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Why Extractors?

Learn about the properties of extractors and dispersers, and their applications in removing randomness in various settings. Discover how to make randomized constructions explicit and explore the fascinating world of computational randomness.

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Why Extractors?

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  1. … 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?

  2. Santa Clause and his (Un)- Biased Elves The Story of Randomized Computations and Weak Random Sources

  3. Distributed Computations • Cryptography The Computational Tasks of Santa (and Atnas) Clause Sampling, Simulations, Algorithms (e.g. Approximated TSP).

  4. A coin please … Santa’s Source of Randomness

  5. A coin please … But the North Pole is no Fairyland …

  6. Thanks Erwin but I’ve grown attached to my elves … Pure Randomness in Nature? Hey Santa, you can use my cat !!

  7. 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].

  8. 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, …]

  9. 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)

  10. 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].

  11. 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:

  12. 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, …]

  13. 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]

  14. (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)

  15. 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.

  16. 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

  17. 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

  18. 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.

  19. 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 …

  20. (read only) Space S (i.e. 2sconfiguration) A (read once) input random string Weak Sources in Computations • Space bounded computations:

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