Deterministic Extractors for Small Space Sources Jesse Kamp

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

Randomness Extractors X Ext Uniform 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.

Independent Sources Ext Uniform

Bit-Fixing Sources ? 1 ? ? Ext 0 1

Small Space Sources n Space s source: min-entropy k source generated by width 2 s branching program. 0. 3, 0 0. 1, 0 n+1 layers 0. 1 1 0. 5, 0. 8, 1 0. 1, 0 , 1 0 / , 1 -1 1/ 1, 1 width 2 s , 0 0. 1, 0 1 1 0 0 1

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

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

Bit Fixing Sources can be modelled by Space 0 sources 0. 5, 1 1, 1 0. 5, 0 ? 0. 5, 0 1 ? 1, 0 1, 1 0. 5, 0 ?

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

c Independent Sources: Space n/c sources. width 2 n/c 011101001101010 0010101111

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

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

We reduce to another model n Total Entropy k independent sources:

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

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

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

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

Some Additive Number Theory n [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, …, XC, | E[ ( X 1 X 2 … XC)] | < 2 -cp

Vazirani’s XOR lemma n 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. X 1 n X 2 X 3 X 4 | E[ ( X 1 X 2 … XC)] | < 2 -cp ) lsbm (X 1 X 2 … XC) is 2 m/2 – cp close to uniform lsb(X 1 X 2 X 3 X 4)

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

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

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]

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

Somewhere Random Source Def: [TS 96] Has some uniformly random row. r t

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

Condensers [BKSSW], [Raz], [Z] Elements in a prime field n A A B B (1. 1) (2 n/3) C C AC+B

Iterating the condenser n A B (1. 1)t (2/3)tn C

Mergers [Raz], [Dvir, Raz] 0. 9 C 99% of rows in output have entropy rate 0. 9

Condense + Merge [Raz] 1. 1 Condense 99% of rows in output have entropy rate 1. 1 Merge C

This process maintains alignment 1. 1 (1. 1)2 C C 2

Bottom Line: lsb(X 1 Y 1 Z 1) X 1 Y 1 Z 1 [BGK] Ct (1. 1)t n/dt

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

Recap W X Y Z X 1 Y 1 Z 1 lsb(X 1 Y 1 Z 1) [BGK] Ct (1. 1)t r sqrt(r) Arbitrary number of sources

Solution Entropy: n /4 of these have rate /4 X 1 (1. 1)t /4 /2 of these have rate /2 Y 1 Z 1 lsb(X 1 Y 1 Z 1) [BGK] Ct

Final Entropy: n /2 of these have rate /2 If n-0. 01 # rows << length of row

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

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

Questions?