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Extractors: applications and constructions

Randomness. Extractors: applications and constructions. Avi Wigderson IAS, Princeton. Cryptography. Applications : Analyzed on perfect randomness. Probabilistic algorithms. Game Theory. Unbiased, independent. biased, dependent. Reality : Sources of imperfect randomness.

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Extractors: applications and constructions

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  1. Randomness Extractors: applications and constructions Avi Wigderson IAS, Princeton

  2. Cryptography Applications: Analyzed on perfect randomness Probabilistic algorithms Game Theory Unbiased, independent biased, dependent Reality: Sources of imperfect randomness Stock market fluctuations Radioactive decay Sun spots Extractors: original motivation Extractor Theory

  3. Applications of Extractors • Using weak random sources in prob algorithms [B84,SV84,V85,VV85,CG85,V87,CW89,Z90-91] • Randomness-efficient error reduction of prob algorithms [Sip88, GZ97, MV99,STV99] • Derandomization of space-bounded algorithms [NZ93, INW94, RR99, GW02] • Distributed Algorithms [WZ95, Zuc97, RZ98, Ind02]. • Hardness of Approximation [Zuc93, Uma99, MU01] • Cryptography[CDHKS00, MW00, Lu02 Vad03] • Data Structures[Ta02]

  4. Unifying Role of Extractors Extractors are intimately related to: • Hash Functions [ILL89,SZ94,GW94] • Expander Graphs [NZ93, WZ93, GW94, RVW00, TUZ01,CRVW02] • Samplers[G97, Z97] • Pseudorandom Generators [Trevisan 99, …] • Error-Correcting Codes [T99, TZ01, TZS01, SU01, U02]  Unify the theory of pseudorandomness.

  5. Definitions

  6. Weak random sources Distributions X on {0,1}n with some entropy: • [vN] sources: ncoins of unknown fixed bias • [SV] sources: Pr[Xi+1 =1|X1=b1,…,Xi=bi]  (δ, 1-δ) • Bit fixing: ncoins, some good, some “sticky” • ….. • [Z] k-sources: H∞(X) ≥ k x Pr[X = x]  2-k e.g X uniform with support 2k {0,1}n X

  7. Randomness Extractors(1st attempt) Impossible even if k=n-1 and m=1 “weak” random source X k can be e.g n/2, √n, log n,… X k-source of length n EXT {0,1}n Ext=0 Ext=1 malmost-uniform bits X

  8. (short) “seed” d random bits Extractors [Nisan & Zuckerman `93] X k-source of length n • Ext : {0,1}n x {0,1}d {0,1}m • X has min-entropy k ( X is a k-source) • m ≤ k+d EXT malmost-uniform bits

  9. (short) “seed” d random bits Extractors [Nisan & Zuckerman `93] X k-source of length n  k-source X, | Ext(X,Ud) – Um|1 <   but -fraction of y’s, | Ext(X, y) – Um|1 <  {0,1}n EXT y {0,1}d Ext(X,y) m bits -close to uniform {0,1}m

  10. {0,1}n {0,1}m Ext(x,y) k-source X |X|=2k (X) x y B Extractors as graphs (k,)-extractor Ext: {0,1}n {0,1}d{0,1}m Sampling Hashing Amplification Coding Expanders …  Discrepancy: For all but 2k of the x {0,1}n, | |(X)  B|/2d-|B|/2m |< 

  11. d random bits Probabilistic algorithms with weak random bits k-source of length n Where from? EXT Efficient? Try all possible 2d strings. Take Majority vote m randombits (upto ) Input Probabilistic algorithm Output + Error prob <δ Want: efficient Ext, small d,  , large m

  12. Extractors - Parameters k-source of length n • Goals: minimize d, , maximize m. • Non-constructive & optimal [Sip88,NZ93,RT97]: • Seed length d = log(n-k) + 2 log 1/ + O(1). • Output length m = k + d - 2 log 1/ - O(1). (short) “seed” EXT d random bits m bits -close to uniform

  13. Extractors - Parameters k-source of length n • Goals: minimize d, maximize m. • Non-constructive & optimal [Sip88,NZ93,RT97]: • Seed length d = log n + O(1). • Output length m = k + d - O(1). (short) “seed” EXT d random bits m bits -close to uniform •  = 0.01 • k n/2

  14. Explicit Constructions Non-constructive & optimal [Sip88,NZ93,RT97]: • Seed length d = log n + O(1). • Output length m = k + d - O(1). [...B86,SV86,CG87, NZ93, WZ93, GW94, SZ94, SSZ95, Zuc96, Ta96, Ta98, Tre99, RRV99a, RRV99b, ISW00, RSW00, RVW00, TUZ01, TZS01, SU01, LRVW03,…] New explicit constructions [GUV07, DW08] - Seed length d = O(log n) [even for =1/n] • Output length m = .99k + d

  15. Applications

  16. d random bits Probabilistic algorithms with weak random bits k-source of length n X EXT Efficient! Try all 2d = poly(n) strings. Take Majority vote m randombits (upto ) Input Probabilistic algorithm Output + Error prob <δ The error set B {0,1}m of alg is sampled accurately whp

  17. Extractors as samplers n-bit string x Efficient! k=2m EXT S(x)={ } Ext(X,1) Ext(X,2) Ext(X,nc) m m m For every B  {0,1}m, all but 2k of x  {0,1}n : | |S(x)  B|/nc-|B|/2m |<  Note: x bad with prob < 2k/2n, n arbitrary

  18. Extractors as list-decodable error-correcting codes [TZ] {0,1}D c2 c1 C: {0,1}n {0,1}D d = c log n D =2d = nc c7 c3 z c8 c6 EXT c5 Polynomial rate! Efficient encoding!! Efficient decoding? c9 n-bit string x c4 C(x)= ……… Ext(X,D) Ext(X,1) Ext(X,2) For z  {0,1}D let Bz {0,1}d+1 be the set {(i,zi) : i [D] } List decoding: For every z, at most D2 of x have C(x) fall in (1/2 -)D hamming ball around z 1 bit 1 bit 1 bit

  19. Beating e-value expansion Task:Construct an graph on [N] of minimal degree DEG s.t. every two sets of size Kare connected by an edge. N Any such graph: DEG> N/K Ramanujan graphs: DEG<(N/K)2 Random graphs: DEG < (N/K)1+o(1) Extractors: DEG < (N/K)1+o(1) K linear in N and constant DEG[RVW] We’ll see it for “moderate” K [WZ] K K

  20. |X|=K |X’|=K Extractors as graphs (again) (k,.01)-extractor Ext: {0,1}n {0,1}d{0,1}m 2k= K = M1+o(1)Ext: [N] x [D]  [M] 2d = D < Mo(1) [N] [M] Take G = Ext2 on [N] DEG < (N/K)1+o(1) Many edges between any two K-sets X,X’ |(X)| > .99M

  21. Constructions

  22. Bx {0,1}m random strings G explicit expander of const degree rt r  r1 x x x Alg Alg Alg Expanders as extractors Pr[error] < 1/3 Majority Thm [Chernoff] r1 r2….rt independent(tm random bits) Thm [AKS] r1 r2….rt random G-path (m+ O(t) random bits) then Pr[error] = Pr[|{r1 r2…. rt }Bx}| > t/2] < exp(-t)

  23. Expanders as extractors (k large) G expander graph of const degree on {0,1}m B any subset, δ=|B|/2m S = {r1 r2….rt} arandom G-path (n = m+ O(t) bits) Thm [G]Pr[| δ -|SB|/t| > ] < exp(-2t) Thm [Z]t=cm=2d, Ext : {0,1}n x {0,1}d {0,1}m Ext(r1 r2….rt ; i) = ri is an (k=.99n, )–extractor of d=O(log n) seed

  24. seed d random bits Condensers [RR99,RSW00,TUZ01] X k-source of length n Sufficient to construct such condensers: from here we can use [Z] extractor Con .99k-source of length k

  25. seed d random bits Mergers [T96] k k … k n=ks X= X1 X2 … XS Some block Xi is random. The other Xj are correlated arbitrarily with it. Mer outputs a high entropy distribution. Mer .9k-source k

  26. seed d random bits Mergers [T96] k k … k n=ks X= X1 X2 … XS XiFqkq ~ n100 Some Xi is random Mer .9k-source [LRVW] Mer = a1X1+a2X2+…+asXsaiFq ( d=slog q ) Mer is a random element in the subspace spanned by Xi’s [D] It works! (proof of the Wolf conjecture). [DW] Mer = a1(y)X1+a2(y)X2+…+as(y)XsyFq ( d=log q ) Mer is a random element in the curve through the Xi’s k

  27. The proof Deg(C) = s-1 (Fq)k B B C(x) x1 x1 xi xi x2 x2 xs xs Mer(x) Assume: E [|C(X)  B|] > 2ε & B small Prx[ |C(x)  B|>ε ] >ε low deg Q:(Fq)k  Fq Q(B) 0  Prx[ Q(C(x))  0] >ε  Pr[ Q(xi)  0] >ε Q 0 #

  28. |X|=N Open Problems Find explicit extractors with • Seed length d = log n + O(1). • Output length m = k + d - O(1). Find explicit bipartite graph, of constant deg [N3] [N2] |Γ(X)|≥ N

  29. d random bits Extractors as samplers Given B  {0,1}m Estimate |B|/2m X k-source of length n EXT Efficient! m randombits (upto ) Try all 2d = poly(n) strings. Count the fraction falls in B Any set B  {0,1}m WHP estimation error <

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