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Example: iterated group product g$  Abelian case f(x1,& ,xn)=x1+x2+& +xn g(x, (r1,& ,rn-1)) = x1+r1 x2+r2 & xn-1+rn-1 xn-r1-& -rn-1 General case [Kil88] f(x1,& ,xn)=x1x2 & xn g(x, (r1,& ,rn-1)) = x1r1 r1-1x2r2 r2-1x2r3 & rn-2-1xn-1rn-1 rn-1-1xn   P, P9 P P P+ PE P P cckckckckck ckckcgoggogoggo goggogogogocccccckckckck k ck ckckckgocgoockgocgoockgocgoockgocgoockc         i      W   Example: iterated group product g$  Notions of Simplicity - III  g(  pLow degree: g(x,r) = vector of degree-d poly in x,r over F aka  Randomizing Polynomials [I, Kushilevitz 00,& ] Application: round-efficient MPC Motivating observation: Low-degree functions are easy to distribute! Round complexity of MPC protocols [BGW88,CCD88,CDM00,& ] Semi-honest model t<n/d 2 rounds t<n/2 multiplicative depth + 1 = log d+1 rounds Malicious model Optimal t O(log d) rounds Randomized func. g reduces to a deterministic degree-d func. g g ((x1, r1), & ,(xn,rn)) = g((x1,& ,xn),(r1+& +rn)) Security of reduction is  model independent N; PX PF P9 P PG P P PA P_ P P P gc/gccc$cc-g#cc c  c   c   c   c   c   c c cc!cccckckckck ckckckck0ccc            ^    (      2 Examples g(  >What s wrong with previous examples? Great degree in x (degx=1), bad in r Coming up: Degree-3 encoding for every f Efficient in size of branching program% +  E %eaia eEa&9  c Notions of Simplicity - IV  g(  Small locality: Application: parallel cryptography! [Applebaum, I, Kushilevitz 04,& ] Coming up: encodings with locality 4 degree 3, fully decomposable efficient in size of branching program R & D gg1c ccc%cDc@E      q qParallel Cryptographye   4Cryptography in NC0?J g(o(g(  kMain Primitives  g$   "Surprising Positive Result [AIK04]4# g$g # |Encoding a PRG  g$  Want: f(x) is a PRG g(x,r) is a PRG Problems: output of g may not be pseudorandom g may shrink its input Solution:  perfect randomized encoding respects pseudorandomness, additive stretch, & stretch of g is typically sublinear even if that of f is superlinear most (not all) known constructions give perfectness for free 2 = -  gc + c       0      )     <#Additional Cryptographic Primitives $ $g$ $ jGeneral compiler also applies to: one-way / trapdoor permutations collision-resistant hashing public key / symmetric encryption signatures / MACs commitments & Caveat: decryption / verification not in NC0& & But: can commit in NC0 with decommit in NC0[AND] Applications: coin-flipping, zero-knowledge, & F"  /$ c "ccg%ckccgfckckc0@  ^  = Non-cryptographic PRGs  g$&   Remaining Challenge  g$  ]Manipulating Encodings  g$  4c&From Branching Programs to locality 4  ' 'g( '  Perfect Degree-3 Encoding of BPs! !$&   Proof of Lemma    ] e)Computationally Private Encodings [AIK05]H* "g$gg *   Proof Outline  g(  ,App 1: Relaxed Assumptions for Crypto in NC0J- +gog - -App 2: Parallel Reductions Between Primitives . .g . 2Thm. All are equivalent under poly-time reductions\3$ gc g c 0 $App 3: Secure Multiparty Computation % %g %  ~Summary2 g$&$  dDifferent flavors of randomized encoding Motivated by different applications Round-efficient secure computation Parallel cryptography  Simplest encodings: outputs of form xirjrk+rh Efficient for various  intermediate complexity classes (NC1, NL/poly, modqL/poly) Algebraic approach  Combinatorial approach: information-theoretic garbled circuit Computationally private encodings: efficient for all P (assuming  Easy PRG ) ) Z$ Z: Z0 ZT ZT ZP Z Z Z Z Z)$:&          <TP c @   I   Open Questions2 g$&$   /5 8DFG T=\^^ldne}klmopqrswxy{~U  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (    6T P  X Click to edit Master title style!!   0H   RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S   0 ``  V*    0  `   T*    0t  `   <  H  0޽h ? ̙33y___PPT10Y+D=' = @B + Default Design 0 0,( j  , , NH &g&g P#    r*   C!!CCdd , Nt &g&g  #   r*   C!!CCddd , c $ ?9   8 , N &g&g  @   RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S  , T &g&g P    r*   C!!CCddx , T &g&g  `  H@___PPT9"@ *""   C!!CCddH , 0޽h ? ̙3380___PPT10.X[G  .,I(     <,q       6r   J  u Yuval Ishai g&     6pw   0  H  g     Hz ?F Qf  hTechnion" ( 2 e  H  0޽h ? ̙33y___PPT10Y+D=' = @B + xK0  V(  x  c $ I      S  "<$ 0   H  0޽h ? ̙33_ W ___PPT107 +D ' = @B D ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 1%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 1h%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+   + yK0 0!0(      f _ж_ж?" f `  fYou came to the right place!( 2  =   fD"_ж_ж?"  ]  7Go home!* *return ticket will NOT be paid by organizers28( 2.  8    `l"_ж_ж?"   @(2  %"   f| "_ж_ж?" 0 d 5Interested in unconditional security in cryptography?6( 26 6  "    fP"_ж_ж?" 0 } gInterested in cryptography?( 2  "    f"_ж_ж?" &  bInterested in Science?( 2  "    f"_ж_ж?"  B  l Like traveling, good food, etc.?!( 2! ! B    f_ж_жD)?z   # lx"_ж_ж?" Mm Nno(2    # l"_ж_ж?" z! @(2  8 0 f  "0 f B   f_ж_жD)?0 B  # l_ж_жD)?f B   f_ж_жD)?}  # l""_ж_ж?" Pp Nno(2  B   f_ж_жD)?   # lh&"_ж_ж?" p  Nno(2  B    f_ж_жD)?   !  `*"_ж_ж?"  :9  Nno(2  F 0 f  # 0 Ff B $  `_ж_жD)?0 B %  f_ж_жD)?f F 0 f  & 0 cf B '  `_ж_жD)?0 B (  f_ж_жD)?f  ,  `L0"_ж_ж?"   Oyes(2   .  ` 5"_ж_ж?" ] } Oyes(2   /  `2"_ж_ж?"   Oyes(2   0  `d<"_ж_ж?"    Oyes(2  F 0 f  )  0 f B *  `_ж_жD)?0 B +  f_ж_жD)?f H  0޽h ? ̙33y___PPT10Y+D=' = @B +J e *"@!Yp~(  | - # l@"_ж_ж?" }1,$ 0 Enc(y)N(2eaaa&   ~  s *,C"  "    0G" "6 ? Tw<$ 0 "   Z_ж_ж?" {j B   f_ж_жD>?j  # l\I"_ж_ж?"  @ [x(2a  B   f_ж_жD>? S  # lHM"_ж_ж?"  @ [y(2a     `4Q"_ж_ж?"  I ]f (2c  ~ % 3 rLU"_ж_ж?" }7,$ 0 Enc(y)J(2aaaa&   3l }b .}b,$D 0B " # l_ж_жD>?b # 3 r\"_ж_ж?" } [x(2a  l bK%  /bK% ,$@ 0 !  `_ж_ж?" b % B $ # l_ж_жD>? K  &  f8a"_ж_ж?"  _  ]g (2c  7l Cjc  0j% ,$D 0B ' # l_ж_жD>?j ) 3 re"_ж_ж?" Cc  _r"(2e  7l 2 M3,$D   0B K  `_ж_ж?" d2 L # lj"_ж_ж?" 6 cdecoder (2c  9l    Q]z,$D   0B O  `_ж_ж?"    P # lo"_ж_ж?" O 6  e simulator (2 c   f R  `4t"_ж_ж?" F,$D  0 Dec(g(x,r)) = f(x)(2c@       U Z@y"_ж_ж?" Gm,$D  0 $Sim(f(x)) g(x,r)@(2 cc@      V DA COMPUTER"`  2L$@ 0U 0 X Z_ж_ж?" b % ,$D 0 W c <Acalculator"`M G L$@ 0U 0 Y  f"_ж_жGH[M?" 6 c ,$D 0 TVariants: perfect, stat., comp.  pure IT 8+( 2 g"c + H  0޽h ?Y ̙33q-i-___PPT10I-+D+' = @B D`+' = @BA?%,( < +O%,( < +DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =%(D' =%(D7' =4@BB@BB%()?)?D' =.S7 BBBBBeM -1.11111E-6 -4.81481E-6 L -1.11111E-6 -0.03796 *3>*B ppt_xB ppt_y=@0BBAApBBB <*.D ' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =-s6Bwipe(down)*<3<*MD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =%(DE' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*M%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-o6Bwipe(up)*<3<*QD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(Dd' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*Q%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"G%(D ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Gh%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(++0+-" ++0+" ++0+%" ++0+%" ++0+R" ++0+U" ++0+Y" + rK0 066x:(  xx x c $ Vf  "  x 0,j'? `Z S "( 2A   x H' jJ?Q5  ag0( 2CC   x 0d? H{  Sr"( 2C  dr x@ < ? Y N  x < "?  hinputs2( 2GC  dr x@ < ?{ ,W   x <D"? ;  o random inputs2( 2 GC    x H" jJ?r  af0( 2CC  jB  x B D?rjB  x B D?LLrjB  x B D?rjB x B D?  jB x B D? ff jB x B D?   x 0"? iX  Sx"( 2C  dr x@ < ? c  x <t"? &  hinputs2( 2GC   x 0"? |  Sx"( 2C   x N|"?] g f(x) is encoded by g(x,r)( 2CCCCCCCCCC4    jB x B D?njB x B D?``njB x B D?  jB x B D? tt jB x B D? jB x B D?l l jB x B D? jB x B D? jB x B D?v v jB x B D? jB  x B D?rr jB !x B D? jB "x B D? jB #x B D?|| jB $x B D?bb jB %x B D?JJ jB &x B D? jB 'x B D?~~HjB (x B D?HjB )x B D?HjB *x B D?~~HjB +x B D?HjB ,x B D?HjB -x B D?HjB .x B D?XXHjB /x B D?. . DjB 0x B D? DjB 1x B D?FFDjB 2x B D?..DjB 3x B D?DjB 4x B D? DjB 5x B D? DjB 6x B D?  DH x 0޽h ? ̙33y___PPT10Y+D=' = @B +E} sK0 **P)))(    H ? y, ,$ 0 ,f(x) = f(w) ( 2AAAAAaaaa4    x  c $ VWwf     0 ? `I S "( 2A  %  H ? ` ; Correctness: f(x) can be efficiently decoded from g(x,r).<( 2 GCCCCCGCCCCCCC@  "     H4 ? 3px ,$ 0 F Privacy: $ efficient simulator S s.t. S(f(x)) a" g(x,U) g(x,U) depends only on f(x) 95( 2(  2GC   AAAA AAAACCC CGCCCCCC"             H,3 ?y=,$D 0 ,f(x) `" f(w) ( 2AAAAAaaaa4       H? ?$ ,D,$ 0 ]r, 2AI  2   H ?"`M d,$@ 02   H ?"` ,$@ 02   H ?"`,$@ 02    A*?Wide upward diagonal"@i ,$@ 0   B ? Q,$@ 0  HE ? ,$ 0 Mw 2A  2  H ?"`:  g,$@ 06  HDJ ?[m p{,$ 0 g(w,U)F( 2AAAA  2  H ?"` * ,$@ 02    HaHaA.?Wide downward diagonal"`9 8 x,$@  02  H ?"` ,$@! 06  HQ ?] } ,$" 0 g(x,U)F( 2AAAA    B ? # ,$@# 0   HX ?/?=,$$ 0 ]r, 2AI    H] ? ;,$% 0 Yx( 2AA  &   `a _ж_ж?" bF,$) 0 Ta"" 2g  B   `_ж_жD?m ,$D+ 0   H f ?<\,$ 0 ]r, 2AI  2  H ?"`e:,$@ 02  H ?"`*W,$@ 02   A*?Wide upward diagonal"@ 6,$@ 0  B ?P3 i,$@ 0  Hk ?),$ 0 Mw 2A  2   H ?"`R3 ` ,$@ 06 ! H$p ?s ,$  0 g(w,U)F( 2AAAA  2 " H ?"` 8,$@  02 #   HaHaA.?Wide downward diagonal"`Q< z ,$@  02 $ H ?"` L ,$@  06 % Hw ?u ,$  0 g(x,U)F( 2AAAA   & B ?6 ,$@ 0  ' H ?5U,$ 0 ]r, 2AI   ( HT ?3}S,$ 0 Yx( 2AA  2 ) H ?"`O ,$@ 0H  0޽h ?O   "$& ̙33QQ___PPT10Q+r,DM' = @B DM' = @BA?%,( < +O%,( < +DB' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =4@BBBB%(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' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D+' =4@BBBB%()?)?D' =.G7 BBBBBYM 4.16667E-6 -3.33333E-6 L 0.01979 0.09561 *3>*B ppt_xB ppt_y=@0BBAApBB"""<BQC=<*D-' =4@BBBB%()?)?D' =.I7 BBBBB[M -8.33333E-7 -1.85185E-6 L 0.01979 0.10486 *3>*B ppt_xB ppt_y=@0BBAApBB"""<BHV=<*D4' =A@BBBB0B%()?)?D{' =.C7 BBBBBUM 2.77778E-6 2.96296E-6 L 0.20868 0.1199 *3>*B ppt_xB ppt_y=@0BBAApBB[=Bu=<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+!  ++0+%  ++0+'  ++0+(  +  { rj?(  ~  s *p       00!  "6 ?'B<$ 0   B :@  f_ж_жD)? Y,$D 0 ; ZH2_ж_ж?" hj C,$D 0 Or8Fa  El i ?i,$@ 0 3  f)_ж_ж?"   lxA,(2ai   4  f/ _ж_ж?"    nxB.( 2ai   5 # l4 _ж_ж?"    MAlice(2   6 # l1 _ж_ж?"    KBob(2   7  `6 _ж_ж?"   MCarol(2  B 8  f_ж_жD)?= B 9B  f_ж_жD)?  3 D <  `H? _ж_ж?" jI3 i f(xA,xB)X (2aiaia   =  `E _ж_ж?" s  ,$ 0 gA(xA,r)n ( 2ememe   >  `4M _ж_ж?" s  ,$ 0 gB(xB,r)n ( 2ememe  H  0޽h ? ̙33___PPT10+20D' = @B Df' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(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' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(++0+  ++0+;  ++0+=  ++0+>  +   (  ~  s *       0  "6 ?'B<$ 0     # l8 _ж_ж?"   lxA,(2ai    # l\ _ж_ж?"   nxB.( 2ai     3 r _ж_ж?"  -  MAlice(2     3 rX _ж_ж?"   -  KBob(2      fl _ж_ж?"  0  MCarol(2  B   # l_ж_жD)?J =  B  @ # l_ж_жD)?J 3  l  ,$D 0@  Z _ж_ж?" ` } rRGZ8 Fc  9   fx _ж_ж?"  a  mA+mBJ(2aiai  D   ` _ж_ж?"   xA+r\( 2emem  D   `, _ж_ж?"    xB-r\( 2emem  B @  f_ж_жD)?M ,$D 0H  0޽h ? ̙33___PPT10+3D7' = @B D' = @BA?%,( < +O%,( < +D' =%(%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+  +  @T(  ~  s *4       00  "6 ?'B<$ 0   B @  f_ж_жD)?M ,$D 0   f _ж_ж?"   lxA,(2ai     f _ж_ж?"   nxB.( 2ai    # lP _ж_ж?"  -  MAlice(2    # l _ж_ж?"   -  KBob(2      ` _ж_ж?"  0  MCarol(2  B    f_ж_жD)?J =  B  @  f_ж_жD)?J 3  z    ,$D 0   Z|$ _ж_ж?" ` } &r1RF \ {0} , r2RF8 Fckck  m   `X" _ж_ж?"  a  mA=mB ?h(2aiaiai4    _   `6 _ж_ж?"   r1xA+r2( 2ememem  _   `> _ж_ж?"    r1xB+r2( 2ememem  H  0޽h ? ̙33___PPT10+ᖤD7' = @B D' = @BA?%,( < +O%,( < +D' =%(%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +F/  |t- (  ~  s *O       0P  "6 ?'B<$ 0   B @  f_ж_жD)? ) ,$@ 0   fQ _ж_ж?" 9 TY,$ 0  00000100000 1 \( 2 eem  ~   `dY _ж_ж?" V Tv ,$ 0  00000000000 0 \( 2 eem  l  D 'S Z7 ,$D  0  # # l_ _ж_ж?"   D nts.( 2ai    $ # ld _ж_ж?" 3 D nts+1.( 2ai    % # lh _ж_ж?" cD nts-1.( 2ai   & 3 rl _ж_ж?" - Y.....(2  z  D ( S Z7 ,$D  0 )  fr _ж_ж?"   D lt1.( 2ai   *  fw _ж_ж?" 3 D lt2.( 2ai   +  f<{ _ж_ж?" cD ltm.( 2ai    , # lt _ж_ж?" - Y.....(2  J -  f _ж_жG0Hq?"   ,$D  0 rExponential complexity ( 2a  H  0޽h ?- ̙33rj___PPT10J+sDN' = @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<*"K%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Kv%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*v%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(D2' =%(D' =4@BBBB%()))D' =+4 8?`CB ppt_xBCB1+ppt_w/2B*Y3>B ppt_x<*(D ' =+4 8?XCB ppt_yBCB ppt_yB*Y3>B ppt_y<*(D' =1:Bhidden*o3>+B#style.visibility<*(%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*'D ' =%(D' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bdissolve*<3<*-++0+  ++0+  ++0+  ++0+-  +<= ~ ''38 '(  ~  s *h       0d  "6 ?'<$ 0     Zl _ж_ж?" { h,$D 0 Or8Fa  l g g,$@ 0B   f_ж_жD)?    # l _ж_ж?" g  lxA,(2ai     # l@ _ж_ж?" g B nxB.( 2ai     3 r _ж_ж?" I Z  MAlice(2     3 r$ _ж_ж?" S KBob(2  J   f _ж_ж?"  < f(xA,xB)X (2aiaia  zl T    T ,$@ 0B  B # l_ж_жD)?g g Z   ` _ж_ж?" T   gA(xA,r)n ( 2ememe  l s   2s   ,$D 0@ s v  s v   3 r _ж_жԔ?"  v 3  POT(2a  B   f_ж_жDo?js j B   f_ж_жDo?s  B   f_ж_жDo?j9 j B   f_ж_жDo?9  N s v    s   3 rx _ж_жԔ?"  v 3  POT(2a  B   f_ж_жDo?js j B   f_ж_жDo?s  B   f_ж_жDo?j9 j B   f_ж_жDo?9  N s v    , s  ! 3 r _ж_жԔ?"  v 3  POT(2a  B "  f_ж_жDo?js j B #  f_ж_жDo?s  B $  f_ж_жDo?j9 j B %  f_ж_жDo?9  N s v  & Qs   ' 3 r _ж_жԔ?"  v 3  POT(2a  B (  f_ж_жDo?js j B )  f_ж_жDo?s  B *  f_ж_жDo?j9 j B +  f_ж_жDo?9  N s v  , vs   - 3 r _ж_жԔ?"  v 3  POT(2a  B .  f_ж_жDo?js j B /  f_ж_жDo?s  B 0  f_ж_жDo?j9 j B 1  f_ж_жDo?9  0 3  f _ж_жGH]k?"  % ,$D  0 XDishonest Alice?(2  "l c [ 8c [,$D 07 4 # ll _ж_ж?" c J  gn(0,r)N( 2gog  + 5  ` _ж_ж?" Fc [J  gn(1,r)N( 2gog  4 6  ` _ж_ж?" c  xnN( 2gog   7  ` _ж_ж?"  t,$  0 gn(xn,r)x ( 2gogog  H  0޽h ?3 ̙33G?___PPT10+~yD#' = @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' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(++0+  ++0+  ++0+3  ++0+7  +T   @6(   l    @  ,$D  0 ; Z_ж_ж?S" ?  < Z_ж_ж?S" ? Q  = Z_ж_ж?S" ? F  > Z_ж_ж?S" ? J  ? Z_ж_ж?S" ?   8 Z_ж_ж?S" ?` q 9 Z_ж_ж?S" ?`m q : Z_ж_ж?S" ?`pmq 7  `_ж_ж?S" ?`:-q~  s * I    6 S   <$ 0   H  0޽h ? ̙33VN___PPT10.+gD' = @B D' = @BA?%,( < +O%,( < +D,' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6 $%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6$9%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*69r%(D' =%(D@' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6s%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(+8+0+6  +<  ..55$.(  $x $ c $5 0@   F   $   $ Z@ _ж_ж?" Pc  f(x1,& ,xn) reduces to p1p2 & pm where: pi S5 Each pi depends on a single xj $ distinct s0,s1 S5 s.t. p1p2 & pm = sf(x) N,8 P2\8 P2             t  I       " $ 0s ? Thm [Barrington 86] Every boolean fNC1 can be computed by a poly-length, width-5 branching program.h( 2eaga>a4   F  $ 04 ?  ,$ 0 Encoding iterated group product p1p2p3 & pm p1r1 r1-1p2r2 r2-1p3r3 & rm-1-1pm  k( 2aeg            k lz *i  $ (*iF ,$D 0 $ HP  jJ? g  [g*( 2BB    $ 0 ?p   fr14( 2FN    $ H  jJ?*^ [f*( 2BB  rB  $ B D? rB  $ B D?yy rB  $ B D?)) rB $ B D?ZrB $ B D?  rB $ B D? rB $ B D?_ _ rB $ B D? rB $ B D? rB $ B D?s  s rB $ B D?  rB $ B D?e e rB $ B D?  rB $ B D?y y rB $ B D?  rB $ B D?  rB $ B D?  rB $ B D?S S rB $ B D?}}^rB $ B D?^rB $ B D?mm^rB  $ B D?UU^rB !$ B D?- - ZrB "$ B D?77ZrB #$ B D?  ZrB $$ B D?  ZN ]  %$  p  &$ 0, ?]  fx14( 2FN   '$ 0 ? P  fx24( 2FN   ($ 0$ ?  vxn6( 2FN  N ]  )$  z  *$ 04 ?]  fx14( 2FN   +$ 08 ? P  fx24( 2FN   ,$ 0p ?  vxn6( 2FN   -$ 0 ? H  fr24( 2FN   .$ 0 ? 9  |rm-1H( 2FNN   /$ # l _ж_ж?" `   J& (2   0$ # l _ж_ж?" # U  J& (2   1$ # lP _ж_ж?"  m  N..& (2  : 2$ # l _ж_ж?"  [ ~ p1r1P(2  x 3$ # l _ж_ж?" @ Q q r1-1p2r2 (2  n 4$ # l _ж_ж?" Fi rm-1-1pm| (2  * 5$ 0t   di,$ 0 Every output bit of g depends on just a single bit of x Efficient fully decomposable encoding for every fNC1 8 6  8c1cc o H $ 0޽h ? ̙33  ___PPT10 +DKD ' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D=' =%(D"' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*$%(D@' =A@BB@BB0B%()?)?D' =.O7 BBBBBaM 4.72222E-6 -1.11111E-6 L 4.72222E-6 -0.50092 *3>*B ppt_xB ppt_y=@0BBAApBBB<<*$D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5$%(++0+$  ++0+$  ++0+5$  +S  $.(  ~  s *       0  "6 ?<<$ 0   H  0޽h ? ̙33___PPT10++D[' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;o%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*o%(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' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$6%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6k%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*k{%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*{%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 7%(+8+0+  +f1  "!220!(  0~ 0 s *:     0  0;  "6 ?i<$ 0   lz *i  0 W ,$@ 0 0 HHB  jJ? g  [g*( 2BB   0 0G ?p   fr14( 2FN   0 H4F  jJ?*^ [f*( 2BB  rB 0 B D? rB  0 B D?yy rB  0 B D?)) rB  0 B D?ZrB  0 B D?  rB  0 B D? rB 0 B D?_ _ rB 0 B D? rB 0 B D? rB 0 B D?s  s rB 0 B D?  rB 0 B D?e e rB 0 B D?  rB 0 B D?y y rB 0 B D?  rB 0 B D?  rB 0 B D?  rB 0 B D?S S rB 0 B D?}}^rB 0 B D?^rB 0 B D?mm^rB 0 B D?UU^rB 0 B D?- - ZrB 0 B D?77ZrB  0 B D?  ZrB !0 B D?  ZN ]  "0  p  #0 0P ?]  fx14( 2FN   $0 0U ? P  fx24( 2FN   %0 0LY ?  vxn6( 2FN  N ]  &0  z  '0 0] ?]  fx14( 2FN   (0 0b ? P  fx24( 2FN   )0 0+B#style.visibility<*0%%(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%K%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*20%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0P[%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0[y%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0y%(+p+0+0  ++0+20  +  `'' (   ~   s *        0 "6 ?'<$ 0   8 =C ' =C N     =C Z2   s *3^-Z2   s *^Z2   s *>^Z2   s *^O Z2   s * ^ Z2   s *` ^ Z2   s * ^q Z2   s * ^' Z2   s * ^ Z2   s *8 ^ Z2   s * ^IZ2   s * 2 Z2   s *  Z2   s *C  Z2   s * T Z2   s * Z2   s *e Z2   s * v Z2   s * , Z2   s * Z2   s *= Z2   s * N ZB   s *DԔ 1ZB   s *DԔ  1ZB   s *DԔ 1Z2   s *^Z2   s *Y^Z2   s *  Z2 !  s *^  ZB "  s *DԔ  1ZB #  s *DԔ  ` 1ZB $  s *DԔ  3 1 %   `X _ж_ж?" 0   Mx(2a   &   ` _ж_ж?" 3   Mr(2a  H   0޽h ? ̙33  ___PPT10 +!lD3 ' = @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<* e%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* f%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+ ' + K0 05&(    Z_ж_жP?S" ?+  Z_ж_ж<?S" ?  Z_ж_жd?S" ?Xi  Z_ж_жx?S" ?   Z_ж_ж?S" ?    Z_ж_ж?S" ? ! ~  s *      <X U  k poly-time& 0  gf     < %)  dNC&0 g    <p cM w ] k log-space& 0  gf     < +  {NC1<0 go    <  2  {AC0<0 go    < . + (  {NC0<0 go   z    ^  ,$D 0Z2  s *3^-Z2  s *^Z2  s *>^Z2  s *^O Z2  s * ^ Z2  s *` ^ Z2  s * ^q Z2  s * ^' Z2  s * ^ Z2  s *8 ^ Z2  s * ^IZ2   s * 2 Z2 ! s *  Z2 " s *C  Z2 # s * T Z2 $ s * Z2 % s *e Z2 & s * v Z2 ' s * , Z2 ( s * Z2 ) s *= Z2 * s * N ZB + s *DԔ 1ZB , s *DԔ  1ZB - s *DԔ 1Z2 . s *^Z2 / s *Y^Z2 0 s *  Z2 1 s *^  ZB 2 s *DԔ  1ZB 3 s *DԔ  ` 1ZB 4 s *DԔ  3 1 5 # l _ж_ж?" i-' qHow low can we get?"(2e  H  0޽h ? 33___PPT10n.U$]+iGRDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ K0 Q I @  (  r  S < w'   $l T)  F T,$D 0 E N ?T) I  u Tempting conjecture:0( 2ca   F Zh  ?"`&  ]crypto hardness(2c   G T ? ` B J Z  ?"` J t$ complex function ( 2c   o c $  MTY 8___PPT10N___PPT90( Longstanding open question Hstad 87 Impagliazzo Naor 89 Goldreich 00 Cryan Miltersen 01 Krause Lucks 01 Mossel Shpilka Trevisan 03 Real-life motivation: super-fast cryptographic hardware$ k% 8$ ccigg8g                     =  bl | <   Y,$D 0 I N) ?|  v m [CM]: Yes4 ( 2gc    H9 ?{ B < k[G]: No4( 2gc  H  0޽h ? ̙33___PPT10+` fPD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+L d ## -YR#(  x  c $XC Vf   '   fZD  ?"`}$ ,$@ 0 Of(2c  B  NZ D?  ,$@ 0B  NZ D?/ / ,$@ 0B  NZ D?J J ,$@ 0  <Z@J ?"` I ,$ 0 P& .( 2B  B   NZ D?p0,$@ 0B   NZ D?U p0U ,$@ 0   <Z`O ?"`( f`j ,$ 0 P& .( 2B  B   NZ D? f& ,$@ 0  TS ?"` ,$@  0 I    B  NZ D?$ f&$ ,$@  0B  NZ D? ^ ,$@  0B  NZ D?L ^L ,$@  0  <ZX ?"`a,$  0 P& .( 2B  B  NZ D?T,$@ 0B  NZ D?T,$@ 0@ # 6] ?"`  ,$ 0 f(Uin)T( 2  ( $ 6c ?"` M ,$ 0 Uout>( 2   % Nh ?"` p,$ 0 Y poly-time ( 2 c   & nA 6 confused%2520man2%2520copy"`w L$@ 0U 0 + Hm  jJ?/^2 af0( 2cc  F R 9 ZrB / B D?PrB 0 B D?PrB 1 B D?PrB 4 B D?RrB 5 B D?""H  8 # lt _ж_ж?" ;[ aOWF"(2e  L R :# Z^(rB ; B D?PrB < B D?PrB = B D?PrB > B D?RrB ? B D?""H B 0z ?"`@' Uin>( 2  D E 0 ?"`@ '  y = f(Uin) ( 2&    N H ?"` t Y poly-time ( 2 c   O hA 6 confused%2520man2%2520copy"`4 M U 0J P B GH\F?"` b find xf -1(y)d(2&   ? Q # lD _ж_ж?" q ,$ 0 aPRG"(2e  ! X 0L ?"`  ,$ 0 Uin>( 2  B Y Z_ж_жD?W  H GCHg\F?"`% y ,$@ 0 Pseudorandom or Random?V(2 gcgffc  H  0޽h ?/ P ̙33(}(___PPT10](+D%' = @B D%' = @BA?%,( < +O%,( < +D$' =%(Ds$' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(+h+0+  ++0+  ++0+   ++0+  ++0+  ++0+#  ++0+$  ++0+%  ++0+Q  ++0+X  ++0+  + T DD`=V  pC(   U /   `X _ж_ж?"`  ,$@' 0 AC0D( 2ckk  U .   ` _ж_ж?"` .& ,$@& 0 AC0D( 2ckk     0 ?G M Compile primitives in a  relatively high complexity class (e.g., NC1, NL/poly, L/poly) into ones in NC0.k( 2aeai aaema k ~   s *l Nn      H ?jdQ ,$ 0 H ( 2c  F   0 ? ,$ 0 NC1 cryptography implied by factoring, discrete-log, lattices& essentially settles open questiond aia ea eaea$ d    BD ?"`2 m cOWF$0 c     \A* ?nowack-7-steamroller"`d`,$D 0z J F    ` ,$@ 0   <t ?"`J F  L$0 c     \ W??OWFArial"` z # F    `  ,$@ 0`B  B 0Dfo?# F `B  B 0Dfo?~ F `B   0Dfo?  F `B   0Dfo? F t2   6o?"` J   3 r _ж_жG=Hu??"` ,$D 0 f locality 4 ( 2 c      TA?"`x e! h  $ 0U 0  \    f _ж_ж?"`.&,$@ 0 NC02D( 2ckk  \    f _ж_ж?"` *",$@ 0 NC03D( 2ckk  [    fL _ж_ж?"` *" ,$@ 0 NC0D( 2ckk  [    f| _ж_ж?"`*" ,$@ 0 NC1D( 2ckk  \    fd$ _ж_ж?"` ,$@ 0 NC02D( 2ckk  \    f* _ж_ж?"` ,$@ 0 NC03D( 2ckk  [    f/ _ж_ж?"` "  ,$@ 0 NC0D( 2ckk  [    fx4 _ж_ж?"`  ,$@ 0 NC1D( 2ckk  1    `H: _ж_ж?"  ,$ 0 _PRG  2e  1    ` _ж_ж?" 2,$ 0 _OWF  2e     # lA _ж_ж3f?"`  ,$@ 0 @(2   !   `_ж_ж?"`  ,$@ 0 "   `_ж_ж?"`[  ,$@ 0 $   `+B#style.visibility<* %(DH ' =%(D' =%(D=' =4@BBBB%(D' =2y;BplayFrom(0.0)*<3<* D' =%(D5' =4@BB?BB%()?)?D'  =.Q7 BBBBBcM -3.33333E-6 -2.22222E-6 L 0.60834 -2.22222E-6 *3>*B ppt_xB ppt_y=@0BBAApBB>B<* D' =%(D~' =A@BB BB0B%())))?D' =-o6Bdissolve*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D= ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(DI' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =2g;B stop*<3<* %(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* DR' =%(D^R' =%(D' =K@BB BBPB0B%(/%(.)))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*! %(D' =-o6Bdissolve*<3<*! Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*" %(D' =-o6Bdissolve*<3<*" Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$ %(D' =-o6Bdissolve*<3<*$ Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*& %(D' =-o6Bdissolve*<3<*& Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*' %(D' =-o6Bdissolve*<3<*' Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*) %(D' =-o6Bdissolve*<3<*) Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<** %(D' =-o6Bdissolve*<3<** Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*+ %(D' =-o6Bdissolve*<3<*+ Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =-o6Bdissolve*<3<*, Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*- %(D' =-o6Bdissolve*<3<*- Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*. %(D' =-o6Bdissolve*<3<*. Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*/ %(D' =-o6Bdissolve*<3<*/ Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =-o6Bdissolve*<3<*0 Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*1 %(D' =-o6Bdissolve*<3<*1 Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*2 %(D' =-o6Bdissolve*<3<*2 Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*3 %(D' =-o6Bdissolve*<3<*3 Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =-o6Bdissolve*<3<*4 D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F %(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*I %(D' =-o6Bdissolve*<3<*I Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*L %(D' =-o6Bdissolve*<3<*L Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*O %(D' =-o6Bdissolve*<3<*O D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E %(D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*O D' =1:Bhidden*o3>+B#style.visibility<*O %(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q %(D' =-o6Bdissolve*<3<*Q D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T %(D' =-o6Bdissolve*<3<*T D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R %(D' =-o6Bdissolve*<3<*R D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U %(D' =-o6Bdissolve*<3<*U D@' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*S %(D' =-o6Bdissolve*<3<*S Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*V %(D' =-o6Bdissolve*<3<*V D'  =$ B B`B?<* %(/%,( < +/%,( < +)?++0+/   ++0+.   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+   ++0+$   ++0+&   ++0+'   ++0+*   ++0++   ++0+,   ++0+-   ++0+1   ++0+2   ++0+3   ++0+4   ++0+E   ++0+F   ++0+L   ++0+Q   ++0+R   ++0+S   ++0+T   ++0+U   ++0+V   +;w ~"v"$8@!(    s *( IY    ^Encoding a OWF  g$     H, ?9> vThm. f(x) is a OWF g(x,r) is a OWF Proof: inverter B for g inverter A for fdQ( 2ea&N      6 "  H5 ̙?t ,$ 0 vg(x,r)=z0 ( 2cc      H0: ? ,$D  0 tf(x)=y0( 2cc  pl d"  1" ,$D 0  N? ?  ix8( 2gg  @ d"  0d"   T ?"`  T ?"`Wy@ d"W  ,d"W    HE ?W   gA6( 2ck    3 r_ж_жqo?" )"W   TJ ̙?"`W W Simulator (2 c    T ?"`G ! NN ?d iy8( 2gg  l )  /F ,$@ 0  N ?"` @ )  +)    f_ж_жo?"     T ?"`  A  3 r0T _ж_ж?S" ? C qB4(2ck     NhX ?6) (x,r)8( 2gg&    " Nx\ ?  ez4( 2gc   & 0`  s 'G ,$ 0 TA succeeds whenever B succeeds Dec(z) = Dec(g(x,r)) = f(x) Dec(z) = Dec(Sim(y)) = y  Z5 Z Zccc             Hl   7 ,$D  0 $ NDl ?  zR g(Un,Um) ( 2cckckc4      ( # lv _ж_ж?"    nprob p(2c&   Jl =  8= ,$@  0|  NT{ ?t  yR f(Un) ( 2cckc4    J ) # l _ж_ж?"  =  prob p@(2cc&    2 0  'G,$  0 EA generates a correct input distribution for B Sim(f(Un)) = g(Un,Um) / Z Z Z/c@/      3 B ?t ,$@ 0 vg(x,r)=z0 ( 2cc   l  V   6V  ,$@ 04 4 Z( _ж_ж?"  V =  Dec(g(x,r)) = f(x)(2c@      h 5 Z  _ж_ж?"  9   $Sim(f(x)) g(x,r)@(2 cc@     H  0޽h ? ̙3391___PPT10+<D' = @B D`' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(Du ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(+P+0+  ++0+  ++0+   ++0+&  ++0+2  ++0+3  +^= k %%05\8X%(  8 8 S  T<$ 0   l 6 m 68} p ,$@ 0 58 Z_ж_ж?S" ?:V Jf  8 Z_ж_ж?S" ?6 )F  8 Z_ж_ж?S" ?*6 :F  8 Z_ж_ж?S" ?;6 KF   8 Z_ж_ж?S" ?L6 \F  !8 Z_ж_ж?S" ?]6 mF  "8 Z_ж_жf3?S" ?ZF jV  #8 Z_ж_ж?S" ?WV gf  $8 Z_ж_ж?S" ?Tf dv %8 Z_ж_жf?S" ?Zvj &8 Z_ж_жf?S" ?JvZ '8 Z_ж_ж?S" ?:vJ (8 Z_ж_ж!?S" ?*v: )8 Z_ж_жff?S" ?v* *8 Z_ж_ж̙?S" ?f *v +8 Z_ж_ж?S" ?V *f  ,8 Z_ж_ж3?S" ?F *V  -8 Z_ж_ж̙?S" ?*F :V  .8 Z_ж_жf?S" ?:F JV  /8 Z_ж_жF?S" ?JF ZV  08 Z_ж_ж?S" ?JV Zf  18 Z_ж_жf3?S" ?Jf Zv 28 Z_ж_ж3?S" ?:f Jv 38 Z_ж_ж3f?S" ?*f :v 48 Z_ж_ж?S" ?*V :f  8  f_ж_жԔ?" 6 jl u' < T83 ' ,$D 0 88  `_ж_ж?S" ?   98  `_ж_ж?S" ?   :8  `_ж_ж?S" ?g w6  ;8  `_ж_ж?S" ?[ jc  <8  `_ж_ж?S" ?"  0  >8  `_ж_жf3?S" ? 0  @8  `_ж_ж?S" ? 0  A8  `_ж_жf?S" ?  B8  `_ж_жf?S" ?Z C8  `_ж_ж?S" ? D8  `_ж_ж!?S" ?gw E8  `_ж_жff?S" ?:f Jv F8  `_ж_ж̙?S" ?9 I G8  `_ж_ж?S" ? H8  `_ж_ж3?S" ?s   I8  `_ж_ж̙?S" ?   J8  `_ж_жf?S" ?w   K8  `_ж_жF?S" ?   M8  `_ж_жf3?S" ?  N8  `_ж_ж3?S" ?f v P8  `_ж_ж?S" ?g w  Q8 # l_ж_жԔ?" u' < Y8 B  Vf    H 8 0޽h ? ̙33*"___PPT10+m@D' = @B Di' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8'%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8'2%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*82W%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8Wo%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T8%(D ' =%(D( ' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*T8D' =1:Bhidden*o3>+B#style.visibility<*T8%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*68%(D' =-o6Bdissolve*<3<*68D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8M%(+8+0+8  + \ y (  x  c $$       S   <$ 0     0| ? `I S "( 2a  H  0޽h ? ̙33___PPT10+&D' = @B D;' = @BA?%,( < +O%,( < +D ' =%(%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"C%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C_%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(+8+0+  + D p #(  x  c $x>'0@     0 ?TZ r* -biased generators [Mossel Shpilka Trevisan 03]: superlinear stretch in NC05 Current work: linear stretch in NC03 optimal locality, stretch( 2=(   '(   %(  cccc ccggegcc gckokgg gcgcko o  c  o           U    00 ?S 5  $ PRGs for space-bounded computation @%( 2#cai&    H  0޽h ? ̙33y___PPT10Y+D=' = @B +! 8 ~ 0(  0x 0 c $1 0@     0 H(3 ?G How to encode  complex f by g NC0? Observation: enough to obtain const. degree encoding(( 271( 2 cca e+a _   0 H@ ?M,S  zLocality Reduction: degree 3 poly over GF(2) locality 4 rand. encodingI%(  eaea  I `0xx  0  `$B _ж_ж?" s Dm  f(x) = T1(x) + T2(x) + & + Tk(x)a( 2  - 4 X   4z &    0 9 T`,$D 0p  0  `0] _ж_ж?" &    g(x,r,s) = ( 233  Z 0 # l8g _ж_ж?"  Z  T1(x)+r1d (2   N 0  `p _ж_ж?" z  T2(x)+r2d (2   ` 0 # lxy _ж_ж?" V   Tk(x)+rkh (2    0  f _ж_ж?"  (  f& ((233  :z   j)  0 :,$D 0d 0 # lp _ж_ж̙?"  / )    r1+s1j(2"3333   0 # l< _ж_ж̙?" - / # )   s1  r2 +s2 (23333333333    0 # l _ж_ж̙?" )/ j)   sk-1 rkv (2333333&    0  f _ж_ж?"  (  f& ((233  @ 0  f _ж_жGOH|?" ,$D 0 hComing up&  (2    H 0 0޽h ?0 ̙33C;___PPT10+uXD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+8+0+0  +w ^ )o!o 0 n(  x  c $@       H ?<\ Composition Lemma:<$(  e  @xX8 S W p'   Nx  jJ? W Yf(( 2@B  ~B   N Do?Y  ~B  N Do?Y  ~B  N Do?Y  B  T Do?**B  T Do?B  T Do?~B  N Do?V  ~B  N Do?cV c B  T Do?B  T Do?ssxB i H Do?V  xB j H Do?<S < P8 @ wF @ wN 1a t MA~B u N Do?1~B v N Do?a1a~B w N Do?1~B x N Do?1~B y N Do?1 8 N  jJ?@ w  h encodes B( 2B B$     ~B F N Do?<<~B G N Do?~B H N Do?~B I N Do?~B J N Do?mm~B K N Do?~B L N Do?~B M N Do?WW~B N N Do?  ~B O N Do???~B P N Do?~B Q N Do?~B R N Do?~B S N Do?k k ~B T N Do?  ~B U N Do?  B X T Do?WWFB Y T Do?||FB Z T Do?FB ^ T Do?  CB _ T Do?/ / CB c T Do?[ [ CB d T Do?  CB e T Do?  CB f T Do?CB g T Do?CB h T Do?""C~B k N Do?gg~B l N Do?~B m N Do?# #   r # l _ж_ж?" G ) Wg(2  l m   m  ,$D 0N 1a { d z ~B | N Do?1~B } N Do?a1a~B ~ N Do?1~B  N Do?1~B  N Do?1  H  jJ?m ^k  h encodes N( 2B B B$     xB  H Do?iiWxB  H Do?WxB  H Do?  WxB  H Do?11WxB  H Do?WxB  H Do?WxB  H Do?WxB  H Do?WxB  H Do?  TxB  H Do?llTxB  H Do?TxB  H Do?TxB  H Do?<<TxB  H Do?  TxB  H Do?  TxB  H Do?& & T~B  N Do?i  ~B  N Do?i  ~B  N Do?i  ~B  N Do? f ~B  N Do?\ f \ ~B  N Do? f ~B  N Do? f ~B  N Do? f  ~B  N Do?f  ~B  N Do?f  ~B  N Do?Of O xB  H Do?TxB  H Do?QxB  H Do?P P T   `x _ж_ж?" sT  X f(2   F d j ٨   ڨ H  jJ?df js  g encodes fF ( 2BBBB  ~B ۨ N Do?w ~B ܨ N Do?w ~B ݨ N Do?w ~B ި N Do?|~B ߨ N Do?T|T~B  N Do?t ~B  N Do?nqn~B  N Do?nxB  H Do?y y_ xB  H Do? _ xB  H Do? _ xB  H Do?+ +_ xB  H Do? _ xB  H Do? \ xB  H Do?| |\ xB  H Do? \ xB  H Do? \ xB  H Do?L L\ xB  H Do? \ xB  H Do? \ xB  H Do?6 6\ ~B  N Do?,x,~B  N Do?xl '6  6 ' ,$D 0$   Nt ? '6 V  Concatenation Lemma:<$(  e  @xX7  N   jJ? ;  g(1) encodes f(1)p( 2BJBBBJB  B  T Do? B  T Do? ]]B  T Do? SSB  T Do? B  T Do? B  T Do? B  T Do? ~B  T Do? @@{~B  N Do?  5 ~B  N Do? Q Q5 ~B  N Do?  5 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do? W W2 ~B  N Do?  2 B   T Do? ))B   T Do? nn X 3 rx _ж_ж? " f N V& (2    NH  jJ? -;  g(l) encodes f(l)( 2BJJJBBBJJJB4     B  T Do? B  T Do?   B  T Do? B  T Do? XXB  T Do? B  T Do?   B  T Do? D D ~B  T Do?   {B  T Do?   B  T Do?   ~B ݩ N Do?  5 ~B ީ N Do?  5 ~B ߩ N Do? 3 3 5 ~B  N Do? O O2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do? 8 8 2 l P  Y  P Y,$D 0 Y  f_ж_ж? " P ` v  N8.  jJ?  ; Y m g encodes f2 ( 2@@@  B  T Do? nnB © T Do? B é T Do? B ĩ T Do? B ũ T Do? ccB Ʃ T Do? JJB ǩ T Do? ~B ȩ T Do? {B ֩ T Do? B ש T Do? B ة T Do?   B ٩ T Do? B ک T Do? ffB ۩ T Do? OOB ܩ T Do? ~B  N Do?  5 ~B  N Do?  5 ~B  N Do? g g5 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do? l l2 ~B  N Do?  5 ~B  N Do?  5 ~B  N Do? g g5 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do?  2 ~B  N Do? l l2   3 r = _ж_ж? "  )9  V& (2    3 rA _ж_ж? " v  Z& (2f  H  0޽h ? ̙33___PPT10+VhD' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+;%  wop S(  Px P c $@M Z-j     Q # lLO _ж_ж?" n  H&  2   KQ HQ ?l af (1),( 2   R HtW ?l af (2),( 2  F Q  VR u o5N  0 6 WR Q l2 XR < ?nxB YR H D?JN  ZR nl2 [R < ff?xB \R H D?JN  ]R n l2 ^R < ff?xB _R H D?l2 `R < ff? nPJN  aR ~l2 bR < ff?xB cR H D?JN  dR ~ l2 eR < ff?xB fR H D?JN  gR  ~@l2 hR < ff?xB iR H D?l2 jR < ff?@~pJN  kR  l2 lR < ff?xB mR H D?JN  nR  @l2 oR < ff?xB pR H D?JN  qR @` l2 rR < ff?xB sR H D?xB tR H D?xB uR H D?~xB vR H D?xB wR H D?0 xB xR H D?0 xB yR H D?P@xB zR H D?P@xB {R H D?p` l2 |R < ?`   }R <_ ?06 _s.( 2FB   ~R 0La ? $0 V D( 2B  h R c $A  ??O>  h R c $A  ??i  h R c $A  ??`.  h R c $A  ??y'  h R c $A  ??   h R c $A  ??PE  h R c $A  ??N  h R c $A  ??i  h R c $A  ??P  h R c $A  ?? t>  h R c $A  ??wE  h R c $A  ??  h R c $A ! ?? ! h R c $A " ??Y> " h R c $A # ??YN # h R c $A $ ?? $ h R c $A % ??5 %  R  fl _ж_ж?"    Mt 2  F Q  R n5N  0 6 R Q l2 R < ?nxB R H D?JN  R nl2 R < ff?xB R H D?JN  R n l2 R < ff?xB R H D?l2 R < ff? nPJN  R ~l2 R < ff?xB R H D?JN  R ~ l2 R < ff?xB R H D?JN  R  ~@l2 R < ff?xB R H D?l2 R < ff?@~pJN  R  l2 R < ff?xB R H D?JN  R  @l2 R < ff?xB R H D?JN  R @` l2 R < ff?xB R H D?xB R H D?xB R H D?~xB R H D?xB R H D?0 xB R H D?0 xB R H D?P@xB R H D?P@xB R H D?p` l2 R < ?`   R <| ?06 _s.( 2FB   R 0T ? $0 V D( 2B  h R c $A & ??O> & h R c $A ' ??i ' h R c $A ( ??`. ( h R c $A ) ??y' ) h R c $A * ??  * h R c $A + ??PE + h R c $A , ??N , h R c $A - ??i - h R c $A . ??P . h R c $A / ?? t> / h R c $A 0 ??wE 0 h R c $A 1 ?? 1 h R c $A 2 ?? 2 h R c $A 3 ??Y> 3 h R c $A 4 ??YN 4 h R c $A 5 ?? 5 h R c $A 6 ??5 6  R  f _ж_ж?"    Mt 2   R H̍ ? wf (l)B( 2  ! R # l _ж_жf?"`` 0 w poly-size BPs ( 2c&    +?l [ <  So,: ,$@ 0N  &u 'Q [   (Q Hl  jJ?"` & D( 2@  rB )Q B D? s rB *Q B D? s rB +Q B D?sN   H ,Q   HxB -Q H D?  HxB .Q H D?LMHxB /Q H D?  H 0Q 6 ?"`  N& ( 2B   1Q 64 ?"`Cj R&  ( 2F  N   H 2Q   HxB 3Q H D?  HxB 4Q H D?5 6 HxB 5Q H D?  H 6Q 6 ?"`#  u R&  ( 2F   7Q 6T ?"`#   N& ( 2B  rB 8Q B D?` q` B 9Q  ` D??  B :Q Z D? 6 B ;Q  ` D??  B Q T D? B ?Q T D?5 B @Q T D?LB AQ T D?  B BQ T D?   CQ 6ȱ ?"`   N& ( 2B  B DQ T D? a B EQ T D?  B FQ T D? B GQ T D? B HQ C x_ж_жD??  B IQ # l_ж_жD?  *@ p <  Sp < N  &u P p   P H\  jJ?"` & D( 2@  rB  P B D? s rB  P B D? s rB  P B D?sN   H  P   HxB  P H D?  HxB P H D?LMHxB P H D?  H P 6  ?"`  N& ( 2B   P 6 ?"`Cj R&  ( 2F  N   H P   HxB P H D?  HxB P H D?5 6 HxB P H D?  H P 6 ?"`#  u R&  ( 2F   P 6< ?"`#   N& ( 2B  rB P B D?` q` B P  ` D??  B P Z D? 6 B P  ` D??  B P T D? g B P T D? LB P T D? B P T D?5 B  P T D?LB !P T D?  B "P T D?   #P 6 ?"`   N& ( 2B  B $P T D? a B %P T D?  B &P T D? B 'P T D? B (P C x_ж_жD??  B )P # l_ж_жD?  N  &u Q wp  Q HD  jJ?"` & D( 2@  rB Q B D? s rB Q B D? s rB Q B D?sN   H  Q   HxB  Q H D?  HxB  Q H D?LMHxB  Q H D?  H  Q 6 ?"`  N& ( 2B   Q 68 ?"`Cj R&  ( 2F  N   H Q   HxB Q H D?  HxB Q H D?5 6 HxB Q H D?  H Q 6h ?"`#  u R&  ( 2F   Q 6  ?"`#   N& ( 2B  rB Q B D?` q` B Q  ` D??  B Q Z D? 6 B Q  ` D??  B Q T D? g B Q T D? LB Q T D? B Q T D?5 B Q T D?LB Q T D?  B Q T D?    Q 6 ?"`   N& ( 2B  B !Q T D? a B "Q T D?  B #Q T D? B $Q T D? B %Q C x_ж_жD??  B &Q # l_ж_жD?   Q # lLd _ж_ж?" P S s  H&  2   R # l _ж_ж?"`p <'  c degree 3 *  c  >2 R 3 r _ж_ж??"` ,$D 0 Z coming up...   c  P2 R S ~ _ж_жG8H??"`/ SF ,$D 0 `locality reduction c  K2 R S ~ _ж_жG3H??"`| &e ,$D  0 [ concatenation c  F Q  R Iu5N  0 6 R Q l2 R < ?nxB R H D?JN  R nl2 R < ff?xB R H D?JN  R n l2 R < ff?xB R H D?l2 R < ff? nPJN  R ~l2 R < ff?xB R H D?JN  R ~ l2 R < ff?xB R H D?JN  R  ~@l2 R < ff?xB R H D?l2 R < ff?@~pJN  R  l2 R < ff?xB R H D?JN  R  @l2 R < ff?xB R H D?JN  R @` l2 R < ff?xB R H D?xB R H D?xB R H D?~xB R H D?xB R H D?0 xB R H D?0 xB R H D?P@xB R H D?P@xB R H D?p` l2 R < ?`   R < ?06 _s.( 2FB   R 0  ? $0 V D( 2B  h R c $A  ??O>  h S c $A  ??i  h S c $A  ??`.  h S c $A  ??y'  h S c $A  ??   h S c $A  ??PE  h S c $A  ??N  h S c $A  ??i  h S c $A  ??P  h S c $A  ?? t>  h  S c $A ! ??wE ! h  S c $A " ?? " h  S c $A # ?? # h  S c $A $ ??Y> $ h  S c $A % ??YN % h S c $A & ?? & h S c $A ' ??5 '  S  f! _ж_ж?"    Mt 2  U2l  S8 ,$@ 0 P H% ?< B( 2  /@ T,w QRN  tY Q T, Q 0|) ?"` &w N& ( 2B   Q 0. ?"`T w N& ( 2B   N  tY Q  tYrB Q B D?VrB Q B D?==VrB Q B D?ddVrB Q B D?  VN  $ Q  $xB Q H D?$xB Q H D?$xB Q H D?  $xB Q H D?; < $xB Q H D?  $VN >$ Q >$xB Q H D?$xB Q H D?MM$xB Q H D?$xB Q H D?=>$ Q H ?"` Wt~B Q N D? W ~B Q N D? V; ~B Q N D? WM~B Q N D?; W~B Q N D? Wx~B Q N D?sW= Q 0 8 ?"`3 Y R&  ( 2F   Q 0; ?"`Y R&  ( 2F  XN  tY Q ,  Q 0? ?"` &w N& ( 2B   Q 0TC ?"`T w N& ( 2B  b N  tY Q  tYrB Q B D?VrB Q B D?==VrB Q B D?ddVrB Q B D?  VvN  $ Q  $B Q H D?C"?$B Q H D?C"?$B Q H D?C"?  $B Q H D?C"?; < $B Q H D?C"?  $VN >$ Q >$xB Q H D?$xB Q H D?MM$xB Q H D?$xB Q H D?=>$ Q H ?"` Wt~B Q N D? W ~B Q N D? V; ~B Q N D? WM~B Q N D?; W~B Q N D? Wx~B Q N D?sW= Q 0 M ?"`3 Y H ( 2F   Q 0P ?"`Y R&  ( 2F  XN  tY Q ,w Q 04U ?"` &w N& ( 2B   Q 0XY ?"`T w N& ( 2B  b N  tY Q  tYrB Q B D?VrB Q B D?==VrB Q B D?ddVrB Q B D?  VvN  $ Q  $B Q H D?C"?$B R H D?C"?$B R H D?C"?  $B R H D?C"?; < $B R H D?C"?  $VN >$ R >$xB R H D?$xB R H D?MM$xB R H D?$xB R H D?=>$  R H ?"` Wt~B  R N D? W ~B  R N D? V; ~B  R N D? WM~B  R N D?; W~B R N D? Wx~B R N D?sW= R 0$c ?"`3 Y H ( 2F   R 0f ?"`Y R&  ( 2F   R 3 rLi _ж_ж?"` ` >(2   R # lLn _ж_ж?" ~  H&  2  5 S # l0m _ж_ж?"`%E NC04 N ckk  /l s . S  ,$@ 0N  tY *P s  +P 0@x ?"` &w N& ( 2B   ,P 0{ ?"`T w N& ( 2B   N  tY -P  tYrB .P B D?VrB /P B D?==VrB 0P B D?ddVrB 1P B D?  VN  $ 2P  $xB 3P H D?$xB 4P H D?$xB 5P H D?  $xB 6P H D?; < $xB 7P H D?  $VN >$ 8P >$xB 9P H D?$xB :P H D?MM$xB ;P H D?$xB $ =P H ?"` Wt~B >P N D? W ~B ?P N D? V; ~B @P N D? WM~B AP N D?; W~B BP N D? Wx~B CP N D?sW= DP 0 ?"`3 Y R&  ( 2F   EP 0\ ?"`Y R&  ( 2F  N  tY Q s  Q 0H ?"` &w N& ( 2B   Q 0 ?"`T w N& ( 2B   N  tY Q  tYrB Q B D?VrB Q B D?==VrB Q B D?ddVrB Q B D?  VN  $ Q  $xB Q H D?$xB Q H D?$xB Q H D?  $xB Q H D?; < $xB Q H D?  $VN >$ Q >$xB Q H D?$xB Q H D?MM$xB Q H D?$xB Q H D?=>$ Q H ?"` Wt~B Q N D? W ~B Q N D? V; ~B Q N D? WM~B Q N D?; W~B Q N D? Wx~B Q N D?sW= Q 0 ?"`3 Y R&  ( 2F   Q 0 ?"`Y R&  ( 2F  N  tY Q s  Q 0 ?"` &w N& ( 2B   Q 0< ?"`T w N& ( 2B   N  tY Q  tYrB Q B D?VrB Q B D?==VrB Q B D?ddVrB Q B D?  VN  $ Q  $xB Q H D?$xB Q H D?$xB Q H D?  $xB Q H D?; < $xB Q H D?  $VN >$ Q >$xB Q H D?$xB Q H D?MM$xB Q H D?$xB Q H D?=>$ Q H ?"` Wt~B Q N D? W ~B Q N D? V; ~B Q N D? WM~B Q N D?; W~B Q N D? Wx~B Q N D?sW= Q 0X ?"`3 Y R&  ( 2F   Q 0 ?"`Y R&  ( 2F   SR # lL _ж_ж?"   H&  2  5 S # lD _ж_ж?"`? .  NC04 N ckk    S H| ?~/ ,$ 0 `g(1),( 2    S Hl ?! ,$ 0 `g(2),( 2  0 S H ?  ,$ 0 g(l)B( 2    S H ?|  ,$  0 `h(1),( 2    S H ? ? ,$  0 `h(2),( 2  0 S H ?  ,$  0 h(l)B( 2  >2 S 3 r _ж_ж??"`a  ,$D  0 Z composition    c   S H ?  ,$D 0 Kh( 2    S # l$ _ж_ж?"`Y` j Z locality 4 ( 2 c  H P 0޽h ? ̙33ld___PPT10D+D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(+0+0+R  ++0+R  ++0+R  ++0+S  ++0+S  ++0+S  ++0+S  ++0+S  ++0+S  ++0+S  +/ '' 88<&(  < < 6  0@ z3 Ways to Degree 3     < 0 ? ` M ( 2  * < <! ? ,$ 0 81. Degree-3 encoding using a circuit representation"98 <29F 9  z 0P j < P 0j,$D 0 < 0' ?0   f(x)=1b( 2BBFBBB   < 0T2 ?P   .$ y1, y2 , y3 y1=NAND(x1 , x2)= x1(1-x2)+(1-x1)x2+(1-x1)(1-x2) y2=NAND(x3 , x4) y3=NAND(y1 , y2) 1 =NAND(y3 , x5)r8 <2FfNfBBFfNfBBFfNfFfNfBBFNBFNBBFNBBFNBBBBFNBFNBBBFNBBFNBFFfNfBBFNBFNBBFfNfBBFfNfBBFfNfBBBFfBBFfNfBFNBB r  < <4 ?   \*( 2B    < <. ?@p j PJNote: $! y1, y2 , y3&( 2B FfNfBFfNfBFfNf & z @*   <  ,$@ 0T 0 @   <#   `2  < 0 ?` @ `2  <B 0 ?`  rB < B D? @ l2 < < ?b0 ` T 0 @  <# z `2 < 0 ?` @ `2 <B 0 ?`  rB < B D? @ l2 < < ?b0 ` rB < B D?: m: rB < B D?: m mrB < B D?  mT 0 @  <C#  : `2 < 0 ?` @ `2 <B 0 ?`  rB < B D? @ l2 < < ?b0 ` rB < B D?z mz rB <B B D? mz mrB < B D?  mrB  < B D?Z Z rB !< B D? rB "< B D?   rB #< B D?   rB $< B D?    %< 0< ?@/  vx1D( 2FNF   &< 0ċ ?; *  vx2D( 2FNF   '< 0Ԑ ?   vx3D( 2FNF   (< 0( ?   vx4D( 2FNF   )< 0 ?L;  vy1D( 2FfNfF   *< 0p ?   vy2D( 2FfNfF   +< 0 ?[ @J : J"( 2F   ,< 0 ?   vx5D( 2FNF  T 0 @  -<# h  p`2 .< 0 ?` @ `2 /<B 0 ?`  rB 0< B D? @ l2 1< < ?b0 ` rB 2< B D?[  rB 3< B D? p rB 4< B D?d d  rB 5<B B D? d rB 6< B D? p  7< 0t ?t 6c 0 vy3D( 2FfNfF  xB 8< H D?( *( H < 0޽h ?<< < <<<.</< ̙33___PPT10k+ciD' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(+8+0+<  + )!0@(  @ @ 0 ? ` M ( 2   @ 0x ?  u%Using circuit representation (contd.) &( 2&F & i @ 0 ?f  'q1(x,y)=0 q2(x,y)=0 ... qk(x,y)=0 (8 22BJBFBFfBBBBJBFBFfBBB BJBFBFfBBB&   d @ < HB,?,  \ @ < ?^( 2 Rdeg.-2( 2B   z P& @ &P,$D 0N P& @ P&T  @ B ?p& F ,p(x, y,r)=S ri qi(x,y)J( 2fffff@@        @ 0 ?  @f(x)=0 P(x) is uniform f(x)=1 P(x) 0 given y=y0, otherwise it is uniform Nb8 x2BBBBBBff BBBBBBff   b   @ 0 ?P  zStatistical distance amplified to 1/2 by 2Q(k) repetitions. v<( 2( 2*BJJJB > l  @ < HB,? h|F  @ <  ?rXF Rdeg.-3( 2B   @ 0' ? 0J,$ 0 > works over any field complexity exponential in circuit size ( 2<( 2( 2BFB&FBB ? H @ 0޽h ? ̙33___PPT10+ׁD' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(+8+0+@  +1  @ D(  D D 08 ?f `l M ( 2  i D 0d= ? j,$ 0 ! one polynomial huge field size ( 2( 2( 2BFBFBB "  D 00G ?|v /2. Degree-3 encoding using quadratic characters 0( 20F 0 F j D >j D 0K ?  tFact from number theory:,( 2FfB  h D c $A ?? j  D <T ?u ,$ 0 n Let N=2n, b = length-N truth-table of f, F=GF(q) Define p(x1,& ,xn, r) = nO8 2BBBJBBBBBBBBB BBBBJBBJBBBBB@/       D c $A ??@ S 8 $@ 0H D 0޽h ? ̙33___PPT10+mRD$' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(+p+0+D  ++0+D  +6 22PbdH2(  H H 0Tr ? ` M ( 2   H < w ? 43. Perfect Degree-3 Encoding from Branching Programs5( 25D 5 HF  0  H  0 l2 H < ?nxB H H D?JN  H nl2 H < ff?xB  H H D?JN   H n l2  H < ff?xB  H H D?l2  H < ff? nPJN  H ~l2 H < ff?xB H H D?JN  H ~ l2 H < ff?xB H H D?JN  H  ~@l2 H < ff?xB H H D?l2 H < ff?@~pJN  H  l2 H < ff?xB H H D?JN  H  @l2 H < ff?xB H H D?JN  H @` l2 H < ff?xB  H H D?xB !H H D?xB "H H D?~xB #H H D?xB $H H D?0 xB %H H D?0 xB &H H D?P@xB 'H H D?P@xB (H H D?p` l2 )H < ?`   *H <X ? _s.( 2FB   +H 0 ? $0  _t.( 2FB  h ,H c $A ??O> h -H c $A ??i h .H c $A ??`. h /H c $A ??y' h 0H c $A ??  h 1H c $A ??PE h 2H c $A ??N h 3H c $A ??i h 4H c $A ??P h 5H c $A ?? t> h 6H c $A ??wE h 7H c $A ?? h 8H c $A ?? h 9H c $A ??Y> h :H c $A ??YN h ;H c $A ?? h H < ff?n pB ?H H D? d2 @H < ff?n d2 AH < ff?nd2 BH < ff?~ pB CH H D? d2 DH < ff?~ d2 EH < ff?~d2 FH < ff?~d2 GH < ff? pB HH H D? d2 IH < ff?pB JH H D?d2 KH < ff?pB LH H D?pB MH H D? pB NH H D? pB OH H D? pB PH H D?pB QH H D?d2 RH < ?0 SH <L ? R  _s.( 2FB   TH 0 ?$0 _t.( 2FB  ` UH c $A ?? > ` VH c $A ?? p  ` WH c $A ??.7 ` XH c $A ??'P ` YH c $A ?? 2i ` ZH c $A ??WE ` [H c $A ?? p  ` \H c $A ?? `> ` ]H c $A ?? `N ` ^H c $A ?? 5 " _H <Ħ ?  BP=(G, s , t, edge-labeling)j( 2BBBBBBB  2 `H 0 ?  Gx=subgraph induced by xb( 2BJBBBB4     bH 6 ?p  .mod-q NBP: f(x) = # s-t paths in Gx (mod q)/( 2FFFBBBBBB BBJB@        dH < ?v p ,$ 0 | size = # of vertices circuit-size BP-size formula-size Boolean case: q=2 V( 2BFBBB BBBB V H H 0޽h ? ̙33___PPT10+ymREDO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*dH%(+8+0+dH  +  g}gXf(  X X 6 o U,$@  0 / 1 * * * 0 1 * * 0 0 1 * 0 0 0 1 0*0F  0  X 6 o  U,$D  0 / 1 0 0 * 0 1 0 * 0 0 1 * 0 0 0 1 0*0F  0  X 6$ o U,$@ 0 // 1 * * * 0 1 * * 0 0 1 * 0 0 0 10*B F fF F fB F fF F fB F fF F f B  0  X 6` o  U,$D 0 // 1 0 0 * 0 1 0 * 0 0 1 * 0 0 0 10*F fB F fF F fB F fF  B F fF F f B  0 x X c $ 0@     X 0 ? ` M ( 2   X 0 ? `M M ( 2  /F Mg  X g?PN  0   X  {l2  X < ?nxB  X H D?JN   X nl2 X < ff?xB X H D?JN  X n l2 X < ff?xB X H D?l2 X < ff? nPJN  X ~l2 X < ff?xB X H D?JN  X ~ l2 X < ff?xB X H D?JN  X  ~@l2 X < ff?xB X H D?l2 X < ff?@~pJN  X  l2 X < ff?xB  X H D?JN  !X  @l2 "X < ff?xB #X H D?JN  $X @` l2 %X < ff?xB &X H D?xB 'X H D?xB (X H D?~xB )X H D?xB *X H D?0 xB +X H D?0 xB ,X H D?P@xB -X H D?P@xB .X H D?p` l2 /X < ?`   0X <0 ? _s.( 2FB   1X 0h ? $0  _t.( 2FB  h 2X c $A ??O> h 3X c $A ??i h 4X c $A ??`. h 5X c $A ??y' h 6X c $A ??  h 7X c $A ??PE h 8X c $A ??N h 9X c $A ??i h :X c $A ??P h ;X c $A ?? t> h X c $A ?? h ?X c $A ??Y> h @X c $A ??YN h AX c $A ?? h BX c $A ??5 l2 CX < ?  l2 DX < ff? 4 xB EX H D?4 $l2 FX < ff?$Tl2 GX < ff?Dtl2 HX < ff? 4 +xB IX H D?4 $l2 JX < ff?$T+l2 KX < ff?Dt+l2 LX < ff?d+l2 MX < ff?$ T;xB NX H D?T#D#l2 OX < ff?D t;xB PX H D?t#d#l2 QX < ff?d ;xB RX H D?##xB SX H D? + xB TX H D?T;DxB UX H D?T+DxB VX H D?t;dxB WX H D?t+dl2 XX < ? ; YX <L% ?$  { _s.( 2FB   ZX 0* ?T _t.( 2FB  h [X c $A ?? 4s  h \X c $A ??  " h ]X c $A ?? h ^X c $A ??  h _X c $A ?? h `X c $A ??t; h aX c $A ??   h bX c $A ??}+ h cX c $A ??}; h dX c $A ??+  ) eX <J ?ML G BP=(G, s, t, edge-labeling)j( 2BBBBBBB  : fX 0Q ? MgG Gx=subgraph induced by xb( 2BJBBBB4    6 gX BTY / Vmod-q BP: f(x) = # st paths in Gx mod q.,*FFFBBBBB BBJZ         z Tn  hX Tn ,$D 0L iX 0 n ?TO n  Lemma: $ degree-1 mapping L : x s.t. det(L(x))= f(x).K8 F2FBBF B FfffffffffffZ5       A jX 6$ om   / * * * * -1 * * * 0 -1 * * 0 0 -1 *t0* F B F B F B F  0 r kX  fZ_ж_ж?" m t    lX # l؋ _ж_ж?" @ 4  nsize(BP) (2 &   A z T n mX Tn ,$D 0= nX 6< o]   / 1 $ $ $ 0 1 $ $ 0 0 1 $ 0 0 0 1p0*B F fB F fB F f B  0 ZB oX s *D x ZB pX s *D+  A qX 6H o  / * * * * -1 * * * 0 -1 * * 0 0 -1 *t0* F B F B F B F  0 M rX 6 op  / 1 0 0 $ 0 1 0 $ 0 0 1 $ 0 0 0 10*F fB F fB F f B F f B  0 B sX  `_ж_жD? x tX 0| ?T n  tEncoding based on Lemma: g(x,r1,r2)= R1(r1)L(x)R2(r2)l;8 F2fffffffffffff ;  z n s T$  uX s A '$ ,$@ 02 vX  `_ж_ж?"  # $  wX # l _ж_жG HC}?" n s TV  'det L(x) (= f(x))( 2           Z        xX 6 o / U,$@  0 / * * * * -1 * * 0 0 -1 * 0 0 0 -1 00* F B F F F B F F B F F  0  yX 6 o / U,$D 0 3/ 0 0 0 * -1 0 0 0 0 -1 0 0 0 0 -1 00*F F F B F F F B F F B F F  0 + zX 0 ?i T6) ,$ 0 !Correctness: f(x)=det g(x,r1,r2)"8 F2 FBBBBBFBBBBBBJBBJB@       {X 0 ?/ T ,$ 0 hPrivacy:0 8 F2F  m |X 6H o U,$@ 0 / * * * * -1 * * * 0 -1 * * 0 0 -1 *t0* F B F B F B F  0 z  i }X  / u,$@ 02 ~X  `_ж_ж?S" ?q, X 6 o i 3/ 0 0 0 * -1 0 0 0 0 -1 0 0 0 0 -1 00*F F F B F F F B F F B F F  0 i X 6 oQ u,$@ 0 / 1 $ $ $ 0 1 $ $ 0 0 1 $ 0 0 0 1p0*B F fB F fB F f B  0 m X 6# o / u,$@ 0 / * * * * -1 * * * 0 -1 * * 0 0 -1 *t0* F B F B F B F  0 y X 6, os r,$D 0 / 1 0 0 $ 0 1 0 $ 0 0 1 $ 0 0 0 10*F fB F fB F f B F f B  0 ;z  lu X g~ p,$@ 0 X 6\6 o 7u // 1 0 0 * 0 1 0 * 0 0 1 * 0 0 0 10*F fB F fF F fB F fF  B F fF F f B  0   X # lC _ж_ж?"  l, Z-1(2  /z vu X s_ r,$D 0 X 6H ogu // 1 * * * 0 1 * * 0 0 1 * 0 0 0 10*B F fF F fB F fF F fB F fF F f B  0  X  `U _ж_ж?" 9v6 Z-1(2  3 X # lY _ж_ж?"  ,$  0 U=(2  y X 6|^ o# u,$D$ 0 / 1 0 0 $ 0 1 0 $ 0 0 1 $ 0 0 0 10*F fB F fB F f B F f B  0 i X 6g o u,$@# 0 / 1 $ $ $ 0 1 $ $ 0 0 1 $ 0 0 0 1p0*B F fB F fB F f B  0  X 0p ?,,$ 0 g(x,r1,r2)  8 F2BBFBFNBFNB  H X 0޽h ?wX ̙33PO___PPT10O+KDHJ' = @B DJ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*hX%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* X%(D' =4@BB@BB%()?)?D6' =.7 BBBBB;M 0.0 0.0 L 0.0 -0.33333 E*3>*B ppt_xB ppt_y=0BB<*hXD' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*mX%(D' =-o6Bdissolve*<3<*mXDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*zX%(D(' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*{X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|X%(D' =A@BBBB0B@%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D' =%(D6' =A@BB#BB0B%(/%,( < +))))D)' =+4 8?^CBhiddenBCBvisibleB*o3>+B#style.visibility<*X0Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*xX%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D' =%(DR' =A@BB#BB0B@%(/%,( < +))))D)' =+4 8?^CBhiddenBCBvisibleB*o3>+B#style.visibility<*XDq' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*xX%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*yX%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*uX%(D' =A@BBBB0B@%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D ' =%(D' =%(D>' =A@BB#BB0B%()?)?D' =.M7 BBBBB_M 1.66667E-6 -3.7037E-7 L -0.09462 -3.7037E-7 *3>*B ppt_xB ppt_y=@0BBAApBB""BB<*XD6' =A@BB?BB0B%()?)?D}' =.E7 BBBBBWM -3.05556E-6 -3.7037E-7 L 0.1 -3.7037E-7 *3>*B ppt_xB ppt_y=@0BBAApBBL=B<*XD' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*X%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*}X%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(DN ' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*XD' =1:Bhidden*o3>+B#style.visibility<*X%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-o6Bdissolve*<3<*XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-o6Bdissolve*<3<*X+x+0+X  ++0+X  ++0+X  ++0+X  ++0+X  ++0+X  ++0+X  ++0+xX  ++0+xX  ++0+yX  ++0+zX  ++0+{X  ++0+|X  ++0+X  ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X  ++0+X  ++0+X  +I7 K0 `N(  `D ` 6` om U X ,$D  0 J -1 * * * * 0 -1 * * * 0 0 -1 * * 0 0 0 -1 * 0 0 0 0 -10K*FfJF K x ` c $ 0@     ` 0T ? ` M ( 2   ` 0 ? `M M ( 2   ` # l _ж_ж?" 87;2 S-A(x)= adjacancy matrix of Gx (over F=GF(q)).( 2  h            ` # l _ж_ж?" v7 p,$ 0 0A* = I+A+A2+& = (I-A)-1 ( 2     K ` # l _ж_ж?" 7$  A*s,t =P( 2   &   D  ` 6 om U X ,$D 0 J 0 * * * * 0 0 * * * 0 0 0 * * 0 0 0 0 * 0 0 0 0 00K*FfJF K   ` # l _ж_ж?"  Y ,$ 0 = det (A-I)|t,s( 2    @     #  ` # l _ж_ж?" \ 7V ,$ 0 EL(x)= (A(x)-I)|t,s ( 24     >  ` 0  ?l'A Lemma: $ degree-1 mapping L : x s.t. det(L(x))= f(x). L8 F2FBBF B Ffffffffffff@5        ` 6! o@  / * * * * -1 * * * 0 -1 * * 0 0 -1 * 0*0F  0  ` 0\& ?_'A Proof: H8 F2Ff   ` # l(+ _ж_ж?" * ,$ 0 H(-1)s+t det (I-A)|t,s / det (I-A) %( 2        Z          s   `# u ] Z ,$D  0 ` Z_ж_жff?S" ? s  ` Z_ж_жff?S" ?    ` # l> _ж_ж?" s  `A="(2  > ` # ltC _ж_ж?" f ,$  0 `L="(2  BF js P  ` s jP  `  `G _ж_ж?" M s m  Os(2   `  `|L _ж_ж?" U  Ot(2   `  ` P _ж_ж?"  s P m  Ot(2   `  `S _ж_ж?" j? ' 9  Os(2  H ` 0޽h ? ̙33___PPT10+h_4JD' = @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<*`%(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<* `%(Do' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* `%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*`%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(++0+`  ++0+`  ++0+ `  ++0+ `  ++0+ `  ++0+ `  ++0+`  ++0+`  ++0+`  +"  ] 0 V(      Hy ?MZ,$ 0 l&(  g   H8 X  H~ ?/U  Thm. size-s BP degree 3 encoding of size O(s2) perfect encoding for mod-q BP (capturing L/poly for q=2) large q: useful for (comp.-secure) two-party computation imperfect for nondeterministic BP (capturing NL/poly) j2$ ;!( 2:( 2:!( 2e a eaia+c9cc7cc  hH8H   0޽h ? ̙33___PPT10+BUD<' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =K@BBBBPB0B%(/%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(DC' =%(D' =K@BBBBPB0B%(/%( D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<* D' =+4 8?`CB ppt_yBCB0-ppt_h/2B*Y3>B ppt_y<* D' =1:Bhidden*o3>+B#style.visibility<* %(+p+0+   ++0+   +2 YQpP(  PC P 0 ?@K  [ Q: How many rounds? How many rounds for maximal privacy? How much privacy in 2 rounds? X( 2C( 2( 2C \  P 0 ?]H\  ! 3 rounds suffice t( 2=F@ >  T 6 c ^F.,( 2F@  4 T # Po  T 68 }c B( 2@   T <  ?   ] Is 3 minimal? ( 2F   H T 0޽h ? ̙33~___PPT10^+YDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(+' F P`@H(  ` P`  `_ж_ж̙?S" ? t ,$D 0~ ` s *(      ` 0l) ? ` M ( 2  ~ ;` B. ?h,$D 0 Known: f L encoding in NC0 Goal: f P encoding in NC0 Idea: relax encoding requirement Respects security of most primitives Thm: f P computational encoding in NC04 assuming  easy PRG (min-PRG L) |5 EAAAA AAEAAAAAA&AAAA A C&  Z  phX8l gN  N`t ,$@ 0 N }b >` g B ?`  f_ж_жD>?b  @` # l@8 _ж_ж?" } [x(2a  N bK%  A`   B` Z_ж_ж?" b % B C`  f_ж_жD>? K D`  `< _ж_ж?"  _  ]g (2c   N Cjc  E` o  B F`  f_ж_жD>?j G` # lB _ж_ж?" Cc  _r"(2e  P H` # lLM _ж_ж?" C Nc  Enc(y)N(2eaaa&   z  % |  J` +  ,$D 0 K` 0C ?"6@`NNN?N &   ncomp:8 22   L` 0, ?"0@NNN?N % |  V$( 2    O`  ` _ж_ж?" ,&,$  0 N|  Easy PRG can be based on factoring, discrete-log, lattices? c gc gcgc>  H ` 0޽h ? ̙33___PPT10+܅D2' = @B D' = @BA?%,( < +O%,( < +DT' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;`"M%(D' =%(D_' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;`No%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;`t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*N`%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*J`%(D' =-o6Bdissolve*<3<*J`D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*O`%(+8+0+O`  +n3 o 7/`(  ` ` Bĉ  w     ` B ? bThm.  easy PRG encoding in NC0 for all fP24 E AAAAE4 +    @`N ` <  ?"6@`NNN?NPh$,$D 0 p f P6( 2c   ` <$  ?"6@`NNN?NPb $,$@ 0 *gNC0[ ]\( 2c   ` <  ?"6@`NNN?NP $,$@ 0 gNC0[min-PRG]\( 2c   Z ` <ԯ  ?"6@`NNN?N G. ,$@  0 |gL6( 2c   ` <д  ?"6@`NNN?N) k ,$@  0 hNC04 ( 2c    ` 6 ?"0@NNN?Ne,$@ 0  ` 6 ?"0@NNN?N& ,$@ 0  ` 6 ?"0@NNN?N Rm ,$@  0  ` 6 ?"0@NNN?N m ,$@ 0E  ` 0H ?"0@NNN?NEq ,$ 0 yYao garbled circuit(8 F2g  + ` 0, ?"0@NNN?N #,$  0 _ easy PRG & ( 2 g  ) ` 0 ?"0@NNN?N ,$ 0 ][AIK04]&( 2g  K `  ` _ж_ж?" Pj ,$ 0 yone-time symmetric encryption (2c  l  <  `j 6 ,$D  0) `  f _ж_ж?"  <  }one-time symmetric encryption$(2g  X ` B  ?"6@`NNN?N   NC0[min-PRG]V( 2   H ` 0޽h ? ̙33___PPT10.q\޹+̹D' = @B DN' = @BA?%,( < +O%,( < +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' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(DY' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(++0+`  ++0+`  ++0+`  ++0+`  ++0+`  ++0+ `  ++0+`  ++0+`  ++0+`  +>" p h5(  h[ h 3 r_ж_ж?" - _ ,$ 0 wexist6(2$    h c $ ?"0@NNN?NUt( [perfect$( 2G    h Bl ? Using encoding:85 CG   @` h c $x ?"0@NNN?NT,$ 0 Ycomp.$( 2G   h B  w   ? h s * ?"0@NNN?N@t:  5OWF OWP PRG Hash Sym-Enc PK-Enc Signature Commit NIZKH8 22( 28 226C 6    h s * ?"6@`NNN?NCt=  $Sym-Enc PK-Enc Signature Commit NIZK4( 28 22%C %    h s *h ?"6@`NNN?NC3 u=  $Sym-Enc PK-Enc Signature Commit NIZK4( 28 22%C % F | < h   C h HT$ ?|  v J"( 2c   h H& ?{ B < J"( 2c   h  f* _ж_ж?"  6  \(2$  * h 3 r( _ж_ж?" " c  z L6(2($  F h 3 r 3 _ж_ж?" oW   NC0P(2($$  x h # l8 _ж_ж?" tR ,$D 0 &Assuming  easy PRG 8(2  ? h s *> ?"0@NNN?Nn3 fh  5OWF OWP PRG Hash Sym-Enc PK-Enc Signature Commit NIZKH8 22( 28 226C 6 H h 0޽h ? ̙33A9___PPT10.q\޹+)#De' = @B D ' = @BA?%,( < +O%,( < +DW ' =%(D,' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =A@BBBB0B%())))?D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =A@BBBB0B%())))?D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(++0+h  ++0+h  ++0+h  ++0+h  ++0+h  ++0+h  ++0+h  +` q 3303p43(  p p c $@^ ?"0@NNN?N,$D9 0 P:Proof: given code of min-PRG Construct f P[min-PRG] via known reduction Use code of f to construct g NC0[min-PRG] Note: non-black-box reduction! ( 2a( 2 4( 2EA0"C &.  k  p c $m ?"0@NNN?NOJ(,$  0 Blum Micali 82, Yao 82, Levin 85, Goldreich Krawczyk Luby 88, Hstad Impagliazzo Levin Luby 90, Goldreich Micali 84, Goldreich Goldwasser Micali 84, Goldwasser Micali Rivest 84, Bellare Micali 88, Naor Yung 89, Rompel 90, Naor 89, Impagliazzo Luby 89, & *4( G                                       p 0k   ,$  0 fWhat about NC reductions? Much less is known& . NewZ4$  cg#ce 4  p s *`~ (<$ 0   ~ p s *    ` p`  ANWith_a_boneWith a bone - Click image to download."}= \$@  0U 0   p@ A  HCave_boyCave boy - Click image to download.8 A ! x  U 0 p c AJCave_girlCave girl - Click image to download. W  U 0  p@ A Hcave_manCave man - Click image to download. 0o  U 0  p@ c ARCaveman_typesCaveman types - Click image to download.8 y  U 0  p A Lcave_womanCave woman - Click image to download.t P U 0  p A  VCaveman_escapesCaveman escapes - Click image to download.P U 0R  p 0 ?"6@`NNN?N I ,$@ 0R p 0 ?"6@`NNN?N F ,$@ 0R p 6n ?"6@`NNN?N% m ,$@ 0R p 0 ?"6@`NNN?N 8  ,$@ 0R p@ 6f ?"6@`NNN?N* M  ,$@ 0 p c $܆ ?"0@NNN?N &|  SOWF ( 2C   p c $ ?"0@NNN?Nz 2a  Wmin-PRG ( 2C   p c $ ?"0@NNN?N i }  SPRG ( 2C   p c $X ?"0@NNN?N  VCommit ( 2C   p c $ ?"0@NNN?NST:  WSym-Enc ( 2C   p c $ ?"0@NNN?Ny `  Y Signature ( 2 C   p c $ ?"0@NNN?N= ,$  0 [ Synthesizer ( 2 C  Az    p   ,$@ 0 p 0 ?"6@`NNN?N \  p c $x ?"0@NNN?N   }NC0J( 2GOG   p 6 ?"6@`NNN?N | ,$@ 0= p c $ ?"0@NNN?N ,$ 0 }NC0J( 2GOG  *  p c $X ?"0@NNN?Np W ,$ 0 j Regular OWF ( 2C   !p 0 ?"6@`NNN?N` ,$D  0= "p c $ ?"0@NNN?Nm 5  ,$  0 }NC1J( 2GOG   #p <Y GM??"6@`NNN?NL~ ,$@2  0 $p <m GM??"6@`NNN?N 8 ,$@0  0 %p 0 ?"6@`NNN?N j ,$@1  0= &p c $̷ ?"0@NNN?N` Z ,$4  0 }NC0J( 2GOG  = 'p c $ ?"0@NNN?NZ  T ,$3  0 }NC0J( 2GOG  = (p c $ ?"0@NNN?N  ,$/  0 }NC0J( 2GOG   )p@ A Lcave_womanCave woman - Click image to download."?P`   U 0 +p c $ ?"0@NNN?N   SPRF ( 2C  R ,p 0 ?"6@`NNN?N`9,$@ 03l c`+  /p`c+ ,$@ 0 -p 0 ?"6@`NNN?N` .p c $ ?"0@NNN?Nc+  }NC0J( 2GOG  < 1p  ` _ж_ж?" g ,$ 0 jHILL Viola AIK (2c  0 2p  `` _ж_ж?" @ j,$  0 ^NR (2c  : 3p  f _ж_ж?"  p ,$ 0 bNaor (2c  H p 0޽h ? ̙33d\___PPT10<.]@ts+cDЖ' = @B D' = @BA?%,( < +O%,( < +D' =%(Db' =%(Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-6B'blinds(horizontal)*<3<* pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*,p%(D' =-6B'blinds(horizontal)*<3<*,pD' =A@BBB B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*p3%(D' =-6B'blinds(horizontal)*<3<*p3D' =A@BBB B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pD' =%(D^' =%(D' =A@BBB B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*p3D' =1:Bhidden*o3>+B#style.visibility<*p3%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<* pD' =1:Bhidden*o3>+B#style.visibility<* p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*,pD' =1:Bhidden*o3>+B#style.visibility<*,p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(D&' =%(D#' =%(D' =A@BBB B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*p0%(D' =-6B'blinds(horizontal)*<3<*p0DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*/p%(D' =-6B'blinds(horizontal)*<3<*/pDT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B'blinds(horizontal)*<3<*pDR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-6B'blinds(horizontal)*<3<* pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3p%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*"pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*"pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*!pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*!pD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2p%(D7' =%(D<7' =%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*)pD' =1:Bhidden*o3>+B#style.visibility<*)p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*!pD' =1:Bhidden*o3>+B#style.visibility<*!p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*"pD' =1:Bhidden*o3>+B#style.visibility<*"p%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*+pD' =1:Bhidden*o3>+B#style.visibility<*+p%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*/pD' =1:Bhidden*o3>+B#style.visibility<*/p%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*1p%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*2p%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*3p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*$pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*%pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*#pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*#pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*'pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*&pD' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*p04%(D' =%(Dz' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p0%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p04%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*pM%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*pM~%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p~%(+@+0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+p#  ++0+ p#  ++0+"p#  ++0+"p#  ++0+&p#  ++0+'p#  ++0+(p#  ++0++p#  ++0+1p#  ++0+1p#  ++0+2p#  ++0+2p#  ++0+3p#  ++0+3p#  +0  2h(  h h c $1# ?"0@NNN?N,f ,$@ 0 In case you don t insist on unconditional security& Secure computatoin of every func. f efficiently reduces to deg-3 poly & asuming  easy PRG In particular: Protocols desribed by Ivan imply const. round computationally secure MPC for every f (Known assuming any PRG [BMR90,DI05]; however, current approach is conceptually simpler.)4( 2G( 2( 2( 2( 23EA C.            I t<      )  '   ~ h s * ># q  #  H h 0޽h ? ̙33  ___PPT10 .]@ts+ D~ ' = @B D9 ' = @BA?%,( < +O%,( < +Dl' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h4{%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h{%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*hT%(+R l `6D(  D~ D s *LU# 'G  #   +D S HV# zq<$ 0 #   D 0W# ? `I S "( 2a  H D 0޽h ? ̙33sk___PPT10K+TD' = @B D' = @BA?%,( < +O%,( < +D,' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D)%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D)M%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+DMp%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+Dp%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D_%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+D_%(+8+0++D#  + m ph_P0 (  P~ P s *pm# 'G  #   P 0n# ? k S "( 2a   MP  `w# _ж_ж?" :UZ@ qRandomized encoding"( 2e   OP  `{# _ж_ж?" :R@ xUnconditionally secure MPC"( 2e   PP  `0# _ж_ж?"  Parallel crypto6( 2em  @l qP ]PPI,$D 0 RP  fD# _ж_ж?" qP Lpoly-size NC0 encoding for every fP? \'( 2 g2!     SP  f# _ж_ж?" Qp befficient constant-round protocol for every fP? 62( 21g2,    p TP  f# _ж_ж?"  } ($OWF $OWF in NC0?l( 2   l  Y)  ^P D) ,$D 06 UP  fĢ# _ж_ж?" c   locality 3 for every f? 6( 2g  H VP  f# _ж_ж?" gc   *maximal privacy with minimal interaction? 6+( 2*g*   WP  ft# _ж_ж?"   Y)  6$OWF in NC1 $OWF in NC03?( 2    l D , _P w,$D 0) XP  `# _ж_ж?" D x better encodings?6( 2g  6 YP  `# _ж_ж?"    better const-round protocols?6( 2g  , ZP  `T# _ж_ж?" # , practical hardware?6( 2g  H P 0޽h ? ̙33___PPT10+yD' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*^P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_P%(+] 0 $$(  $^ $ S ,9   #  $ S @K , @  L    H $ 0޽h ? ̙3380___PPT10.Ac 0 T$(  T^ T S ,9   ' T S D , @  L    H T 0޽h ? ̙3380___PPT10.A4 0 d(  dX d C ,9   ' d S i , @  i    H d 0޽h ? ̙3380___PPT10.i 0  4$(  4^ 4 S ,9   ' 4 S ` , @     H 4 0޽h ? ̙3380___PPT10.rp5 0  $(  ^  S ,9   ;   S i , @  i    H  0޽h ? ̙3380___PPT10.g+e 0  (  X  C ,9   ;   S 0 , @  L    H  0޽h ? ̙3380___PPT10.uj 0  (  X  C ,9     S T L , @     H  0޽h ? ̙3380___PPT10.P! : 0 p$$(  $^ $ S ,9   ;  $ S Xi , @  i    H $ 0޽h ? ̙3380___PPT10.qPhg 0 $(  ^  S ,9   ;   S Pi , @  i    H  0޽h ? ̙3380___PPT10.g+ 0 0$(  ^  S ,9   ;   S  , @  L    H  0޽h ? ̙3380___PPT10.A 0 0$(  ^  S ,9   ;   S li , @  i    H  0޽h ? ̙3380___PPT10.A 0 P$(  ^  S ,9   ;   S pL , @  L    H  0޽h ? ̙3380___PPT10.rp5 0 @<$(  <^ < S ,9   ;  < S i , @  i    H < 0޽h ? ̙3380___PPT10.A 0 pH$(  H^ H S ,9   ?  H S ? , @  L    H H 0޽h ? ̙3380___PPT10.uj 0 T$(  T^ T S ,9   ?  T S P[? , @  i    H T 0޽h ? ̙3380___PPT10.uj 0 d*(  d^ d S ,9   ?  d c $|"? , @  ?    H d 0޽h ? ̙3380___PPT10.t\ 0 l*(  l^ l S ,9   ?  l c $(? , @  ?    H l 0޽h ? ̙3380___PPT10.t\ 0  t*(  t^ t S ,9   ?  t c $H/? , @  ?    H t 0޽h ? ̙3380___PPT10.^ ] 0 @|$(  |^ | S ,9   ?  | S Ȗ , @     H | 0޽h ? ̙3380___PPT10.}a 0 `$(  ^  S ,9   ?   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