ࡱ> F=*%AOJFIF``6ExifMM*bj(1r2i``Adobe Photoshop 7.02004:04:23 17:16:18(&HHJFIFHH Adobe_CMAdobed            `"?   3!1AQa"q2B#$Rb34rC%Scs5&DTdE£t6UeuF'Vfv7GWgw5!1AQaq"2B#R3$brCScs4%&5DTdEU6teuFVfv'7GWgw ?:gp@je9NßI)d %)MERIJI$RI$I%)$IOHS%䒘Ɖ$J\&;w$3))I$I$$I)I$JRI$$$%-:IRE:J]$IJI$RI$~O󴓟'%)$rJY<ޟl}.w5RL RI8?wO8KSyvRJ@hۥ.RSPRLRRBt.t߿u%,O (;R2yLHIJ&<QfJRRI$42p?ڒ?/44`Ǐ3<3w'G"IIKngT~ I%)$IKO?NRR I%)$|E}O;X=ht7sI%)$IO(,Photoshop 3.08BIM%8BIM com.apple.print.PageFormat.PMHorizontalRes com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMHorizontalRes 72 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PageFormat.PMOrientation com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMOrientation 1 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PageFormat.PMScaling com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMScaling 1 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PageFormat.PMVerticalRes com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMVerticalRes 72 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PageFormat.PMVerticalScaling com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMVerticalScaling 1 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.subTicket.paper_info_ticket com.apple.print.PageFormat.PMAdjustedPageRect com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMAdjustedPageRect 0.0 0.0 734 576 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T21:15:26Z com.apple.print.ticket.stateFlag 0 com.apple.print.PageFormat.PMAdjustedPaperRect com.apple.print.ticket.creator com.apple.printingmanager com.apple.print.ticket.itemArray com.apple.print.PageFormat.PMAdjustedPaperRect -18 -18 774 594 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T21:15:26Z com.apple.print.ticket.stateFlag 0 com.apple.print.PaperInfo.PMPaperName com.apple.print.ticket.creator com.apple.print.pm.PostScript com.apple.print.ticket.itemArray com.apple.print.PaperInfo.PMPaperName na-letter com.apple.print.ticket.client com.apple.print.pm.PostScript com.apple.print.ticket.modDate 2003-07-01T17:49:36Z com.apple.print.ticket.stateFlag 1 com.apple.print.PaperInfo.PMUnadjustedPageRect com.apple.print.ticket.creator com.apple.print.pm.PostScript com.apple.print.ticket.itemArray com.apple.print.PaperInfo.PMUnadjustedPageRect 0.0 0.0 734 576 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PaperInfo.PMUnadjustedPaperRect com.apple.print.ticket.creator com.apple.print.pm.PostScript com.apple.print.ticket.itemArray com.apple.print.PaperInfo.PMUnadjustedPaperRect -18 -18 774 594 com.apple.print.ticket.client com.apple.printingmanager com.apple.print.ticket.modDate 2004-04-23T19:54:21Z com.apple.print.ticket.stateFlag 0 com.apple.print.PaperInfo.ppd.PMPaperName com.apple.print.ticket.creator com.apple.print.pm.PostScript com.apple.print.ticket.itemArray com.apple.print.PaperInfo.ppd.PMPaperName US Letter com.apple.print.ticket.client com.apple.print.pm.PostScript com.apple.print.ticket.modDate 2003-07-01T17:49:36Z com.apple.print.ticket.stateFlag 1 com.apple.print.ticket.APIVersion 00.20 com.apple.print.ticket.privateLock com.apple.print.ticket.type com.apple.print.PaperInfoTicket com.apple.print.ticket.APIVersion 00.20 com.apple.print.ticket.privateLock com.apple.print.ticket.type com.apple.print.PageFormatTicket 8BIMxHH@Rg(HH(dh 8BIM``8BIM&?8BIM x8BIM8BIM 8BIM 8BIM' 8BIMH/fflff/ff2Z5-8BIMp8BIM 8BIM8BIM@@8BIM8BIM=try2nullboundsObjcRct1Top longLeftlongBtomlongRghtlongslicesVlLsObjcslicesliceIDlonggroupIDlongoriginenum ESliceOrigin autoGeneratedTypeenum ESliceTypeImg boundsObjcRct1Top longLeftlongBtomlongRghtlongurlTEXTnullTEXTMsgeTEXTaltTagTEXTcellTextIsHTMLboolcellTextTEXT horzAlignenumESliceHorzAligndefault vertAlignenumESliceVertAligndefault bgColorTypeenumESliceBGColorTypeNone topOutsetlong leftOutsetlong bottomOutsetlong rightOutsetlong8BIM8BIM $`JFIFHH Adobe_CMAdobed            `"?   3!1AQa"q2B#$Rb34rC%Scs5&DTdE£t6UeuF'Vfv7GWgw5!1AQaq"2B#R3$brCScs4%&5DTdEU6teuFVfv'7GWgw ?:gp@je9NßI)d %)MERIJI$RI$I%)$IOHS%䒘Ɖ$J\&;w$3))I$I$$I)I$JRI$$$%-:IRE:J]$IJI$RI$~O󴓟'%)$rJY<ޟl}.w5RL RI8?wO8KSyvRJ@hۥ.RSPRLRRBt.t߿u%,O (;R2yLHIJ&<QfJRRI$42p?ڒ?/44`Ǐ3<3w'G"IIKngT~ I%)$IKO?NRR I%)$|E}O;X=ht7sI%)$IO8BIM!UAdobe PhotoshopAdobe Photoshop 7.08BIMHhttp://ns.adobe.com/xap/1.0/ adobe:docid:photoshop:17faae6c-96db-11d8-8a98-d1ec14d5eb44 XICC_PROFILEHADBEmntrRGB XYZ acspMSFTnone-ADBE cprtLdescHkwtptbkptrTRCgTRCbTRCrXYZ gXYZ bXYZ4textCopyright (c) 1999 Adobe Systems Incorporated. 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Because progs are hard to understand (bugs,HALTING,& ) Maybe compiler is already obfuscator? (e.g.,  closed source considered unreadable) Because in crypto we can do anything :) Some commercial candidates. X%&-I,   WHY SHOULD WE CARE?PInteresting in its own right. Constructing OWF-based PK crypto [DH76] (Arguably central problem of crypto.) Software protection. Digital rights management (DRM) & z@$: MAIN RESULT (informal)LThm [BGI+01]: General-purpose obfs, even under very weak defs, do not exist.^M v4  $ 6  DEFINING OBFs FDef: O:PP  totally fails on P if $DG F G   $ $*  TASTE OF PROOF $ $*  TASTE OF PROOF   '#MEANING OF RESULT&Proved: No general-purpose obf exists.$'  P@MEANING OF RESULT&Proved: No general-purpose obf exists.$'  I?PROBLEM W/ THIS ARGUMENTCQ: If Alice writes new prog P, how can she know O is secure on P?nDC)+' TALK PLANPart I:  Scrambling/Obfuscating Programs  A negative result [BGI+01].2G=>\`Y{ff` R>&- {p_/̴>?" dd@$|?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>> (  X  C "A reka"p@@p  0x "`  T Click to edit Master title style! !  0 @  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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S   TD8bZbZ P:   n*  <<<ZZxx   T;bZbZ  :  p*  <<<ZZxxH  0޽h ? 3380___PPT10.+sp (     NpT_T_ P   v* 777RRmm  NT_T_    x* 777RRmm  T T_T_ P:   v* 777RRmm  T\T_T_  :  x* 777RRmmH  0޽h ? 3380___PPT10.30C ZR  (  r  S    x  c $` 0@    0m XNew Techniques in Cryptography(2 H  0޽h ? 33___PPT10i.x)+D=' = @B +y   nfp(  x  c $     c $p<$ 0     ZAothello2P  H  0޽h ? 33___PPT10.'*`m_+,D'' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+8+0+ +    L(   ~  s *xB   r  S LC0   "  0Dk ZIgnore actual code  only care about functionL. 2 $ $  00K 0g  D Very common: 2 f  0o"` XB  0D XB  0D    0tP  =Input 2  0T  >Output 2  0W   e(i.e., input/output relation), 2$H  0޽h ? 33  ___PPT10 ./*į+D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(D ' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* #D' =1:Bhidden*o3>+B#style.visibility<* #%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* #GD' =1:Bhidden*o3>+B#style.visibility<* #G%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* G_D' =1:Bhidden*o3>+B#style.visibility<* G_%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(++0+  ++0+  ++0+  +  P Z(  ~  s *   "  0(k ZIgnore actual code  only care about functionL. 2 $ $   0 n ,$ 0 <Common Intuition: No loss in generality since general code is useless anyway: can t be understood.\c 2<$)   0V k ,$ 0 -Sometimes: Formal Justification (HALTING,SAT)L. 2   0 `,$ 0 \"Can we justify it in cryptography?#(2#$f  0o"` XB  0D XB  0D    0ܩ  =Input 2  0  >Output 2H  0޽h ? 33  ___PPT10 ./*į+gD ' = @B D ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*++0+  ++0+  ++0+ +  $l(  $x $ c $    $ S  @  n $ 0 p ,$ 0 PImplication: Problem actually hard ) scheme unbreakable (before sun collapses)PQ G6Y1 yQJ $ 0,   ,$ 0 ZIf common intuition holds (code useless) it s2.$$$ $ 0r  P,$ 0 ( bad for crypto: limits on reductions good for crypto: can  scramble programs ZR$$  $ 0r pP,$ 0 TShow that if 9 a scheme-breaking alg then 9 a problem-solving (e.g. factoring) alg.ZU GG),!,H $ 0޽h ? 33___PPT10v.1*+6?Dj ' = @B D% ' = @BA?%,( < +O%,( < +D8' =%(%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$(R%(++0+$r ++0+$r ++0+$r ++0+$r +   (j(  (~ ( s *7r  r  ( c $D>rP`<$ 0 r  ( 0@r P` ,$ 0 O This implies: ( 0TRr  P0 ,$ 0 - positive results: more powerful reductions ,.$ ( 0X]r 0P`,$ 0 /Surprisingly, in many cases intuition is false..0)$# ( 0\hr   ,$ 0 /Get new (believed unobtainable) crypto schemes.60$)  ( 0arP P~,$ 0  negative results: some schemes can t be obtained>W 2$DH ( 0޽h ? 33`X___PPT108.1*+"D' = @B Dw' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(.%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(+P+0+(r ++0+(r ++0+(r ++0+(r ++0+(r ++0+ (r +   ,,(  ,~ , s *r  r  , c $Lr r *|Tn , 0Dr P  Part II:  Zero Knowledge on the Internet  A positive result [B01].JF $:6~V.6  , 0r    Z light talk  almost no proofs / formal defs . $")|T  , 0Lr    LPart III: Some subsequent results [BGGL01,B02,BL02,L02,BLV03,KOS03,PR03,P04]6M *:6~V.H , 0޽h ? 33___PPT10i.1*+D=' = @B +  0r(  0~ 0 s *r  r  0 s *$r` r *|T 0 0r @,$ 0 "lQ: Can we take arbitrary prog P and convert to P s.t.H7,Fea9X 0 0 p` ,$ 0 1. P has same function as P 2. P is not much slower/bigger than P 3. P is  completely unintelligible jvv v!:n~.V. 0 0   `,$ 0 fProcedure to convert P P is called  obfuscator .f4  $|T  0 ZAmorsereg<$D 0  0 C &Amorseo ,$D 0H 0 0޽h ? 33___PPT10.1*+0D' #= @B D{' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* 0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* 0D' =-g6B fade*<3<* 0D' =%(DX' =%(D{' =4@BBB B%()))D' =-6B-randombar(horizontal)*<3<* 0D' =1:Bhidden*o3>+B#style.visibility<* 0%(D}' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-6B-randombar(horizontal)*<3<*0D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(++0+0 ++0+0 ++0+0 +   D"(  Dx D c $-   r D S -0    D 0 ;  P,$ 0 \Diffie&Hellman (76): Maybe can obtain public key enc. by  obfuscating a private key enc. scheme?Vb& $ $TH D 0޽h ? 33___PPT10.>*`Qo+PQFDO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(+8+0+D +  0H6(  H~ H s *J   x H c $PP  H H 0޽h ? 33___PPT10i.>*`Qo+D=' = @B +  @L(  Lx L c $^   r L S T_@0   N L 0T`   [BGI+01] Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, Yang  On the (Im)possibility of Obfuscating Programs , CRYPTO 2001.4 yt  (>:zH L 0޽h ? 33___PPT10i.?*c+D=' = @B +  UMPP(  Px P c $}   r P S ~   P P 0  T1. P can be efficiently recovered from O(P) (i.e., complete recovery of source code)fU#G )xBb P 0   62. P is hard to learn (i.e., can t recover P using BB access to its function)dO%!xBb P 0 ` 0 {EThm [BGI+01]: 8 O 9 P s.t. O totally fails on P. (assuming OWF exist)FvEG EG $-73 {S+ P 0 "`P,$ 0 8$*  TASTE OF PROOFH P 0޽h ? 33___PPT10.@*+-<D8' = @B D' = @BA?%,( < +O%,( < +D*' =%(D ' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*P$D' =1:Bhidden*o3>+B#style.visibility<*P$%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*PD' =1:Bhidden*o3>+B#style.visibility<*P%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*PD' =1:Bhidden*o3>+B#style.visibility<*P%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*PD' =1:Bhidden*o3>+B#style.visibility<*P%(DB' =A@BB@BB0B%()?)?D' =.Q7 BBBBBcM -3.33333E-6 3.33333E-6 L -3.33333E-6 -0.56111 *3>*B ppt_xB ppt_y=@0BBAApBBB<*PD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*P+P+0+P ++0+P ++0+P ++0+P ++0+P ++0+P ++  }`T(  T~ T s *tP    T 0 0p,$ 0 Pf: Show function family {Pa,b} s.t. O totally fails (code recovery + hard to learn) on random member:iG $ $%!Ex@Bb T 0 R {EThm [BGI+01]: 8 O 9 P s.t. O totally fails on P. (assuming OWF exist)FvEG EG $-6Y1 yQ T 0  @p ,$ 0 `Define$%xBbZ  T 0   ` ,$ 0 Pa,b(b,x)=b xBb  T 0)   ,$ 0 VbxBb  T 0+   ,$ 0 tb=0 , x=a, xBb  T 0<  h ,$ 0 a,bLxBb2  T 0x6  h ,$  0 b=1 , x(0,a)=bL xBb T 0R t P ,$  0 a0*xBb T 0\ t  ,$  0 o otherwise0  xBbr T 6"` p P ,$D 0N T 0_ p,$  0 nClaim: 8O for random a,b w.h.p. O totally fails on Pa,b8vGG G%H T 0޽h ? 33___PPT10.@*+ȩD|' = @B D7' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(+0+0+T ++0+T ++0+ T ++0+ T ++0+ T ++0+ T ++0+ T ++0+T ++0+T ++0+T +z K0 (  "8        0d  @p  `Define$%xBb.  0  `  Pa,b(b,x)=b xBb  0    VbxBb  0h    tb=0 , x=a, xBb   0D  h  a,bLxBb   0  h  b=1 , x(0,a)=bL xBb   0  t P  a0*xBb   0 t   o otherwise0  xBbtr   6"` p P   0 p nClaim: 8O for random a,b w.h.p. O totally fails on Pa,b8vGG G%  B "`P     0$ R {EThm [BGI+01]: 8 O 9 P s.t. O totally fails on P. (assuming OWF exist)FvEG EG $-6Y1 yQ|  0; 0p Pf: Show function family {Pa,b} s.t. O totally fails (code recovery + hard to learn) on random member:iG $ $%!Ex@BbH  0޽h ? 33 ___PPT10 .@*+D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(DQ ' =%(D' =4@BB@BB%()?)?Dg' =./7 BBBBBAM 0.0 4.81481E-6 L 0.0 -0.5088 *3>*B ppt_xB ppt_y=@0BBAApBBBy@<*D$' =A@BB@BB0B%()?)?Dk' =.37 BBBBBEM 0.0 -4.44444E-6 L 0.0 -0.50555 *3>*B ppt_xB ppt_y=@0BBAApBBBl<*D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+ +)  p!X(  X X 0T] .,$ 0 MPf:. X 0r   ,$ 0 ~bTo recover a,b from P =O(Pa,b) - output P (1,P )2 G  ( H" X 0  ,$ 0 For random a,b can t distinguish bet Pa,b and all-zero function using BB access.R (h( X 0x P@ `Define$%xBb& X 0 @  Pa,b(b,x)=b xBb X 0  x VbxBb X 0 x tb=0 , x=a, xBb X 0 H  a,bLxBb X 0p H b=1 , x(0,a)=bL xBb X 0  P T a0*xBb X 0 T o otherwise0  xBblr X 6"`P p  X 0t   nClaim: 8O for random a,b w.h.p. O totally fails on Pa,b8vGG G% X s *"`  ,$D 0u X 0  ,$ 0 GNote: In paper, rule out OBFs for programs with bounded input length.>H,$+b"  X 0D   ,$ 0 |Black-box access is useless:&$b"\ !X 0   ,$ 0 FCan recover source from obf d code:&$#$b"H X 0޽h ? 33___PPT10i.@*+LD' = @B D' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D0' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-g6B fade*<3<* XDh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*!X%(D' =-g6B fade*<3<*!XDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*X+P+0+X ++0+X ++0+X ++0+X ++0+ X ++0+!X +Q*  $`(  r  S O   r  S `P  Z  0Q P,$ 0 ZMaybe  virtually general-purpose obf exists?..$ " l pP0  $pP0 ,$D  02  6Y"`p0  :   0t\ P  = Counter Ex.  Nl jb   jb ,$D  0   hBCDEFJA8c Z ]"(#Vh&6Bzg]pD9CV[ hq tMY K8Wbeouu{A  &58YJTQSKG7/:g[mb8 qBC}hhbLV\_h|\1B_eoz'&Xth%ALxyqfjR;B;Bg  xnjfNLMBHE;Taqk]_ar{}uou&GK\vqMx\KM`KHBB:F7 (2;DP\hrzT4vqje_^~JBMB\B_ZhfV>  @                                                                               `"`jb $  0i }  F Useful progs (DES,RSA,AES,SHA,& )*$   0b P ,$ 0 a/Similar to critique of NP-completeness results.00 l    #  ,$D  0P  !  BC DEFccA8c##|}-RRh~o J Jvm^[X`dub2%.:ANh n2<[nmPm\tHbeFzndZSK-A-AXDdxV[hF6nH=W@Jd xY$8+)*6 BUd,AY&nRx@ ^ j o e > L- S\ bG R6 n1 v0 - G ] \ G = J l @ g ' / ~ J1 MY N S T S \ rv v   * U zv n w  -P$k!9 A_$-<@M)bn..hQJSmW[SIIRR ?!tQNEQpIa %./8H]nq_R<3 n^mAmA:R<gI`Jvy]\ -Ov   @ RS #      " =@ S x ` $     ^ $          V a)  ) ` i  z u m _ O @   v 1 * f A / \ d U C 6 W d j z v f h & P A I Q x V 5    [ X K R `   { w n a h ve jb ^^ R\ K I R8 D: D J N W \ ` b e u        [ L = -  n n b B J Y j R K H @ 8 0 $             Kr 0y n  >  } p e t\ ^S IJ <D 28 $8 8 E S - Xw q F  ov{n1~[HdA2^'-Jd}gm8~nQA2(-6I I - Pq|U8Av%Ld-0.6Dmmh.[.%"O! mxpt@                                                                                                                                                                                      `   " 0o @ \O secure8 CGH  0޽h ? 33)!___PPT10.+3+T/De' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bwipe(left)*<3<*$D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+ ++0+ +.   pX`((  Xx X c $   x X c $  & X 0؜ P ZMaybe  virtually general-purpose obf exists?..$ "   X 0 P  a/Similar to critique of NP-completeness results.00 X 0, "``,$ 0 ,PROBLEM W/ THIS ARGUMENT8 jP  XjP   X # tBCDEFJ8c Z ]"(#Vh&6Bzg]pD9CV[ hq tMY K8Wbeouu{A  &58YJTQSKG7/:g[mb8 qBC}hhbLV\_h|\1B_eoz'&Xth%ALxyqfjR;B;Bg  xnjfNLMBHE;Taqk]_ar{}uou&GK\vqMx\KM`KHBB:F7 (2;DP\hrzT4vqje_^~JBMB\B_ZhfV>  @                                                                               `"`jb *  X 6 }  F Useful progs (DES,RSA,AES,SHA,& )*$ 2 X <"`p0  :  X 6 P  = Counter Ex.  x L    X#   P  X  BC DEFccA8c##|}-RRh~o J Jvm^[X`dub2%.:ANh n2<[nmPm\tHbeFzndZSK-A-AXDdxV[hF6nH=W@Jd xY$8+)*6 BUd,AY&nRx@ ^ j o e > L- S\ bG R6 n1 v0 - G ] \ G = J l @ g ' / ~ J1 MY N S T S \ rv v   * U zv n w  -P$k!9 A_$-<@M)bn..hQJSmW[SIIRR ?!tQNEQpIa %./8H]nq_R<3 n^mAmA:R<gI`Jvy]\ -Ov   @ RS #      " =@ S x ` $     ^ $          V a)  ) ` i  z u m _ O @   v 1 * f A / \ d U C 6 W d j z v f h & P A I Q x V 5    [ X K R `   { w n a h ve jb ^^ R\ K I R8 D: D J N W \ ` b e u        [ L = -  n n b B J Y j R K H @ 8 0 $             Kr 0y n  >  } p e t\ ^S IJ <D 28 $8 8 E S - Xw q F  ov{n1~[HdA2^'-Jd}gm8~nQA2(-6I I - Pq|U8Av%Ld-0.6Dmmh.[.%"O! mxpt@                                                                                                                                                                                      `   X 0l @ \O secure8 CGH X 0޽h ? 33___PPT10.+3+yD' = @B DC' = @BA?%,( < +O%,( < +Dz' =%(%(D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*X'D' =1:Bhidden*o3>+B#style.visibility<*X'%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* XD' =1:Bhidden*o3>+B#style.visibility<* X%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XD5' =4@BB@BB%()?)?D' =.Q7 BBBBBcM -3.33333E-6 1.90751E-6 L -3.33333E-6 -0.19977 *3>*B ppt_xB ppt_y=@0BBAApBBB0̽<*XD5' =4@BB@BB%()?)?D' =.Q7 BBBBBcM -3.61111E-6 4.45087E-6 L -3.61111E-6 -0.20185 *3>*B ppt_xB ppt_y=@0BBAApBBBMν<*X++0+X ++0+X ++0+X ++0+ X ++0+X +-   MEPP(  P P  nBCDEFJ8c Z ]"(#Vh&6Bzg]pD9CV[ hq tMY K8Wbeouu{A  &58YJTQSKG7/:g[mb8 qBC}hhbLV\_h|\1B_eoz'&Xth%ALxyqfjR;B;Bg  xnjfNLMBHE;Taqk]_ar{}uou&GK\vqMx\KM`KHBB:F7 (2;DP\hrzT4vqje_^~JBMB\B_ZhfV>  @                                                                               `"`jb r x P c $`    P 0 } @ F Useful progs (DES,RSA,AES,SHA,& )*$ H   P  BC DEFccA8c##|}-RRh~o J Jvm^[X`dub2%.:ANh n2<[nmPm\tHbeFzndZSK-A-AXDdxV[hF6nH=W@Jd xY$8+)*6 BUd,AY&nRx@ ^ j o e > L- S\ bG R6 n1 v0 - G ] \ G = J l @ g ' / ~ J1 MY N S T S \ rv v   * U zv n w  -P$k!9 A_$-<@M)bn..hQJSmW[SIIRR ?!tQNEQpIa %./8H]nq_R<3 n^mAmA:R<gI`Jvy]\ -Ov   @ RS #      " =@ S x ` $     ^ $          V a)  ) ` i  z u m _ O @   v 1 * f A / \ d U C 6 W d j z v f h & P A I Q x V 5    [ X K R `   { w n a h ve jb ^^ R\ K I R8 D: D J N W \ ` b e u        [ L = -  n n b B J Y j R K H @ 8 0 $             Kr 0y n  >  } p e t\ ^S IJ <D 28 $8 8 E S - Xw q F  ov{n1~[HdA2^'-Jd}gm8~nQA2(-6I I - Pq|U8Av%Ld-0.6Dmmh.[.%"O! mxpt@                                                                                                                                                                                      `    P 0D `@ \O secure8 CG P <   P <$ 0  l 0  P0 ,$D  0n P 0Ԕ"`0  P 0 0p v assured progs$  P 0  P,$ 0 A: Maker should provide well-defined set of  assured secure progs.$DB=R P 0  PG,$ 0 pProblem: in many metrics, counter ex. close to  useful .290$8 pP PpP2 P 6$"`p@ :  P 0' P = Counter Ex.  H P 0޽h ? 33___PPT10.+3+ :D ' = @B D' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*PD%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-s6Bwipe(left)*<3<*PD' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*PD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*PD' =-g6B fade*<3<*PD' =%(D/' =4@BB#BB%()?)?D' =.K7 BBBBB]M -3.33333E-6 -2.31214E-6 L -0.29166 0.04994 *3>*B ppt_xB ppt_y=@0BBAApBBUUB0<<*P++0+P ++0+P ++0+P +B  YQ (  ~  s *L     c $$S  *|Tn  0_ P  Part II:  Zero Knowledge on the Internet  A positive result [B01].JF $:6~V.  0i    GPart III: Some subsequent results [BGGL01,B02,L02,BLV03,KOS03,PR03,P04]6H %:6~V.H  0޽h ? 33___PPT10i.1*+D=' = @B +  q i   (  ~  s *|     0  `P0,$ 0 rRecall: Central crypto activity  Construct scheme S s.t.>:,5  0Ș `P,$ 0 )9alg A breaks S ) 9alg B factors integers*GGG<  0 P ,$ 0 &Standard Pf: B uses A as BB subroutineX' vp  0t  P ,$ 0 TQ: Can B gain anything by using A s code?^+vn  0 @ P ,$ 0 jIntuition: NO  don t know anything about adversary.66 v%95 }U-  0 P P,$ 0 T[B01]:Intuition is false  obtain results previously proven impossible to obtain w/ black-box pf.Xb $! $5 }U-H  0޽h ? 33___PPT10.>*`Qo+HD;' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+  +o+    ` Q (  ~  s *0O   Z  0*`Qo+^ץ$Db' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D ' =%(D ' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ +[6   u m p  (  ~  s *    &  0L P  JRoughly: Proof with  no added value :B&`9Y1  0# Pp,$ 0 jA central crypto thm of 80 s [GMW86,FS89,BCY89,GK96]:66"`9Y1  00 @0p ,$ 0 )Anything can be proven in zero knowledge. *)9Y1J  08 @ P ,$ 0 FA central question of 90 s [DNS98]:6$`9Y1-  0D  P,$ 0 /Is knowledge leaked in a concurrent execution?00 $ `9Y1  0 = "`,$ 0 I CONCURRENT ZK  $  0|H &RAlice proves X true (e.g., G 3-colorable) to Bob. Bob learns only that X is trueS  $6  0$\  0,$ 0 R(a.k.a.  zero-knowledge on the internet )**`9Y1  0^ 0 ,$ 0 ( (using only O(1) communication rounds).D)  H  0޽h ? 33))___PPT10f).>*`Qo+$ID%' = @B Du%' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D.' =%(D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DB' =A@BB@BB0B%()?)?D' =.Q7 BBBBBcM 5.55112E-17 9.82659E-7 L 5.55112E-17 -0.52162 *3>*B ppt_xB ppt_y=@0BBAApBBBʈ<* DB' =A@BB@BB0B%()?)?D' =.Q7 BBBBBcM -3.33333E-6 3.12139E-6 L -3.33333E-6 -0.52162 *3>*B ppt_xB ppt_y=@0BBAApBBBʈ<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +{r ! ! D(   ~   s *      0L pP  FA central question of 90 s [DNS98]:6$`9Y1   0( P@ /Is knowledge leaked in a concurrent execution?00 $ `9Y1   08  ,$ 0 YAlice5 }U-   0 @0 `,$ 0 hBob1*$$5 }U-   0  P ,$ 0 hBob2*$$5 }U-   0p  p,$ 0 hBob3*$$5 }U-   0   ,$ 0 f& *$$5 }U-   0 0p,$ 0 Bobn*$$5 }U-   6f8c"` Q,$D  0   6"8c"`0 ` ,$D  0   6f8c"``` ,$D  0   08c"`p0 0 ,$D   0   6L8c"`0 0  ,$D  0   08c"` 0  ,$D   0   6Z8c"` @ ,$D   0   6>8c"` @ ,$D  0   6Z8c"` P ,$D  0   6"8c"`  ,$D 0   6W8c"`P ,$D  0   6"8c"`` ` ,$D  0r !  0 Pp ,$ 0 \Known: Coordinated  Bob may learn something.H/$`9Y1H   0޽h ? 33o_g____PPT10G_.>*`Qo+ՅD \' = @B D[' = @BA?%,( < +O%,( < +D.' =%(%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* DP ' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%( D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(|D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(pD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(dD9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(XD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(LD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*! %(D*' =%(DN*' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<*! D' =1:Bhidden*o3>+B#style.visibility<*! %(++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+!  ++0+!  +9   ~p (  ~  s *^     0`i pP  FA central question of 90 s [DNS98]:6$`9Y1  0c P@ /Is knowledge leaked in a concurrent execution?00 $ `9Y1s  0x  ,$ 0 3Thm [RK99]: Anything can be proven in concurrent ZKX4 $1`9Y1k  0xb  ,$ 0 !# rounds: O~(log n) [KPR00,PRS02]Z"  `9Y1   0  P  ,$ 0 vThm [CKPR01]: Protocols w/ black-box proofs require W~(log n) rounds.G $ D79Y1~  0   ` ,$ 0 >Thm [B01]: Anything can be proven in O(1)-round concurrent ZK.X?<`9Y1$  0P  v`,$ 0 Uses (inherently) non-BB proof8$ $$`9Y1  0 "`,$ 0 `$*  TASTE OF PROOF $  <"` l$D 0 U 0U 0 skip U 0U 0  0Ľ ,$ 0 Y!(concurrent = bounded concurrent)" 2"H  0޽h ? 33 ++___PPT10*.>*`Qo+D6'' = @B D&' = @BA?%,( < +O%,( < +DR' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DR' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*"D' =1:Bhidden*o3>+B#style.visibility<*"%(D>' =A@BB@BB0B%()?)?D' =.M7 BBBBB_M -3.61111E-6 -2.08092E-6 L -0.01892 -0.54381 *3>*B ppt_xB ppt_y=@0BBAApBBqBv7<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +V, " ; 3  # (  ~  s *   ^  0 P0,$ 0 2Tool: Witness Indistinguishable (WI) proofs [FS89]^3$$ `9Y1  0  `d ,$ 0 zWeaker property than ZK:(`9Y1  0<1 Pd` ,$ 0 (XWhen proving a statement X of form AB only required to hide from Bob if A or B is true.Y G# 9Y1   0P<  P ,$ 0 wWhat we need to know:($$`9Y10  0F  (,$ 0 (Anything can be proven in O(1)-round WI.:) 9Y1$ ! 0(  ,$ 0 *Unlike ZK, WI composes concurrently [FS89],+#9Y1J # 0h^ p >Thm [B01]: Anything can be proven in O(1)-round concurrent ZK.X?<`9Y1H  0޽h ? 33!!___PPT10!.>*`Qo+6D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(DY' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dx' =A@BB BB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =%(D ' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<*!D' =1:Bhidden*o3>+B#style.visibility<*!%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+! ++0+! +4 # rj(  ~  s *     00 @pP,$ 0 YAlice$5 }U-  08 pP,$ 0 WBob$5 }U-z     @ ,$D  0`B   0D `  `B   0D ` *   0 Pp  WIP X true or KC(r)<5nN G5 }U->  0 t0,$ 0 *Our Proof System: To prove statement X do:F+`9Y1p  0  t ,$ 0 ,KC(r) = length of min-sized TM M s.t. M()=rJ-$ `9Y1  0d  P ,$ 0 Z( KC(r)<5n=|r|/2 means r is  compressible )T.(`9Y1l y@  @,$D 0N    @`B B 0D `     0$   r 2R {0,1} 10n\GN5 }U-  0@ ' y  ^  `9Y1J  0  p >Thm [B01]: Anything can be proven in O(1)-round concurrent ZK.X?<`9Y1  0<  ,,$  0 H~A random r is  incompressible w.h.p. and so protocol is sound.n@ $$$$9Y1  03|"`b ,$D  0pl 0 p^,$D   0(  0  0 TNext: show no info leaked in 2 executions& 2+$%9Y1@      6"` @U 0U 0 :skip n  0|"` H  0޽h ? 33/ ' ___PPT10 .>*`Qo+zD' = @B DF' = @BA?%,( < +O%,( < +Dy' =%(%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* DY' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dx' =A@BB BB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(DC' =%(D' =%(D' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+ +> K0   8( (  (O ( 0e Pp 5Suppose Bob learns f(X) after 2 concurrent sessions.^6 `9Y1y ( 0r @P,$ 0 =We show f(X) is easy to compute (even w/o talking to Alice!)L> 2`9Y1 ( 0x `,$ 0 \Algorithm to compute f(X) will use Bob s code!N/$$`9Y1 ( 0Xx  @pp,$ 0 YAlice$5 }U- ( 0$  p ,$ 0 hBob1*$$5 }U-z y@ ( )@ ,$D  0N   ( @`B  (B 0D `  !( 0   r=Bob1()V $$5 }U- "( 0D ' y  ^  `9Y1 #( 0  0,$ 0 hBob2*$$5 }U- l P   .(W  ,$@,  0fB &(B 6DP    '( 6|  j  "r =Bob2(p-dialog)V$$ 5 }U-O -( 0P +@{,$  0 f(X)=Bob3(dialog)V$$5 }U- /( 0|"` {,$D 0B 2( s *D8c@@,$@ 0 3( 0p @P`,$ 0 eSample execution:`9Y1Jl `   6(`  ,$@   0XN   ( `  `B ( 0D `  `B ( 0D ` * ( 0 Pp  WIP X true or KC(r)<5nNG5 }U-n 4( 03|"`b dl   7( ,$D   0rN   )(  `B *( 0D `  `B +( 0D ` D ,( 0 Pp  2WIP X true or KC(r )<5nN G5 }U-n 5( 03|"`  ` 8( s *"` PH ( 0޽h ? 33&&___PPT10%.>*`Qo+ D,$' = @B D#' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D ' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3(%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#(%(D' =-g6B fade*<3<*#(D ' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-u6Bwipe(right)*<3<*(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6(%(D' =-s6Bwipe(left)*<3<*6(D9' =4@BBBB%(,D' =1:Bvisible*o3>+B#style.visibility<*.(%(D' =-u6Bwipe(right)*<3<*.(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7(%(D' =-s6Bwipe(left)*<3<*7(D' =%(DX' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-(%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*/(%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*/(D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*/(D' =-g6B fade*<3<*/(D' =%(D' =%(D/' =4@BB*BB%()?)?D' =.K7 BBBBB]M 3.33333E-6 4.73988E-6 L 3.33333E-6 0.07213 *3>*B ppt_xB ppt_y=@0BBAApBBBx=<*8(++0+( ++0+( ++0+( ++0+( ++0+#( ++0+-( ++0+3( +  K0 !!t_(  tO t 0< Pp 5Suppose Bob learns f(X) after 2 concurrent sessions.^6 `9Y1f t 0I ` \Algorithm to compute f(X) will use Bob s code!N/$$`9Y1 t 0S  @pp YAlice$5 }U- t 0\  p  hBob1*$$5 }U-F y@ t )@ N   t @`B  tB 0D `   t 04i   r=Bob1()V $$5 }U-  t 0$W ' y  ^  `9Y1  t 0|  0 hBob2*$$5 }U-F P    t W  `B tB 0DP   t 0c  j  "r =Bob2(p-dialog)V$$ 5 }U- t 0 +@{ f(X)=Bob3(dialog)V$$5 }U-f t 0|"` {RB t s *D8c@@ t 0 @P` eSample execution:`9Y1$F `   t `  XN   t `  `B t 0D `  `B t 0D ` * t 00 Pp  WIP X true or KC(r)<5nNG5 }U-n t 03|"`b >F   t  rN   t  `B t 0D `  `B t 0D ` D t 0  Pp  2WIP X true or KC(r )<5nN G5 }U-n t 03|"`  8 @ !t@M t 0p P@ =We show f(X) is easy to compute (even w/o talking to Alice!)L> 2`9Y1h  t s *"`RPBH t 0޽h ? 33  ___PPT10 .>*`Qo+D ' = @B D' = @BA?%,( < +O%,( < +D$' =%(D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(D' =%(Do' =%(D' =4@BB@BB%()?)?Dk' =.37 BBBBBEM 0.0 -3.12139E-6 L 0.0 -0.07214 *3>*B ppt_xB ppt_y=@0BBAApBBBx<*!t+p+0+t ++0+t +^:  K0 }&/d(  dE d 0 P@ =We show f(X) is easy to compute (even w/o talking to Alice!)L> 2`9Y1 d 0|  @P0,$ 0 :Compute (w/o Alice!) string monolog indisting from dialog.V;$$ `9Y1 d 0  @pp YAlice$5 }U- d 0#  p  hBob1*$$5 }U-F y@  d )@ N    d @`B  dB 0D `  d 0&   r=Bob1()V $$5 }U- d 0T: ' y  ^  `9Y1 d 0tC  0 hBob2*$$5 }U-F P   d W  `B dB 0DP   d 0*  j  "r =Bob2(p-dialog)V$$ 5 }U- d 0] +@{ f(X)=Bob3(dialog)V$$5 }U-f d 0|"` {RB d s *D8c@@ d 0H @P` eSample execution:`9Y1 d 0u P0,$ 0 $Thus Bob3(monolog)=Bob3(dialog)=f(X)~%$$ $$ $ `9Y12 d 0 +  {,$  0 =Bob3(monolog)V$$ 5 }U- d 0|"`  ,$D  0 d 00o @P`,$  0 fLook ma, no Alice!`9Y1  d 0  P ,$   0 U?23`9Y1 !d 0p   ,$   0 U?23`9Y1 "d 0X  ,$   0 UX23`9Y1>F   #d  rN   $d  `B %d 0D `  `B &d 0D ` D 'd 0 Pp  2WIP X true or KC(r )<5nN G5 }U-n (d 03|"`  $F `   )d `  XN   *d `  `B +d 0D `  `B ,d 0D ` * -d 0P Pp  WIP X true or KC(r)<5nNG5 }U-n .d 03|"`b ` /d s *"` PH d 0޽h ? 33ia___PPT10A.>*`Qo+.rDU' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(DU ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*dD' =1:Bhidden*o3>+B#style.visibility<*d%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-s6Bwipe(left)*<3<*dD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-s6Bwipe(left)*<3<*dD' =%(D' =%(D/' =4@BB*BB%()?)?D' =.K7 BBBBB]M 3.33333E-6 4.73988E-6 L 3.33333E-6 0.07213 *3>*B ppt_xB ppt_y=@0BBAApBBBx=<*/dD' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*dD' =1:Bhidden*o3>+B#style.visibility<*d%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-g6B fade*<3<*dD' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"d%(D' =-s6Bwipe(left)*<3<*"dD' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =-s6Bwipe(left)*<3<* dD' =%( DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!d%(D' =-s6Bwipe(left)*<3<*!d++0+d ++0+d ++0+d ++0+d ++0+d ++0+ d ++0+!d ++0+"d +" K0 %'|(  |E | 0P P@ =We show f(X) is easy to compute (even w/o talking to Alice!)L> 2`9Y1 | 0,  @pp YAlice$5 }U- | 0.  p  hBob1*$$5 }U-F y@ | )@ N   | @`B |B 0D `   | 01   r=Bob1()V $$5 }U-  | 06 ' y  ^  `9Y1  | 0lN  0 hBob2*$$5 }U-F P    | W  `B  |B 0DP   | 05  j  "r =Bob2(p-dialog)V$$ 5 }U- | 0g +@{ f(X)=Bob3(dialog)V$$5 }U-RB | s *D8c@@^ | 0v P0 $Thus Bob3(monolog)=Bob3(dialog)=f(X)~%$$ $$ $ `9Y1 | 0S +  { =Bob3(monolog)V$$ 5 }U-f | 0|"`   | 0 @P` fLook ma, no Alice!`9Y1 | 0  P  U?23`9Y1 | 0\    U?23`9Y1 | 0   UX23`9Y1>F   |  rN   |  `B | 0D `  `B | 0D ` D | 0 Pp  2WIP X true or KC(r )<5nN G5 }U-n | 03|"`  $F `    | `  XN   !| `  `B "| 0D `  `B #| 0D ` * $| 0 Pp  WIP X true or KC(r)<5nNG5 }U-n %| 03|"`b  8 @0 '|@0T | 0 P@0 :Compute (w/o Alice!) string monolog indisting from dialog.V;$$ `9Y1h &| s *"`IP9H | 0޽h ? 33U M ___PPT10- .>*`Qo+D' = @B DL' = @BA?%,( < +O%,( < +D' =%(%(D ' =%(D7' =4@BB@BB%()?)?D' =.S7 BBBBBeM -3.33333E-6 -3.12139E-6 L -3.33333E-6 -0.07214 *3>*B ppt_xB ppt_y=@0BBAApBBBx<*'|D6' =A@BB BB0B%(D' =-g6B fade*<3<*|D' =1:Bhidden*o3>+B#style.visibility<*|%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*|D' =1:Bhidden*o3>+B#style.visibility<*|%(+p+0+| ++0+| +G3  $+l 6(  lL l 0  P0[ :Compute (w/o Alice!) string monolog indisting from dialog.V;$$ `9Y1 l 0   @pp YAlice$5 }U- l 08,  p  hBob1*$$5 }U-F y@ l )@ N   l @`B lB 0D `   l 0.   r=Bob1()V $$5 }U-  l 03 ' y  ^  `9Y1  l 0K  0 hBob2*$$5 }U-F P    l W  `B  lB 0DP   l 03  j  "r =Bob2(p-dialog)V$$ 5 }U- l 0de +@{ f(X)=Bob3(dialog)V$$5 }U-RB l s *D8c@@ l 0] P0,$ 0 OUUsing some tools (pseudorandom gens, PCP thm), can ensure |Bob1|,|Bob2|,|p-dialog|*`Qo+LD' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D:' =%(D' =%(D+' =4@BB#BB%()?)?D' =.G7 BBBBBYM -4.44444E-6 3.2948E-6 L -0.03402 0.04971 *3>*B ppt_xB ppt_y=@0BBAApBB`BL<<*%lD0' =4@BBBB%(D' =0l9  CBBB*<3<*%lD' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*lD' =1:Bhidden*o3>+B#style.visibility<*l%(D+' =%( Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(l%(D' =-g6B fade*<3<*(lD+' =4@BB#BB%()?)?D' =.G7 BBBBBYM -4.44444E-6 3.2948E-6 L -0.03402 0.04971 *3>*B ppt_xB ppt_y=@0BBAApBB`BL<<**lD0' =4@BBBB%(D' =0l9 *CBBB*<3<**lD' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*)lD' =1:Bhidden*o3>+B#style.visibility<*)l%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*+l%(D' =-g6B fade*<3<*+lD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'l%(++0+l ++0+l ++0+(l ++0+)l ++0++l +* $ A9(  ~  s *'     c $0.  *|TV  0< P  Part II:  Zero Knowledge on the Internet  A positive result [B01].2F? I want now to dicuss a bit the meaning of this impossibility result. What we prove is that there doesn t exist a general purpose obfuscator that takes any program and scrambles it securely. However, a manufacturer of commercial obfuscator might claim that we still didn t rule out the possibility that his obfuscator is  virtually general purpose . By this I mean that it is secure for all the programs that come up in practice, Although it will of course fail for our somewhat contrived counterexample. Let s visualize this hypothetical (or not so hypothetical, see slashdot) obfuscator manufacturer s argument as follows: We have the set of  useful programs that come up in practice (don t worry if you don t know all these acronyms). Then there is our counter example, which admittedly is not in this set. Then, the manufacturer claims that the set of programs for which his obfuscator is secure contains all the useful programs and hence his obfuscator is  virtually general purpose . This claim is similar to criticism of NP-completeness results. When we prove that the TSP problem is NP-complete, we prove that it is hard in the worst case, and that no algorithm can solve it on the crazy instances that come from our reduction. However, it may very well be that there is a simple heuristic that solves TSP on all the instances that actually come up in practice. I want to comment that in fact there is a difference between the case of obfuscation and the case of TSP, and actually in our case our negative result means more than may seem at first sight. ,%H  0޽h ? 3380___PPT10.+`eE  0 (  p (  ^  S     @    c $Ȕ@  0  @   Programs are hard to understand. One example is this othello playing game from the IOCCC. Another example is the fact that we seem unable to write bug-free programs, and so apperantly We don t understand even our own programs. Understanding some else s program is even harder. This is especially true if the program is not given to us in a high level language with extensive comments and documentation. In general, it seems that the  natural state of computer programs is complete unreadability and unless the programmer takes active steps to maintain the program readable, it will deteriorate to this unfortunate state. We also have theoretical evidence that programs are hard to understand. It is known to be impossible to decide if a program enters into an infinite loop. It is also widely believed that we can t tell whether a program has a non-zero output fast than trying all possible inputs. This is essentially the NP vs P problem, which we are still far from resolving. Indeed, one reason that we are so far from resolving this question is that we don t really understand programs. -- Ioccc 87 winner, Roemer B. Lievaart - othello playing game 5q + H  0޽h ? 3380___PPT10.-J  0 P H (  ^  S     @ B  c $@  0  @  The idea of software obfuscation is as follows: we said that experience shows us that programs can be extremely hard to understand if we don t take active steps to make them more readable. Thus it seems that if we took active steps to obscure our programs and make them *unreadable* then they should be really completely  scrambled . The question we are concerned here is whether indeed there is such a way to transform any program into a functionally equivalent unreadable form. That is, we ask whether a program P can be converted into a program P that (1) computes the same function (2) has roughly the same running time, but (3) is  completely unreadable . For example, such a transformation will take a program like this which has comments and meaningful variable names, and convert it into the following program, which at least for me seems completely unintellgible. The procedure that converts the program P into the unreadable program P is called an  obfuscator .\ hH  0޽h ? 3380___PPT10.- 79 0  ,(  ,d , c $    @   , s *@  0  @   In order to prove that no information is leaked, we must define more formally what we mean by this notion. To prove that no information is leaked, we assume, towards a contradiction, that Bob learned some information f(x) after interacting with Alice. We will show that in fact Bob did not learn f(x), as he could have computed f(x) by himself, without any interaction with Alice whatsoever. In more pictorial terms, we will show that anything that Bob learned from the dialog with Alice, he could have learned by just having a  monolog withh himself. The sense in which our proof of security will be non-black-box, is that in the course of showing that f(X) is easy to compute, we will use the code of algorithm that described Bob s strategy. This is our general plan. hopefully, things will be somewhat clearer once you see the details. As I said we assume that Bob learns f(X) after two concurrent sessions with Alice. Let s us consider a typical interaction between Alice and Bob. The reason that I don t say that r and r are chosen at random is because Bob may be malicious and so may in fact use any efficient strategy to compute r and r . Since r is the first message Bob must compute not based on anything he heard from Alice, but r can be computed based on the partial dialog that Bob and Alice had up to that point. We denote the transcript of all messages Bob heard from Alice by  dialog and at the end of the interaction we assume that Bob is able to learn f(X) by running some efficient algorithm Bob_3 on the string dialog. We ll now move to showing that bob could have in fact computed f(X) without talking to Alice. LqTH , 0޽h ? 3380___PPT10.,`Ru$ 0 .&0(  0^ 0 S       0 c $ 0   Hi, I m Boaz Barak from IAS and I will present some of the results from my thesis. These results are in one area which I ve been interested in  the power of using the actual code of programs in cryptography.H 0 0޽h ? 3380___PPT10.s1- ? 0 x p `T (  Td T c $    @ d  T s *  0  @   Loosely speaking the difference is due to the fact that you can t test security. Thus, you can always run a TSP heuristic and if it works you win, and if not then its not so bad. However, if you run a heuristically secure obfuscator how do you know if it is actually secure? One solution is that you don t, and you can just hope you ll be lucky. This is the solution the DVD consortium adopted when they obfuscated their encryption algorithm, and unsurprisingly, it failed. A more acceptable solution is that the manufacturer of the obfuscator should provide some guarantees or assurances of its security. This means that the manufacturer should give a cleanly-defined set of  assured program for which he has reason to believe that the obfuscator will be secure. For the obfuscator to be  virtually general purpose this set should contain the standard useful programs people normally obfuscate in cryptography, and of course it should not contain our counter example. If the manufacturer gives such a set then Alice can verify whether or not her program P falls in it. The problem is that we considered several natural candidates for such sets, and all these candidates fail to separate between the useful programs and our counter example. Roughly speaking, the reason is that our counter example can be made to be very close to useful standard programs under many natural metrics. H T 0޽h ? 3380___PPT10.+`eE$@ 0 \4(  \d \ c $     \ s *8 0    H \ 0޽h ? 3380___PPT10.+`eE 0 ` (  `^ ` S       ` c $\ 0    However, fortunately Richardson and Kilian showed that every statement can be proven in zero knowledge in a way that retains Security even if the protocol is executed several times concurrently. We call such a protocol a  concurrent zero knowledge protocol . (In parenthesis I will note that the technical term for the notion that I ll use is actually bounded concurrent zero knowledge, but it s not worth getting into the difference between these terms here.) After Richardson and Kilian s result, the next natural question was to try to obtain a concurrent ZK protocol with reduced round complexity. Prabakharan, Rosen and Sahai gave a protocol with number of rounds that is roughly logarithmic in the input length. In fact, it was thought at some point that every concurrent zero knowledge protocol must a number of rounds that is roughly logarithmic, due to a result of Canetti et al, that proved that this holds for any protocol for which the zero property is proven using a black box reduction. However, in my FOCS 01 paper, I showed that anything can be proven in concurrent zero knowledge using a constant-number of rounds. The security proof of my protocol uses the adversary s code in a non-black-box way, and in fact Canetti et al s result shows that this must be the case. At least in my eyes, the fact that this protocol an inherently non-black-box protocol is much more interesting than the fact that it is secure when executed several times concurrently. This was thought to be essentially the best possible sincet$p  \)H ` 0޽h ? 3380___PPT10.#2"A 0 h2(  hd h c $     h s *d6 0   In order to show this, we show that it is possible to efficiently compute without talking to Alice a string monolog that is indistinguishable from the string dialog. Once we show this we re done, because we can then feed this string monolog into the algorithm Bob_3. Because Bob_3 won t be able to tell aprt monolog from dialog, it will still output f(X). In the course of computing this string monolog we will make our non-black-box use of Bob s code. At first, the task seems rather hard. We have to compute something that looks like the transcript of talking with Alice, without having any access to Alice. In particular, if X is the statement  G is 3-colorable , how will we be able to simulate the WI proofs without knowing the 3-coloring? ,/+H h 0޽h ? 3380___PPT10.,`Ru$B 0 .&p(  pd p c $     p s *H 0   rNow it turns out that using some tools, and in particular cryptographic pseudorandom generators, and a variant of the PCP theorem, we can ensure in fact that the descriptions of the algorithms Bob_1, Bob_2 and the length of the partial dialog, will be all of length shorter than n. This means that in fact the Kolmogorov complexity of both r and r is less than 2n. Furthermore, since we have the actual code for Bob_1 and Bob_2 we have actual witness/proof that this fact is true. Thus using this code we can run the WI proof for the OR statements, this time using the second part of the OR. The concurrent WI property of the proof system means that the transcript we generate this way is indistinguishable from the transcript generated in a real interaction with Alice. This finishes this sketch or  taste of proof. 7 H p 0޽h ? 3380___PPT10.,`Ru$$C 0 x4(  xd x c $     x s *lN 0    H x 0޽h ? 3380___PPT10.,`Ru$$D 0 4(  d  c $      s * _ 0    H  0޽h ? 3380___PPT10.,`Ru$  0 Z R P (  ^  S     L   c $_ 0   In computer science, to treat a program as a  black box means to consider the program as a box which we feed inputs to and obtain outputs from. When we treat a program as as black box we ignore the actual code of the program and only care about the function it computes. Treating programs as black boxes is very common in all areas of computer science. For example, almost all programming languages include mechanisms such as function calls to facilitate using other programs as black boxes. When developing algorithms for a big problem we often divide it into smaller problems (or smaller Instances of the same problem) and use algorithms for the smaller problems as black-boxes. Treating programs as black boxes is also very common in complexity. For example, when we say that the TSP problem is NP-complete, we mean that if this problem had an efficient algorithm, then every other problem in NP would also have an efficient algorithm. The way we prove this statement is by showing that any problem in NP can be solved efficiently if we just have access to a black box that solves TSP.HH H  0޽h ? 3380___PPT10.3@Dr 5 0   ` (  ^  S        c $x 0   x & In all these cases, it may seem that we lose some information when we treat our components as black boxes. Possibly we could learn more from the code of the program than we can learn just by using it as a black box. However, the common intuition is that we don t lose any meaningful information. The reason is this phenomenon described earlier that we don t understand general code. Therefore, unless we are in a very special situation where we are assured that program given is extensively documented and  annotated , it seems that we can t really hope to use the code in any meaningful way. As I said before, in some cases, such as the Halting and Satisfiability problem, we can even give formal evidence that seeing the actual code is no better than just having black box access. The main question I will be discuss in this talk is whether this intuition is also justified in cryptography. That is, there are several settings in cryptography where the notion of treating programs as black boxes naturally arises, and it is interesting to know whether this is all that we can do, or perhaps seeing the actual code of the program is more powerful in these settings. H  0޽h ? 3380___PPT10.3F 0 &p(  ^  S       c $( 0   zBefore going into this issue, let me give a high level overview of one central activity of modern cryptography. This activity is constructing so called  provably secure cryptographic scheme. The idea is as follows: we construct a scheme for some cryptographic task such as voting, auctions, or playing poker over the phone. We then rigorously define what it means for this scheme to be secure, and then give a reduction between the task of solving some hard problem such as factoring large integers, to the task of breaking the security of our scheme. This means that if there exists a  scheme breaking algorithm, then there Exists a factoring algorithm with similar efficiency. We call such schemes  provably secure because if the problem is actually hard (say takes exponential-time to solve) then it will imply that our scheme is unbreakable, at least in the lifetime of this universe. Now we can examine the implications of the intuition described before, that understanding code is hard, to modern cryptography. On one hand there is a negative implication, since if the intuition holds then we are  stuck with black-box reductions, and we can t get more power by actually looking at the code. In particular, there have been some negative results showing limitations on what can be proven in crypto using black-box reductions. If the intuition holds then these negative results actually show limitations on what can be proven period (regardless of the reduction type). REPHRASE On the other hand, there are also positive implication for the intuition. At least intuitively, crypto uses hard problems, and so it seems that if understanding code is a hard problem then we can utilize this hard problem for cryptographic purposes. In particular, people have suggested to use this problem for the task of  scrambling or  obfuscating code. >Z> H  0޽h ? 3380___PPT10.3a 0 (  ^  S       c $P 0   rIn this talk I will examine whether or not this common intuition that we  can t understand code Holds up when is applied in cryptography. Somewhat surprisingly and un-intuitively, in many cases it turns out that the intuition doesn t hold. On one hand this implies positive results for crypto. We can have more powerful reductions or security proofs by using the code of programs. Thus we obtain new cryptographic schemes which were previously proven impossible to obtain using black-box reductions, and hence were conjectured not to exist. On the other hand we also get negative results in crypto. In particular, we give impossibility results for the task of  scrambling / obfuscating code. H  0޽h ? 3380___PPT10.3p% 0 <4(  ^  S     .  c $4 0   I will first present the negative result on software obfuscation. Then I will present a positive result on zero-knowledge in asynchronous networks. At the end of the talk I will briefly survey some other results that use and extend these techniques. I will try as much as possible to avoid technical proofs and definitions, and I will even sometimes cheat slightly and hide some details in the proofs that I do give. However there will be some slides that may be hard to understand without background. These slides are marked with asterixs, and there will be very few of those. So, if you lose me in one of these you should be able to catch up soon. pH  0޽h ? 3380___PPT10.3=&|  0 (  ^  S       c $ 0   Why might someone have the idea that such creatures exist? Of course one reason is the intuition which I expanded on earlier, that programs are hard to understand. In fact, one may think that we already have obfuscators, namely compilers. Indeed, it seems very hard to understand compiled code, and the difference between  open source and  closed source is considered to be major. Thus, maybe if we just analyze our compilers better, perhaps using throwing to them some additional optimizations and transformations, we can get obfuscators. Another reason, which is actually the reason I thought obfuscators may exist, is that in crypto we can do anything. By this I mean that, although obfuscator are certainly a strong object, they don t seem more unbelievable than all the crazy things we can already do in crypto, such as public key encryptions, digital signatures, and playing poker over the phone. Finally, there are companies out there that are actually making and selling obfuscators. And, perhaps more importantly, there also people that buy them. Thus, at least some people seem to believe that obfuscator exist. I would like to note that when Diffie and Hellman wrote their seminal paper that put forward the notion of public key encryption and digital signatures, they didn t have candidate constructions for these notions. However, one way in which they suggested one might obtain such schemes is by obfuscating *private key encryption schemes. I don t want to get into the details, but if you think about it for a while, one can see that intuitively an obfuscation of a private key encryption should yield a public key encryption scheme. I just want to note that this comment suggests that at least in 1976, obfuscator didn t seem a more far fetched notion than public key encryption. Z ,uH  0޽h ? 3380___PPT10.3P&   0 4 ,  (  ^  S     &   c $ 0   VA second natural question is of course, why should someone care whether or not obfuscator exist. A first answer is that at least to me this seems a fundamentally interesting question. Another answer is that if they exist, perhaps we would be able to revive Diffie and Hellman s suggestion of obtaining public key encryption from private key encryption or one-way functions. Today we are in the rather unsettling situation that if someone comes up with an efficient algorithm for the factoring and discrete log problems, or alternatively, if quantum computers become a reality, then significant chunks of modern cryptography lose their foundation. Thus, it is an extremely important open problem to come up with other candidate public key encryption schemes. Another obvious application of obfuscators is for the task of software protection  hiding the details of your algorithm from someone that looks at your software. Finally, most commercial obfuscators today seem to be used for the task of digital rights management  protecting copyrighted material etc.. ,Z,, H  0޽h ? 3380___PPT10.3&2  0 B(  ^  S       c $@  0   8The main result of this part is that general purpose software obfuscators do *not* exist, even if we use a very weak definition of security. By  general purpose I mean an obfuscator that takes any program and should output a Scrambled version of this program. This result is joint work with Goldreich, Impagliazzo, Rudich, Sahai, Vadhan and Yang. b%   H  0޽h ? 3380___PPT10.3`'V  0   f (  ^  S        c $ 0   \ H Before proving such a result we need to formally define what does it mean for An obfuscator to be secure, or at least what it means for an obfuscator to be insecure. We use the following definition. Let O be a candidate obfuscator. That means that O is an algorithm that takes A program as input and outputs a different program. We say that O totally fails on some program P if P can be efficiently recovered from O(P). Note that this means that it is possible to invert the obfuscator and recover the entire Original source code from the supposedly obfuscated version. If this happens then the obfuscator definitely failed. However one may complain that this definition does not capture all possible ways in which an obfuscator can fail. Indeed, if P contained comments Then even the trivial obfuscator that just throws away the comments will not fail totally on P. However, because we Are shooting for a negative result, the stronger notion of failure we require, the better  it will just mean our impossibility result will be stronger. Condition # 2 is meant to avoid trivialities. For example, if P is a program that prints its own source code, then, No matter how you obfuscate P, it will always be possible to recover the source from the obfuscated version by simple executing it. Thus, we require that it will be impossible to recover the source code of P just by executing it. The main theorem of this part is that for every candidate obfuscator O there exists a program P on which O totally fails. I will now give a  taste of the proof of this theorem. H  0޽h ? 3380___PPT10.3%'  0 0(  ^  S       c $ 0   MTo prove the theorem we will give a family of functions, parametrized by a pair of strings \alpha,\beta of length, say 1000, Such that for every O if we choose alpha and beta at random then O will totally fail on P_{\alpha,\beta}. For every \alpha,\beta we define the program P_{\alpha,\beta} as follows: It takes as input a bit b and a string x. If b=0 and x=\alpha, then P_{\alpha,\beta} outputs \beta. I will now turn to proving this claim. If b=1, then P_{\alpha,\beta} interprets the string x as a description of a program! It then runs this program On the input (0,\alpha). If the output is equal to \beta then P_{\alpha,\beta} outputs the pair (\alpha,\beta). In all other cases P_{\alpha,\beta} outputs 0. As I said above, I will claim that every obfuscator fails w.h.p. on a P_{\alpha,\beta} if alpha,\beta are chosen at random. ,9 ?H  0޽h ? 3380___PPT10.3@^D' 0 @(  ^  S       c $L 0   I will first show the second condition (i.e., that a random P_{\alpha,\beta} is hard to learn). I claim that black box access to P_{\alpha,\beta} is completely useless, since there is only negligible probability that one can obtain a non-zero answer from that black-box. Indeed, suppose that someone gives you a black-box that computes P_{\alpha,\beta} but you don t know \alpha or \beta. Then, if you feed the box a query of the form (0,x) you will get a zero answer unless you happened to guess \alpha, which We can assume never happens, since it will only happen with low probability. Now, if you only get zero answers For queries of the form 0,x then you have no information about \beta, and so you have no way of coming up with a program that outputs \beta on input \alpha. This means that you will also get zero answers on queries of the form (1,x). Thus with very high probability you will only get zero answers from the black box. I will now show the first condition. Namely, that it is possible to efficiently recover the source of P_[\alpha,\beta} From any P which is an obfuscated version of P_{\alpha,\beta}. First, note that once we know \alpha and \beta Then it is easy to recover the code of P_{\alpha,\beta}, and hence we need to show that we can recover \alpha and \beta from P . However, this is quite easy  to get \alpha and \beta run P on input 1 and *its own code*. Since P is a program that on input 0,\alpha outputs \beta, it follows that P (1,P )=\alpha,\beta. This finishes the proof. I just want to comment that this argument used crucially the fact that we could feed P as input to itself. This is no longer true if we want to rule out obfuscators for programs with bounded input length. I don t have time to even hint how we overcome this obstacle , but actually in some sense that part is the technical heart of the paper. H  0޽h ? 3380___PPT10.3P'6' 0 PF(  ^  S       c $Q 0   <Now that we finished with the negative result on software obfuscation, we can move on to Part II of this talk. This will be a positive result on zero knowledge proofs in ansynchronous networks, where we ll use some non-black Techniques to obtain this result.  MH  0޽h ? 3380___PPT10.3(J & 0   `Z (  ^  S        c $ s 0   P  Recall that I said that a central activity in modern cryptography is to construct  provably secure schemes. These are schemes where we know how to convert any algorithm A that breaks the security into an algorithm B that say, factors large integers. The standard way to prove this implication, that was used in all works prior to the one I ll now describe, is that B was actually a generic algorithm that was given access to a black-box. Then, one proved that if this black box breaks the scheme then B can factor integers. A natural question is whether we can gain in power by considering an algorithm B that doesn t just use A as a black box but actually uses the code of A. The intuition is that this wouldn t help us. The reason is that A is the algorithm that is used by our adversary that tries to break the scheme, and since we can t really assume anything about the structure of the algorithm our adversary will use, it seems that we ll not gain anything by looking at its code. Somewhat surprisingly, it turned out that this intuition is false. In my focs 2001 paper I was able to construct a scheme with a security Proof involving a non-black-box use of the adversary s algorithm. This scheme satisfies some properties which were previously proved unobtainable by any scheme with a black-box security proof. The scheme is for a zero-knowledge proof system, and I will now explain what this term means. )aH  0޽h ? 3380___PPT10.3;+(( 0 ldp(  ^  S     ^  c $ 0   >Zero Knowledge proof system were introduced by Goldwasser, Micali and Rackoff. Roughly speaking, a zero knowledge proof system is a proof that doesn t leak extra information. Usually in mathematics, we hope that if Alice proves a mathematical theorem to Bob then after listening to the proof, Bob Willgain some new knowledge about the theorem beyond the one bit of information that it is true. In a zero knowledge proof, we try to achieve exactly the opposite goal. Namely, Alice tries to prove the theorem to Bob in a way that will convince Bob that the theorem is true, but without him learning any additional information about the statement proven. This is of course a very rough and informal description, and in a few slides I will hint on how this requirement is formalized. BTW, To obtain zero-knowledge proofs, Goldwasser Micali and Rackoff had to consider a generalization of the static mathematical proofs we usually think of, to a process that is interactive and probabilistic. One example to think about is that X is a statement of the form  the graph G is 3-colorable , Alice knows a 3-coloring of G, but wants to prove that G is a 3-colorable without leaking any information to Bob about the coloring. Actually it turns out that because 3-coloring is NP-complete, this example is quite general. The first motivation for thinking about zero knowledge proofs is that just from their description, they seem fascinating and paradoxical objects, And hence interesting to study. A more practical motivation is that, as Fiat and Shamir showed, such systems are actually very useful for identification protocols. In an identification protocol Alice proves to Bob that she knows some secret or password, without giving Bob information that will enable him to impersonate her in the future.. Another motivation is for more protocols solving more general cryptographic tasks such as electronic voting schemes, auctions, or playing poker over the phones. Zero-knowledge proofs are many times a central component of such protocols. Roughly speaking, they are used to allow one participant to prove to the other participants that behaved according to the rules without compromising his secret inputs. /  H  0޽h ? 3380___PPT10.39H() 0 0((  ^  S     "  c $ 0   Mathematics undergrads probably conjectured this for centuries, but only in the late 80 s Goldreich Micali and Wigderson proved that basically every statement can be proven in zero-knowledge. They did it by presenting a zero-knowledge proof system for 3-coloring.. Others have improved their work to show that even though any such proof system must be interactive, the number of communication rounds needed can be a constant independent of the input length (i.e., 4). In the the late 90 s, Dwork Naor and Sahai raised the question of whether such zero knowledge proofs remain secure if they are executed several time concurrently with one another. This was dubbed as the question of  zero knowledge on the internet by Kilian, Petrank and Rackoff. Z  e H  0޽h ? 3380___PPT10.3c(7 0 "(  ^  S       c $0z 0   The reason is that this can arise if Alice is a web server that actually talks to many  Bob s simultanously. The worry is that if all these Bob s are coordinated by some malicious adversary then he may be able to learn some non-trivial infromation about the statement proven. In fact this worry is substantiated, as it was shown that there are zero knowledge protocols that are secure in a standalone settings, but are completely insecure even if we allow only two concurrent sessions. ,_  H  0޽h ? 3380___PPT10.3Ю(  0 v n  (  ^  S     h   c $ 0    I will now give a taste of the proof of this theorem. An essential tool used is the notion of Witness Indistinguishability or WI, as I will call it. WI is a weaker requirement from a proof system than zero knowledge. Instead of requiring that we do not leak any information, we only require That we don t leak information of a specific type. Namely, when Alices proves to Bob a statement that is an OR statement of the form A or B (for example to prove that either G is 3-colorable or G is 3-colorable), we only require that Bob does not learn if A was true or B is true. We ll need two facts about WI proofs: The first fact is that anything can be proven in constant-round WI. Indeed, this follows immediately from the fact that WI is a weaker property than zero knowledge and everything can be proven in constant-round zero-knowledge. The second fact is that unlike zero knowledge, standalone WI security implies concurrent WI security. Note that these two facts together imply that anything can be proven in constant-round concurrent WI. Note also that to prove that either G is 3-colorable or G is 3-colorable, Alice needs to know a 3-coloring for one of the graphs, since this by itself assures that the OR statement is true. ,fqH  0޽h ? 3380___PPT10.3@X(+ 0 LD (  ^  S     >   c $x  0   f We are now ready to describe our proof system. Suppose that Alice wants to prove to Bob that a statement X is true, where X is described by an n-bit long string. As a first step, Bob chooses a string r of length 10n at random and sends it to Alice. Then, Alice proves to Bob using a WI proof that either X is true or the Kolmogorov complexity of r is at most 5n. The Kolmogorov complexity of a string r is the length of the smallest Turing machine that describes it. Roughly speaking, if KC(r)<|r|/2 then it means that r can be described succinctly and hence is  compressible in some sense. Bob accepts the proof as a valid proof for X if the WI proof is convincing. It is well known that a random string r will be incompressible and have very high kolmogorov complexity with very high probability. Hence the protocol is sound in the sense that if X is false then the OR will also be false with very high probability, and then Bob will reject by the soundness of the WI system. Of course, because r will likely be incompressible, and even if it wasn t there is no way for Alice to know a short description for r, Alice will use the fact she knows that X is true to run the WI proof system. So, if X is the statement  G is 3-colorable and Alice knows a 3-coloring for G, she will use this 3-coloring for the WI proof system of the second part. We know turn to show that this protocol is concurrent zero knowledge. Actually we ll restrict ourselves to showing that it remains zero knowledge under two concurrent executions, since this will already give a god taste of the proof. PC $ o 9H  0޽h ? 3380___PPT10.3 V(0 0 TL(  ^  S     F  c $L 0   We have finished the two main parts of this talk. 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