shafi goldwasser, madhu sudan and vinod vaikuntanathan ... · distributed computing with imperfect...
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Distributed Computing With Imperfect Randomness
Shafi Goldwasser, Madhu Sudan and Vinod Vaikuntanathan
Presentation by: Daniel Thomas
Donnerstag, 19. Mai 2011
1. Randomness
2
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1. Randomness
3
QuickSort
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1. Randomness
4
QuickSort
P
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1. Randomness
5
QuickSortP
< P >= P
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1. Randomness
6
QuickSortP
< P >= P
What If Already Sorted?
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1. Randomness
7
QuickSortP
< P >= P
What If Already Sorted?
→ Randomized QuickSort
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1. Randomness
8
Real Randomness?(Strong Randomness)
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1. Randomness
9
Atomic Decay?
Real Randomness?(Strong Randomness)
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1. Randomness
10
Atomic Decay?
Real Randomness?(Strong Randomness)
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1. Randomness
11
Pseudorandomness?(Weak Randmoness?)
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1. Randomness
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Pseudorandomness?(Weak Randmoness?)
Get Randomness From System?
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Setting
13
1. Randomness
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Setting
14
Agreement?
1. Randomness
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15
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
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1. Randomness, Overview
2. Byzantine Agreement
4. Randomness In A Distributed Environment
3. Probabilistic BA Protocol
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2. Byzantine Agreement (single fault case)
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2. Byzantine Agreement (single fault case)
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v(1)
v(3)
v(2)
v(4)
Traitors
Loyal
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2. Byzantine Agreement (single fault case)
20
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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2. Byzantine Agreement (single fault case)
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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
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3. Probabilistic Byzantine Agreement Protocols
22
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
Donnerstag, 19. Mai 2011
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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4. Randomness In A Distributed Environment
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Formalizing Our Randomness Model
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4. Randomness In A Distributed Environment
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Formalizing Our Randomness Model
Strong Randomness: Um
Weak Randomness: (k, δ)-weak source
No Randomness
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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?
26
4. Randomness In A Distributed Environment
Donnerstag, 19. Mai 2011
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?
27
4. Randomness In A Distributed Environment
Extractors!
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Extractors
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4. Randomness In A Distributed Environment
ExtractorDeterministic Function
Ext: {0,1} ➝ {0,1}
Weak Random Source{0,1}
0100110011011101011
Almost unbiased & independent bits{0,1}
01001110
k
k m
m
Donnerstag, 19. Mai 2011
Extractors
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4. Randomness In A Distributed Environment
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
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2 Source Extractors
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4. Randomness In A Distributed Environment
Weak Random Source0100110011011101011
Weak Random Source0100111011010011011←independent→
(k, δ)-two source Extractor)Deterministic Function
Ext: ({0,1} ) ➝ {0,1}
Almost unbiased & independent bits {0,1}
01001110
k m
m
2
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 1
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 0
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 1
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 11 0
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 11 00 0
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 11 00 01 0
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 11 00 01 00 1
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
0 10 01 11 00 01 00 1
0
1
10
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A Very Simple Extractor
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4. Randomness In A Distributed Environment
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?
4. Randomness In A Distributed Environment
Is this enough?
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4. Randomness In A Distributed Environment
Is this enough?No, as we have faulty players!What if they try to “poison” therandomness?
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4. Randomness In A Distributed Environment
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
Almost unbiased & independent bits {0,1}
01001110
(κ, τ)-immune extractorDeterministic Function
Ext: ({0,1}) ➝ {0,1}k+1
m
m
κ + 1 block sources
m
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4. Randomness In A Distributed Environment
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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4. Randomness In A Distributed Environment
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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4. Randomness In A Distributed Environment
... each player sends each other player a string from its weak random source ...
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4. Randomness In A Distributed Environment
... each player can use I-Ext to generate a string very close to Um ...
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4. Randomness In A Distributed Environment
... 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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