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Error-Free Multi-Valued Consensus with Byzantine Failures

Error-Free Multi-Valued Consensus with Byzantine Failures. Guanfeng Liang Electrical and Computer E ngineering University of Illinois at Urbana-Champaign Joint work with Nitin Vaidya. Multi-Valued Byzantine Consensus.

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Error-Free Multi-Valued Consensus with Byzantine Failures

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  1. Error-Free Multi-Valued Consensus with Byzantine Failures Guanfeng Liang Electrical and Computer Engineering University of Illinois at Urbana-Champaign Joint work with NitinVaidya

  2. Multi-Valued Byzantine Consensus N nodes each given an L-bit input value, want to compute their L-bit outputs such that • Synchronous • Fault-free nodes must agree • If all fault-free nodes have identical inputs values  agree on this value • Up to f < N/3 Byzantine failures

  3. Related Work • Consensus: • Ω(N2) for 1-bit agreement[Dolev and Reischuk, J.ACM’85] • O(N1.5) for 1-bit with randomized [King and Saia, PODC’10] • O(NL) for large L with hashing [Fitziand Hirt, PODC’06] • Broadcast: • O(NL) for large L [Beerliova-Trubiniova and Hirt, TCC’08] Error-free May error May error Error-free

  4. Related Work • Multicast with Byzantine faults • System level diagnosis

  5. Our Work • Error-free consensus within 4x optimal • Can be improved to 2x optimal

  6. Overview of the Algorithm • Divide L-bit input into many generations of D bits • Consensus one generation at a time • Exchange information efficiently with coding • Identify a cliqueof Snodes that “trust” each other, and appear to have identical inputs: If not found, terminate with default output • If found: Try to agree with the inputs in the clique • Any misbehave will be detected, then update “trust”:If X did “bad” things to Y, Y will not trust X any more • Repeat for next generation • Memory of “trust” across generations

  7. Code used for info exchange (n,k) MDS (maximum distance separable) code • n: length; k: dimension • k data symbols  n coded symbols • Any k coded symbols  k data symbols • Any m ≥ k locations consist a(m,k) MDS code (also has dimension k)

  8. S - f 3 1 i … 2 n

  9. N 3 1 i … 2 n Encode with (N, S-f) MDS code

  10. 3 1 i … 2 n

  11. 1 1 3 0 1 i … 1 0 2 n Same as the local one? If not  inputs must be different

  12. 1 1 3 0 1 i … 1 0 2 n

  13. 1 1 1 1 3 0 1 1 i … 1 1 0 1 2 n

  14. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n Broadcast the 1-bit flags Find a “clique” of S nodes match with each other

  15. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n If not found: Good nodes have different inputs

  16. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n If clique of S nodes is found: Try to “agree” using packets from the clique

  17. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n At most f bad nodes  At least S - f good nodes in the clique  Good nodes share ≥S – fpackets identically

  18. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n Nodes in the clique: The code has dimension S - f  All have same input

  19. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n Nodes not in the clique: Either all have identical codeword of (S, S - f) Or someone is not a codeword

  20. 1 1 1 1 1 0 1 1 1 1 1 1 0 1 3 0 1 1 0 0 1 0 1 i … 1 1 1 0 1 1 1 0 1 0 0 0 0 1 2 n Either all decode to the input of good nodes in the cliqueOr someone can’t decode  misbehavior detected

  21. Overview of the Algorithm • Divide L-bit input into many generations of D bits • Consensus one generation at a time • Exchange information efficiently with coding • Identify a clique of S nodes that trust each other, and appear to have identical inputs: If not found, terminate with default output • If found: Try to agree with the inputs in the clique • Any misbehave will be detected, then update “trust”: If X did “bad” things to Y, Y will not trust X any more • Repeat for next generation • Memory of “trust” across generations

  22. Complexity of the Algorithm • Many generations without failure • Few expensive generations with failure • Total cost dominated by the failure-free generations • For large L, communication complexity is

  23. Property of the Algorithm • S = N – f : original consensus with • Stronger property satisfied when • S < N – f : If S fault-free nodes have same input output is input of a fault-free node • S > N/2 : If S fault-free nodes have same input output is the majority input

  24. Summary of Results • Error-free multi-valued Byzantine consensus with complexity < 3NL • Order optimal, 4x optimal • Same complexity for many consensus of small inputs, instead of one very long one • Can be improved to 1.5NL (2x optimal)

  25. Future Work & Latest Results • Is 1.5NL the best we can do? • Generalize to other network models: point-to-point, wireless, etc. • Point-to-Point network model: max # of bits transmit on each link per unit time independent with other links • Capacity of Byzantine agreement: max # of bits agreed per unit time • Achieve at least 1/2 of capacity using Random Linear Codes

  26. Thank you!

  27. Flow of the Algorithm • Fast generation (no failure) • Fast generation …… • Fast generation in which failure is detected • Expensive operation to learn new info about failure • Fast generation • Fast generation …… • Fast generation in which failure is detected • Expensive operation to learn new info about failure • Only fast generations hereon Failures identified after a small number of generations

  28. Failure models • Crash failure – fail by stopping (“do no harm”) • Byzantine failure – arbitrary, potentially harmful, behavior

  29. Known results • Need N ≥ 3f + 1 nodes to tolerate f failures • Need Ω(N2) messages

  30. 1-bit value • Each message at least 1 bit • O(N2) bits “communication complexity” to agree on just 1 bit value

  31. Larger values (L bits) • Upper bound: Agree on each bit separately  O(N2 L) bits communication complexity • Lower bound: Need Ω(N L) bits to agree on L bits

  32. Effort To Improve complexity • L = 1: O(N1.5) with randomized algorithm [King and Saia, PODC’10] • Large L: O(N L) with hashing[Fitzi and Hirt, PODC’06] Both probabilistically correct = Not error-free

  33. Modification • Try to agree on small pieces (D bits) our of L bits data in each “round” • If X misbehaves with Y in a given round, avoid using XY links in the next round (for next rounds D bits of data) • Repeat

  34. Algorithm structure • Fast round (as in the example)

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