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Deterministic Extractors for Small Space Sources Jesse Kamp, Anup Rao, Salil Vadhan, David Zuckerman

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Randomness Extractors Defn: min-entropy(X)k if x Pr[X=x] · 2 -k. No “deterministic” (seedless) extractor for all X with min-entropy k: 1.Can add seed. 2.Can restrict X. ExtX Uniform

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Independent Sources Ext Uniform

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Bit-Fixing Sources ? 1 ? ? 0 1 Ext

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Small Space Sources Space s source: min-entropy k source generated by width 2 s branching program. n+1 layers / , 0 1-1/ , 0 1,1 0.1,0 0.8,1 0.1,0 0.3,0 0.5,1 0.1,1 0.1,0 1 width 2 s

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Related Work [Blum]: Markov Chain with a constant number of states [Koenig, Maurer]: related model [Trevisan, Vadhan]: considered sources sampled by small circuits requires complexity theoretic assumptions.

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Small space sources capture: Bit fixing sources space 0 sources General Sources with min-entropy k space k sources c Independent sources space n/c sources

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Bit Fixing Sources can be modelled by Space 0 sources ? 1 ? ? ,1 0.5,0 1,11,01,1

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General Sources are Space n sources Pr[X 2 = 1|X 1 =1], 1 Pr[X 1 = 0], 0 Pr[X 1 = 1], 1 Pr[X 2 = 0|X 1 =1], 0 n layers width 2 n X = X 1 X 2 X 3 X 4 X 5 …..………….. Min-entropy k sources are convex combinations of space k sources

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c Independent Sources: Space n/c sources width 2 n/c

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Our Main Results Min-EntropySpaceErrorOutput Bits k = n 1-c n 1-4c 2 -n c 99% k k = ncncn 2 -n/polylog(n) 99% k c = sufficiently small constant > 0

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Outline Our Techniques Extractor for linear min-entropy rate Extractor for polynomial min-entropy rate Future Directions

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We reduce to another model Total Entropy k independent sources:

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X|State 5 = V Y| State 5 = V Y The Reduction X V These two distributions are independent! Expect the min-entropy of X|State 5 = V, Y|State 5 = V to be about k – s.

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Can get many independent sources W X Y Z If we condition on t states, we expect to lose ts bits of min-entropy.

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Entropy Loss Let S 1, …, S t denote the random variables for the state in the t layers. Pr[X = x] Pr[X=x|S 1 =s 1,…,S t =s t ] Pr[S 1 =s 1,…,S t =s t ] X|S 1 =s 1,…,S t =s t has min-entropy < k – 2ts ) Pr[S 1 = s 1,…,S t =s t ] < 2 -2ts Union bound: happens with prob < 2 -ts

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The Reduction Every space s source with min-entropy k is close to a convex combination of t total entropy k-2ts sources. W X Y Z

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Some Additive Number Theory [Bourgain, Glibichuk, Konyagin] ( >0) ( integers C=C( ), c=c( )): non-trivial additive character of GF(2 p ) and every independent min-entropy p sources X 1, …, X C, | E[ ( X 1 X 2 … X C )] | < 2 -cp

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Vazirani’s XOR lemma Z GF(2 n ) a random variable with |E[ (Z)]| < for every nontrivial , then any m bits of Z are 2 m/2 close to uniform. | E[ ( X 1 X 2 … X C )] | < 2 -cp ) lsb m (X 1 X 2 … X C ) is 2 m/2 – cp close to uniform X 1 X 2 X 3 X 4 lsb(X 1 X 2 X 3 X 4 )

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More than an independent sources extractor Analysis: (X 1 X 2 ), (X 3 X 4 ), (X 5 X 6 X 7 ), X 8 are independent sources. X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 lsb m (X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 )

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Small Space Extractor for n entropy If the source has min-entropy n, /2 fraction of blocks must have min-entropy rate . Take (2/ ) C( /2) blocks ) C( /2) blocks have min-entropy rate /2. lsb( )

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Result Theorem: ( > 0, > 0) efficient extractor for min-entropy k n space n output length = (n) error = 2 - (n) Can improve to get 99% of the min-entropy out using techniques from [Gabizon,Raz,Shaltiel]

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For Polynomial Entropy Rate Black Boxes: Good Condensers: [Barak, Kindler, Shaltiel, Sudakov, Wigderson], [Raz] Good Mergers: [Raz], [Dvir, Raz] White Box: Condensing somewhere random sources: [Rao]

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Somewhere Random Source Def: [TS96] Has some uniformly random row. t r

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Aligned Somewhere High Entropy Sources Def: Two somewhere high-entropy sources are aligned if the same row has high entropy in both sources.

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Condensers [BKSSW],[Raz],[Z] A B C nn A B C AC+B (1.1) (2n/3) Elements in a prime field

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Iterating the condenser A B C nn (1.1) t (2/3) t n

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Mergers [Raz], [Dvir, Raz] 0.9 99% of rows in output have entropy rate 0.9 C

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Condense + Merge [Raz] 1.1 99% of rows in output have entropy rate 1.1 Condense Merge C

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This process maintains alignment 1.1 C (1.1) 2 C2C2

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Bottom Line: (1.1) t CtCt [BGK] X1X1 Y1Y1 Z1Z1 lsb(X 1 Y 1 Z 1 ) n/d t

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Extracting from SR-sources [Rao] r sqrt(r) r We generalize this: Arbitrary number of sources

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Recap (1.1) t CtCt [BGK] X1X1 Y1Y1 Z1Z1 lsb(X 1 Y 1 Z 1 ) sqrt(r) r Arbitrary number of sources W X Y Z

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Solution Entropy: n 2 of these have rate /2 4 of these have rate /4 CtCt (1.1) t [BGK] X1X1 Y1Y1 Z1Z1 lsb(X 1 Y 1 Z 1 )

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Final Entropy: n 2 of these have rate /2 If n # rows << length of row

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Result Theorem: (Assuming we can find primes) ( ) efficient extractor for min-entropy n 1- space n 1-4 output length n (1) error 2 -n (1) Can improve to get 99% of the min-entropy out using techniques from [Gabizon,Raz,Shaltiel]

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Future Directions Smaller min-entropy k? Non-explicit: k=O(log n) Our results: k=n 1- (1) Larger space? Non-explicit: (k) Our results: (k) only for k= (n) Other natural models?

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Questions?

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