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Agreement: Byzantine Generals

UNIVERSITY of WISCONSIN-MADISON Computer Sciences Department. Agreement: Byzantine Generals. CS 739 Distributed Systems. Andrea C. Arpaci-Dusseau. Paper: “The Byzantine Generals Problem”, by Lamport, Shostak, Pease, In ACM Transactions on Programming Languages and Systems , July 1982

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Agreement: Byzantine Generals

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  1. UNIVERSITY of WISCONSIN-MADISONComputer Sciences Department Agreement: Byzantine Generals CS 739Distributed Systems Andrea C. Arpaci-Dusseau • Paper: • “The Byzantine Generals Problem”, by Lamport, Shostak, Pease, In ACM Transactions on Programming Languages and Systems, July 1982 • why we need agreement • assumptions • algorithm steps through examples • Bigger Picture: How to handle malicious components • Play variant of Mafia/Werewolf

  2. C1 C2 C3 Motivation • Build reliable systems in presence of faulty components • Common approach: • Send request (or input) to some “f-tolerant” server • Have multiple (potentially faulty) components compute same function • Perform majority vote on outputs to get “right” result majority(v1,v2,v3) f faulty, f+1 good components ==> 2f+1 total

  3. What is a Byzantine Failure? • Three primary differences from Fail-Stop Failure • Component can produce arbitrary output • Fail-stop: produces correct output or none • Cannot always detect output is faulty • Fail-stop: can always detect that component has stopped • Components may work together maliciously • With fail-stop failures: How many components are needed to be f-tolerant??

  4. C1 C2 C3 Assumption for F-tolerant Servers • Good (non-faulty) components must use same input • Otherwise, can’t trust their output result either • For majority voting to work: • All non-faulty processors must use same input • If input is non-faulty, then all non-faulty processes use the value it provides • Must agree on value of input A B

  5. Byzantine Generals • Algorithm to achieve agreement among “loyal generals” (i.e., working components) given m “traitors” (i.e., faulty components) • Agreement such that: • All loyal generals decide on same plan(important even when input is faulty! Why?) • Small number of traitors cannot cause loyal generals to adopt “bad plan” • Terminology • Let v(i) be information communicated (I.e., input observed) by ith general • Each general combines values v(1)...v(n) to form plan

  6. Agreement Conditions • Rephrase agreement conditions: • Loyal generals decide on same plan ifAll loyal generals use same method for combining information (and see same inputs) • Small number of traitors can’t hurt loyal generals ifUse robust function for decision, such as majority function of values v(1)...v(n)

  7. Key Step: Agree on inputs • Generals communicate v(i) values to one another: 1) Every loyal general must obtain same v(1)..v(n) 1’) Any two loyal generals use same value of v(i) • Traitor i will try to trick loyal generals into using different v(i)’s 2) If ith general is loyal, then the value he sends must be used by every other general as v(i) • How can each general send his value to n-1 others? • A commanding general must send an order (order: Use v(i) as my value) to his n-1 lieutenants such that: IC1) All loyal lieutenants obey same order IC2) If commanding general is loyal, every loyal lieutenant obeys the order he sends • Interactive Consistency conditions

  8. commander attack L1 L2 retreat Impossibility Result • With only 3 generals, no solution can work with even 1 traitor (given oral messages) What should L1 do? Is commander or L2 the traitor???

  9. Option 1: Loyal Commander commander attack attack L1 L2 retreat What must L1 do? By IC2: L1 must obey commander and attack

  10. Option 2: Loyal L2 commander retreat attack L1 L2 retreat What must L1 do? By IC1: L1 and L2 must obey same order --> L1 must retreat Problem: L1 can’t distinguish between 2 scenarios

  11. General Impossibility Result • No solution with fewer than 3m+1 generals can cope with m traitors • < see paper for details >

  12. Oral Messages • Assumptions • A1) Every message sent is delivered correctly • What if it is not? • A2) Receiver knows who sent message • What scenarios is this true for? • A3) Absence of message can be detected • How can this be done?

  13. Oral Message Algorithm • OM(m), m>0 • Commander sends his value to every lieutenant • For each i, let vi be value Lieutenant i receives from commander; act as commander for OM(m-1) and send vi to n-2 other lieutenants • For each i and each j not i, let vj be value Lieut i received from Lieut j. Lieut i computes majority(v1,...,vn-1) • OM(0) • Commander sends his value to every lieutenant

  14. C A A A L2 L3 L1 A R A R Example: Bad Lieutenant • Scenario: m=1, n=4, traitor = L3 OM(1): C OM(0):??? L2 L3 L1 Decision?? L1 = m (A, A, R); L2 = m (A, A, R); Both attack!

  15. Example: Bad Commander • Scenario: m=1, n=4, traitor = C C A A OM(1): R L2 L3 L1 A OM(0):??? L2 R L3 L1 A A R A Decision?? L1=m(A, R, A); L2=m(A, R, A); L3=m(A,R,A); Attack!

  16. L3 L3 L6 L6 L1 L1 L2 L2 L4 L4 L5 L5 A R R A A A Bigger Example: Bad Lieutenants • Scenario: m=2, n=3m+1=7, traitors=L5, L6 C A A A A A A Messages? m(A,A,A,A,R,R) ==> All loyal lieutenants attack! Decision???

  17. A x A R R A Messages? L3 L6 L1 L2 L4 L5 R A A R A A,R,A,R,A Bigger Example: Bad Commander+ • Scenario: m=2, n=7, traitors=C, L6 C L3 L6 L1 L2 L4 L5 Decision???

  18. Decision with Bad Commander+ • L1: m(A,R,A,R,A,A) ==> Attack • L2: m(A,R,A,R,A,R) ==> Retreat • L3: m(A,R,A,R,A,A) ==> Attack • L4: m(A,R,A,R,A,R) ==> Retreat • L5: m(A,R,A,R,A,A) ==> Attack • Problem: All loyal lieutenants do NOT choose same action

  19. Next Step of Algorithm • Verify that lieutenants tell each other the same thing • Requires rounds = m+1 • OM(0): Msg from Lieut i of form: “L0 said v0, L1 said v1, etc...” • What messages does L1 receive in this example? • OM(2): A • OM(1): 2R, 3A, 4R, 5A, 6A (doesn’t know 6 is traitor) • OM(0): 2{ 3A, 4R, 5A, 6R} • 3{2R, 4R, 5A, 6A} • 4{2R, 3A, 5A, 6R} • 5{2R, 3A, 4R, 6A} • 6{ total confusion } • All see same messages in OM(0) from L1,2,3,4, and 5 • m(A,R,A,R,A,-) ==> All attack

  20. Signed Messages • Problem: Traitors can lie about what others said; how can we remove that ability? • New assumption: Signed messages (Cryptography) • A4) a. Loyal general’s signature cannot be forged and contents cannot be altered • b. Anyone can verify authenticity of signature • Simplifies problem: • When lieutenant i passes on signed message from j, receiver knows that i did not lie about what j said • Lieutenants cannot do any harm alone (cannot forge loyal general’s orders) • Only have to check for traitor commander • With cryptographic primitives, can implement Byzantine Agreement with m+2 nodes, using SM(m)

  21. Signed Messages Algorithm: SM(m) • Commander signs v and sends to all as (v:0) • Each lieut 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)

  22. A:0 R:0 What next? A:0:L1 L2 L1 R:0:L2 SM(1) Example: Bad Commander • Scenario: m=1, n=m+2=3, bad commander C L2 L1 V1={A,R} V2={R,A} Both L1 and L2 can trust orders are from C Both apply same decision to {A,R}

  23. A:0:L1 A:0:L3 R:0:L3:L1 A:0:L2 L2 L1 L2 L3 L1 R:0:L3 SM(2): Bad Commander+ • Scenario: m=2, n=m+2=4, bad commander and L3 C Goal? L1 and L2 must make same decision A:0 x A:0 L2 L3 L1 V1 = V2 = {A,R} ==> Same decision

  24. Other Variations • How to handle missing communication paths • < see paper for details>

  25. Assumptions • A1) Every message sent by nonfaulty processor is delivered correctly • Network failure ==> processor failure • Handle as less connectivity in graph • A2) Processor can determine sender of message • Communication is over fixed, dedicated lines • Switched network??? • A3) Absence of message can be detected • Fixed max time to send message + synchronized clocks ==> If msg not received in fixed time, use default • A4) Processors sign msgs such that nonfaulty signatures cannot be forged • Use randomizing function or cryptography to make liklihood of forgery very small

  26. Importance of Assumptions • “Separating Agreement from Execution for Byzantine Fault Tolerant Services” - SOSP’03 • Goal: Reduce replication costs • 3f+1 agreement replicas • 2g+1 execution replicas • Costly part to replicate • Often uses different software versions • Potentially long running time • To provide A2, protocol assumes cryptographic primitives, such that one can be sure “i said v” in switched environment • What is the problem??

  27. Conclusions • Problem: To implement a fault-tolerant service with coordinated replicas, must 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

  28. Byzantine Werewolves • Werewolf/Mafia: Psychological party game • Two groups: Werewolves and villagers • Multiple rounds of night and day • Night: Werewolves kill a villager • Day: All vote on who to kill…hopefully a werewolf • Werewolves trick villages into bad decisions (killing one of their own) • Werewolves lie, act in collision • Werewolves have more information than villagers (I.e., who is a werewolf) • Traditional: Only 1 village “seer” has any info • My variant: Every villager can ask one question per round • Traditional: More psychological…

  29. Byzantine-Werewolf Game Rules • Everyone secretly assigned as werewolf or villager • 3 werewolves, rest are “seeing” villagers • I am moderator • Night round: • “Close your eyes”; make noises to hide activity • “Werewolves, open your eyes”: 3 can see who is who • “Werewolves, pick someone to kill” (Not first round) • Silently agree on villager to kill by pointing • “Werewolves, close your eyes” • For all: “NAME, open your eyes” “Pick someone to ask about” • Useless for Werewolves, but hides their identity… • Point to another player • Moderator signs thumbs up for werewolf, down for villager • “NAME, close your eyes”

  30. Rules: Day Time • Day Time: “Everyone open your eyes; its daytime” • “NAME, you have been killed by the werewolves!” • They are now out of the game • Agreement time: Everyone talks and votes on who should be “decommissioned” • Villagers try to decommission werewolf • Werewolves try to trick villagers with bad info • Pairwise communication • Variant: Signed messages; leave a note with everyone telling them what you know; can show this note to others • When you’ve made up your mind, tell moderator • Moderator: Uses majority voting to determine who is decommissioned “Okay, NAME is dead” • Person is out of game (can’t talk anymore) and shows card • Repeat cycle until All werewolves dead OR werewolves >= villagers

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