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D!:m(b^SqK* 5?@~$ Z"**"QFA;B#h֭(i/-(T˨;` 7nL2e1q)BqlT#bb8A/UQ+[_=Y $]*6瓯@DSymbolsTrue Random Number Generators Secure in a Changing Environment??0BBoaz Barak, Ronen Shaltiel, Eran Tromer Weizmann Institute, Israelt$True Random Number Generators (TRNG)%%(SA TRNG is a device that outputs a sequence of independent bits (i.e., coin tosses).(TNDesign of TRNG2Previous Constructions of Entropy Extraction Phase33( Tailor-made design for a  nice source X (Intel RNG, IBM-4758, & ) Use cryptographic hash functions (PGP, Linux, Apache& ) N(" Example: Intel RNG Intel RNG Extractor* Is Source Really Independent??"Proof using physics: ???? (probably not) Statistical tests: passed all except for  minor deviations in tests involving spectral analysis [JK99]NN Using Crypto Hash Functions Problems w/ using Crypto-hashRelying on unproven properties of hash functions to obtain supposedly true random bits. Assumptions are not even explicitly stated. Not an explicit goal of either hash designers or cryptanalysts (In contrast to collision resistance)JF3C& Natural Security Def    Our ContributionWe give an explicit model and definition for the entropy extraction problem. We prove unconditionally that a known simple construction satisfies the definition. We implemented and tested the above solution. Our FrameworkOutline of Security DefDefining Allowed Modificationst-Resilient Extractors Main Qualitative Result lBetter Efficiency  Pairwise Indp. 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B6CDElFvd @ 6*yP*xH0  $  **T,V8<@`#" ` /D ~ "  :BbCDE8FBd @ 7P\Zb$bJJ$CZ7 @`#" ` J  # JBCDEpFz s71[$T0 g$N~I%1s!+QIaCQ+!:<@`#" `Y\   gU $ "  gUf %  "BCDE(F4 @ JJ\r<le0 &Jn 0e9or\EoL9nJ&Ll<E\\\~c-S0*$n*J0,S~T~6\96T~.d,nd.-c9\\@`@`#" `hJ %. &  B CDEF @ $$$JBlfO<  I0ZZ* J*!NQr{C{lQH'0JL@`#" ` & @ '  BaCDE`Fj @  q$0xNr7aa1~Zl<$k   24@`#" ` V (  BHC6DE$F.d @  H$6$66*HHHH@`#" `~ )  ZBClDEHFRd @ +ZZN0B`l+lfTg<g*BT+Z+Z&(@`#" `ZU *  JBCUDE@FJd @ wrqY/%UUMqr<wYq<wrwr"$@`#" `FgV +  BSCZDE$F.d @  ;ZSTG<5*#0;Z;Z@`#" `( , bBCDEF ??%yIVg g%wgTI0%0TfwH0%g 0DBJ0 g$kB<`+<Oksg2bsO+@`#" `] * -  BCDEF @ ::hh<T'rE]ioi7Kg'nsfBg$7$0BZ~Br$ 7$mNm z ?im7u]9fB*hhvx@`#" `  .  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BBC0DE F* @ B0  0*BB@`#" `o 2 / T? @? #" `W2 0 N? @? #" `.=2 1 T?d @? #" `B+2 2 N? @? #" `.^X2 3 T?d @? #" `Rr02 4 N? @? #" `\  I  5 " I  6  jB~C`DEPFZd@ `ZSB#$*MHZ`Z7T`B~*x*ZB1NZ``*,@`#" `r   7  JBC6DE@FJd@  xN $06$$Nx"$@`#" `   8  JB<CDE@FJd@ 6Z0~6~<Z6B0***0B6Z6Z"$@`#" `  I n 9  *BCDE0F:d@  r H6 r r r @`#" `   :  rBCDETF^@  r`N5*e_ /$ HZr )A) r r,0@`#" ` Y  ;  BCDE`Fj@ B w$HZrwZ`rZB0H24@`#" `q + v <  2B+CDEFd@ ,,Z* 6HZl ~*Z~+l+Z+H6`0~ lZ H60`  6%H%Z%l~[\@`@`#" `  b    =# "  2 > H?? #" `  2 ? N?d? #" `  2 @ H?? #" `7% w 2 A N?d? #" `R5 d  B <T  "*p   T Click to edit Master title style! ! C 0W  " `     W#Click to edit Master subtitle style$ $ D 6<\  #" `` `   Z* E 64`  #" ``    \* F 6De  #" `` `   \*B  s *޽h ? h___PPT10i. +D=' = @B +$  0(  x  c $4p B*p   x  c $q C     H  0޽h ? 33___PPT10i.R8 +D=' = @B +S #   @(  @x @ c $ C `}   x @ c $Ⱥ D `    f @ 08c"`P P  @ 0  p  8TRNG^B @ 6D  p @ 0pm  ` ,$ 0 X(0110100110111010110&   @ 0 p @v,$ 0 (Results generalize to case of sequence, also it s always possible to apply pseudo-RNG to block)<b 2MC  @ 0D p ,$ 0 OIn this talk: focus on case that TRNG outputs a block of bits (e.g., 256 bits).DP 2  #   H @ 0޽h ? 3f  ___PPT10 .R X+WD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(DV' =%(D' =A@BBBB0B@HB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(DF' =+4 8?pCB#0088e4 [5]BCB#99ff99 [6]B*z3>!Bstyle.color= `B<*@D-' =+4 8?pCB#0088e4 [5]BCB#99ff99 [6]B*a3>Bfillcolor<*@D' =1:B solid*a3>Bfill.type<*@D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(++0+@m  ++0+ @m  ++0+ @m  + #   -9 (  x  c $ C `}     0  OA TRNG can be split into two phases: entropy generation and entropy extraction.PP 2%     c8  `p $` p@  P   P   < 8c"` P  :  " VADhdisk2     0  @P  :E. GenfB  6D PP P n   08c"`` p   0  . C0 0 0 1 1 0 0 1 2fB  6D  P  P z  <G8c"` PP P   0   P @  7ExtfB  6D P P P n  08c"`   0   = 0 1 0 1 1 2    0 =,$ 0 y)We focus on the entropy extraction phase.4* 2   Z ' c 2A typing  X * c 0A  diode   l `  @ - @ p,$D 0 + B GCHϸ333> "`` @ :  , 00    nHigh Entropy Source: X< 23333H  0޽h ?+ h  ___PPT10 .R`Qq+d9DO ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*-D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*-D' =-g6B fade*<3<*-D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*-D' =1:Bhidden*o3>+B#style.visibility<*-%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+  + # B(  r  S <C `}     <( D  H  0޽h ? h___PPT10i.R0+D=' = @B + #   <(  r  S 8 C `}     TA Dintel   2  033Ԕ"` ` ,$D 0l p @@`  @p @` ,$D 0n  08c"`p @`    6 p @@`  F000101001010001001 2l      ,$D 0n   08c"`    6   > 0101101001 2 H  0޽h ? h___PPT10.RpcI+<Dc' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(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' =%(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' =%(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<* +dK # +(  x  c $d' C0   F PNp^   Pp t  68c"`PNp^   6<) k[  C0 0 0 1 1 0 1 1 2lB   <D p>>   BG8c"`>>    6 .  ~. 7ExtlB   <D >P>t   68c"`P^p  61 k [ 70 1 2B  s *DԔ2,$D 0B  s *DԔ" ,$D  0  <7 Ԕ"`Pp,$D 0 :   <: Ԕ"`0Pp,$D 0 :   <> Ԕ"`Pp,$D 0 :   <A Ԕ"`Yy,$D 0 :   0@@  0 ,$ 0 XVon-Neumann Operator( 2    0I * `0 ,$ 0 pr00 No output 01 output 0 10 output 1 11 No output: 2                   0U  0 ,$ 0 UIf each bit is independently 1 w.p. p then Pr[ output = 1] = Pr[ output = 0] = p(1-p)|V 2       + 4  0` # 0,$ 0 Output length: m= 2p(1-p)n nL 2  G  H  0޽h ? he:]:___PPT10=:.RpcI+_5Dq7' = @B D,7' = @BA?%,( < +O%,( < +D' =%(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<*D' =%(D' =%(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$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D$' =%(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<*D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D$' =%(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<*D' =%(D' =%(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' =-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<*%(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<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  +T # 800 (   r   S , C `      S ( D0 `<$ 0   :   0  ,$ 0 bNote: There are natural sources on which von-Neumann completely fails (e.g., if sampling too fast)(c 2^"l `   ` ,$D 0*   6| 1  Suppose that source is indp. in  ideal conditions. Does this hold for all RNGs  out there in varying (& possibly adversarial) environments?P 21E,0@t"   6p"`` H   0޽h ? h  ___PPT10 .UP(m+!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' =%(D' =%(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<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+p+0+   ++0+   + # b Z @ (  x  c $x C0   8 Ppp  pPp t  68c"`Ppp   6< k[>  C0 0 0 1 1 0 1 1 2lB  <D p``  BG8c"```   6  0P ; hlB   <D `P`t   68c"`Pp   6 P,@2 70 1 2  0  0,$ 0 jMotivation: Don t want to assume anything about source s structure. Intuitively, should work for every high-entropy source. T~ 2  U      0  ,$ 0 4h:{0,1}n {0,1}m (n>m) 2         0X 0 ,$ 0 uE.g., h = SHA-1 or MD5 : 2     <T  "P \,(Linux, Apache, SSL,& )H  0޽h ? h___PPT10.RpcI+$3D' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+  ++0+  ++0+  +} # `$(  r  S 4 C `}    r  S  D   H  0޽h ? h___PPT10i.UPn:+D=' = @B +  # skp(  x  c $T  C0    F Ppp   P` n  08c"`Ppp   0 k[>  C0 0 0 1 1 0 1 1 2fB  6D p``z  <G8c"```   0  0P ; hfB   6D `P`n   08c"`Pp   0 P,@2 70 1 2d   0#   ,$ 0 Def 1: h is an extractor w/ params k,e if for every (efficiently samplable) r.v. X over {0,1}n if Ent(X) k then h(X)~eUmV{ 2  +Gb  L   04 &`  4h:{0,1}n {0,1}m (n>m) 2        0> `   uE.g., h = SHA-1 or MD5 : 2   >  <C GH "`  ,$D 0 Ent(X) is the min-entropy of X<  0H @,$ 0 @Bad News: Def 1 is impossible to obtain. (even for k=n-1 and e=)B 2      0S  0,$ 0 ;? 2,3H  0޽h ? h___PPT10.RpcI+GDT' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D?' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+   ++0+  ++0+  ++0+  ++0+  +f #  B(      0 ,$ 0 zThm [NZ]: 8 efficiently computable h , 9 efficiently sampleable X s.t. 1) Ent(X) n-1 2) The first bit of h(X) is fixed."{ 2   G    G       G       Z3  T   0` ,$ 0 XProof: W.l.o.g Pr[h1(Un)=0] . Sample X as follows: 1. Let x {0,1}n 2. If h1(x)=0, output x. Otherwise, goto 1 X is uniform over h1-1(0)  a set of size at least 2n/2. 2        G            #             ,_<H   0޽h ? h\T___PPT104.Uz`+o-?D' = @B DS' = @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<* %(+p+0+   ++0+   +} # $$(  $r $ S . C `}   r $ S  D `   H $ 0޽h ? h___PPT10i. Upҿ+D=' = @B +9 #  }  0 (  0x 0 c $`L C0   F Ppp  0 pPp n 0 08c"`Ppp  0 0S k[>  C0 0 0 1 1 0 1 1 2fB 0 6D p``z 0 <G8c"```  0 0X  0P ; hfB  0 6D `P`n  0 08c"`Pp  0 0] P,@2 70 1 20  0 0^  G,$ 0 h is chosen at random from a collection H and made public. The same choice h can be used for many RNG s.i 2 ' $       b  0 0i  ,$ 0 4h:{0,1}n {0,1}m (n>m) 2      H 0 0޽h ? h\T___PPT104.RpcI+ ED' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(+p+0+ 0  ++0+ 0  +& #    ( (  (r ( S Lz C `    ( `ADdevilfinish  B ( 0DpP P,$D  0Y ( 0|  ,$  0 X (r.v. over {0,1}n)p 2  B ( 0Dp0 ,$D  0   ( 04  t,$ 0 v h HP 2  $ r  ( 0H ` ,$  0 0X (r.v. over {0,1}n)p 2    ( 0  0H ,$ 0 @We require that X 2X X is the set of allowed modifications to X We require that Ent(X )k for all X 2X 2i 2  G $ $      G     G $ Q l @  (@ ,$D 0  ( 0 Pp (FCollection H is secure if h(X ) Um$ 2  $    G    n"  ( 0p"`@  ( 0̯ 0,$  0 ;? 2,3H ( 0޽h ? h___PPT10a.!Uv+RoD' = @B D' = @BA?%,( < +O%,( < +D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(left)*<3<*(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(left)*<3<*(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-s6Bwipe(down)*<3<* (D7' =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' =%(D' =%(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<*(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(++0+(  ++0+ (  ++0+ (  ++0+ (  ++0+(  +- # RJ8(  8x 8 c $hƞ C `    8 fADdevilfinish  XB 8 0DpP P% 8 0Ȟ   X (r.v. over {0,1}n)p 2  XB 8 0Dp0  8 08ў  t v h HP 2  $ > 8 0מ `  0X (r.v. over {0,1}n)p 2    8 0  0H  @We require that X 2X X is the set of allowed modifications to X We require that Ent(X )k for all X 2X 2i 2  G $ $      G     G $ Q d  8 0h pP "Collection H is secure if h(X) Um# 2  $    G    f"  8 0p"`@   8 0 W"`,$D 0"  8 6 "`  ,$D 0 8 6 J,$ 0 ffPossible choices: Allow adv. to change bounded # of bits in X Allow adv. to change sampling speed of X. Allow adv. to bias bits of X Allow adv. to xor X with a fixed value. &  2 2 +( 8 0 W"`@P@,$D  0z @ 0  8 @ 0 ,$D  0n" 8 0p"`@ 0  8 0 p =We want one definition that unifies all these requirements. (> 2<  H 8 0޽h ? h]U___PPT105.!Uv+~D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* 8D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* 8D' =-g6B fade*<3<* 8DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8?%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8?j%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8j%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*8D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*8D' =-g6B fade*<3<*8+8+0+8  +& # 2 *  @ (  @x @ c $ C `    @ fADdevilfinish  B @ 0DpP P,$D  0 @ 0$ 4,$  0 )X (family of 2t r.v. of ent. at least k)* 2$  ,B @ 0Dp0 ,$D  0  @ 0*  t,$ 0 v h HP 2  $ J @ 00 ` ,$  0 X (X 2X) 2  G:l 0 p @0 p,$D 0H  @ 0|9 p   Collection H is a t-resilient extractor with parameters n>k>m and e If h(X)eUmT 2  $         G    Kn"  @ 0p"`0 p  @ 0H P,$ 0 ~4This captures all modifications from previous slide..5 24 @ 0M ``@,$  0 ;? 2,3H @ 0޽h ? h___PPT10a.!Uv+i D' = @B D' = @BA?%,( < +O%,( < +D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-s6Bwipe(left)*<3<*@D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-s6Bwipe(left)*<3<*@D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-g6B fade*<3<*@D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-s6Bwipe(left)*<3<*@D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-s6Bwipe(left)*<3<*@D' =%(D' =%(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<*@DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(++0+@  ++0+@  ++0+@  ++0+ @  ++0+@  + #    DG (  Dr D S |d C `}   8 D 0f P0 ,$ 0 Thm 1: For every choice of n>k>m and e 9 t-resilient extractor H s.t.J 2C $B D 0w  0,$ 0 d@t= Q( ( k-m+log(1/e) )(|h|/n) )! 2 ,, $    $G$$ $ , D 0  `,$ 0 EProof (similar to [TV]): Show that an l-wise independent hash function collection, for appropriately chosen l, is an extractor with wanted parameters. 2$*tl P   DP ,$D 0 D 0 `@  bAllows to obtain any desired level of entropy loss or resiliency (at expense of less efficient h).c 2  n" D 0p"`P H D 0޽h ? h  ___PPT10| .8UM|+YلID ' = @B Dc ' = @BA?%,( < +O%,( < +D2' =%(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' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* DD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* DD' =-g6B fade*<3<* DDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(++0+D  ++0+D  ++0+D  +  #   H(  HX" H 0 0r H S  C `}    H 0d   * H is pairwise indp. if for every distinct x1,x22{0,1}n, if hH then the r.v. h(x1) and h(x2) are uniform and indp. random variables. 2 G *b) # H 0 # ~Thm: For every choice of n>k>m and e if H:{0,1}n{0,1}m is a pairwise indp. collection then it is a t-resilient extractor where  2 6<7  H 0tТ , 0  4t= (k-m)/2 - 2log(1/e) - 1p 2 $   ,H H 0޽h ? h___PPT10i.;UF+D=' = @B +-  # yq@P (  Pr P S ܢ C `}    P 0ޢ c Suppose source contains k=512 bits of entropy and we want to extract m=256 bits with stat. distance e=2-35. m 2  (       L P 0  0 ,$ 0 6Using pairwise collection: get 57-resilient extractor.D7 2    )J P 0L  0 ,$ 0 6Using 16-wise collection: get 667-resilient extractor.d7 2       P 0d 9 0 c/Note: results are independent of source length.0 20H P 0޽h ? h\T___PPT104.>U3+~zD' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(+p+0+P  ++0+P  + # ? 7 P T (  Tr T S | Cr `@   B T 0   We implemented H = {hA} where hA:{0,1}n{0,1}m hA(x) = Ax and A is a nm Topelitz matrix.] 2 GCb  * T 0   VH is not pairwise independent but can be proven to satisfy same extraction properties.6W 2 T F T 0D$  ,$ 0 Note: For better efficiency on a PC, we worked over the field GF(216) and used some tricks to implement multiplication in that field faster.r 28Gv T 0- @  ,$ 0 NFor n=768, m=256, processed input at rate of 56Mbit/Sec on 1.7Ghz Pentium XeonxO 2 < T 0p6   ,$ 0 FPassed DIEHARD statistical tests on input a 90MB file of english text.$G 2F9  T 0< 0` ;? 2,3B  T s *Do  ,$@ 0H T 0޽h ? h_ W ___PPT107 .?Ur+ TDc ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =-g6B fade*<3<* TD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-g6B fade*<3<*TDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(++0+T  ++0+T  ++0+T  +rl` `cv0] % I!AJPRc 7 2 ? 0# Oh+'0T hp    ,8@?True Random Number Generators Secure in a Changing EnvironmentBoazRanRipplenBoazen61zMicrosoft PowerPointner@)?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Pictures+Current UserSummaryInformation( UPowerPoint Document(#DocumentSummaryInformation8