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I. Motivation AOriginal Motivation [SV84,Vaz85,VV85,CG85,Vaz87,CW89,Zuc90,Zuc91]&B- Randomization is pervasive in CS Algorithm design, cryptography, distributed computing, & Typically assume perfect random source. Unbiased, independent random bits Unrealistic? Can we use a  weak random source? Source of biased & correlated bits. More realistic model of physical sources. (Randomness) Extractors: convert a weak random source into an almost-perfect random source. !Z:Z(Z0Z#ZOZ]ZZ!:(0#O  E Applications of ExtractorsDerandomization of BPP [Sip88,GZ97,MV99,STV99] Derandomization of logspace algorithms [NZ93,INW94,RR99,GW02] Distributed & Network Algorithms [WZ95,Zuc97,RZ98,Ind02]. Hardness of Approximation [Zuc93,Uma99,MU01] Cryptography [CDHKS00,MW00,Lu02] Data Structures [Ta02] Z)#6The Unifying Role of ExtractorsNExtractors can be viewed as types of: Hash Functions Expander Graphs Samplers Pseudorandom Generators Error-Correcting Codes Unify the theory of pseudorandomness. T'X(' This Tutorial\Is framed around connections between extractors & other objects. We ll use these to: Gain intuition for the definition. Describe a few applications. Hint at the constructions. Many omissions. For further reading: N. Nisan and A. Ta-Shma. Extracting randomness: a survey and new constructions. Journal of Computer & System Sciences, 58 (1):148-173, 1999. R. Shaltiel. Recent developments in explicit constructions of extractors. Bulletin of EATCS, 77:67-95, June 2002. S. Vadhan. Course Notes for CS225: Pseudorandomness. http://eecs.harvard.edu/~salil VZ\Z&ZUZZZV\&P(`j  zi OutlineI. Motivation II. Extractors as extractors III. Extractors as hash functions IV. Extractors as expander graphs V. Extractors as pseudorandom generators VI. Extractors as error-correcting codes VII. Concluding remarks & open problemsII. Extractors as Extractors Weak Random SourcesWhat is a source of biased & correlated bits? Probability distribution X on {0,1}n. Must contain some  randomness . Want: no independence assumptions ) one sample Measure of  randomness Shannon entropy: No good: Better [Zuckerman `90]: min-entropy .ZvZZZ Z3Z. D &  , 2 Min-entropysDef: X is a k-source if H1(X) k. i.e. Pr[X=x] 2-k for all x Examples: Unpredictable Source [SV84]: 8 i2[n], b1, ..., bi-12 {0,1}, Bit-fixing [CGH+85,BL85,LLS87,CW89]: Some k coordinates of X uniform, rest fixed (or even depend arbitrarily on others). Flat k-source: Uniform over S {0,1}n, |S|=2k Fact [CG85]: every k-source is convex combination of flat ones.KZ<ZZZZ@Z        %>       , Extractors: 1st attempt$  A function Ext : {0,1}n ! {0,1}m s.t. 8 k-source X, Ext(X) is  close to uniform.S   #Extractors [Nisan & Zuckerman `93]$  hDef: A (k,e)-extractor is Ext : {0,1}n {0,1}d ! {0,1}m s.t. 8 k-source X, Ext(X,Ud) is e-close to Um. i       RDefinitional Details Ut = uniform distribution on {0,1}t Measure of closeness: statistical difference (a.k.a. variation distance) T =  statistical test or  distinguisher metric, 2 [0,1], very well-behaved Def: X, Y e-close if D(X,Y)e.$<JO    1The Parameters,The min-entropy k: High min-entropy: k = n-a, a =o(n) Constant entropy rate: k = W(n) Middle (hardest) range: k = na, 0 " Y$o4Application: BPP w/a weak source [Zuckerman `90,`91]5! !III. Extractors as Hash Functions Strong extractors6Output looks random even after seeing the seed. Def: Ext is a (k,e) strong extractor if Ext0(x,y)=yExt(x,y) is a (k,e) extractor i.e. 8 k-sources X, for a 1-e0 frac. of y2{0,1}d Ext(X,y) is e0-close to Um Optimal: d= log(n-k)+O(1), m= k-O(1) Can obtain strongness explicitly at little cost [RSW00].ZZ&ZZ9Z1         0 ,^ $Extractors as Hash FunctionsExtractors from Hash FunctionsLeftover Hash Lemma [ILL89]: universal (ie pairwise independent) hash functions yield strong extractors output length: m= k-O(1) seed length: d= O(n) example: Ext(x,(a,b))=first m bits of ax+b in GF(2n) Almost pairwise independence [SZ94,GW94]: seed length: d= O(log n+k) <hi*L      >( 6IIb. Extractors as Extractors Composing ExtractorsWe have some nontrivial basic extractors. Idea: compose them to get better extractors Original approach of [NZ93] & still in use. &,VX.Increasing the Output [WZ93]& #Increasing the Output Length [WZ93]$ Proof of Key Lemma Increasing the Output [WZ93] 8An Application [NZ93]: Pseudorandom bits vs. Small Space(9 # #) Output looks uniform to observer.$$#Shortening the Seed|Ext2 may have shorter seed (due to shorter output). Problem: Ext1 only guaranteed to work when seed independent of source. }Z  0  #  Block Sources [CG85]&iDef: (X1,X2) is a (k1,k2) block source if X1 is a k1-source is a k2-source *Z@Z       * <+The [NZ93] ParadigmXAn approach to constructing extractors: Given a general source X Convert it to a block source (X1,X2) can use part of the seed for this may want many blocks (X1,X2 , X3,...) Apply block extraction (using known extractors, e.g. almost pairwise independence) Still useful today  it  improves extractors, e.g. [RSW00] How to do Step 2?? get a block by randomly sampling bits from source... harder as min-entropy gets lower. (Z?" ZIZT" ZQZXZ(  7    T /X.OutlineI. Motivation II. Extractors as extractors III. Extractors as hash functions IV. Extractors as expander graphs V. Extractors as pseudorandom generators VI. Extractors as error-correcting codes VII. Concluding remarks & open problemsP$!IV. Extractors as Expander Graphs Expander Graphs*Measures of Expansion: Vertex Expansion: Every subset S of size an has at least b|S| neighbors for constants a > 0, b > 1. Eigenvalues: 2nd largest eigenvalue of random walk on G is l for constant l < 1. (equivalent for constant-degree graphs [Tan84,AM85,Alo86])p>  '>>  YExtractors & Expansion [NZ93] Connect x{0,1}n and y{0,1}m if Ext(x,r)=y for some r {0,1}d Short seed low degree Extraction expansionrZ @1Extractors vs. Expander GraphsMain Differences: Extractors are unbalanced, bipartite graphs. Different expansion measures (extraction vs. e-value). Extractors graphs which  beat the e-value bound [NZ93,WZ93] Extractors polylog degree, expanders constant degree. Extractors: expansion for sets of a fixed size Expanders: expansion for all sets up to some size . 7 @ .7 (    B-Extractors vs. Expander GraphsMain Differences: Extractors are unbalanced, bipartite graphs. Different expansion measures (extraction vs. e-value). Extractors graphs which  beat the e-value bound [NZ93,WZ95] Extractors polylog degree, expanders constant degree. Extractors expansion for sets of a fixed size Expanders expansion for all sets up to some size - 7 @ -7  (    B->Expansion Measures  Extraction"  " hExtractors: Start w/min-entropy k, end e-close to min-entropy m ) measures how much min-entropy increases (or is not lost) Eigenvalue: similar, but for  2-entropy (w/o e-close) }8"  ; -} .FExpansion Measures  The Eigenvalue.$ "   Let G = D-regular, N-vertex graph A = transition matrix of random walk on G = (adj. mx)/D Fact: A has 2nd largest e-value l iff " prob. distribution X || A X - UN ||2 l|| X - UN ||2 Fact: e-value measures how random step increases 2-entropy Z Z8Z & 5PU'+9Extractors vs. Expander GraphsMain Differences: Extractors are unbalanced, bipartite graphs. Different expansion measures (extraction vs. e-value). Extractors graphs which  beat the e-value bound [NZ93,WZ95] Extractors polylog degree, expanders constant degree. Extractors: expansion for sets of a fixed size Expanders: expansion for all sets up to some size - 7@ -7 ( 7  :   The DegreeConstant-degree expanders viewed as  difficult . Extractors typically nonconstant degree,  elementary Optimal: d log (n-k) truly random bits. Typically: k = dn or k = nd d=W(log n) Lower min-entropies viewed as hardest. Contradictory views? Easiest extractors highest min-entropy k = n O(1) d=O(1) constant degree Resolved in [RVW01]: high min-entropy extractors & constant-degree expanders from same, simple  zig-zag product construction.P16|26 ()      k>G W"High Min-Entropy Extractors [GW94]#  Zig-Zag Product [RVW00]( $ Extractors vs. Expander GraphsMain Differences: Extractors are unbalanced, bipartite graphs. Different expansion measures (extraction vs. e-value). Extractors graphs which  beat the e-value bound [NZ93,WZ95] Extractors polylog degree, expanders constant degree. Extractors: expansion for sets of a fixed size Expanders: expansion for all sets up to some size 8- 7@6 a -7 ( 6 : Randomness Conductors [CRVW02] Six parameters: n, m, d, k, e, D For every k k and every input k-source, output is e-close to a (k+D)-source. m = k+D : extractor with guarantee for smaller sets. D = d :  Lossless expander [TUZ01] Equivalent to graphs with vertex expansion (1-e)degree! Explicitly: very unbalanced case w/polylog degree [TUZ01], nearly balanced case w/const. deg [CRVW02]#Z2RZ(6Z2$ZZ   ++   2  / ,o? (V. Extractors as Pseudorandom Generators  'Pseudorandom Generators [BM82,Y82,NW88](zGenerate many bits that  look random from short random seed.Hardness vs. RandomnessAny function of high circuit complexity PRGs [BM82,Y82,NW88,BFNW93,IW97,...] Current state-of-the-art [SU01,Uma02]: Thm [IW97]: If E=DTIME(2O(l)) requires circuits of size 2W(l), then P=BPP.(!    ,*MHExtractors & PRGs Thm [Trevisan `99]: Any  sufficiently general construction of PRGs from hard functions is also an extractor. 4pD63,Extractors & PRGs Extractors & PRGs Extractors & PRGs Analysis (intuition) &Analysis ( formal )When does this work?When PRG has a  black-box proof: for any function f and any statistical test T, i.e. if PRG construction  relativizes [Mil99,KvM99] Almost all PRG constructions are of this form. Partial converse: If EXT is an explicit extractor and f has high description (Kolmogorov) complexity relative to T, then EXT( f , ) is pseudorandom for T. UZ6ZZ' 709,p  I ConsequencesSimple (and very good) extractor based on NW PRG [Tre99] (w/ subsequent improvements [RRV99,ISW00,TUZ01]) More importantly: new ways of thinking about both objects. Benefits for extractors: Reconstruction paradigm: Given T distinguishing Ext(x,Ud) from Um,  reconstruct x w/short advice (used in [TZS01,SU01]) New techniques from PRG literature. Benefits for PRGs: Best known PRG construction [Uma02], finer notion of optimality. Distinguishes information-theoretic vs. efficiency issues. To go beyond extractor limitations, must use special properties of hard function or distinguisher (as in [IW98,Kab00,IKW01,TV02]).jZ<ZZZZ1?   ' ,s(VI. Extractors as Error-Correcting Codes Error-Correcting CodesnClassically: Large pairwise Hamming distance. List Decoding: Every Hamming ball of rel. radius -e in {0,1}D has at most K codewords. Many PRG [GL89,BFNW93,STV99,MV99] and extractor [Tre99,...,RSW00] constructions use codes. [Ta-Shma & Zuckerman `01]: Extractors are a generalization of list-decodable codes.<\2T !" 9P8% dLRStrong 1-bit Ext s List-Decodable Codes**     -RStrong 1-bit Ext s List-Decodable Codes**     RStrong 1-bit Ext s List-Decodable Codes**     Extractors & CodesMany-bit extractors list-decodable codes over large alphabets (size 2m) [TZ01]  Reconstruction proof in PRG view $ Decoding algorithm in code view Trevisan s extractor has efficient decoding alg. [TZ01]. several applications (data structures for set storage [Ta02]...) Idea [Ta-Shma, Zuckerman, & Safra `01]: Exploit codes more directly in extractor construction.B`2,QB(7P "V>"Extractors from CodesExisting codes give extractors with short output. Q: How to get many bits? Use seed to select m positions in encoding of source. Positions independent: works but seed too long. Dependent positions? [Tre99] gives one way. `ZZ3*j$Dependent projectionsNaive: consecutive positions,&The ReconstructionFEasy case: P always correct. Advice: first consecutive i m positions.*  'Dealing with ErrorsFQ: How to deal with errors? Use error-correcting properties of code Suffices to reconstruct  most positions. -e errors requires list-decoding ) additional advice But one incorrect prediction can ruin everything! Idea: error-correct with each prediction step Need  consecutive to be compatible w/decoding, reuse advice. Reed-Muller code: ECC(x) =low-degree poly. Fm! F [Ta-Shma,Zuckerman, & Safra `01]: consecutive = along line. [Shaltiel-Umans `01]: consecutive = according to linear map which  generates Fmn {0} [Umans `02]: PRG from this.)/>2),E)>                       "8 tz">#VII. Concluding Remarks (Towards OptimalityRecall: For every k n, 9 a (k,e)-extractor w/ Seed d= log(n-k)+O(1) & Output m = k+d-O(1) Thm [...,NZ93,WZ93,GW94,SZ94,SSZ95,Zuc96,Ta96,Ta98,Tre99, RRV99, ISW00,RSW00,RVW00,TUZ01,TZS01,SU01,LRVW02] For every k n, 9 an EXPLICIT (k,e)-extractor w/ Seed d= O(log(n-k)) & Output m = .99k Seed d= O(log2(n-k)) & Output m = k+d-O(1) Not there yet! Optimize up to additive constants. In many apps, efficiency D=2d Often entropy loss k+d-m significant, rather than m itself. Dependence on error e N1Z-Z&ZtZZ h       ^) ConclusionsThe many guises of randomness extractors extractors, hash fns, expanders, samplers, pseudorandom generators, error-correcting codes translating ideas between views very powerful! increases impact of work on each object The study of extractors many applications many constructions:  information theoretic vs.  reconstruction paradigm optimality important L)s)s*Some Research Directions`Exploit connections further. Optimality up to additive constants. Single, self-contained construction for all ranges of parameters. ([SU01] comes closest.) Study randomness conductors. When can we have extractors with no seed? important for e.g. cryptography w/imperfect random sources. sources with  independence conditions [vN51,Eli72,Blu84,SV84, Vaz85, CG85,CGH+85,BBR85,BL85,LLS87,CDH+00] for  efficient sources [TV02] %<[<2*0!cD+Further ReadinggN. Nisan and A. Ta-Shma. Extracting randomness: a survey and new constructions. Journal of Computer & System Sciences, 58 (1):148-173, 1999. R. Shaltiel. Recent developments in explicit constructions of extractors. Bulletin of EATCS, 77:67-95, June 2002. S. Vadhan. Course Notes for CS225: Pseudorandomness. http://eecs.harvard.edu/~salil many papers... ZZZZZP(Jk {k  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@$x?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    67 `0 7 T Click to edit Master title style! !  07 ` 7 RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0D7 `` 7 X*  07 `  7 Z*  07 `  7 Z*H  0޽h ? 3̙f Default Designx  t(  t t N0=yYyY N  > x* ---DDZZ t N3<yYyY   = z* ---DDZZ t T<yYyY N  = x* ---DDZZ t TϟyYyY   = z* ---DDZZH t 0޽h" ? ̙3380___PPT10.Yp @ P(  r  S 7 > r  S 7 }  7 $   <$7O y .to be posted at http://eecs.harvard.edu/~salil//PH  0޽h ? ̙33p  p(  px p c $ 3  H p 0޽h ? ̙33  ph$(  hr h S `:`0   r h S 4;`  H h 0޽h ? 3̙f  `l$(  lr l S e`0   r l S |f`  H l 0޽h ? 3̙f  Pt$(  tr t S h~`0   r t S <`  H t 0޽h ? 3̙f  @|$(  |r | S `0   r | S `  H | 0޽h ? 3̙f  0x:(  xr x S `0    x S @`0  "`H x 0޽h ? 3̙f  $(  r  S lp  r  S  `    H  0޽h ? 3̙f  2* (  r  S `0     S |`  " 0Pp   ^A txp_fig` f ` X  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$X=\cases{ \mbox{uniform on $\zo^n$} & w.p. $.1$\cr 0^n & w.p. $.9$} \Rightarrow \H(X) \geq .1n$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH494.8758PICTUREFILESIZE 49722    ^Atxp_figR XA 8 0  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\H(X) \eqdef \Exp_{x\getsr X}\left[\log_2\left(\frac{1}{\pr{X=x}}\right)\right]$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH3308PICTUREFILESIZE 40826    ^Atxp_figL . &  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\Hmin(X) \eqdef \min_{x}\left[\log_2\left(\frac{1}{\pr{X=x}}\right)\right]$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH3428PICTUREFILESIZE 38942H  0޽h ? 3̙f   P H  (  r  S 6   r  S <4     ^A txp_fig\&   SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$k/n \leq \pr{X_i=1 | X_1=b_1,\ldots,X_{i-1}=b_{i-1}} \leq 1-k/n$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH512.8758PICTUREFILESIZE 24450H  0޽h ? 3̙f  \T (  r  S $@`0  @ r  S %@C @ J   0+@ =  pImpossible! 9 set of 2n-1 inputs x on which first bit of Ext(x) is constant ) flat (n-1)-source X, bad for Ext.Hq        6,@Ԕ P  UEXT6( (  6x3@ԔP k-source of length nP  B@ԔX p  sm almost-uniform bitsBdB  <DԔ  dB  <DԔP   @ H  0޽h ? 3̙f   *(  r  S Z@`0  @ r  S [@T @ dB  <DԔ @   B`@Ԕ @V  rd random bitsH   B`c@  >  seed x  0s@  ! fKey point: seed can be much shorter than output. Goals: minimize seed length, maximize output length.1 05   -  6j@Ԕ  UEXT6( (  6X@ԔNP k-source of length nP  B@Ԕ p  sm almost-uniform bitsBdB  <DԔ  dB  <DԔ   H  0޽h ? 3̙fN      (  r  S @`0  @ r  S h@ @ b   ^A txp_fig]Ny   f SOURCEJ\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} \begin{eqnarray*} \Delta(X,Y)&\eqdef& \max_{T}\left|\pr{X\in T}-\pr{Y\in T}\right|\\ &=& \frac{1}{2} \sum_x \left|\pr{X=x}-\pr{Y=x}\right| \end{eqnarray*} \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH428.8758PICTUREFILESIZE 85854H  0޽h ? 3̙f  $(  r  S t@`0  @ r  S H@ @ H  0޽h ? 3̙f"  RJ (  r  S (@  @ r  S @A\ @ Al ~: ~:,$D 0   ^Atxp_figW Kv SOURCEl\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $${N \choose K} \cdot 2^M \leq 2^{\delta KD}$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH163.8758PICTUREFILESIZE 158141t ~: ~:,$D 0  0@ ~: KPf sketch: Probabilistic Method. Show that for random Ext, Pr[Ext not (k,e)-extractor] < 1. Use capital letters: N=2n, M=2m, ... For fixed flat k-source X and T {0,1}m, # choices of X and T = R]0xk0 Z ,       <C   ^ (Chernoff)    ^Atxp_fig l  z  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\Pr_\Ext\left[\left|\Pr_{X,U_d}\left[\Ext(X,U_d)\in T\right] -\frac{|T|}{M}\right|>\eps\right] \leq 2^{-\Omega(KD)}$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH4748PICTUREFILESIZE 54622m  <@z K,$D 0 ( log n except high min-ent.) p H  0޽h ? 3̙f___PPT10+ƶD' = @B DF' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+C +  0(  x  c $.CN  C x  c $.C C H  0޽h ? 3̙f  *0 (  r  S 1C  C    Bl6C    ? accept/reject   6:CԔP   FRandomized AlgorithmdB   <DԔ@ ~@   6X?CԔ ~  Ginput x$dB  <DԔ 6 6 [   BDDCp `  rerrs w.p. dF Fl p P )p P,$D 0  HKCp   x 2( )0   <+B#style.visibility<**%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(+ $(  r  S Cp C r  S pC `   C H  0޽h ? 3̙f  p$(  r  S |C`0  C r  S PC` C H  0޽h ? 3̙f  `a(  r  S ЗC`0  C ^2  6oGvNX ^2  6o\    <Cm D{0,1}n"    <Ce D{0,1}m" f   ^Atxp_fig7 r SOURCEV\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$h_y(\cdot)\eqdef \Ext(\cdot,y)$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1638PICTUREFILESIZE 11062l ]1 ]1,$D 0]t J]  J] ,$D 0f2  6oJ `B  0Do`B  0Do  `B   0Do(I3   <C])c 'flat k-source, i.e. set of size 2k 2m|(    <ԹC 1 PBFor most y, hy maps sets of size K almost uniformly onto range. C  5XB  0DoEH  0޽h ? 3̙f~___PPT10^++WDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+  P2(  r  S C`0  C r  S @C` C R  C *Aj0079104Zj \ kR  C *Aj0079100w e R  C *Aj0079099   H  0޽h ? 3̙f @0(  x  c $Cp C x  c $C `   C H  0޽h ? 3̙f  0$(  r  S C`0  C r  S  ! 6KԔ  XEXT18(   ( " 6KԔ VX bk-source> dB $ <DԔUH H dB % <DԔH H W z o  / Z ,$D 0rB ( BDԔv0 0  ) HD KԔoe  ud2 bitsR Ul 3 rP  -b3 P ,$D 0  HdKԔ3 h P   m2-bit outputX   & <KԔ $\  hEXT2H(  (rB ' BDԔ@\ @g rB * BDԔ@r@$ + HpKoP ,$D 0 B 8l `  2` ,$D 0Z 0 C *Aj0079104  Z 1 C *Aj0079100``Y H  0޽h ? 3̙f  ___PPT10 +ľDL ' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(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<*/%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(+p+0+K ++0++K +   .(  r  S /K`0  K T   64K<   Prop: If Ext1 is a (k,e)-extractor & Ext2 a (k-m1-O(1),e)-extractor, then Ext is a (k,3e)-extractor. Key lemma: (X,Z) (correlated) random vars. 6nx.    d   ^A txp_figp SOURCET\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\H(X|Z)\geq \H(X)-\H(Z) \geq k-s$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH3078PICTUREFILESIZE 14622  Bp_KK   X a k-source and |Z|=s`   BjK8    7w.p. 1-e over zZ, X|Z=z is a (k-s-log(1/e))-source.8 %  Bt}K%  E b Compare w/Shannon entropy: *^  6o c ^  6o+U H  0޽h ? 3̙f  7/ (  r  S ĄK`0  K   6`Kr o Key lemma: (X,Z) (correlated) random vars, Proof: Let BAD = { z : Pr[Z=z] e 2-s}. Then 0x2P    P%7   ^Atxp_fig* e% H @  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} \begin{eqnarray*} z\notin BAD \Rightarrow \forall x \pr{X|_{Z=z} = x} &\eqdef& \pr{X=x | Z=z}\\ &\leq& \frac{\pr{X=x}}{\pr{Z=z}}\\ &\leq& \frac{2^{-k}}{\eps\cdot 2^{-s}} = 2^{-(k-s-\log(1/\eps))} \end{eqnarray*} \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH600:PICTUREFILESIZE 186818   ^Atxp_fig ]   SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\pr{Z\in BAD} \leq |BAD|\cdot (\eps\cdot 2^{-s}) \leq \eps$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH3548PICTUREFILESIZE 17170   BK< X a k-source and |Z|=s`    BHK) } 7w.p. 1-e over zZ, X|Z=z is a (k-s-log(1/e))-source.8 %^   6oc H  0޽h ? 3̙f`  =550 (  E   <M ,$D 0 Pf of Prop: Z1 e-close to uniform (because Ext1 an extractor) w.p. 1-e over z Z1 X|Z1=z a (k1-m1-O(1))-source (by Key Lemma) Z2 |Z1=z e-close to uniform (because Ext2 an extractor) ) (Z1,Z2) 3e-close to uniform.  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(  r  S 8M  M   S M& hm<$ 0 M   6TMԔG    m m1-bit outputD dB  <DԔ     6MԔ [  cd1 bits@  6MԔ_  XEXT18(   (  6 MԔ?k w bk-source> dB   <DԔ VVG dB   <DԔwVV_l  @  ? @ ,$D 0   <TMԔ@  wd2 bitsT   <MԔ e `zzzz ,   <8MԔ   hEXT2H(  (lB  <DԔKK  S BC DEFo   @[ K lB  <DԔpp  0XM 0* 5Idea: use output of one extractor as seed to another.46 /Rz  Z   Z,$D 0`  S 0Aj0079104 `  S 0Aj0079100ZH  0޽h ? 3̙f@ 8 ___PPT10 .+BD ' = @B Ds ' = @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<*5%(D%' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5|%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+O +  (  x  c $O!  O   c $M& )Im<$ 0 O   6OԔG    m m1-bit outputD dB  <DԔ     6OԔ [  cd1 bits@  6 OԔ_  XEXT18(   (  6"OԔ?k w BX1$ dB   <DԔ VVG dB   <DԔwVV_z  @    ? @ ,$D 0   6'OԔ@  ud2 bitsR    6D.OԔ e `X2B  63OԔ   hEXT2H(  (lB  <DԔKK  c BC DEFo   @[ K lB  <DԔpp  0:O ;* qQ: When does this work? & z   ^Atxp_figr/ SOURCEl\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} $$\forall x_1 \hspace{.5em} X_2|_{X_1=x_1}$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1386PICTUREFILESIZE6902H  0޽h ? 3̙f  :(  r  S $XO`0  O   S ^O`( O " 0PpH  0޽h ? 3̙f  F(  x  c $pwO`0  O   c $>`0 O "`H  0޽h ? 3̙f $(  r  S >p O r  S (> `   > H  0޽h ? 3̙fi  OG (  ~  s *A<  < ~  s *!<T9 <   <O{3 y9 Informally: Sparse graphs w/ very strong connectivity.$:+  0O  i Goals: Minimize the degree. Maximize the expansion. Random graphs of degree 3 are expanders [Pin73], but explicit constructions of constant-degree expanders much harder [Mar73,...,LPS86,Mar88]. /  /*FR2  s *  R2  s *6MR2  s *v R2   s * 6 R2   s *]R2   s *m &V R2   s *]  R2   s *&- ^B  6D1M ^B  6D1F ^B  6D1  ^B  6D1}q ^B  6D1 ^B  6D1@ ^B  6D1q F ^B  6D1 ^B  6D1q  ^B  6D18? ^B  6D1} ^B  6D1" yl   p z ,$D 0:  3 B]CgDEdF.jJ}YSi>83a9dg4Q]83 VXIk }Y@        `   <OX GS*z fk    fk ,$D 0J  3 B{ChDEpF2jJC%uj)oN'v?qeh$G-1pyz{ aC@         `Bk   BOf` n Neighbors(S)F H  0޽h ? 3̙f___PPT10+ED~' = @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<* %(+*  P"H"CY!(  l  Vh  U Vh ,$D 0l2  <ZfxVh rB * BD1& rB + BD1d rB , BD1E rB - BZD1?rB . BD16 rB / BD1<rB 0 BD1rB 1 BD1rB 2 BD1erB 3 BD1% lB 6 <DlB 7 <D d'l  X W X,$D 0 ! <XOB6 CK&r2  BZ1 hXrr   BZjJ>,~  s *O  O ~  s *O 2 O  " <xO x [N] {0,1}nP  # <OL   x [M] {0,1}mP  ) BO18X AD$^2  6Z0`d2  <Z1d2  <Z1@d2   <Z1d2   <Z1 d2   <Z1fd2   <Z1d2   <Z1Fvd2  <Z1d2  <Z1&Vd2  <Z1d2  <Z16d2  <Z1v^2  6Z d2  <Z1& d2  <Z1f d2  <Z1 d2  <Z1Fv d2  <Z1 d2  <Z1&V d2  <Z1 d2  <Z16 d2  <Z1 ^2  6ZV ^2  6Z ^2  6Z6 f jB $ BD1HjB % BZD1jB & BD1jB ' BZD1A ( # ZB`CDEF10<`x``0\@  X l f   V f ,$D 0 5 <O   v (1-e) MD r 4 BZjJf V  F 6pOԔQ c zn-bit k-sourceP G 6OԔ.g (  mm almost-uniform bits< H BOԔmq T d-bit seed.  dB I <DԔW|W J 6$OԔ?= IEXT*( (dB K <DԔlPP?dB L <DԔ<PW X <O 5x Y <O}8  5yH  0޽h ? 3̙f___PPT10+?D' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*V%(+  pF(  x  c $P P`0  P   c $lP` P " 0PpH  0޽h ? 3̙f  `F(  x  c $4P`0  P   c $4<` P " 0PpH  0޽h ? 3̙f   b Z PAA((  (Tz  Vh  -(  Vh ,$D 0f2 .( 6ZfxVh lB /( <D1& lB 0( <D1d lB 1( <D1E lB 2( <ZD1?lB 3( <D16 lB 4( <D1<lB 5( <D1lB 6( <D1lB 7( <D1elB 8( <D1% fB 9( 6DfB :( 6D d)z  X !(  X,$D 0 "( <9PB6 CK&l2 #( <Z1 hXlr $( <ZjJ>,~ ( s *P  P  ( <<w x [N] {0,1}nP  ( <?PL   x [M] {0,1}mP  ( BGP18X AD$X2 ( 0Z0`^2 ( 6Z1^2  ( 6Z1@^2  ( 6Z1^2  ( 6Z1 ^2  ( 6Z1f^2  ( 6Z1^2 ( 6Z1Fv^2 ( 6Z1^2 ( 6Z1&V^2 ( 6Z1^2 ( 6Z16^2 ( 6Z1vX2 ( 0Z ^2 ( 6Z1& ^2 ( 6Z1f ^2 ( 6Z1 ^2 ( 6Z1Fv ^2 ( 6Z1 ^2 ( 6Z1&V ^2 ( 6Z1 ^2 ( 6Z16 ^2 ( 6Z1 X2 ( 0ZV X2 ( 0Z X2  ( 0Z6 f dB %( <D1HdB &( <ZD1dB '( <D1dB (( <ZD1 H )( # ZB`CDEF10<`x``0\@  X Bz f   *(  f ,$D 0 +( <pQP   x (1-e) MF  ,( NZ8SPjJf V  >  ;( 6\PԔQ c zn-bit k-sourceP <( 6XcPԔ.g (  mm almost-uniform bits< =( BiPԔmq T d-bit seed.  dB >( <DԔW|W ?( 6@oPԔ?= IEXT*( (dB @( <DԔlPP?dB A( <DԔ<PWH ( 0޽h ? 3̙f  ph@(  x  c $TzP` P x  c $({P`0  P `  c $A u??$T Z  u`  c $A ?? z  H  0޽h ? 3̙f  0 F(   x   c $ȎP`0  P    c $P` P " 0PpH   0޽h ? 3̙f   0(  x  c $P`0  P x  c $ȦP0P P H  0޽h ? 3̙f8  $(  x  c $P  P   B|Po$$  > jr  BZH*1]$   <<yO s length n, (n-a)-sourcezz J  " J ,$D 0FT |  # | ZB   s *Do4 $4    << Jn1,    BP |  @n2"    6 z S # S,$D 0T S # S`B B 0Dolf  6oD f  6os=  <P   Jd1, `B B 0Do<h  BPD  TEXT14$ `B  0DoD Jf  6oSS  <lP Jm1, f  6o  g  B+B#style.visibility<*"%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(+ $  33(  x  c $,T  T   B To$$  > 8F |   | ZB  s *Do4 $4   <T Jn1,   B T |  @n2" F S  S`B  B 0Dolf   6oD f   6os=   <T   Jd1, `B  B 0Do<h  B0TD  TEXT14$ `B  0DoD Jf  6oSS  <\T Jm1, f  6o  g  B,$T  K TEXT24$ f  6o   <)TUM Jd2, `B B 0Do w w`B  0Do< D `B B 0Dog jr  BZH*1]$   </TyO s length n, min-entropy n-a`&  08T,$D 0 $Problem: Lose a bits of min-entropy.R% z 4gkr   g4kr ,$D 0N 4gt   4gt   BDATo   > ZB  s *Do     <+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(+  F(  x  c $rT`0  T   c $xT` T " 0PpH  0޽h ? 3̙f8    x(   x   c $ T  T x   c $TR T    6H<Ԕ7LeI O n-bit input(     6d<Ԕ P m-bit output(     BTԔ X d-bit seed2  dB   <DԔ= R =   6TԔ%R # ICON*( (dB   <DԔR %dB   <DԔ" H   0޽h ? 3̙f P$(  Pr P S PTp T r P S T `   T H P 0޽h ? 3̙f\    ,(  , , 6(TԔ;  9PRG(~ , s *T`0  T ~ , s *T~aN T ( , 6TԔ G  $m bits indistinguishable fr. uniform`%  s , <` =  c Distributions X, Y computationally indistinguishable if for all efficient T (circuit of size m),d!    , BTԔ- N T d-bit seed.  x  , <A ??l$ XB  , 0DoN ;XB  , 0Do H , 0޽h ? 3̙f   0J(  0x 0 c $dT`0  T x 0 c $T` T  0 0T ^0  > 2 $ 0 6To@0 X  Jf : {0,1}l{0,1} circuit complexity kX& 0 6(To@ X  "@PRGf : {0,1}O(l){0,1}m m kW(1)! 0 BTo  > H 0 0޽h ? 3̙f  40(  4x 4 c $V`0  V x 4 c $HV V H 4 0޽h ? 3̙fx   (  p(  px p c $@ V`0  V  p 6TԔ h  fPRGf,(( p BVԔ  T d-bit seed.  XB p 0DottXB p 0Doh yyU  p 6tVԔo  GEXT((  p B|TԔ  T d-bit seed.  XB  p 0Do77XB  p 0Don <<[   p 6h VԔO m  statistically close to UmXXB  p 0Do/ /  p 6'VԔ[ |n bits w/min-entropy kJ p 6<.VԔO + m  comp. indisting. from UmX  H p 0޽h ? 3̙f   p h t (  tx t c $7V`0  V  t 6@9VԔ h  fPRGf,(( t B>VԔ  T d-bit seed.  XB t 0DottXB t 0Doh yyU  t 6CVԔo  GEXT((  t B:VԔ T d-bit seed.  XB  t 0Do77XB  t 0Don <<[   t 6\NVԔO m  statistically close to UmXXB  t 0Do/ /  t 6TVԔ K >  |n bits w/min-entropy kJ t 6H\VԔO + m  comp. indisting. from UmX  @ t 0lV  ~Step I: View extractors as  randomness multipliers [NZ93,SZ94]0@ - H t 0޽h ? 3̙f{   + # px (  xx x c $|gV`0  V  x 6TrVԔ.@  fPRGf,(( x B\wVԔ 9 B T d-bit seed.  XB x 0DoB/XB x 0Do  x 6|VԔ5 GEXT((  x BlVԔ B T d-bit seed.  XB  x 0DoH::5XB  x 0Do??   x 6 VԔ   statistically close to UmXXB  x 0Do  x 6@VԔN  |n bits w/min-entropy kJ x 6`VԔ 2  comp. indisting. from UmX   x 0XV  /Step II: View hard function as an input to PRG.V0 | x 6VԔ Zf : {0,1}log n{0,1} circuit complexity k ~. ,XB x 0Do@H x 0޽h ? 3̙fY   `&|(  | | 6lVԔj & KPRG,(( | BXVԔF ~ T d-bit seed.  XB | 0Do~N N kXB | 0Do&S S   | 6VԔ +  comp. indisting. from Um\   | 6$VԔ  (Zf : {0,1}log n{0,1} circuit complexity k . ,XB | 0Do , | 0V U :PH@___PPT9" t f from dist. of min-entropy k whp f has circuit complexity k  O(1) (even  description size k  O(1)) Statistical closeness computational indistinguishability  relative to any distinguisher (1) holds  relative to any distinguisher ."  n,#qK 0Pp | BV `0  V  | <Vjd A min-entropy  | <`V^X  Istatistically close | BV3e EEXT&( d2 | <jJ d2 | <jJ. b d2 | <jJKi  | s B,CDEFjJ$K,"f(@   n $| c BCDEFop}oE@  K   &| c BCDEFoH[*@ @  Qe H | 0޽h ? 3̙f6  Pv(    6WԔj & KPRG,((  BWԔF ~ T d-bit seed.  XB  0Do~N N kXB  0Do&S S    6p WԔ +  comp. indisting. from Um\    6WԔ  (Zf : {0,1}log n{0,1} circuit complexity k . ,XB  0Do    0W U :PH@___PPT9"  Fix a statistical test T {0,1}m. Suppose T distinguishes PRGf (Ud) from Um. PRG correctness proof ) circuit C of size k0 s.t. CT f. But w.h.p., f|C is a (k-k0-O(1))-source (by Key Lemma), so undetermined if k0 k. )(              ( ,; @`   BPW `0  W    <LQWjd A min-entropy    <UW^X  Istatistically close   BYW3e EEXT&( d2  <jJ d2  <jJ. b d2  <jJKi   s B,CDEFjJ$K,"f(@   n  s BCDEFop}oE@  K    s BCDEFoH[*@ @  Qe   <_W '  h2 reconstruction paradigm XB @ 0Do l& H  0޽h ? 3̙fX  @H(  Hx H c $hgW`0  W x H c $(hW` W ` H c $A ??g|/k H H 0޽h ? 3̙f  0L0(  Lx L c $rW  W x L c $sWP W H L 0޽h ? 3̙f  d$(  dr d S ,Wp W r d S W `   W H d 0޽h ? 3̙f%  0 Te(  Tx T c $W  W x T c $SW #: W  T 64WԔ   9ECC$ T 6DWԔ  mn-bit message xB  T 6WԔCU D-bit codeword ECC(x)TXB  T 0Doc c CXB  T 0Do$c c H T 0޽h ? 3̙f  :X((  XdB X <Do Z X BWo   _d-bit y< X BPW   W  X 6ԨWԔZ ?EXT $  X 6(WԔ2  en-bit xBXB X 0Do:XB  X 0Do $X <W <  &X 6(WԔHx 5ECC$ 'X 6WԔ  en-bit xB (X 6WԔ;Aa M  k D=2d bitsF XB )X 0Do;XB *X 0Dol * o :X* S ,$D 0 !X # tBCDEFJo..dJFJ7#rD\zLb=]# ##iFG-H-%Zr:2cY7|W"$@               `* o^# %X <WQ  e-close to Ud+1b  fB +X 6Do wG -X 6`WԔ/}J.  Ks.l Ih  9XI ,$D 0 ,X 6TWԔo  Ks. 0X c BCDE(Fo J + f4M  @      2X 6WԔ|I Ks. 7X 0W h ,$ 0 &The Correspondence: ECC(x)y = EXT(x,y)'(H X 0޽h ? 3̙f___PPT10+$X/D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9X%(+  TL>(  4  6Z-',$D 0 `Claim: EXT (k,e) extractor ECC has < 2k codewords in any (-e)-ball Pf: Suppose 2k codewords within distance (-e) of z2{0,1}D. Feed extractor uniform dist. on corresponding msgs. Consider statistical test T={ ys : y2{0,1}d, s=zy } Pr[extractor outputT] > +e Pr[uniform distributionT] = . H0Z202<0Z2k0Z2A0 Z2  PP, " R3D  B Z }M  Z dB ) <Do Z * B48Zo * _d-bit y< + 6p>ZԔZ ?EXT $  , 6ZԔ 2 en-bit xBXB - 0DoLXB . 0Do- / <TIZ&  <  0 6$LZԔHx 5ECC$ 1 6PZԔ   en-bit xB 2 6 6XrZ ],$ 0 ECC(x)y = EXT(x,y)(H  0޽h ? 3̙f  O\$(  \ \ 60Z~ ,$D 0 Claim: ECC has < e2k codewords in any (-e)-ball EXT (k,2e) extractor Pf: Suppose on k-source X, output 2e-far from Ud+1. $ P : {0,1}d {0,1} s.t. PrX,Y[P(Y)=EXT(X,Y)] > +2e. EX[dist(P,ECC(X))] < -2e. But only e2k codewords in (-e)-ball around P. t0Z270Z20Z260Z2        P ZC  :\ BZ Y  Z dB ;\ <Do Z <\ BZo? K _d-bit y< =\ 6ZԔZ ?EXT $  >\ 6ZԔ-2? en-bit xBXB ?\ 0DomXB @\ 0DoN A\ <ZGA <  B\ 6`ZԔHx 5ECC$ C\ 6ZԔ. @ en-bit xB D\ 6ZԔnAa  k D=2d bitsF XB E\ 0DonXB F\ 0DoOz * o G\ 1*  ,$D 0 H\ 3 zBCDEFJo..dJFJ7#rD\zLb=]# ##iFG-H-%Zr:2cY7|W"$@               `* o^# I\ < ZQ  e-close to Ud+1b  fB J\ 6Do wG K\ 6ZԔb}Ja Ks. L\ 6ZԔro q Ks. M\  BCDE(Fo J + f4M  @    d N\ 6[Ԕa|I` Ks.j O\ 0[ ],$ 0 ECC(x)y = EXT(x,y)(H \ 0޽h ? 3̙f  `0(  `x ` c $ [`0  [ x ` c $X [` [ H ` 0޽h ? 3̙f  ,(  r  S [`0  [ r  S ![` [   6p"[Ԕ8 9ECC$  6%[Ԕe}w en-bit xB   6+[Ԕbt ZECC(x)8XB   0Do66bXB   0Do668   BP1[o  Nseed y,   BC+DEFVU4f+@  Q QeX  0aBzpdl  bn   bn,$D 0    BCDEFR < @    m   BOCDEF'_'#N9O@   VCe   B]CgDEF@ S(]g@   Jt`  0ee`  0&c^k`  0*nbn`  0rll   BCIDEF Cz;I@   $m2l  lJ  l J ,$D 0  69[Ԕ   ^ m-bit output6  ZB  s *D t& ZB  s *D^t ZB B s *D6tQ ZB B s *Dt ZB B s *DzlJ d  <oD&% H  0޽h ? 3̙f___PPT10+F3D~' = @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<*%(+#  _W/(  r  S dO[  [ r  S O[xf =   0n[ v ,$D 0 3Analysis attempt (reconstruction): Goal: given T {0,1}m which distinguishes Ext(x,Ud) from Um, reconstruct x with few (i.e. k) bits of advice. J$ q ! $   W=  6W[Ԕ8 9ECC$   6[[Ԕe}w en-bit xB   6ta[Ԕba > XB   0Do66bXB   0Do668   Be[o  Nseed y,  3 BC+DEFjJVU4f+@  Q >GX  0bJS  6 k[Ԕ   Xm bits6XB @ 0DjJeO XB @ 0DjJe d  <oD&% l   / ,$D 0T   ^Atxp_fig    P SOURCE4\documentclass{slides}\pagestyle{empty} \begin{document} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\poly}{{\mathrm{poly}}} \newcommand{\polylog}{{\mathrm{polylog}}} \newcommand{\loglog}{{\mathop{\mathrm{loglog}}}} \newcommand{\zo}{\{0,1\}} \newcommand{\pr}[2][]{\Pr_{#1}\left[#2\right]} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \newcommand{\Hmin}{\mathrm{H}_{\infty}} \renewcommand{\H}{\mathrm{H}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Samp}{\mathrm{Samp}} \newcommand{\Supp}{\mathrm{Supp}} \newcommand{\Ecc}{\mathrm{Ecc}} $$\Pr_y\left[P(\Ecc(x)_{y}\Ecc(x)_{y+1}\cdots \Ecc(x)_{y+i-1})=\Ecc(x)_{y+i})\right] > \frac{1}{2}+\frac{\eps}{m}$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH639.8758PICTUREFILESIZE 60206 ! 6[Ԕ  > ` " 0P  Z % s *  lr ' <Zo@   ) # BC0DE4Fo  mR!<W1&N*T}{0 @    [  * <[  3P + <[   bECC(x)@ , <آ[   5ie - B[  7 From T, get next-bit predictor P : {0,1}i! {0,1} s.t.8!    . <[-LF 9[Yao82]H  0޽h ? 3̙f___PPT10+q.D' = @B DF' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(+8+0+[ +0l  4z4T)@3(  r  S о[`0  [ r  S [D(H  [   6[Ԕ  > X  0q R  s *  dr  <ZoCa   3 BC0DE4Fo  mR!<W1&N*T}{0 @    x|    <[S s 3P   <t[ bECC(x)@   B[V@Y * P correct for +e fraction of positions.H+    <0[R  5i   6l[Ԕ   > X  0 E   <[ @   bECC(x)@  <[ ]  5id  B[ 3N @ Repeatedly applying P, reconstruct all of ECC(x) & hence x.A     6jJ @ ,$D 0 { \  # { \ ,$D 0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <[s{ \  3P  0jJ | ,$D 0 { \  # {  ,$D 0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <]s{ \  3P ̔ 0jJ  ,$D 0 { \  ͔# { D ,$D 0lr Δ <Zok   ϔ 3 BC0DE4Fo  mR!<W1&N*T}{0 @     f  Д <0]s{ \  3P є 0jJ 4 ,$D 0 { \  Ҕ# {  ,$D 0lr Ӕ <Zok   Ԕ 3 BC0DE4Fo  mR!<W1&N*T}{0 @     f  Ք < ]s{ \  3P ֔ 0jJ 0 p ,$D  0 { \  ה# { L ,$D  0lr ؔ <Zok   ٔ 3 BC0DE4Fo  mR!<W1&N*T}{0 @     f  ڔ <]s{ \  3P ۔ 0jJ l ,$D  0 { \  ܔ# {  ,$D  0lr ݔ <Zok   ޔ 3 BC0DE4Fo  mR!<W1&N*T}{0 @     f  ߔ <]s{ \  3P  0jJ ,$D 0 { \  # { 4 ,$D  0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <]s{ \  3P  0jJ $ ,$D 0 { \  # { p ,$D 0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <8]s{ \  3P  0jJ ` ,$D 0 { \  # { < ,$D 0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <"]s{ \  3P  0jJ \ ,$D 0 { \  # { x ,$D 0lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   < (]s{ \  3Pl : {  ){ : ,$D 0  0jJL ,$D 0t : {  (: { ,$D 0`  0jJ @ : {  ': { T { \  # : { lr  <Zok    3 BC0DE4Fo  mR!<W1&N*T}{0 @     f   <P.]s{ \  3P`  0jJ  `  0jJ P `  0jJ `   0jJ  `  0jJ @ `  0jJ< | `  0jJx ` ! 0jJ H  0޽h ? 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CT>)SE#:¨Yo+}읾?g|KfexidD"`-SlyQ@'*x127֯gظ֥g @7%i(o7m$D? > "(;,Q:LekE>0YN"zm1@ Y ̸͍eihٴƋPW~l@~0MXJdq\´ȷ\ Z"X|% xh)1S_r5s+S0x9;A&TeޟE;Zf݊Jwy;7 [3llJ~ٺQ Uj"bf ydsaNo ]侘u0rۧ&g}9dH~BuVKF"6 ut\_~bk[+_kfK%} oqޏS-V#avhqrDނUxBk\g!_L\sf}Ă9vi$EH+PZۺg_y:̣M xٗd$:rC98/.jpMxgO[r53Ϟr  /pNȏ h}f5됶r Nq1܅#T]VBLb"9&f&f?la<7/]?{OF:\B\~|p![P,\)3Q5_˧,Y]![,~h‚:/ԂQn?`<{=P ׃Cm-{Z3ʬ˶ؤ(AM]B10㓙Rnh,z;}!i:񭽝iHwDhO["Z`Z$ EzZږ/@);1-(}ZmS}_ѐz'h}U FUͻtsW*;_8m+ڨhJ@nTFŨMO?Grt0ʌ8oWRȟPxzfL>9104SR2uKoNgPn $0(AD,.EFpsɕp׵u `IM6"p8Z;R=>?kDtɍP[%"0i7&`E}5U9A;-=?- %\*e^.}Oh+'0| hp  No Slide TitleSalil P. Vadhan Salil Vadhanan350Microsoft PowerPointP@0hq@ @p7@pcY GXg  $*  --$--'@"Arial-. 3'2 Randomness Extractors$("."Systempi-@"Arial-. 3$2 @& their Many Guisesr! )' .-@"Arial-. 2  Salil Vadhan .-@"Arial-. "2 kHarvard University     .-@"Arial-. *2 to be posted at http://.-@"Arial-.  2 eecs.-@"Arial-.  2 ..-@"Arial-. 2 harvard  .-@"Arial-.  2 L..-@"Arial-.  2 Tedu.-@"Arial-.  2 /~.-@"Arial-. 2 salilr.-՜.+,0     @ On-screen Show -s/[G: STimes New RomanArialSymboltimescmsy10 Helvetica Wingdings MT Extramsbm10SystemDefault DesignMicrosoft Equation 3.0*Randomness Extractors & their Many GuisesI. MotivationBOriginal Motivation [SV84,Vaz85,VV85,CG85,Vaz87,CW89,Zuc90,Zuc91]Applications of Extractors The Unifying Role of ExtractorsThis TutorialOutlineII. Extractors as ExtractorsWeak Random Sources Min-entropyExtractors: 1st attempt$Extractors [Nisan & Zuckerman `93]Definitional Details The ParametersThe Optimal ExtractorThe Optimal Extractor5Application: BPP w/a weak source [Zuckerman `90,`91]"III. Extractors as Hash FunctionsStrong extractorsExtractors as Hash FunctionsExtractors from Hash FunctionsIIb. Extractors as ExtractorsComposing ExtractorsIncreasing the Output [WZ93]$Increasing the Output Length [WZ93]Proof of Key LemmaIncreasing the Output [WZ93]9An Application [NZ93]: Pseudorandom bits vs. Small SpaceShortening the SeedBlock Sources [CG85]The [NZ93] ParadigmOutline"IV. Extractors as Expander GraphsExpander GraphsExtractors & Expansion [NZ93]Extractors vs. Expander GraphsExtractors vs. Expander Graphs"Expansion Measures — Extraction&Expansion Measures — The EigenvalueExtractors vs. Expander Graphs The Degree#High Min-Entropy Extractors [GW94]Zig-Zag Product [RVW00]Extractors vs. Expander GraphsRandomness Conductors [CRVW02])V. Extractors as Pseudorandom Generators(Pseudorandom Generators [BM82,Y82,NW88]Hardness vs. RandomnessExtractors & PRGsExtractors & PRGsExtractors & PRGsExtractors & PRGsAnalysis (intuition)Analysis (“formal”)When does this work? Consequences)VI. Extractors as Error-Correcting CodesError-Correcting Codes.Strong 1-bit Ext’s  List-Decodable Codes.Strong 1-bit Ext’s  List-Decodable Codes.Strong 1-bit Ext’s  List-Decodable CodesExtractors & CodesExtractors from CodesDependent projectionsThe ReconstructionDealing with ErrorsVII. Concluding RemarksTowards Optimality ConclusionsSome Research DirectionsFurther Reading  Fonts Used Design TemplateEmbedded OLE Servers Slide TitlesG$_d` ]Salil VadhanSalil Vadhan  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-.012345689:;<=>@ABCDEFORoot EntrydO)PicturesCurrent User?SummaryInformation(/PowerPoint Document(~`DocumentSummaryInformation87