The Byzantine Generals Problem

1
The Byzantine Generals Problem

• By L Lamport, R Shostak, M Pease
• COP 6614 Paper Presentation
• Presented by Aasavari Bhave
• Sep 20, 2005

2
Outline

• Introduction
• Impossibility Results
• A Solution with Oral Messages
• A Solution with Signed Messages
• Observation
• Communication Paths
• Conclusion
• References

3
What is Byzantine Generals Problem

• Reaching a consensus amongst distributed
  components (Generals of an army) by sending
  information to each other when one or more
  components send conflicting information.
• Abstraction of fault tolerant distributed system
  having one or more malfunctioning nodes.
• Details help building a reliable system
  operating reasonably correct in the presence of
  errors.

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Problem Domain

• Each division of Byzantine army is directed by a
  general - loyal (most) or traitor (few).
• Generals communicate with each other using oral
  or signed messages.
• All generals decide upon plan of action Attack or
  Retreat by looking at messages from others.
• All loyal generals must decide upon the same
  plan.
• Few traitors cannot cause loyal generals to opt a
  bad plan.

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Interactive Consistency Conditions

• A commanding general orders n-1 lieutenants
  generals.
• IC1 - All loyal lieutenants obey the same order.
• IC2 - If the commanding general is loyal then all
  loyal lieutenants obey the order he sends.
• If commanding general is loyal IC2 -gt IC1.
• Each commander uses a function Majority
  f(v1,v2,..,v n-1) to make a decision.

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Impossibility Results

• No solution exists with oral messages in presence
  of m traitors amongst maximum of 3m generals.

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Impossibility Results

• No solution with fewer than 3m1 generals can
  cope with m traitors with oral communication
  Can be proved by contradiction.
• With signed communication Byzantine generals
  problem can be solved with any number of
  traitors.

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A Solution With Oral Messages

• OM(m), m0
• The commander sends his value to each lieutenant.
• Each lieutenant uses the value he receives or
  uses a default value if he does not receive any
  value.

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Solution Cntd

• OM (m), m gt 0
• Commander sends his value to every lieutenant.
• For each i let lieutenant i receives the value Vi
  from commander. Lieutenant i acts as commander in
  call OM(m-1) and sends Vi to n-2 lieutenants.
• 3. For each i and j ltgt i, Lieutenant i received
  value Vj (from call to OM (m-1) in step 2 ) from
  lieutenant j. Lieutenant i uses function
  Majority(V1,,Vn-1) to make a decision (Default
  value is used if no majority is reached).

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Example - Oral Messages

• Majority Achieved

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Example - Oral Messages

• Lack of Majority leads to using a default value

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A Solution With Signed Messages

• Assumptions
• A loyal generals signature cannot be forged.
• Any alteration to the signature can be detected.
• Anyone can verify the authenticity of a generals
  signature.
• Notation
• Message xi value x signed by general i.
• Message vij value v signed by i and value vi
  signed by j.

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Solution Cntd

• SM (m) Vi ?
• Commander signs and sends his value to every
  lieutenant.
• For each i
• If lieutenant i receives message v0 and he has
  not yet received any order then
• Vi v
• Lieutenant i sends v0i to all other
  lieutenants.
• If lieutenant i receives message v0j1..jk and
  v is not in the set Vi then
• Add v to Vi
• If k lt m then he sends the message v0j1.jki
  to all lieutenants other than j1jk.
• For each i when lieutenant i will not receive any
  more messages be obeys the order Choice(Vi).

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Example Signed Messages
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Observation

• Both OM(m) and SM(m) are very expensive in terms
  of number of messages and amount of time
  required.
• It assumes a complete graph of n generals/nodes
  which is not true in practice.
• Combining messages will reduce the total number
  of separate messages sent.

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Observation Cntd

• In general case (m gt 1) the solution takes m1
  iterations.
• In each iteration a process sends to a subset of
  other processes the value it received in previous
  iteration.
• It involves sending O(N m1) messages.
• Fischer and Lynch 1982 proved that in no less
  than f1 iterations a deterministic solution can
  be obtained.

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Observation Cntd

• Dolev and Strong 1983 algorithm uses f1
  iterations but sends only O(N2) messages.
• The Byzantine generals solution is applicable
  only when there is a great threat.
• Detailed knowledge of fault model is very useful.
• A solution without signature(oral messages) is
  impractical.

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Communication Paths

• IC2 imposes that a loyal lieutenant must obey
  loyal commander. If every message from commander
  to lieutenant is relayed through a traitor then
  there is no guarantee that lieutenant gets the
  actual commanders order.
• IC1 imposes that all loyal lieutenants follow the
  same order. Hence if 2 lieutenants communicate
  via a traitorous intermediaries, IC1 cannot be
  guaranteed.
• Byzantine generals problem is solvable if the
  sub-graph formed by the loyal generals is
  connected.

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Communication Paths Cntd

• The Byzantine generals problem is solvable in
  presence of m traitors if the graph of generals
  is 3m-regular.
• A 3m-regular graph must have 3m1 or more
  generals.
• A variation of OM(m) - OM(m,p) is proposed for
  graph of generals which is a p-regular graph.

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Modified Oral Message Solution

• OM(m,p)
• Let N be set of p lieutenants, neighbors of
  commander. Commander sends his value to each
  element of N.
• For each i in N let vi be the value he received
  from commander. Lieutenant i sends vi to each
  other lieutenant k as
• If m1 , send the value along path i-k.
• If m gt 1 then act as commander and invoke
• OM (m-1,p-1). Exclude original commander from G.
• For each k and i in N (i ltgt k), let vi be the
  value lieutenant k received. . k uses the
  majority value.

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Conclusion

• Byzantine generals problem creates an abstraction
  of faulty nodes in a distributed systems.
• The faulty nodes (due to hardware failure etc.)
  having inconsistent behavior while interactive
  with other components is a typical Byzantine
  fault.
• Two Solutions to the problem are proposed
• Exchange of unsigned forgettable oral messages
  Works only when number of faulty elements is less
  than 1/3rd of total number of elements.
• Exchange of signed messages Works with any
  number of faulty elements.It is more practical.
• The solutions are costly but implementing them
  ensures extremely reliable distributed system.
• Variations of solutions can be made even if the
  graph of generals is not a complete graph and
  sub-graph of loyal generals is connected.

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References

• L Lamport, R Shokstak, M pease, The Byzantine
  Generals Problem, 1982
• G Coulouris, J Dollimore, T Kindburg,
  Distributed Systems, Addison Wesley, 3rd
  edition.