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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