lec11 - CS151 Complexity Theory Lecture 11 May 3 2011...

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CS151 Complexity Theory Lecture 11 May 3, 2011
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May 3, 2011 2 Min-entropy General model of physical source w/ k < n bits of hidden randomness Definition : random variable X on {0,1} n has min-entropy min x –log(Pr[X = x]) min-entropy k implies no string has weight more than 2 -k {0,1} n 2 k   strings string sampled uniformly from  this set
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May 3, 2011 3 Extractor Extractor: universal procedure for “purifying” imperfect source: E is efficiently computable truly random seed as “catalyst” seed source string near-uniform {0,1} n 2 k   strings E t bits m  bits
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May 3, 2011 4 Extractor “(k, ε)-extractor” for all X with min-entropy k: output fools all circuits C : |Pr z [C(z) = 1] - Pr y, x X [C(E(x, y)) = 1]| ≤ ε – distributions E(X, U t ), U m ε -close” (L 1 dist ≤ 2 ε ) Notice similarity to PRGs output of PRG fools all efficient tests output of extractor fools all tests
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May 3, 2011 5 Extractors Using extractors use output in place of randomness in any application alters probability of any outcome by at most ε Main motivating application: use output in place of randomness in algorithm how to get truly random seed? enumerate all seeds, take majority
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May 3, 2011 6 Extractors Goals: good : best : short seed O(log n) log n+O(1) long output m = k Ω(1) m = k+t–O(1) many k’s k = n Ω(1) any k = k(n) seed source string near-uniform {0,1} n 2 k   strings E t bits m  bits
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May 3, 2011 7 Extractors random function for E achieves best ! but we need explicit constructions many known; often complex + technical optimal extractors still open Trevisan Extractor: insight: use NW generator with source string in place of hard function this works (!!) proof slightly different than NW, easier
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May 3, 2011 8 Trevisan Extractor Ingredients: ( δ > 0, m are parameters) error-correcting code C:{0,1} n {0,1} n’ distance (½ - ¼m -4 )n’ blocklength n’ = poly(n) – (log n’, a = δlog n/3) design: S 1 ,S 2 , …,S m {1…t = O(log n’)} E(x, y)=C(x)[y |S 1 ]◦C(x)[y |S 2 ]◦…◦C(x)[y |S m ]
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May 3, 2011 9 Trevisan Extractor E(x, y)=C(x)[y |S 1 ]◦C(x)[y |S 2
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