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why extractors

… Extractors, and the closely related “Dispersers”, exhibit some of the most “random-like” properties of explicitly constructed combinatorial structures. In turn, extractors and dispersers have many applications in “removing randomness” in various settings, and in making randomized constructions explicit …

Why Extractors?

santa clause and his un biased elves

Santa Clause and his (Un)- Biased Elves

The Story of Randomized Computations and Weak Random Sources

the computational tasks of santa and atnas clause

Distributed Computations

  • Cryptography

The Computational Tasks of Santa (and Atnas) Clause

Sampling, Simulations, Algorithms (e.g. Approximated TSP).

deterministic extraction

EXT

almost uniform output

source of biased correlated bits

Assumeb1 b2 …bi … are i.i.d. 0/1 variablesand bi =1with someprobabilityp < 1then translate

01 1

10 0

Deterministic Extraction

Other “easy” sources: markov chains[vN51,Eli72,Blu84],two independent sources[SV84,Vaz85,CG85] , bit-fixing sources[CGH+85,BBR85,BL85,LLS87,CDH+00],some efficiently samplable sources[TV00].

can this work for all sources

If b1 b2 …bi … are 0/1 variabless.t. bi =1with prob.

p = p(b1 b2 …bi-1)[½-, ½+]

cannot deterministically extract even a single bit !!

A single SV-Source is sufficient to simulate BPP

Can this Work for all Sources?
  • Can use even weaker sources [ChorGo88, CohenWi89, …]
extractors 93

Distribution on {0,1}nw/

k“bits of randomness”

EXT

mbits distance

from uniform

dtruly random bits

Extractors[ , 93]
  • Xhas min-entropy k if xPr[X = x]  2-k(i.e. no likely elements).
  • Nonconstructive & optimal [NZ,RT]: extract all the randomness (i.e. mk+d) using d log ntruly random bits ( =.01)
where does the seed come from
Where Does the Seed Come From?
  • If “truly” random bits exist but expensive ...
  • Sometimes we can just enumerate over all 2d seeds:Let A be some probabilistic procedure and e an element from the weak random source.Run A(Ext(e,0…0)) , … , A(Ext(e,1…1))“combine” the outputs (majority, median, best, …).
  • In particular: can simulate BPP using a weak source [Zuc90].
weak sources in space bounced computations

G

x,y

x, Ext(x,y)

Weak Sources in (Space Bounced) Computations
  • Thm [NZ93] Let A be a (randomized) space S machine(i.e. A can be in 2s configurations). If A uses poly(S) random bits it can be fully derandomized in space O(S).
  • Basic idea: Let A read a random2Sbit string x. Since A remembers at most S bits, x still contains (roughly) Sbits of entropy (independent of A’s state). Can recycle:
applications of extractors
Applications of Extractors
  • Randomized algorithms w/ weak random sources.
  • Pseudorandom generators [NZ93,RR99,STV99]
  • Randomness efficient sampling and deterministic amplification [Zuc97]
  • Hardness of approximation [Zuc96,Uma99]
  • Exposure-resilient cryptography [CDHKS00]
  • Superconcentrators, sorting & selecting in rounds, highly expanding graphs [WZ93]
  • Leader election [Zuc96, RZ98], List decodable error correcting codes [TZ00], and more [Sip88,GZ97, …]
constructions of extractors
Constructions of Extractors
  • The “early days” [Zuc,NZ,WZ,GW,SZ,SSZ,NT,Zuc,TaS]Mainly hashing and various sorts of compositions.Some extractors:
    • [Zuc97] Fork =  (n)can extractm=(1-) k bitsusing d =O (log n/)
    • [NT98] For allk can getm=k andd = poly (log n/)

Other results in the high min-entropy case [GW], low min-entropy case [GW,SZ], dispersers [SSZ,TaS]

constructions of extractors cont

(Some) constructions of PRG from hard functions  extractors

Ha yes ... and there is a very nice one based on the NW generators

Constructions of Extractors (cont.)
  • The “new age” [Tre99,RRVa,RRVb,ISW,RSW,RVW,TUZ]
  • Some more extractors [RSW]: for allk,
    • m= (k)andd = log n polyloglog nor
    • m=k/log kandd = O(log n)
dispersers sipser 88

S, |S|=K=2k

|(S)| >

(1-) M

D =2d

Dispersers [Sipser 88]

N=2n

M =2m

Difference from Expanders:

  • Typically M << N (farewell constant degree).
  • Expansion to almost the entire right hand side.
extractors imply dispersers

Ext(x,0…0)

S

T

x

Ext(x,1…1)

Extractors imply Dispersers

N=2n ={0,1}n

  • In fact we have the stronger property that S, |S|=K=2k and T,

M =2m ={0,1}m

a construction in search of many applications wz

N

N

Y

X

A Construction in Search of Many Applications [WZ]
  • If G is a disperser (with  < 1/2) then X, Y s.t.|X|=|Y|=K have at least onecommon neighbors.

M

G

G

  • Using similar ideas, [WZ93] get Superconcentrators, highly expanding graphs, and much more
depth 2 superconcentrators

N

N

Y

X

Depth 2 Superconcentrators
  • X, Y, ts.t.|X|=|Y|=tthere existstvertex-disjoint paths between X andY.
  • [WZ] A construction with N log2N edges.
  • [RT] More carefully gives N log2N/loglog N edges. And this is essentially the only possible construction.
some conclusions
Some Conclusions
  • Need randomness to extract randomness.
  • Weak random sources appear naturally in computations.
  • Expanders, Extractors and Dispersers are closely related combinatorial objects.
  • Extractors are fascinating and very useful objects. Go home and build your own extractor …
weak sources in computations

(read only)

Space S (i.e. 2sconfiguration)

A

(read once)

input

random string

Weak Sources in Computations
  • Space bounded computations: