1. Copyright 2006 Koren & Krishna ECE655/ByzGen.1 UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Fault Tolerant Computing ECE 655 Byzantine Failures

2. Copyright 2006 Koren & Krishna ECE655/ByzGen.2 Failure Types  Fail-Stop: Fails by stopping. No output is produced.  Consistent Output: All users see the same wrong output.  Byzantine Failure: No consistency is guaranteed: different users may see different values of the output.

3. Copyright 2006 Koren & Krishna ECE655/ByzGen.3 Original Motivation  A computer system uses sensor inputs to control some process.  Is there a way to ensure that the output of a sensor is seen consistently by all the functional processors despite the sensor exhibiting Byzantine failure?  The problem was originally uncovered at NASA Langley and studied by contractors from SRI International in 1978.

4. Copyright 2006 Koren & Krishna ECE655/ByzGen.4 Byzantine Generals Problem  The Byzantine army is besieging a city, each division commanded by its own general. The commander-in-chief coordinates the divisions.  There could be traitors among these commanders.  The c-in-c has two alternatives:  Attack  Retreat  If the actions taken by each division is not consistent with that of the others, the army will be defeated.  Commands are sent in the form of two-party oral messages. Assume messages are not lost.

5. Copyright 2006 Koren & Krishna ECE655/ByzGen.5 Requirements for Success  The decision algorithm must satisfy the following conditions of interactive consistency:  IC1 All loyal generals must agree on the same plan of action.  IC2. If the c-in-c is loyal, the loyal generals must obey his order.  It is NOT an aim of the algorithm to identify the traitors.

6. Copyright 2006 Koren & Krishna ECE655/ByzGen.6 Traitors’ Aim  One or more of the generals (including the c-in-c) could be a traitor.  The aim of the traitors is to cause the Byzantine army to be defeated by violating the conditions for victory.

7. Copyright 2006 Koren & Krishna ECE655/ByzGen.7 Impossibility Result  Try to solve the problem with the c-in-c and two divisional generals, A and B.  Case 1. The c-in-c is a traitor: tells A to attack and B to retreat.  Case 2. The c-in-c is loyal but A is a traitor. He tells both A and B to attack; however, A tells B that his orders from the c-in-c are to retreat. What should B do?  Conclusion: The problem cannot be solved for a 3- node system with one Byzantine failure.

8. Copyright 2006 Koren & Krishna ECE655/ByzGen.8 Byzantine Generals Algorithm  General Approach:  The c-in-c sends his order to each of the divisional generals.  Each divisional general recursively uses the Byzantine Generals algorithm to disseminate to his colleagues the order he received from the c-in-c.  The key algorithm parameter is m, the maximum number of traitors to be allowed for.

9. Copyright 2006 Koren & Krishna ECE655/ByzGen.9 Algorithm OM(0)  Algorithm OM(0)  The c-in-c sends his order to each divisional general.  Each divisional general obeys the order he receives from the c-in-c.

10. Copyright 2006 Koren & Krishna ECE655/ByzGen.10 Algorithm OM(m)  Step 1. The c-in-c sends his order to each divisional general.  Step 2. Each divisional general uses OM(m-1) to disseminate the order he got from the c-in-c. At the end of this step, each divisional general has a vector containing: (a) The order he received from the c-in-c and (b) The order disseminated by every other divisional general.  Step 3. Each divisional general follows the majority decision from the vector obtained in Step 2.

11. Copyright 2006 Koren & Krishna ECE655/ByzGen.11 Claim  If the total number of generals is N>=3m+1, running OM(m) will ensure that the interactive consistency conditions are satisfied.  If the total number of generals is N<3m+1, no algorithm exists that can ensure the interactive consistency conditions are satisfied for this fault model.

12. Copyright 2006 Koren & Krishna ECE655/ByzGen.12 Proof of Correctness  Induction proof: induction basis of m=0 is obvious.  Suppose the result holds for up to m=M. Consider what happens if m=M+1. N>=3m+1 is a condition.  Case 1. The c-in-c is a traitor: There are up to m-1 traitors among the N-1 divisional generals. Now, use the induction hypothesis.  Case 2. The c-in-c is loyal: The c-in-c sent a consistent order to everyone. There are up to m traitors among the N-1 divisional generals.

13. Copyright 2006 Koren & Krishna ECE655/ByzGen.13 Modification: Signed Messages  Orders are sent by means of signed messages.  Assumptions about signed messages:  A loyal general’s signature cannot be forged.  Anyone can authenticate a general’s signature.

14. Copyright 2006 Koren & Krishna ECE655/ByzGen.14 Signed Messages Algorithm  Step 1. The c-in-c sends a signed order to the divisional generals.  Step 2. When a divisional general receives an order, he  Adds it to a vector of copies of this order, V.  If this order has fewer than m distinct signatures on it, he then »Adds his own signature to the order »Sends the order (augmented with his signature) to every divisional general who has not signed it.  Step 3. After all generals have been heard from (or a timeout has passed), decide on course of action. [If the c-in-c is unmasked as a traitor, use a pre-agreed default action.]