Shafi Goldwasser, Madhu Sudan and Vinod Vaikuntanathan

Distributed Computing With Imperfect Randomness

Shafi Goldwasser, Madhu Sudan and Vinod Vaikuntanathan

Presentation by: Daniel Thomas

Donnerstag, 19. Mai 2011

1. Randomness

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1. Randomness

3

QuickSort


1. Randomness

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QuickSort

P


1. Randomness

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QuickSortP

< P >= P


1. Randomness

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QuickSortP

< P >= P

What If Already Sorted?


1. Randomness

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QuickSortP

< P >= P

What If Already Sorted?

→ Randomized QuickSort


1. Randomness

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Real Randomness?(Strong Randomness)


1. Randomness

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Atomic Decay?



1. Randomness

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Atomic Decay?



1. Randomness

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Pseudorandomness?(Weak Randmoness?)


1. Randomness

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Pseudorandomness?(Weak Randmoness?)

Get Randomness From System?


Setting

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1. Randomness


Setting

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Agreement?

1. Randomness


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Faulty Player

Agreement?

Setting

1. Randomness


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Faulty Player

Agreement?

Weak Random Source

Weak Random Source

Weak Random Source

1. Randomness

Setting


1. Randomness, Overview

2. Byzantine Agreement

4. Randomness In A Distributed Environment

3. Probabilistic BA Protocol


2. Byzantine Agreement (single fault case)

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v(1)

v(3)

v(2)

v(4)

Traitors

Loyal



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A. The non-faulty generals compute exactly the same vectorB. The element of this vector corresponding to a given non-faulty general is the private value/opinion of this general

v(1)

v(3)

v(2)

v(4)

Traitors

Loyal



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Works if n >= 3m + 1And for m+1 rounds are required.

v(1)

v(3)

v(2)

v(4)

Traitors

Loyal


3. Probabilistic Byzantine Agreement Protocols

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Deterministic AlgorithmRandomized Algorithm

A Simple and Efficient Randomized Byzantine Agreement Algorithm, B.Chor And B.A Coan

m+1 rounds O(m/log n) rounds


3. Probabilistic Byzantine Agreement Protocols

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Deterministic AlgorithmRandomized Algorithm

A Simple and Efficient Randomized Byzantine Agreement Algorithm, B.Chor And B.A Coan

m+1 rounds O(m/log n) rounds

Randomized Algorithm Expects Players To Have Real Random Source!



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Formalizing Our Randomness Model



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Formalizing Our Randomness Model

Strong Randomness: Um

Weak Randomness: (k, δ)-weak source

No Randomness


Randomness Model

Each player (sensor) has access to its own weak source that is independent of the sources of all other players

Is there a way to get real randomness (a string of unbiased and independent random bits) by combining weak random sources?

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Randomness Model

Each player (sensor) has access to its own weak source that is independent of the sources of all other players

Is there a way to get real randomness (a string of unbiased and independent random bits) by combining weak random sources?

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Extractors!


Extractors

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ExtractorDeterministic Function

Ext: {0,1} ➝ {0,1}

Weak Random Source{0,1}

0100110011011101011

Almost unbiased & independent bits{0,1}

01001110

k

k m

m


Extractors

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ExtractorDeterministic Function

Ext: {0,1} ➝ {0,1}

Weak Random Source{0,1}

0100110011011101011

Almost unbiased & independent bits {0,1}

01001110

k

k m

m

Not Possible

In General


2 Source Extractors

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Weak Random Source0100110011011101011

Weak Random Source0100111011010011011←independent→

(k, δ)-two source Extractor)Deterministic Function

Ext: ({0,1} ) ➝ {0,1}


01001110

k m

m

2


A Very Simple Extractor

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0 1



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0 10 0



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0 10 01 1



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0 10 01 11 0



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0 10 01 11 00 0



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0 10 01 11 00 01 0



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0 10 01 11 00 01 00 1



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0 10 01 11 00 01 00 1

0

1

10



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0 10 01 11 00 01 00 1

0

1

10

Not a (k, δ)-two source Extractor!(do you see why?)


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Agreement?


Is this enough?


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Is this enough?No, as we have faulty players!What if they try to “poison” therandomness?


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Multi-source extractors whose output is random even if an arbitrary subset of the input sources don’t send a weak random string (e.g. faulty players) but all sources are independent

Immune Extractors


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4. Randomness In A Distributed Environment(κ, τ) Immune Extractors, as a picture

(k, δ) source (k, δ) source(k, δ) source(k, δ) source


01001110

(κ, τ)-immune extractorDeterministic Function

Ext: ({0,1}) ➝ {0,1}k+1

m

m

κ + 1 block sources

m


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Let Ext be any (k, δ) two source extractor Let X₁², X₁³, ... denote t distinct blocks from the source X₁ Let X₂, X₃, ... be one block each from the other t sources I-Ext({X₁} , X₂, ..., Xt+1) = ⊕ Ext(X₁, Xi)

I-Ext: A(t, t-1)-immune extractor

ii=2

t+1 t+1

i=2

i


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Let Ext be any (k, δ) two source extractor Let X₁², X₁³, ... denote t distinct blocks from the source X₁ Let X₂, X₃, ... be one block each from the other t sources I-Ext({X₁} , X₂, ..., Xt+1) = ⊕ Ext(X₁, Xi)

Theorem: I-Ext is a (t, t-1) immune extractor!

I-Ext: A (t, t-1)-immune extractor

ii=2

t+1 t+1

i=2

i

i


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... each player sends each other player a string from its weak random source ...


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... each player can use I-Ext to generate a string very close to Um ...


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... every player has an almost random string ...


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5. Conclusion

The paper mainly showed how it is possible to for each player of a network to extract a almost unbiased, uniform random string by sharing their (mutually independent) random sources, even in presence of faulty or even malicious parties.


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?Donnerstag, 19. Mai 2011

Shafi Goldwasser, Madhu Sudan and Vinod Vaikuntanathan

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