)*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñ¿'ñ =ñ%ñ(ñôDñ)'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñÏDñ6'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñÏDñé'ñ =ñ%ñ(ñÄ Dñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñg'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*DÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*DÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñÁ'ñ ·=ñ%ñ(ñÿÿÿÿDñ'ñ =ñ%ñ(ñDñ7'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*"Dÿÿÿÿÿÿÿÿ%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(left)*ñ<3ñ<ñû*"DÿÿÿÿÿÿÿÿDñ¿'ñ =ñ%ñ(ñôDñ6'ñ =ñA@BñBñ  Bñ°Bñ0Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*DÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñÏDñ)'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*"DÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*"Dÿÿÿÿÿÿÿÿ%ñ(ñÏDñé'ñ =ñ%ñ(ñÄ Dñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Dÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ =ñ%ñ(ñÄ Dñ7'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Dÿÿÿÿÿÿÿÿ%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(left)*ñ<3ñ<ñû* DÿÿÿÿÿÿÿÿDñƒ'ñ =ñ%ñ(ñž Dñ+'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*&Dÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*&Dÿÿÿÿÿÿÿÿ+0+0+ D  ++0+ D  ++0+ D  ++0+D  ++0+D  ++0+D  ++0+D  ++0+D  ++0+D  ++0+D  +îï € ºð² ð HðJð( ð ðHðr ðH S ð€LO ¿ÿC(ðÒÿ ` ðà    ð žð¢ ðH ƒ ð0€,P ¿ƒ¿Àÿð`Àð,$ñ 0 ðüŸ ŠThm [Yao]: If X is unpredicatble with adv. eð then X is pseudorandom w/ param eð =n¢eð¡êT 2 ÿÿþ ÿÿþ  ý]þ ‡ ý]þ  ý]þ ‡ ý]þ ý]þG ý]þÇ ý]þªH '𞢠ðH ƒ ð0€ì_ ¿ƒ¿ÀÿðÀÀ` ð,$ñ 0 ð$Ÿ šLoss of factor of n due to hybrid argument  useless for constant advantage eð¡nN 2  ý]þ  ÿÿþ‡ ÿÿþÇ ÿÿþð¢ ðH ƒ ð0€4h ¿ƒ¿Àÿðð ÀPj ð,$ñ 0 ð럚aThis loss can be crucial for some applications (e.g., extractors, derandomizing small-space algs)¡:b 2/ 2Ǐý]þª,B  ð¹ðl ðàP ð0 ð Hð€ àð`ð,$ñD 0ðn" ðH ƒ ð0…‡¿ÀÿÿËԔÿ"ñ¿`ðàP ð0ðÇ¢ ðH ƒ ð0€Ðn ¿ƒ¿Àÿð € í ð_ŸšCan we do better?¡2 2 ÿÿþÇ ÿÿþðH ðH ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆüŠôº___PPT10‹Ôë.ZhÃ`ÓQò+*çDñ'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ» 'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ)'ñ =ñ%ñ(ñDñÑ'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Hÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ)'ñ =ñ%ñ(ñDñÑ'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Hÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ)'ñ =ñ%ñ(ñDñÑ'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Hÿÿÿÿÿÿÿÿ%ñ(ñDñW'ñ ·=ñ%ñ(ñÿÿÿÿDñÿ'ñ =ñ%ñ(ñDñ§'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñ)ñ)ñ)ñ)ñ€?Dñø'ñ µ=ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Hÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ µè=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû* HÿÿÿÿÿÿÿÿDñ'ñ µè=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû* HÿÿÿÿÿÿÿÿDñŸ'ñ µè=ñ-ñg6ñBñ fade*ñ<3ñ<ñû* Hÿÿÿÿÿÿÿÿ+š+0+H  ++0+H  ++0+H  +îÏï € ð`ð Xð©ð( ð ðXðx ðX c ð$€ìoK ¿ÿC(ˆðÒÿ ` ðà  K  ð žðè¢ ðX ƒ ð0€„qK ¿ƒ¿ÀÿðÐ` pð,$ñ 0 ðTŸ ’IT Fact [TZS]: If X is IT-unpredictable with const. adv. then H(X)=Wð(n) ¡ŠJ 2 ÿÿþ ÿÿþ  ý]þ,  ý]þ‡ ý]þ ý]þÃð ¢ ðX ƒ ð0€0}K ¿ƒ¿Àÿð`` Íð,$ñ 0 ðwŸš)We obtain the following imperfect analog:¡2* 2) Ç ðJ¢ ð X ƒ ð0€dK ¿ƒ¿Àÿðp`@ ð,$ñ 0 ð¶Ÿ ªThm 2: If X is unpredictable by SAT-gate circuits with const. adv. then HMet(X)=Wð(n) ¡ÄV 2 ÿÿþ  ý]þ  ÿÿþ  ý]þ ý]þ ý]þ‡ ý]þ ý]þª$E𜢠ð X ƒ ð0€ðK ¿ƒ¿Àÿðð Ðð,$ñ 0 ðŸšLIn paper: A variant of Thm 2 for nonuniform online logspace.¡ZM 2  ÿÿþ(  ÿÿþ Ç ý]þª> ðH ðX ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆ^ ŠV º___PPT10‹6 ë.ZhÃ`ÓQò+f(ZoDñ* 'ñ ¹ÿÿÿÿ=ñ @Bñ Dñå 'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Xÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Xÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Xÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Xÿÿÿÿÿÿÿÿ%ñ(ñó+à+0+XK  ++0+XK  ++0+ XK  ++0+ XK  +îŠ(ï € ðpð\ð”ð( ð ð\ð¢ ð\ ƒ ð0€Ô€ ¿ƒ¿Àÿð``@ ð¶Ÿ ªThm 2: If X is unpredictable by SAT-gate circuits with const. adv. then HMet(X)=Wð(n) ¡ÄV 2 ÿÿþ  ý]þ  ÿÿþ  ý]þ ý]þ ý]þ‡ ý]þ ý]þª$EðJ¢ ð\ ƒ ð0€4ˆ ¿ƒ¿Àÿð°`@Pð,$ñ 0 ð¶Ÿ ¢Proof: Suppose that HMet(X)<dðn We ll construct a SAT-gate predictor P s.t.¡ÄR 2 ÿÿþ   ý]þ ý]þ ý]þ‡ ý]þ  ý]þ%ý]þ ý]þª,3ð¢ ð\ ƒ ð0€• ¿ƒ¿ÀÿðP`p ð,$ñ 0 ðnŸ @Pri,X[ P(X1,& ,Xi-1)=Xi ] = 1  dð¡ø! 2 ý]þ ý]þçÿ(ý]þ ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ(ý]þ ý]þ‡ ý]þ ý]þªðV¢ ð\ ƒ ð0€Š ¿ƒ¿Àÿð À ð,$ñ 0 ðŸ TWe have that maxD2CminH(Y)ždðn biasD(X,Y)žeð¡0+ 2  ý]þ ý]þçÿO ý]þçÿO ý]þçÿ ý]þ ý]þçÿO ý]þçÿÏ ý]þçÿN ý]þçÿ ý]þ ý]þçÿ ý]þG ý]þ‡ ý]þ ª ð†B ð\ s ð*D¿ÀÿÿËԔÿð Àp ð,$ñD 0𚢠ð \ ƒ ð0€àµ ¿ƒ¿Àÿð Àm ð,$ñ 0 ðŸ \i.e., 9D s.t. 8Y If H(Y)ždðn then biasD(X,Y)ž eð¡V/ 2G ý]þ ý]þ  ý]þG ý]þ ý]þ  ý]þG ý]þ‡ ý]þ ý]þ  ý]þ ý]þçÿ ý]þG ý]þ ý]þ‡ ý]þƒ ª>  ð ¢ ð \ ƒ ð0€Æ ¿ƒ¿Àÿðà Àà€ð,$ñ 0 ðvŸ nAssume: 1) |D-1(1)| < 2dðn *2) PrX[ D(X)=1 ] = 1¡Ê8 2 ÿÿþ ý]þ ý]þ ý]þ ý]þŽ ý]þ ý]þ ÿÿþ ý]þ ý]þçÿ ý]þª%ð4ðl ð Ѐ°  ð\ðp €P ð,$ñD 0ð¶â ð \ ³ ðB€àÓ ¿GžH*yƒ¿Àÿ ð Ѐ°  ð<Ÿ¡ª ðúðN ðÀPp  ð \ ðˆð° ð`À ðn ð \ ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ðÀPp ðn2 ð\ ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ð À° ð ðn2 ð\ ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ðð€ð  ðŒ¢ ð\ ƒ ð0€üÖ ¿ƒ¿ÀÿðÀcÀÐ ðTŸš{0,1}n¡2 2F ý]þN ý]þð·¢ ð\ ƒ ð0€pÛ ¿ƒ¿Àÿðà  ðOŸšD¡2 2F ý]þN ý]þð·¢ ð\ ƒ ð0€ôß ¿ƒ¿ÀÿðPà M  ðOŸšX¡2 2F ý]þN ý]þðH ð\ ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ðð \ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆ2Š*º___PPT10‹ ë.`hÃPÜÿ†+µâN†DñÆ'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ2'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*\ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*\ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*\ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*\ÿÿÿÿÿÿÿÿ%ñ(ñDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* \ÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* \ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ ·=ñ%ñ(ñÿÿÿÿDñ¿'ñ =ñ%ñ(ñDñg'ñ =ñ4@BñBñ7 Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*\ÿÿÿÿÿÿÿÿ%ñ(ñDñ'ñ è=ñ+ñã4ñ 8?ñfCñBñ#ppt_w*0.70BñCñèBñ#ppt_wBñ*ñY3ñ>ñBñ ppt_w<ñû*\ÿÿÿÿÿÿÿÿDñ'ñ è=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_hBñCñèBñ#ppt_hBñ*ñY3ñ>ñBñ ppt_h<ñû*\ÿÿÿÿÿÿÿÿDñŸ'ñ è=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*\ÿÿÿÿÿÿÿÿ++0+\  ++0+\  ++0+\  ++0+ \  ++0+ \  +î|ï € G ð? €ð `ñðà ð( ð ð`𔢠ð ` ƒ ð0€\Í ¿ƒ¿Àÿð€À ð,$ñ 0 🠠Define predictor P as follows: P(x1,& ,xi)=0 iff Pr[ D(x1,& ,xi,0,Un-i-1)=1] > œ ¡"Q 2  ý]þ   ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ  ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ ý]þçÿ  ý]þ ª-!ð†¢ ð` ƒ ð0€`× ¿ƒ¿Àÿðpà  ð&Ÿ N1) |D-1(1)| < 2dðn 2) PrX[ D(X)=1 ] = 1¡š( 2 ý]þ ý]þ ý]þ ý]þŽ ý]þ ý]þ ý]þçÿ ý]þªðøðL ðÀPp  ð`# ð ˆð@@ ðððn ð` ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ðÀPp ðn2 ð` ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ð À° ð ðn2 ð` ƒ ð0…‡¿ÀÿÿË|ÿ"ñ¿`ðð€ð  ðŒ¢ ð` ƒ ð0€à ¿ƒ¿ÀÿðÀcÀ' ðTŸš{0,1}n¡2 2F ý]þN ý]þð·¢ ð` ƒ ð0€Üä ¿ƒ¿Àÿðà å ðOŸšD¡2 2F ý]þN ý]þð·¢ ð` ƒ ð0€„é ¿ƒ¿ÀÿðPß €  ðOŸšX¡2 2F ý]þN ý]þðœ ð` à ðH€ ë ‡¿ƒ¿Àÿ C(ˆð ÿÐnðà   ð žð™¢ ð` ƒ ð0€ã ¿ƒ¿Àÿðó À “ ð,$ñ 0 ðŸš]Note that P does not depend on X and can be constructed w/ NP oracle. (approx counting [JVV])¡Œ^ 2   ý]þ  ÿÿþ   ý]þ& ð¢ ð` ƒ ð0€q ¿ƒ¿ÀÿðSÀ€Àð,$ñ 0 ðˆŸ hClaim: 8x2D, P predicts at least (1-dð)n indices of x¡5 2 ÿÿþG ý]þ ý]þG ý]þ ý]þ  ý]þ  ý]þ‡ ý]þ ý]þ   ý]þ$ý]þðH ð` ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ðð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆÝŠÕº___PPT10‹µë.ghÃð.œM+OA°xDñá'ñ ¹ÿÿÿÿ=ñ @Bñ Dñœ'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* `ÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*`ÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*`ÿÿÿÿÿÿÿÿ%ñ(ñ+š+0+ `q  ++0+`q  ++0+`q  +î=ï € ð ð#'dð€ð( ð ðdðè¢ ðd ƒ ð0€”q ¿ƒ¿Àÿð` ý ðˆŸ hClaim: 8x2D, P predicts at least (1-dð)n indices of x¡5 2 ÿÿþG ý]þ ý]þG ý]þ ý]þ  ý]þ   ý]þ‡ ý]þ ý]þ  ý]þ$ý]þðì¢ ðd ƒ ð0€l!q ¿ƒ¿Àÿð0Ð ðŒŸ `P(x1,& ,xi)=0 iff Pr[ D(x1,& ,xi,0,Un-i-1)=1] > œ ¡î1 2 ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ  ý]þ ý]þçÿ ý]þ ý]þçÿ ý]þ ý]þçÿ  ý]þ ª !ðRB ðd s ð*D¿ÀÿÿËԔÿð00 0ðÌðl ð¢Õà  ðdð`m  ° ð,$ñD 0ðfB ðd “ ð6D¿ÀÀÀÀËԔÐÑÿð0 ` À à ðfB ðd “ ð6D¿ÀÀÀÀËԔÐÑÿð0 ÐP à ðfB ðd‚ “ ð6D¿ÀÀÀÀËԔÐÑÿð °   ` ðfB ðd‚ “ ð6D¿ÀÀÀÀËԔÐÑÿð` ð° ðfB ðd‚ “ ð6D¿ÀÀÀÀËԔÐÑÿðP@ ððfB ð d “ ð6D¿ÀÀÀÀËԔÐÑÿð0 0ÐðfB ðdB “ ð6D¿ÀÿÿËԔÐÑÿð 0 @ðfB ðd “ ð6D¿ÀÿÿËԔÐÑÿð @° ððfB ð d “ ð6D¿ÀÿÿËԔÐÑÿð° ð0 ÐðfB ð dB “ ð6D¿ÀÿÿËԔÐÑÿð  Ð0 ° ðhR ð d s ð*…‡ÀÀÀ¿Àÿ"ñ¿`ð…£ÕÓðfB ð d “ ð6D¿ÀÿÿËԔÐÑÿð  ° 0 ` ðhR ðd s ð*…‡ÀÀÀ¿Àÿ"ñ¿`ð¢Ãòó ðhR ðd s ð*…‡ÀÀÀ¿Àÿ"ñ¿`ð²ú L ðhR ðd s ð*…‡ÀÀÀ¿Àÿ"ñ¿`ðÛ × Ë W ðfB ðdB “ ð6D¿ÀÿÿËԔÐÑÿðÐ ` 0 à ðhR ðd s ð*…‡ÀÀÀ¿Àÿ"ñ¿`ðž W p P ð†¢ ðd ƒ ð0€ˆ0q ¿ƒ¿Àÿð#` Ãð,$ñ 0 ðòŸš2Proof: Suppose P fails to predict x in m indices. ¡€3 2 ÿÿþ  ý]þ  ý]þ ý]þ (ý]þð„¢ ðd ƒ ð0€¬2m, obtaining a contradiction.¡p3 2 ý]þ ý]þ  ý]þðš2 ðd ƒ ð0…‡¿ÀÿË8cÿ"ñ¿`ðÐ €p ð,$ñD 0ðš2 ðd ƒ ð0…‡¿ÀÿË8cÿ"ñ¿`ð@€p ð,$ñ@ 0ðš2 ðd ƒ ð0…‡¿ÀÿË8cÿ"ñ¿`ð°P @pð,$ñ@ 0ðÛ¢ ð d ƒ ð0€Fq ¿ƒ¿Àÿð) pPpð,$ñ 0 ðGŸš1¡* 2ÿ3þÿ3þðð¢ ð!d ƒ ð0€üJq ¿ƒ¿Àÿð) àp ð,$ñ  0 ð\Ÿšž2¡> 2Gÿ3þÿ3þÿ3þðð¢ ð"d ƒ ð0€lPq ¿ƒ¿Àÿðp @ · ð,$ñ  0 ð\Ÿšž2¡> 2Gÿ3þÿ3þÿ3þðð¢ ð#d ƒ ð0€pUq ¿ƒ¿ÀÿðépP0 ð,$ñ  0 ð\Ÿšž4¡> 2Gÿ3þÿ3þÿ3þðð¢ ð$d ƒ ð0€tZq ¿ƒ¿Àÿð0 à wð,$ñ  0 ð\Ÿšž4¡> 2Gÿ3þÿ3þÿ3þðð¢ ð%d ƒ ð0€x_q ¿ƒ¿ÀÿðYð ð,$ñ  0 ð\Ÿšž8¡> 2Gÿ3þÿ3þÿ3þð¢ ð&d ƒ ð0€|dq ¿ƒ¿Àÿð Pçð,$ñ 0 ðoŸšž2m¡P 2Gÿ3þÿ3þÿ3þ ÿ3þð” ð'd s ð*…‡¿Àÿ"ñ¿`ð€  ð ð,$ñD 0ðH ðd ƒ ð0ƒ“ŽŸ‹”Þœh¿ÿ ?ð h®ÿÿÿŠèÌìÿ™™ˆä™ÿ™¯áÿˆ€#Šœ#º___PPT10‹|#ë.khÃo¬+^H†¶DñX!'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ!'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñ4'ñ ·=ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@BñBñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ+'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*dÿÿÿÿÿÿÿÿDñ+'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*dÿÿÿÿÿÿÿÿDñ+'ñ =ñ4@BñBñ  Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*dÿÿÿÿÿÿÿÿ%ñ(ñDñŸ'ñ Ð=ñ-ñg6ñBñ fade*ñ<3ñ<ñû*dÿÿÿÿÿÿÿÿDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*"dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*!dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*#dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*$dÿÿÿÿÿÿÿÿ%ñ(ñDñA'ñ ·=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@BñBñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ 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