Byzantine Agreement Consensus in the Presence of Uncertainties

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Byzantine AgreementConsensus in the Presence of
Uncertainties

• CoCoA Talk 28 9 May 2003

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The Consensus/Agreement Problem

• Each process in the network begins with an
  initial value and is supposed to output a value
  of the same type.
• All processes must agree outputs must be the
  same even though the inputs can be arbitrary.
• The problem arises in many distributed computing
  applications, e.g processes attempting to reach
  agreement, or identifying a component as faulty.

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Lets attack!
Lets retreat!
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Byzantine Agreement Problem

• Process might exhibit arbitrary behavior.
• Butthey can only affect the system components
  over which they are supposed to have control
• Sample problem

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We take all this for granted

• The network is an n-node connected undirected
  graph with processes 1,,n, where each process
  knows the entire graph.
• Every message that is sent is delivered
  correctly.
• The receiver of a message knows who sent it.
• There can be at most f failures.

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Correctness Conditions

• AgreementNo two processes decide on different
  values.
• ValidityIf all processes start with the same
  initial value v ? V then v is the only possible
  decision value.
• TerminationAll processes eventually decide.

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Conditions in the presence of failures

• AgreementNo two non-faulty processes decide on
  different values.
• ValidityIf all non-faulty processes start with
  the same initial value v ? V then v is the only
  possible decision value for a non-faulty
  process.
• TerminationAll non-faulty processes eventually
  decide.

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attack
attack
attack
retreat
he said retreat
he said retreat
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Impossibility Result
0
0
1
0
1
1
10
a
a1
11
a2
a
12
0
a
1
a3
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Byzantine Agreement in General Graphs

• Condition for complete graphs
• Byzantine agreement in a tree?
• ConnectivityThe connectivity of a graph G,
  conn(G), is defined to be the minimum number of
  nodes whose removal results in either a
  disconnected graph or a trivial 1-node graph.

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Connectivity

• c-connectivityA graph G is said to be
  c-connected if conn(G) c.
• Mengers Theorem A graph is c-connected if and
  only if every pair of nodes in G is connected by
  at least c node-disjoint paths.

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• The Byzantine agreement problem can be solved in
  an n-node network graph G, tolerating f faults,
  if and only if the following conditions hold
• n gt 3f
• conn(G) gt 2f

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1
2
4
3
17
3
4
1
2
4
2
3
1
18
0
0
0
19
1
1
1
20
1
0
0
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Exponential Information Gathering
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Algorithm EIGByz

• Round 1 Process i broadcasts val(l) to all
  processes, including i itself. Then process i
  records the incoming information.
• 1. If a message with value v ? V arrives at i
  from j, then i sets its val(j) to v
• If no message with a value in V arrives from i to
  j, then i sets val(j) to null.
• Round k, 2 k f 1Process i broadcasts all
  pairs (x,v) and processes the incoming
  information.
• If xj is a level k node label in T, where x is a
  string of process indices and j is a single
  index, and a message saying that val(x) v ? V
  arrives at I from j, then I sets val(xj) to v.
• If xj is a level k node label and no message with
  a value in V for val(x) arrives at I from j, then
  I sets val(xj) to null.

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EIGByz continued

• At the end of f1 rounds, process i adjusts its
  val assignment so that any null value is replaced
  by the default v0
• Process i works from the leaves up decorating
  each node with an additional newval.
• For each leaf labeled x, newval(x) val(x). For
  each non-leaf node labeled x, newval(x) is
  defined to be the newval held by a strict
  majority of the children node of x.
• Process is final decision is newval(l)

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Invariant Assertions

• Lemma 1 After f1 rounds of the EIGByz
  algorithm, the following holds. If i,j, and k are
  all nonfaulty processes, with i?j, then val(x)i
  val(x)j for every label x ending in k
• All nonfaulty processes agree on the value
  relayed directly form nonfaulty processes.

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• Lemma 2 After f1 rounds of the EIGByz
  algorithm, the following holds. Suppose that x is
  a label ending with the index of a nonfaulty
  process. Then there is a value v ?V such that
  val(x)i newval(x)i v for all nonfaulty
  processes i.
• All nonfaulty processes agree on the newvals
  computed for nodes whose labels end with
  nonfaulty process indices