Reaching Approximate Agreement in an Asynchronous Environment

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Reaching Approximate Agreementin an
Asynchronous Environment

• And what does it have to do with the Witness
  Protection Program

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Todays Lecture

• What approximately happened at Byzantine
• The Byzantine Agreement Problem
• BA - Known Results
• The Approximate Byzantine Agreement Problem
• Formal Definition
• Previous Work
• Our Algorithm, Two Versions.

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What approximately happened at Byzantine

• May 29th, 1453
• The Turks are besieging the city of
  Constantinople, A.K.A Byzantine.
• The Muslim generalsAre trying to coordinate an
  attack.

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The Byzantine Agreement Problem

• Introduced by Lamport, Pease and Shostak, 80-82
• A world of n processes/generals, t of them are
  faulty/traitors. The generals are trying to
  coordinate.
• Can this be solved?
• Depends on the model.
• Computational Bounds / Cryptography
• Network Topology
• Synchronous VS. Asynchronous

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BA - Known Results

• Synchronous 3t1 deterministic algorithm
• Asynchronous No Deterministic Algorithm Exists
  FLP 85
• Randomized Algorithms Exists Benor, Bracha and
  more.

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The Approximate Byzantine Agreement Problem

• Introduced by Dolev et al, 82
• How Approximate?
• Each process has a Real initial value
• A predetermined Epsilon.
• All processes must halt within Epsilon from
  each-other.

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Formal Definition

• Agreement - All non-faulty processes halt with
  values within Epsilon of each other
• Validity - The value of each non-faulty process
  must be within the range of the initial values of
  non-faulty processes.
• Termination All non-faulty processes must
  eventually halt.

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Previous Work

• Dolev. 82
• The family of algorithms.
• Trimming Functions.
• 3t1 Synchronous
• 5t1 Asynchronous
• Fekete 86,94
• Work on asynchronous, failure-by-omission.
• Proven Asymptotically optimal algorithms.

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Our Algorithm

• Using Reliable Broadcast with FIFO channels.
• Correctness If a non-faulty process p
  broadcasts m, all non-faulty processes will
  accept m from p.
• Unforgeability if a non-faulty process doesnt
  broadcast m, no non-faulty process accept m from
  p
• Relay If a non-faulty process accepts m from p,
  all non-faulty processes eventually accept m from
  p.
• Using Reliable Broadcast we can lower the
  requirement to 4t1

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Our Algorithm, ngt4t
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Our Algorithm, Cont

• The range of the non-faulty processes is cut in
  half in every round intuition.
• Note that each pair of processes have in common
  atleast 2t1 values.
• The worst the adversary can do, is pick a side.
  
• After the trimming, theres enough in common.

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Our Algorithm, cont

• To reach 3t1 we do the following
• Each process broadcasts its value
• Collect values
• Report what youve heard to all processes
• Collects others reports.
• When sufficient reports are obtained,
• Trim the values, and calculate a new value

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Our Algorithm, cont Whats sufficient?

• When we have n-t witnesses.
• A witness for process q is a process p, whose
  first n-t values were also explicitly heard by q.
• Common witnesses - Quorums

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Our Algorithm, Cont Initialization Termination

• Each round we trim the range in half.
• Initial declarative round, where the bounds are
  set.
• We run for log(range) rounds.

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Our Algorithm, Cont

• The range of the non-faulty processes is cut in
  half in every round
• Every two processes have at least n-t common
  values.
• The Median of the common values is remains after
  trimming in both processes.
• Thus, after averaging, the range is cut.

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Our Algorithm, Conclusion.

• We have devised a t-resilient algorithm, where
  ngt3t, and thus is Optimal
• Convergence rate is bound by the non-faulty
  processes initial range.
• The Witness concept may be useful for other
  problems as well.