Byzantine Agreement

Gábor Mészáros

CEU Budapest, Hungary

Gábor Mészáros Byzantine Agreement

1453 AD, Byzantium

Gábor Mészáros Byzantine Agreement

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Byzantine Generals Problem

ExampleThe "Simple Majority" strategy is not 1-resilient.

Gábor Mészáros Byzantine Agreement

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

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

No 1-resilient P in K3

Proof

Gábor Mészáros Byzantine Agreement

No 1-resilient P in K3

Proof

Gábor Mészáros Byzantine Agreement

No 1-resilient P in K3

Proof

Gábor Mészáros Byzantine Agreement

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

Attributes

SynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

Attributes

SynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

AttributesSynchronous

ReliableAuthenticatedBroadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

AttributesSynchronousReliable

AuthenticatedBroadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

AttributesSynchronousReliableAuthenticated

Broadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

AttributesSynchronousReliableAuthenticatedBroadcast on the edges

Presence of faulty participants

Gábor Mészáros Byzantine Agreement

Generalized Byzantine Generals Problem II. - Hypergraphs

Communication ModelH = (V ,E ) hypergraph.

AttributesSynchronousReliableAuthenticatedBroadcast on the edgesPresence of faulty participants

Gábor Mészáros Byzantine Agreement

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

Other Possible Generalizations

Variants

Asynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes...

Gábor Mészáros Byzantine Agreement

Other Possible Generalizations

VariantsAsynchronous communication

General HypergraphsCorruptible subsetsRandom processes...

Gábor Mészáros Byzantine Agreement

Other Possible Generalizations

VariantsAsynchronous communicationGeneral Hypergraphs

Corruptible subsetsRandom processes...

Gábor Mészáros Byzantine Agreement

Other Possible Generalizations

VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsets

Random processes...

Gábor Mészáros Byzantine Agreement

Other Possible Generalizations

VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes

...

Gábor Mészáros Byzantine Agreement

Other Possible Generalizations

VariantsAsynchronous communicationGeneral HypergraphsCorruptible subsetsRandom processes...

Gábor Mészáros Byzantine Agreement

THANK YOU!

Gábor Mészáros Byzantine Agreement