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Byzantine Agreement
Gábor Mészáros
CEU Budapest, Hungary
Gábor Mészáros Byzantine Agreement
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1453 AD, Byzantium
Gábor Mészáros Byzantine Agreement
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Distibuted Systems
Communication System Model
G = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
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Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 5: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/5.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)
E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 6: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/6.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
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Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 8: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/8.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of Interest
CryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 9: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/9.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of InterestCryptography
Data CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 10: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/10.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of InterestCryptographyData Compression
Distributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 11: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/11.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of InterestCryptographyData CompressionDistributed Computing
...
Gábor Mészáros Byzantine Agreement
![Page 12: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/12.jpg)
Distibuted Systems
Communication System ModelG = (V ,E ) simple graph
V : nodes - participants (finite state machines)E : edges - communication channels
Description of the communication mechanism
Different Attributes - Different Fields of InterestCryptographyData CompressionDistributed Computing...
Gábor Mészáros Byzantine Agreement
![Page 13: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/13.jpg)
Byzantine Generals Problem
Attributes
SynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronous
ReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliable
AuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticated
Point-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-Point
Presence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many steps
Agreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")
Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
AttributesSynchronousReliableAuthenticatedPoint-to-PointPresence of faulty participants ("traitors") which can behavearbitrarily ("Byzantine failures").
GoalsGiven the set of initial assessments xi ∈ {0, 1} of each Gi ∈ L ⊂ V (G )("loyal generals") calculate decisions di ∈ {0, 1} satisfying:
Termination: each process terminates in finitely many stepsAgreement: di = dj∀Gi ,Gj ∈ L (the set of "loyal generals")Nontriviality: xi = c ∈ {0, 1}∀Gi ∈ L⇒ di = c
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
DefinitionA protocol P is t-resilient if it tolerates byzantine failure of at most t faultyparticipants.
QuestionHow many byzantine failures can a network tolerate?
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
DefinitionA protocol P is t-resilient if it tolerates byzantine failure of at most t faultyparticipants.
QuestionHow many byzantine failures can a network tolerate?
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
ExampleThe "Simple Majority" strategy is not 1-resilient.
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
Theorem (Lamport, Pease, Shostak, 1980)There exists t-resilient protocol ⇔ t < n
3 .
LemmaNo 1-resilient protocol P exists on K3.
Gábor Mészáros Byzantine Agreement
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Byzantine Generals Problem
Theorem (Lamport, Pease, Shostak, 1980)There exists t-resilient protocol ⇔ t < n
3 .
LemmaNo 1-resilient protocol P exists on K3.
Gábor Mészáros Byzantine Agreement
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No 1-resilient P in K3
Proof
Gábor Mészáros Byzantine Agreement
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No 1-resilient P in K3
Proof
Gábor Mészáros Byzantine Agreement
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No 1-resilient P in K3
Proof
Gábor Mészáros Byzantine Agreement
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t-resilient ⇔ t < n3
Corollary of the Lemma - ReductionA t ≤ n
3 -resilient protocol is 1-resilient in K3.
Constructions for t < n3 (sketch)
1 Exponential data trees - "x told me, that y told him, that..." - fill()and resolve() -not efficient
2 Efficient (polinomial) Broadcast- firefly effect, echoes...
Gábor Mészáros Byzantine Agreement
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t-resilient ⇔ t < n3
Corollary of the Lemma - ReductionA t ≤ n
3 -resilient protocol is 1-resilient in K3.
Constructions for t < n3 (sketch)
1 Exponential data trees - "x told me, that y told him, that..." - fill()and resolve() -not efficient
2 Efficient (polinomial) Broadcast- firefly effect, echoes...
Gábor Mészáros Byzantine Agreement
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t-resilient ⇔ t < n3
Corollary of the Lemma - ReductionA t ≤ n
3 -resilient protocol is 1-resilient in K3.
Constructions for t < n3 (sketch)
1 Exponential data trees - "x told me, that y told him, that..." - fill()and resolve() -not efficient
2 Efficient (polinomial) Broadcast- firefly effect, echoes...
Gábor Mészáros Byzantine Agreement
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t-resilient ⇔ t < n3
Corollary of the Lemma - ReductionA t ≤ n
3 -resilient protocol is 1-resilient in K3.
Constructions for t < n3 (sketch)
1 Exponential data trees - "x told me, that y told him, that..." - fill()and resolve() -not efficient
2 Efficient (polinomial) Broadcast- firefly effect, echoes...
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
Attributes
SynchronousReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
Attributes
SynchronousReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronous
ReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronousReliable
AuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronousReliableAuthenticated
Not necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronousReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)
Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronousReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
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Generalized Byzatine Generals Problem I. - Graphs
Communication ModelG = (V ,E ) simple (not necessarily complete) graph with connectivitynumber k(G ) := k
AttributesSynchronousReliableAuthenticatedNot necessarily Point-to-Point (communication on edges only)Presence of faulty participants
GoalUnanimity between the non-faulty processors
Gábor Mészáros Byzantine Agreement
![Page 43: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/43.jpg)
Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient ⇔ t < n3 and t < k
2 .
Theorem (Kumar,2002)
Given S ⊂ 2V (G) set of corruptible subsets in G = (V ,E ) unanimity isattainable ⇔
no union S1 ∪ S2 of any pair S1, S2 ∈ S contains a cut of G ,no union S1 ∪ S2 ∪ S3 of any triple S1, S2, S3 ∈ S covers V (G ).
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient ⇔ t < n3 and t < k
2 .
Theorem (Kumar,2002)
Given S ⊂ 2V (G) set of corruptible subsets in G = (V ,E ) unanimity isattainable ⇔
no union S1 ∪ S2 of any pair S1, S2 ∈ S contains a cut of G ,no union S1 ∪ S2 ∪ S3 of any triple S1, S2, S3 ∈ S covers V (G ).
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient ⇔ t < n3 and t < k
2 .
Theorem (Kumar,2002)
Given S ⊂ 2V (G) set of corruptible subsets in G = (V ,E ) unanimity isattainable ⇔
no union S1 ∪ S2 of any pair S1, S2 ∈ S contains a cut of G ,
no union S1 ∪ S2 ∪ S3 of any triple S1, S2, S3 ∈ S covers V (G ).
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient ⇔ t < n3 and t < k
2 .
Theorem (Kumar,2002)
Given S ⊂ 2V (G) set of corruptible subsets in G = (V ,E ) unanimity isattainable ⇔
no union S1 ∪ S2 of any pair S1, S2 ∈ S contains a cut of G ,no union S1 ∪ S2 ∪ S3 of any triple S1, S2, S3 ∈ S covers V (G ).
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient iff t < n3 and t < k
2 .
Proof ("⇐")
1 For each Gi ,Gj ∈ V (G ), (GiGj) 6∈ E (G ) fix disjoint pathsP1,P2, ...,Pk between the nodes ("delivery channels").
2 Send messages from Gi to Gj via P1,P2, ...,Pk and consider majorityof the 0 - 1 messages. t < k
2 guaranties reliability.3 Emulate the solution of the original BA problem.
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient iff t < n3 and t < k
2 .
Proof ("⇐")
1 For each Gi ,Gj ∈ V (G ), (GiGj) 6∈ E (G ) fix disjoint pathsP1,P2, ...,Pk between the nodes ("delivery channels").
2 Send messages from Gi to Gj via P1,P2, ...,Pk and consider majorityof the 0 - 1 messages. t < k
2 guaranties reliability.3 Emulate the solution of the original BA problem.
Gábor Mészáros Byzantine Agreement
![Page 49: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/49.jpg)
Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient iff t < n3 and t < k
2 .
Proof ("⇐")1 For each Gi ,Gj ∈ V (G ), (GiGj) 6∈ E (G ) fix disjoint paths
P1,P2, ...,Pk between the nodes ("delivery channels").
2 Send messages from Gi to Gj via P1,P2, ...,Pk and consider majorityof the 0 - 1 messages. t < k
2 guaranties reliability.3 Emulate the solution of the original BA problem.
Gábor Mészáros Byzantine Agreement
![Page 50: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/50.jpg)
Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient iff t < n3 and t < k
2 .
Proof ("⇐")1 For each Gi ,Gj ∈ V (G ), (GiGj) 6∈ E (G ) fix disjoint paths
P1,P2, ...,Pk between the nodes ("delivery channels").2 Send messages from Gi to Gj via P1,P2, ...,Pk and consider majority
of the 0 - 1 messages. t < k2 guaranties reliability.
3 Emulate the solution of the original BA problem.
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem I. - Graphs
Theorem (Dolev, 1982)
G = (V ,E ) is t-resilient iff t < n3 and t < k
2 .
Proof ("⇐")1 For each Gi ,Gj ∈ V (G ), (GiGj) 6∈ E (G ) fix disjoint paths
P1,P2, ...,Pk between the nodes ("delivery channels").2 Send messages from Gi to Gj via P1,P2, ...,Pk and consider majority
of the 0 - 1 messages. t < k2 guaranties reliability.
3 Emulate the solution of the original BA problem.
Gábor Mészáros Byzantine Agreement
![Page 52: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/52.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
Attributes
SynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
Attributes
SynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
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Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
AttributesSynchronous
ReliableAuthenticatedBroadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
![Page 55: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/55.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
AttributesSynchronousReliable
AuthenticatedBroadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
![Page 56: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/56.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
AttributesSynchronousReliableAuthenticated
Broadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
![Page 57: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/57.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
AttributesSynchronousReliableAuthenticatedBroadcast on the edges
Presence of faulty participants
Gábor Mészáros Byzantine Agreement
![Page 58: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/58.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Communication ModelH = (V ,E ) hypergraph.
AttributesSynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants
Gábor Mészáros Byzantine Agreement
![Page 59: GáborMészáros · Byzantine Agreement GáborMészáros CEU Budapest, Hungary Gábor Mészáros Byzantine Agreement](https://reader030.vdocuments.net/reader030/viewer/2022040302/5e80c6d7b9d8d544876bd3d4/html5/thumbnails/59.jpg)
Generalized Byzantine Generals Problem II. - Hypergraphs
Theorem (Fitzi, Maurer, 2000)H = (V ,E ) 3-uniform complete hypergraph is t-resilible ⇔ n ≤ 2 · t + 1.
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
Variants
Asynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes...
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
VariantsAsynchronous communication
General HypergraphsCorruptible subsetsRandom processes...
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
VariantsAsynchronous communicationGeneral Hypergraphs
Corruptible subsetsRandom processes...
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsets
Random processes...
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes
...
Gábor Mészáros Byzantine Agreement
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Other Possible Generalizations
VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes...
Gábor Mészáros Byzantine Agreement
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THANK YOU!
Gábor Mészáros Byzantine Agreement