Extractors: Optimal Up to Constant Factors - PowerPoint PPT Presentation

About This Presentation
Title:

Extractors: Optimal Up to Constant Factors

Description:

Extractors: Optimal Up to Constant Factors Avi Wigderson IAS, Princeton Hebrew U., Jerusalem Joint work with Chi-Jen Lu, Omer Reingold, Salil Vadhan. – PowerPoint PPT presentation

Number of Views:32
Avg rating:3.0/5.0
Slides: 33
Provided by: ome73
Learn more at: https://www.math.ias.edu
Category:

less

Transcript and Presenter's Notes

Title: Extractors: Optimal Up to Constant Factors


1
Extractors Optimal Up to Constant Factors
  • Avi Wigderson
  • IAS, Princeton
  • Hebrew U., Jerusalem
  • Joint work with
  • Chi-Jen Lu, Omer Reingold, Salil Vadhan.
  • To appear STOC 03

2
Original MotivationB84,SV84,V85,VV85,CG85,V87,CW
89,Z90-91
  • Randomization is pervasive in CS
  • Algorithm design, cryptography, distributed
    computing,
  • Typically assume perfect random source.
  • Unbiased, independent random bits
  • Can we use a weak random source?
  • (Randomness) Extractors convert weak random
    sources into almost perfect randomness.

3
Extractors Nisan Zuckerman 93
k-source of length n
EXT
m almost-uniform bits
  • X has min-entropy k (? is a k-source) if ?x
    PrX x ? 2-k (i.e. no heavy elements).

4
Extractors Nisan Zuckerman 93
k-source of length n
EXT
m bits ?-close to uniform
  • X has min-entropy k (? is a k-source) if ?x
    PrX x ? 2-k (i.e. no heavy elements).
  • Measure of closeness statistical difference
    (a.k.a. variation distance, a.k.a. half L1-norm).

5
Applications of Extractors
  • Derandomization of error reduction in BPP Sip88,
    GZ97, MV99,STV99
  • Derandomization of space-bounded algorithms
    NZ93, INW94, RR99, GW02
  • Distributed Network Algorithms WZ95, Zuc97,
    RZ98, Ind02.
  • Hardness of Approximation Zuc93, Uma99, MU01
  • Cryptography CDHKS00, MW00, Lu02 Vad03
  • Data Structures Ta02

6
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.

7
Extractors as graphs
(k,?)-extractor Ext 0,1n ? 0,1d
?0,1m
Sampling Hashing Amplification Coding Expanders
?
Discrepancy For all but 2k of the x? 0,1n,
?(X) ? B/2d - B/2m lt ?
8
Extractors - Parameters
k-source of length n
(short) seed
EXT
d random bits
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).

9
Extractors - Parameters
k-source of length n
(short) seed
EXT
d random bits
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).
  • ? 0.01
  • k ? n/2

10
Explicit Constructions
  • A large body of work ..., NZ93, WZ93, GW94,
    SZ94, SSZ95, Zuc96, Ta96, Ta98, Tre99, RRV99a,
    RRV99b, ISW00, RSW00, RVW00, TUZ01, TZS01, SU01
    ( those I forgot )
  • Some results for particular value of k (small k
    and large k are easier). Very useful example
    Zuc96 k?(n), dO(log n), m.99 k
  • For general k, either optimize seed length or
    output length. Previous records RSW00
  • dO(log n (poly loglog n)), m.99 k
  • dO(log n), m k/log k

11
This Work
  • Main Result Any k, dO(log n), m.99 k
  • Other results (mainly for general ?).
  • Technical contributions
  • New condensers w/ constant seed length.
  • Augmenting the win-win repeated condensing
    paradigm of RSW00 w/ error reduction à la
    RRV99.
  • General construction of mergers TaShma96 from
    locally decodable error-correcting codes.

12
Condensers RR99,RSW00,TUZ01
A (k,k,?)-condenser
k-source of length n
Con
(?,k)-source of length n
  • Lossless Condenser if kk (in this case, denote
    as (k,?)-condenser).

13
Repeated Condensing RSW00
k-source length n
(?0,k)-source length n/2
(t.?0,k)-source length O(k)
14
1st Challenge Error Accumulation
  • Number of steps tlog (n/k).
  • Final error gt t.?0 ? Need ?0 lt 1/t.
  • Condenser seed length gt log 1/?0 gt log t.
  • ? Extractor seed length gt t.log t which may be as
    large as log n.loglog n (partially explains seed
    length of RSW00).
  • Solution idea start with constant ?0. Combine
    repeated condensing w/ error reduction (à la
    RRV99) to prevent error accumulation.

15
Error Reduction
Con0 w/ error ? condensation ? seed length d
  • Con has condensation 2? seed length 2d.
  • Hope error ? ?2.

Only if error comes from seeds!
16
Parallel composition
Con0 seed error ? condensation ? seed length
d source error ? entropy loss ?
kd-k
  • Con seed error O(?2) condensation 2? seed
    dO(log 1/?)
  • source error ? entropy loss ?1

17
Serial composition
Con0 seed error ? condensation ? seed d
source error ? entropy loss ? kd-k
  • Con seed error O(?) condensation ?2 seed 2d
  • source error ?(11/?) entropy loss 2?

18
Repeated condensing revisited
  • Start with Con0 w/ constant seed error ? ? 1/18
    constant condensation ? constant seed length d
    (source error ?0 entropy loss ?0).
  • Alternate parallel and serial composition
    loglog n/k O(1) times.
  • ? Con w/ seed error ? condenses to O(k) bits
    optimal seed length dO(log n/k)
  • (source error (polylog n/k). ?0
  • entropy loss O(log n/k).?0).
  • Home? Not so fast

19
2nd Challenge No Such Explicit Lossless
Condensers
  • Previous condensers with constant seed length
  • A (k,?)-condenser w/ nn-O(1) CRVW02
  • A (k,?(k),?)-condenser w/ nn/100 Vad03 for
    k?(n)
  • Here A (k,?(k),?)-condenser w/ nn/100 for any
    k (see them later).
  • Still not lossless! Challenge persists

20
Win-Win Condensers RSW00
  • Assume Con is a (k,?(k),?)-condenser then
    ?k-source X, we are in one of two good cases
  • Con(X,Y) contains almost k bits of randomness ?
    Con is almost lossless.
  • X still has some randomness even conditioned on
    Con(X,Y).
  • ? (Con(X,Y), X) is a block source CG85. Good
    extractors for block sources already known (based
    on NZ93, )
  • ? Ext(Con(X,Y), X) is uniform on ?(k) bits

21
Win-Win under composition
  • More generally (Con,Som) is a win-win condenser
    if ?k-source X, either
  • Con(X,Y) is lossless. Or
  • Som(X,Y) is somewhere random a list of b sources
    one of which is uniform on ?(k) bits.
  • Parallel composition generalized
  • Con(X,Y1?Y2) Con(X,Y1)?Con(X,Y2)
  • Som(X,Y1?Y2) Som(X,Y1) ? Som(X,Y2)
  • Serial composition generalized
  • Con(X,Y1?Y2) Con(Con(X,Y1),Y2)
  • Som(X,Y1?Y2) Som(X,Y1) ? Som(Con(X,Y1),Y2)

22
Partial Summary
  • We give a constant seed length,
    (k,?(k),?)-condenser w/ nn/100 (still to be
    seen).
  • Implies a lossless win-win condenser with
    constant seed length.
  • Iterate repeated condensing and (seed-) error
    reduction loglog n/k O(1) times.
  • Get a win-win condenser (Con,Som) where
  • Con condenses to O(k) bits and
  • Som produces a short list of t sources where
    one of which is a block source (t can be made as
    small as log(c)n).

23
3rd Challenge Mergers TaShma96
  • Now we have a somewhere random source X1,X2,,Xt
    (one of the Xi is random)
  • t can be made as small as log(c)n
  • An extractor for such a source is called merger
    TaS96.
  • Previous constructions mergers w/ seed length
    dO(log t . log n) TaS96 .
  • Here mergers w/ seed length dO(log t) and
    seed length dO(t) (independent of n)

24
New Mergers From LDCs
  • Example mergers from Hadamard codes.
  • Input, a somewhere k-source XX1,X2,,Xt (one of
    the Xi is a k-source).
  • Seed, Y is t bits. Define Con(X,y) ?i?Y Xi
  • Claim With prob ½, Con(X,Y) has entropy k/2
  • Proof idea
  • Assume wlog. that X1 is a k-source.
  • For every y, Con(X,y) ? Con(X,y?e1) X1.
  • ? At least one of Con(X,y) and Con(X,y?e1)
    contains entropy ? k/2.

25
Old Debt The Condenser
  • Promised a (k,?(k),?)-condenser, w/ constant
    seed length and nn/100.
  • Two new simple ways of obtaining them.
  • Based on any error correcting codes (gives best
    parameters influenced byRSW00,Vad03).
  • Based on the new mergers Ran Raz.
  • Mergers ? CondensersLet X be a k-source. For any
    constant t XX1,X2,,Xt is ? a somewhere k/t
    source.
  • ? The Hadamard merger is also a condenser w/
    the desired parameters.

26
Some Open Problems
  • Improved dependence on ?. Possible direction
    mergers for t blocks with seed length f(t)
    O(log n/?).
  • Getting the right constants
  • d log n O(1).
  • m k d - O(1).
  • Possible directions
  • .lossless condenser w/ constant seed
  • lossless mergers
  • Better locally decodable codes.

27
New Mergers From LDCs
  • Generally View the somewhere k-source
    XX1,X2,,Xt ?(?n)t as a t?n matrix.
  • Encode each column with a code C ?t ? ?u.











  • Output a random row of the encoded matrix
  • dlog u (independent of n)

28
New Mergers From LDCs
  • C ?t ? ?u is (q,?) locally decodable erasure
    code if
  • For any fraction ? of non-erased codeword
    symbols S.
  • For any message position i.
  • The ith message symbol can be recovered using q
    codeword symbols from S.
  • Using such C, the above mergers essentially turn
    a somewhere k-sources to a k/q-source with
    probability at least 1- ?.

29
New Mergers From LDCs
  • Hadamard mergers w/ smaller error
  • Seed length dO(t.log 1/?). Transform a
    somewhere k-sources X1,X2,,Xt to a (k/2,
    ?)-source.
  • Reed-Muller mergers w/ smaller error
  • Seed length dO(t? .log 1/?). Transform a
    somewhere k-sources X1,X2,,Xt to a (?(?k),
    ?)-source.
  • Note seed length doesnt depend on n !
  • Efficient enough to obtain the desired extractors.

30
Error Reduction
  • Main Extractor Any k, m.99 k, dO(log n).
  • Caveat constant ?.
  • Error reduction for extractors RRV99 is not
    efficient enough for this case.
  • Our new mergers RRV99,RSW00 give improved
    error reduction.
  • ? Get various new extractors for general ?.
  • Any k, m.99 k, dO(log n), ?exp(-log n/log(c)n)
  • Any k, m.99 k, dO(log n (logn)2 log 1/? ),

31
Source vs. Seed Error Cont.
Defining bad inputs
0,1n
0,1n
(ka)-source X
heavy outputs lt ?2k ? bad inputs lt 2k (a
fraction 2-a)
32
Source vs. Seed Error Conclusion
  • More formally for a (k,k,?)-condenser Con,
    ?(klog 1/?)-source X, ?set G of good pairs
    (x,y) s.t.
  • For 1-? density x?X, Pr(x,Y)?Ggt1-2?.
  • Con(X,Y)(X,Y)?G is a (k-log 1/?) source.
  • ? Can differentiate source error ? and seed error
    ?.
  • Source error free in seed length!
  • ? (Con(x,y1),Con(x,y2)) has source error ? and
    seed error O(?2).
  • Dont need random y1,y2 (expander edge).
Write a Comment
User Comments (0)
About PowerShow.com