This document discusses the randomized Byzantine generals problem and its solution proposed by Michael Rabin in 1983. The key points are: - The Byzantine generals problem models reaching consensus in a distributed system where some processes may be unreliable or malicious. - Rabin proposed a randomized algorithm where processes agree on a common value through multiple rounds of exchanging signed messages. This algorithm ensures agreement with high probability within a bounded number of rounds. - The algorithm uses authentication techniques like digital signatures to ensure traitors can only lie about other traitors, not impersonate others. It also uses a "lottery" procedure for processes to randomly select a coordinator in each round. - Rabin's randomized algorithm guarantees consensus

1. RANDOMIZED BYZANTINE GENERALS
by
Michael O.~Rabin*
Presented by
Kishor Datta Gupta
L20421951
COSC 5302

2. Motivation
• One of the Fundamental Problem in distributed system is
Consensus problem
• A faulty processor can sends inconsistent information
to the other processors. Which will violate the properties
of distributed system
• Influenced by popularity of dining philosophers problem
Leslie Lamport associated story of generals to increase
the attention for solving fault tolerant system

3. History
• In 1975 Two generals problems (sometimes called the
Chinese Generals Problem) introduced by E. A.
Akkoyunlu, K. Ekanadham, and R. V. Huber .
• In 1978 “Two Generals Paradox” names by Jim Gray
• In 1980 Leslie Lamport published “Reaching
Agreement in the Presence of Faults” (with Marshall
Pease and Robert Shostak) this paper shows This paper
shows that "Byzantine" faults, in which a faulty processor
sends inconsistent information to the other processors,
can defeat any traditional three-processor algorithm

4. History (cont)
• In April 1982 Leslie Lamport wrote Byzantine Generals
and Transaction Commit Protocols (with Michael
Fischer) and proved that current Byzantine problem will
lead more failure
• In July 1982, Leslie Lamport solved this problem in a
paper written with Marshall Pease and Robert Shostak
and name “Byzantine General problem” established
• 1982 Ben-Or, M., Another advantage of free choice:
Completely asynchronous agreement protocols,
Abstract.

5. History (cont)
• July 1983 A weaker Byzantine problem Leslie Lamport
• In 1983 Michael O.~Rabin proposed “Random Byzantine
General” solutions

6. Background
Consensus Problem :
The consensus problem requires agreement among a
number of processes (or agents) for a single data value.
Some of the processes (agents) may fail or be unreliable in
other ways, so consensus protocols must be fault
tolerant or resilient. The processes must somehow put forth
their candidate values, communicate with one another, and
agree on a single consensus value

7. Consensus Problem(cont)
• Consensus Problem example
Suppose process p1 ,p2, p3 decided that a data value is
X, if there is another p4 decide that value is not X then
the problem create is called Consensus problem.
• Practical Example: mutual exclusion, election,
transactions

8. Consensus Problem
• Every processor has an input x є X
• Termination: Eventually every non-faulty processor
must decide on a value y.
• Agreement: All decisions by non-faulty processors must
be the same.
• Validity: If all inputs are the same, then the decision of a
non-faulty processor must equal the common input (this
avoids trivial solutions).

9. Consensus Problem(cont)
Consensus Problem Failure Reasons
• Link Failure
• Processor Failure
• Byzantine fault
Byzantine fault: If one or more processor became
malicious or faulty that they send wrong messages.

10. Byzantine Fault
Characteristics
• Most difficult
• Most common
Difference between process failure and Byzantine Failure is
in Byzantine Failure process send wrong information while
in process failure processor doesn’t send any thing. That’s
why Byzantine failure is difficult to detect than Processor
failure

11. Byzantine Failures Show

12. Byzantine Problem
• “We imagine that several divisions of the Byzantine army
are camped outside an enemy city, each division
commanded by its own general. The generals can
communicate with one another only by messenger. After
observing the enemy, they must decide upon a common
plan of action. However, some of the generals may be
traitors, trying to prevent the loyal generals from
reaching agreement.”----LESLIE LAMPORT, ROBERT
SHOSTAK, and MARSHALL PEASE

13. General Byzantine
Problem
o Each division of Byzantine army is directed by its own general.
o There are n Generals, some of which are traitors.
o All armies are camped outside enemy castle, observing enemy.
o Communicate with each other by messengers.
o Requirements:
• A: All loyal generals decide upon the same plan of action
• B: A small number of traitors cannot cause the loyal generals to
adopt a bad plan
Note: We do not need to identify the traitors.

14. Naïve Solution
All generals send message to all other generals . Majority
Result will be taken as decisions
Failure of the solutions:
o Traitors may send different values to different
generals.
o Loyal generals might get conflicting values from
traitors

15. Reduction
Interactive Consistency Conditions:
o IC1: All loyal lieutenants obey the same order.
o IC2: If the commanding general is loyal, then every loyal
lieutenant obeys the order he sends
Note: If General is loyal, IC2 => IC1.

16. Example Scenario
Conditions
• 3 generals, 1 traitor among them.
• Message: Attack / Retreat
For LIEUTENANT1 Who is traitor? COMMANDAR or LIEUTENANT2?
In Fig 1 LIEUTENANT1 has to attack to satisfy IC2. Fig 2 LIEUTENANT1
attacks, LIEUTENANT2 retreats. IC1 violated.
So it’s a impossibly situation.

17. Limit of traitors
Proof by contradiction.
o Assume there is a solution for 3m Albanians with m
traitors.
o Reduce to 3-General problem.
So we can decide that no solutions with fewer than
3m+1 generals can cope with m traitors.

18. Solution by Oral Message
Assumptions
A1 – Each message that is sent is delivered correctly.
Assures: Traitors cannot interfere with communication as
third party.
A2 – The receiver of a message knows who sent it.
Assures : Traitors cannot send fake messages
A3 – The absence of a message can be detected.
Assures: Traitors cannot interfere by being silent.
Note: Default order to “retreat” for silent traitor

19. Oral Message Algorithm
• Algorithm OM(0).
• (1) The commander sends his value to every lieutenant.
• (2) Each lieutenant uses the value he receives from the
commander, or uses the value
RETREAT if he receives no value.

20. Oral Message
Algorithm(cont)
Here m >0
o Commander sends his value to every Lieutenant (vi)
o Each Lieutenant acts as commander for OM(m-1) and sends
vi to the other n-2 lieutenants (or RETREAT)
o For each i, and each j<>i, let vj be the value lieutenant i
receives from lieutenant j in step (2) using OM(m-1).
Lieutenant i uses the value majority (v1, …, vn-1).

21. Explanation When
Lieutenant is traitor

22. When Commander is
traitor

23. Issue With Oral Message
• Need 3f+1 nodes to tolerate f failures which is expensive
• Difficult because traitors can lie about what others said
Time Complexity O(n^m)

24. Signed Message Solution
New Assumptions:
(a) A loyal general's signature cannot be forged, and any
alteration of the contents of his signed messages can be
detected.
(b) Anyone can verify the authenticity of a general's
signature.
Step:
• Each lieutenant maintains a set V of properly signed
orders received so far.
• The commander sends a signed order to lieutenants

25. Signed Message
Algorithm
• A lieutenant receives an order from someone (either
from commander or other lieutenants),
o Verifies authenticity and puts it in V.
o If there are less than m distinct signatures on the order
• Augments orders with signature
• Relays messages to lieutenants who have not seen the order.
• When lieutenant receives no new messages, and use
choice(V) as the desired action.

26. Explanation

27. Advantage and Issue
Advantage
• Need f+2 nodes
• Easier because traitors can only lie about other traitors
Issue:
• communication overheads
• Signatures
What if not all generals can reach all other generals
directly?

28. Missing Path
if the communication graph is 3m-regular, and there are at most m traitors,
the problem can still be solved.

29. Reliability
Achieving reliability in the face of arbitrary malfunctioning is
a difficult problem, and its solution seems to be inherently
expensive
To avoid expense it is often assumed that a computer may
fail to respond but will never respond incorrectly.

30. Asynchronous System
No algorithm can guarantee to reach consensus in an
asynchronous system, even with one process crash failure

31. RANDOMIZED BYZANTINE GENERALS
Based on:
• n processes with up to t faulty ones, t +1computing
phases are necessary for reaching Byzantine
Agreement.
• BG problem has no solution in the asynchronous case.
N=Number of process, T=faulty , G = Generals

32. Why Randomized
• Certain to achieve Byzantine Agreement
• The expected number of rounds required to do so is 4,
independent of N and T.
• The total expected number of messages =c*n*t, where
c = small constant
• In some case employs a fixed number R of rounds. To
reach Byzantine agreement,. but there is a probability
2^-R of error.

33. Stage
1. Basic Concepts
2. Authentication
3. Lottery
4. The Agreement Protocol
5. Correctness
6. Bounded Expected Time & Errorless Solutions

34. Basic Concept
Assumptions:
• Every Gi can directly exchange messages with every
other Gj. Where i,j ϵ n
• Every Gi agreed on a common value in the message
• If one Gi proved faulty for not having the common value
it will remain faulty even it later have the common value

35. Basic Concept
• If all the proper processes have the same initial message
then the system will be called proper, otherwise faulty.
• The processes reach agreement on the common
message by exchanging information
• We assume that each process Gi has a local phase-
clock p(i), and that pi assigns p (i) := p (i) + 1 at the end
of each phase.

36. Authentication
• A public directory containing for each participant B a
public key' KB.
• When the participant B needs to authenticate a
message M, he employs a secret key DB to compute
another message 0B(M) =N.
• Every other user can recover M form N by use of KB,
Note: The public-key directory is part of the data in each Pi,
and must be incorporated by a Non-faulty "dealer" at ,the
creation of the processes Gi

37. LOTTERY
• Algorithm employ a lottery procedure by which the
proper Gi's can agree on a randomly chosen s =0 ,I.
• Using Shamirs A method for obtaining digital signatures
and public-key cryptosystems,
• The lottery procedure admits a parameter m<N, so that
Lottery(m) is the m-th lottery round.
• End of the execution of Lottery(m) by the proper Gi, at
least all the proper processes will share Sm.

38. THE AGREEMENT
PROTOCOL(BAP)
Structure:
Polling : Gi polls the other processes on their value of the
message.
Lottery: Gi decide common random value
Decision: using s Gi determines whether to adopt the
plurality candidate version of the message obtained
through Polling or his current version of the message
These 3 happen in R time. R = desired reliability

39. BAP(Polling)

40. BAP(Lottery)

41. BAP(Decision)

42. BAP

43. Correctness
If Initial message of every proper process = M,
Proper System: End of BAP we shall have message (i)=
M for all proper Gi
Faulty System: With probability at ,least 2^-R , all proper
processes Gi will have the same value for message(i).
Note: Probability for not reaching agreement in R rounds is
2^-R

44. Bounded expected time
As long as t <n/10 we can reach an agreement ignoring
n and t
It requires 0(n^2) message
Using a wakeup call it could be reduces to 0(nt)

45. Advantage
• Both applied for Synchronized and asynchronous
version
• Better than Ben-or tossing approach
• With some condition it is a robust solution for the
distributed commit problem
• It is more simpler than other existing approach

46. Disadvantage
• Synchronized system simplify the algorithm but also
arises issue for coordinated agreement
• For Byzanatine Consensus problem also need to
implement wakeup protocol
• It only resilient under appropriate conditions

47. Practical Implementations
• For bitcoin peer to peer this solutions could be
implemented
• For distributed commit problem this solution can be
implemented and some cases its implemented

48. Conclusion
Randomize solution has some upper hand also has some
limit. However in 1984 this solutions creates such effect
which later used to implement various extended solutions
for Byzantine agreement

49. References
• [1] Leslie Lamport, Robert Shostak and Marshall Pease.
TheByzantine Generals Problem. ACM Transactions
onProgramming Languages and Systems, 4(3): 382-401,
July 1982.
• [2] “AWS S3 Availability Event”,
http://status.aws.amazon.com/s3-20080720.html .
Amazon Rec. 10/2013
• https://en.wikipedia.org/wiki/Consensus_(computer_scie
nce)
• http://marknelson.us/2007/07/23/byzantine/

50. References
• http://lamport.azurewebsites.net/pubs/pubs.html#implem
entation
• http://www.cs.cornell.edu/courses/cs6410/2009fa/lecture
s/24-byzantine.pdf
• http://www.di.univaq.it/~proietti/slide_algdist2010/5-
Consensus.ppt
• http://lamport.azurewebsites.net/pubs/weak-byz.pdf
• http://cs-www.bu.edu/faculty/homer/537/talks/Sachin-
byzagree-lecture.pdf

51. References
• http://www.cs.sfu.ca/CourseCentral/401/qshi1/slides/cmp
t_401_byzantine_generals.ppt
• http://webcourse.cs.technion.ac.il/236357/Winter2007-
2008/ho/WCFiles/Anna-
The%20Byzantine%20Generals%20Problem.ppt
• http://www.cs.toronto.edu/~demke/469F.07/Lectures/Lect
ure18.ppt
• http://www-inst.eecs.berkeley.edu/~cs162/fa12/hand-
outs/Original_Byzantine.pdf