Download presentation

Presentation is loading. Please wait.

Published byLinda Keetch Modified over 2 years ago

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

2
Extractors: original motivation Unbiased, independent Probabilistic algorithms Cryptography Game Theory Applications: Analyzed on perfect randomness biased, dependent Reality: Sources of imperfect randomness Stock market fluctuations Sun spots Radioactive decay 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: n coins of unknown fixed bias [SV] sources : Pr[X i+1 =1|X 1 =b 1,…,X i =b i ] ( δ, 1-δ) Bit fixing: n coins, some good, some “sticky” ….. [Z] k-sources: H ∞ (X) ≥ k x Pr[X = x] 2 -k e.g X uniform with support 2 k {0,1} n X

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

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

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

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

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

12
Extractors - Parameters E XT k -source of length n m bits -close to uniform 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). d random bits (short) “seed”

13
Extractors - Parameters E XT k -source of length n m bits -close to uniform 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). d random bits (short) “seed” = 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
Probabilistic algorithms with weak random bits k-source of length n X m random bits E XT d random bits Probabilistic algorithm Input (upto ) Output Error prob < δ ++ Try all 2 d = poly(n) strings. Take Majority vote Efficient! The error set B {0,1} m of alg is sampled accurately whp

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

18
Extractors as list-decodable error-correcting codes [TZ] Polynomial rate! Efficient encoding!! Efficient decoding? n-bit string x Ext(X,1) E XT Ext(X,2) Ext(X,D) 1 bit 1 bit 1 bit C(x)= ……… For z {0,1} D let B z {0,1} d+1 be the set {(i,z i ) : i [D] } List decoding: For every z, at most D 2 of x have C(x) fall in (1/2 - )D hamming ball around z c2c2 c1c1 c3c3 {0,1} D c8c8 c7c7 c6c6 c5c5 c4c4 c9c9 z d = c log n D =2 d = n c C: {0,1} n {0,1} D

19
Beating e-value expansion Task: Construct an graph on [N] of minimal degree DEG s.t. every two sets of size K are connected by an edge. 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] N K K

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

21
Constructions

22
Expanders as extractors Algxx rr {0,1} m random strings Thm [Chernoff] r 1 r 2 …. r t independent (tm random bits) Thm [AKS] r 1 r 2 …. r t random G-path (m+ O(t) random bits) Algxx rtrt xx r1r1 Majority G explicit expander of const degree BxBx Pr[error] < 1/3 then Pr[error] = Pr[|{r 1 r 2 …. r t } B x }| > 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|/2 m S = { r 1 r 2 …. r t } a random G-path (n = m+ O(t) bits) Thm [G] Pr[| δ - |S B|/t | > ] < exp(- 2 t) Thm [Z] t=cm=2 d, Ext : {0,1} n x {0,1} d {0,1} m Ext(r 1 r 2 …. r t ; i) = r i is an (k=.99n, )–extractor of d=O(log n) seed

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

25
Mergers [T96] d random bits seed Mer X 1 X 2 … X S.9k -source Some block X i is random. The other X j are correlated arbitrarily with it. Mer outputs a high entropy distribution. X= n=ks k k … k k

26
Mergers [T96] d random bits seed Mer X 1 X 2 … X S. 9k-source X= n=ks k k … k k X i F q k q ~ n 100 Some X i is random [LRVW] Mer = a 1 X 1 +a 2 X 2 +…+a s X s a i F q ( d=slog q ) Mer is a random element in the subspace spanned by X i ’s [D] It works! (proof of the Wolf conjecture). [DW] Mer = a 1 (y)X 1 +a 2 (y)X 2 +…+a s (y)X s y F q ( d=log q ) Mer is a random element in the curve through the X i ’s

27
The proof Assume: E [|C(X) B|] > 2 ε & B small x1x1 x2x2 xixi xsxs x1x1 x2x2 xixi xsxs C(x) (F q ) k B Mer(x) B Pr x [ |C(x) B|> ε ] >ε Pr x [ Q(C(x)) 0 ] >ε Deg(C) = s-1 Pr [ Q(x i ) 0 ] >ε Q 0 # low deg Q:(F q ) k F q Q(B) 0

28
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 [N 3 ] [N 2 ] |X|=N |Γ(X)|≥ N

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

Similar presentations

OK

Umans Complexity Theory Lecturess Lecture 11: Randomness Extractors.

Umans Complexity Theory Lecturess Lecture 11: Randomness Extractors.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download a ppt on natural disasters Ppt on p&g products brands tide Ppt on hard gelatin capsules Ppt on diversity in living organisms Ppt on circular linked list Ppt on chemical properties of metals and nonmetals Ppt on 14 principles of henri fayol Ppt on group development stages Ppt on linked list in java Download seminar ppt on android