Zero-Fixing Extractors for Sub-Logarithmic Entropy Joint with Igor Shinkar Gil Cohen.
8
Zero-Fixing Extractors for Sub-Logarithmic Entropy Joint with Igor Shinkar Gil Cohen
-
Upload
darcy-lynch -
Category
Documents
-
view
218 -
download
1
Transcript of Zero-Fixing Extractors for Sub-Logarithmic Entropy Joint with Igor Shinkar Gil Cohen.
Zero-Fixing Extractors forSub-Logarithmic Entropy
Joint with Igor Shinkar
Gil Cohen
0
Assumption. of the bits are jointly uniform. The rest are fixed.
A random variable with such distribution is called an -bit-fixing source [Vaz’85, BenBraRob’85, ChoGolHasFriRudSmo’85].
0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 1
* For any-bit-fixing source , is close to uniform.
Ext𝑥∈ {0 ,1 }𝑛 𝐸𝑥𝑡 (𝑥 )∈ {0 ,1 }𝑚
* Maximize (clearly, ).
* is computable in -time.
Goal. Extract the randomness from bit-fixing sources.
The Bit-Fixing Extractors Problem
The Bit-Fixing Extractors Problem
Two easy facts:
* If then most functions are - bit-fixing extractors with output bits.
1 𝑛
[Vaz’85,BBR’85,CGHFRR’85]
𝑛2√𝑛
[KamZuc’06]
𝑝𝑜𝑙𝑦 log𝑛
[GabRazSha’06,
Rao’09]
Theorem [KamZuc’06]. For any k, there is an efficient and simple extractor with output bits.
Theorem [ResVad’10]. Space-bounded streaming algorithms cannot extract bits for .
* If then most functions are not -bit-fixing extractors.
log𝑛?
1 𝑛
[Vaz’85,BBR’85,CGHFRR’85]
𝑛2√𝑛
[KZ’06]
𝑝𝑜𝑙𝑦 log𝑛
[GRS’06,
Rao’09]
log log𝑛
Theorem 1
One can extract bits.
Theorem 2
( log∗𝑛 )2/3
No algorithm can extract more than bits. Running-time .
Our Contribution
log𝑛
Theorem 1. The threshold, if exists, for extracting all the randomness is not at .
Even when the fixed bits are set to 0 !
Theorem 2. When k is small enough, only a logarithmic amount of the entropy is accessible, information-theoretically.
Theorem 3. There exists an efficient extractor for -zero-fixing sources for with output bits.
Theorem 2. When k is small enough, only a logarithmic amount of the entropy is accessible, information-theoretically.
Proof sketch for the impossibility result
𝑒1,… ,𝑒𝑛 The all-zeros vector
Weight 2 vectors
By the pigeonhole principle, there exist weight 1 vectors on which is constant.
If we found an -zero-fixing source on which is symmetric.
Proof sketch for the impossibility result
The all-zeros vector
Weight 2 vectors
By the pigeonhole principle, there exist weight 1 vectors on which is constant.
If we found an -zero-fixing source on which is symmetric.
Consider the complete graph on vertices corresponding to the “surviving” indices. Each edge is colored by . 𝑓 (𝑒 𝑖+𝑒 𝑗 )𝑖
𝑗
By the (multi-colored variant of) Ramsey theorem, there exists a monochromatic clique of size in this graph.
If , we found an -zero-fixing source on which is symmetric.
𝑒1,… ,𝑒𝑛
Proof sketch for the impossibility result
To find an -zero-fixing source on which is symmetric, we consider the 3-uniform hypergraph on the “surviving” vertices .
𝑓 (𝑒 𝑖+𝑒 𝑗 )𝑖𝑗
𝑖𝑗
𝑘
𝑓 (𝑒𝑖+𝑒 𝑗+𝑒𝑘)By the (multi-colored variant of) Ramsey theorem for hyper- graphs, there exists a monochromatic clique of size in this hypergraph.
If , we found an -zero-fixing source on which is symmetric.
We repeat this process times and find an -zero-fixing source on which is symmetric, assuming . Thus, for to be an extractor it must be that .
* What is the smallest function that allows for the extraction of all the entropy from bit-fixing / zero-fixing sources?
Open Problems
Thank you!
* Design an efficient bit-fixing extractor with output bits for , or even for .
* Is there a threshold behavior? If so, is it at ?
