All-to-All Gradecast using Coding with Byzantine Failures

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1 All-to-All Gradecast using Coding with Byzantine Failures John Bridgman Vijay Garg Parallel and Distributed Systems Lab (PDSL) at The University of Texas at Austin Presented at SSS 2012 October 4, All-to-All Gradecast using Coding with Byzantine Failures 1/28

2 Motivating Scenario Want to compute some global function For example, the average of the values at each process No faults: everyone transmit and compute No problems All-to-All Gradecast using Coding with Byzantine Failures 2/28

3 With Faulty Processes n processes At most t faulty processes johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 3/28

4 Example 64 A 73 B 32 C 100 F 20 E XX D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

5 Example 64 A B C F E D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

6 Example A B C F 32 E 32 D 32 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

7 Example A 93 B 0 C 8 F 12 E 1 D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

8 Example (64,73,32,0,20,100) (64,73,32,93,20,100) A B (64,73,32,8,20,100) C (64,73,32,12,20,100) F (64,73,32,1,20,100) E D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

9 Example (64,73,32,0,20,100) (64,73,32,93,20,100) A B (64,73,32,8,20,100) C (64,73,32,12,20,100) F (64,73,32,1,20,100) E D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

10 Example A B C (64,73,32,1,20,100) (64,73,32,1,20,100) (64,73,32,1,20,100) F (64,73,32,1,20,100) E D (64,73,32,1,20,100) johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

11 Example A B C F E D johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

12 Example (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (65,23,33, 5,23,110) (64,73,32, 1,20,100) (64,73,32,12,20,100) (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (35,33,31, 9,13, 10) (64,73,32, 1,20,100) (64,73,32,12,20,100) F A (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (43,83,23,10,13,11) (64,73,32, 1,20,100) (64,73,32,12,20,100) E B (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (60,30,39,10,53,60) (64,73,32, 1,20,100) (64,73,32,12,20,100) D C (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (61,53,39, 9,20,100) (64,73,32, 1,20,100) (64,73,32,12,20,100) johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

13 Example (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (65,23,33, 5,23,110) (64,73,32, 1,20,100) (64,73,32,12,20,100) (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (35,33,31, 9,13, 10) (64,73,32, 1,20,100) (64,73,32,12,20,100) F A (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (43,83,23,10,13,11) (64,73,32, 1,20,100) (64,73,32,12,20,100) E B (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (60,30,39,10,53,60) (64,73,32, 1,20,100) (64,73,32,12,20,100) D C (64,73,32,93,20,100) (64,73,32, 0,20,100) (64,73,32, 8,20,100) (61,53,39, 9,20,100) (64,73,32, 1,20,100) (64,73,32,12,20,100) johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 4/28

14 Complexity 2n 2 messages mn 2 + mn 3 message bits O(mn 3 ) Can we do better when t << n? n : number of nodes m : message size t : maximum faulty nodes johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 5/28

15 Error Correcting Codes Send message over some channel with probability p to change symbols 1-p 1 1 p Alice p Bob p Alice wants to send message M Pick a code based on p Encode M to get S Send S Bob decodes S to get M johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 6/28

16 Systematic Codes Called systematic if the first part of S is M Encoding gives: (M) (M, parity) There are many such codes Example: Reed-Solomon codes [Reed and Solomon 1960] johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 7/28

17 How is this useful here? How is this useful here? Second broadcast sends redundant information At most t of the values different between processes Faulty channel indistinguishable from faulty process johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 8/28

18 Method Alice Bob [241, 86, 35, 35] [241, 86, 35, 40] All-to-All Gradecast using Coding with Byzantine Failures 9/28

19 Method Alice Bob [241, 86, 35, 35] [39, 78] [241, 86, 35, 40] Encode Reed-Solomon code over Galois field size 256: (35x x x x 254 )%( x + x 2 ) = x johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 9/28

20 Method Alice [39, 78] Bob [241, 86, 35, 35] [241, 86, 35, 40] All-to-All Gradecast using Coding with Byzantine Failures 9/28

21 Method Alice Bob [241, 86, 35, 35] [241, 86, 35, 40] [241, 86, 35, 40][39, 78] All-to-All Gradecast using Coding with Byzantine Failures 9/28

22 Method Alice Bob [241, 86, 35, 35] [241, 86, 35, 40] [241, 86, 35, 40][39, 78] Decode [241, 86, 35, 35] All-to-All Gradecast using Coding with Byzantine Failures 9/28

23 Basic Method At most t differences between processes Encode to tolerate t corruptions Only send parity Reduces message size johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 10/28

24 Example XX 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

25 Example XX 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

26 Example [241, 86, 35, 35] [241, 86, 35, 35] 1 2 [241, 86, 35, 40] 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

27 Example [241, 86, 35, 35] [39, 78] [241, 86, 35, 35] [39, 78] 1 2 [241, 86, 35, 40] [82, 30] 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

28 Example [39, 78] [39, 78] 1 2 [82, 30] 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

29 Example [241, 86, 35, 35] [39, 78] [39, 78] [82, 30] [22, 77] 1 2 [241, 86, 35, 35] [39, 78] [39, 78] [82, 30] [0, 136] [241, 86, 35, 40] [39, 78] [39, 78] [82, 30] 3 [121, 159] 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

30 Example [241, 86, 35, 35] [241, 86, 35, 35] [241, 86, 35, 40] [241, 49, 35, 35] 1 2 [241, 86, 35, 35] [241, 86, 35, 35] [241, 86, 35, 40] [241, 86, 129, 35] [241, 86, 35, 35] [241, 86, 35, 35] [241, 86, 35, 40] [157, 86, 35, 40] 3 4 johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 11/28

31 Method Application: Gradecast Gradecast is a broadcast algorithm that can tolerate Byzantine faults [Feldman, Micali 1988] Requires t < n/3 When P i gradecasts a value, every process P j receives v j [i] and a confidence level confidence j [i] {0, 1, 2} johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 12/28

32 Gradecast Properties Confidence is greater than zero implies same value P i, P j non-faulty and P k : confidence j [k] > 0 and confidence i [k] > 0 = value j [k] = value i [k] Process v i confidence i value i P 1 20 (2, 2, 2, 1) (20, 38, 10, 5) P 2 38 (2, 2, 2, 1) (20, 38, 10, 5) P 3 10 (2, 2, 2, 0) (20, 38, 10, ) P 4 XX XX XX johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 13/28

33 Gradecast Properties Confidence differs by at most one P i, P j non-faulty and P k : confidence i [k] confidence j [k] 1 Process v i confidence i value i P 1 20 (2, 2, 2, 1) (20, 38, 10, 5) P 2 38 (2, 2, 2, 1) (20, 38, 10, 5) P 3 10 (2, 2, 2, 0) (20, 38, 10, ) P 4 XX XX XX johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 14/28

34 Gradecast Properties Non-faulty source gets maximum confidence If P k is non-faulty, P i non-faulty: confidence i [k] = 2 and value i [k] = P k s initial value Process v i confidence i value i P 1 20 (2, 2, 2, 1) (20, 38, 10, 5) P 2 38 (2, 2, 2, 1) (20, 38, 10, 5) P 3 10 (2, 2, 2, 0) (20, 38, 10, ) P 4 XX XX XX johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 15/28

35 Algorithm Setup P i :: Inputs: v i : Input value for P i Common knowledge: n : The number of processes t : Maximum number of faulty processes Variables: V i [1..n] : Vector received in Step 2, initially Vecc i [1..2t + 1] : error correction vector for V i X i [1..n][1..n] : Matrix of decoded values in Step 3 Y i [1..n] : Vector of values computed in Step 3 Yecc i [1..2t + 1] : Error correction vector for Y i Z i [1..n][1..n] : Matrix of decoded values in Step 4 value i [1..n] : Vector of output values confidence i [1..n] : Vector of confidence levels johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 16/28

36 Algorithm Step One Though Three // Step 1: Initial Broadcast for j : 1 to n do P i.send(p j, v i ); end // Step 2: Receive, Encode, Broadcast encoding for j : 1 to n do V i [j] = P i.receive(p j ); end Vecc i = encode(v i ); for j : 1 to n do P i.send(p j, Vecc i ); end // Step 3: Receive encoding and process, encode result // and broadcast encoding for j : 1 to n do X i [j] = decode(v i, P i.receive(p j )); end j let Y i [j] = x if x s.t. {k : X i [k][j] = x} n t otherwise Y i [j] = Yecc i = encode(y i ); for j : 1 to n do P i.send(p j, Yecc i ); end johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 17/28

37 Algorithm Step Four // Step 4: Receive encoding, decode and compute final value for j : 1 to n do Z i [j] = decode(y i, P i.receive(p j )); end for j : 1 to n do if max x {k : Z i [k][j] = x} 2t + 1 then value i [j] = arg max x {k : Z i [k][j] = x} ; confidence i [j] = 2; elseif max x {k : Z i [k][j] = x} > t then value i [j] = arg max x {k : Z i [k][j] = x} ; confidence i [j] = 1; else value i [j] = ; confidence i [j] = 0; end end Output value i and confidence i. johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 18/28

38 Property (1) Theorem All non-faulty processes with positive confidence about process k have identical value[k]. Formally, P i, P j G, k : confidence i [k] > 0 confidence j [k] > 0 implies value i [k] = value j [k]. johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 19/28

39 Property (2) Theorem For any two non-faulty processes, the difference in their confidence levels for any process P k can differ by at most 1. Formally, P i, P j G, k : confidence i [k] confidence j [k] 1. johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 20/28

40 Property (3) Theorem If P k is non-faulty, then, all non-faulty processes P i have the value sent by process P k and their confidence level on this value is 2. Formally, P i, P k G : (confidence i [k] = 2) (value i [k] = v k ). johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 21/28

41 Gradecast Application: Byzantine Agreement Simple Byzantine Agreement algorithm by [Ben-Or, Dolev, Hoch 2010] that uses gradecast with O(mn 3 ) message bit complexity Can use our version of gradecast to get O(mtn 2 ) message bit complexity n : number of nodes m : message size t : maximum faulty nodes johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 22/28

42 Gradecast Application: Approximate Agreement [Ben-Or, Dolev, Hoch 2010] also give an approximate agreement algorithm using gradecast Can use our modified algorithm O(mkn 3 ) to O(mktn 2 ) reduction in message bit complexity n : number of nodes m : message size t : maximum faulty nodes k : rounds in approximate agreement johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 23/28

43 Related Work [Liang and Vaidya 2011] O(mn) bits for broadcast if m = Ω(n 6 ) otherwise O(nm + n 4 m 1/2 + n 6 ) [Friedman, Mostéfaoui, Rajsbaum and Raynal 2007] A mapping from a distributed agreement problem to a coding problem johnfbiii@utexas.edu, garg@ece.utexas.edu All-to-All Gradecast using Coding with Byzantine Failures 24/28

44 What about crash faults? Processes can only crash Translates to erasures Needs one half the parity length All-to-All Gradecast using Coding with Byzantine Failures 25/28

45 Conclusions Used Forward Error Correcting Codes to reduce message bit complexity of fault tolerant broadcast All-to-All Gradecast using Coding with Byzantine Failures 26/28

46 Future Work Apply to other distributed algorithms like simulated authentication [Srikanth and Toueg 1987] and interactive consistency [Hélary, Hurfin, Mostéfaoui, Raynal, and Tronel 2000] Derive lower bounds using coding theory results All-to-All Gradecast using Coding with Byzantine Failures 27/28

47 Thank you All-to-All Gradecast using Coding with Byzantine Failures 28/28

Deterministic Consensus Algorithm with Linear Per-Bit Complexity

Deterministic Consensus Algorithm with Linear Per-Bit Complexity Guanfeng Liang and Nitin Vaidya Department of Electrical and Computer Engineering, and Coordinated Science Laboratory University of Illinois