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The byzantine generals problem

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Introduction about the Byzantine Generals Problem

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The byzantine generals problem

1. 1. The Byzantine Generals Problems LESLIE LAMPORT, ROBERT SHOSTAK, and MARSHALL PEASE ! Present by: Nguyen Thi Mai & Nguyen Van Luong
2. 2. Motivation A reliable computer system must be able to cope with a failure of one or more of its components A failed computer behaviour in this case: Sending conﬂicting messages to different parts of the system Not sending some of the messages
3. 3. Motivation All generals must agree upon a common battle plan Communicate only be messenger Some of generals are traitors who try to confuse the others
4. 4. Outline Motivation Oral Messages algorithm Signed Messages algorithm Conclusion
5. 5. Formally 1. All loyal lieutenants obey the same order 2. If the commander is loyal, then every loyal lieutenant obeys the order he sends
6. 6. Oral Message algorithm Assumptions: Every message that is sent is delivered correctly A receiver of a message knows who sent it The absence of a message can be detected
7. 7. Oral Message algorithm A recursive deﬁnition, with a base case for m=0, and a recursive step for m > 0: Algorithm OM(0) : 1.The commander sends his value to every lieutenant. 2.Each lieutenant uses the value he receives from the commander. Algorithm OM(m), m > 0 1.The commander sends his value to each lieutenant. 2.For each i, let vi be the value lieutenant i receives from the commander. Lieutenant i acts as the commander in Algorithm OM(m-1) to send the value vi to each of the n-2 other lieutenants. 3.For each i, and each j ≠ i, let vi be the value lieutenant i received from lieutenant j in step 2 (using Algorithm OM(m-1)). Lieutenant i uses the value Majority(v1, v2, … vn).
8. 8. Oral Message algorithm Lemma 1: For any m and k, Algorithm OM(m) satisﬁes (2) if there are more than 2k+m generals and at most k traitors Theorem 1: For any m, algorithm OM(m) satisﬁes conditions 1 and 2 if there are more than 3m generals, and at most m traitors.
9. 9. Oral Message algorithm Example: Bad Lieutenant Scenario: m=1, n=4, traitor = L3 OM(1): C A A A L2 L1 L3 C OM(0):??? L1 L2 A A Decision?? R L3 R L1 = m (A, A, R); L2 = m (A, A, R); Both attack!
10. 10. Oral Message algorithm Example: Bad Commander Scenario: m=1, n=4, traitor = C OM(1): C A R L2 L1 OM(0):??? A L3 A L1 R A L2 A R L3 A Decision?? L1=m(A, R, A); L2=m(A, R, A); L3=m(A,R,A); Attack!
11. 11. Signed Message algorithm More assumptions: A loyal general’s signature cannot be forged, and any alteration of the contents of his signed message can be detected Anyone can verify the authenticity of a general’s signature => There exists an algorithm that copes with m traitors for any number of generals (n≥m+2)
12. 12. Signed Message algorithm 1. Commander signs v and sends to all as (v:0) 2. Each lieutenant i: A) If receive (v:0) and no other order 1) Vi = v 2) send (V:0:i) to all B) If receive (v:0:j:...:k) and v not in Vi 1) Add v to Vi 2) if (k<m) send (v:0:j:...:k:i) to all not in j...k 3. When no more msgs, obey order of choice(Vi)
13. 13. Signed Message algorithm choice(V): • • If V={v} then choice(V)= v choice(Empty)=Default
14. 14. Signed Message algorithm SM(1) Example: Bad Commander Scenario: m=1, n=m+2=3, bad commander A:0 C R:0 L2 L1 What next? A:0:L1 L2 L1 R:0:L2 V1={A,R} V2={R,A} Both L1 and L2 can trust orders are from C Both apply same decision to {A,R}
15. 15. Signed Message algorithm SM(2): Bad Commander+ Scenario: m=2, n=m+2=4, bad commander and L3 A:0 L1 A:0:L1 L1 A:0:L2 C A:0 L2 L2 A:0:L3 L3 R:0:L3 V1 = V2 = {A,R} ==> Same decision x L3 Goal? L1 and L2 must make same decision R:0:L3:L1 L2 L1
16. 16. Conclusion Problem: T implement a fault-tolerant service with coordinated replicas, must o agree on inputs Byzantine failures make agreement challenging: Produce arbitrary output, can’t detect, collude User different agreement protocol depending on assumptions: Oral messages: Need 3f+1 nodes to tolerate f failures Difﬁcult because traitors can lie about what others said Signed messages: Need f+2 nodes Easier because traitors can only lie about other traitors
17. 17. “Question???”