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Set 10: Consensus with Byzantine Failures CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS CSCE 668 Fall 2011 Prof. Jennifer Welch 1 Consensus with Byzantine Failures 2 How many processors total are needed to solve consensus when f = 1 ? Suppose n = 2. If p0 has input 0 and p1 has 1, someone has to change, but not both. What if one processor is faulty? How can the other one know? Suppose n = 3. If p0 has input 0, p1 has input 1, and p2 is faulty, then a tie-breaker is needed, but p2 can act maliciously. Set 10: Consensus with Byzantine Failures CSCE 668 Processor Lower Bound for f = 1 3 Theorem (5.7): Any consensus algorithm for 1 Byzantine failure must have at least 4 processors. Proof: Suppose in contradiction there is a consensus algorithm A = (A,B,C) for 3 processors and 1 Byzantine failure. p1 A p0 B p2 C Set 10: Consensus with Byzantine Failures CSCE 668 Specifying Faulty Behavior 4 Consider a ring of 6 nonfaulty processors running components of A like this: A B C p1 p2 1 p0 1 0 p3 0 1 A 0 p5 p4 C B This execution probably doesn't solve consensus (it doesn't have to). But the processors do something -- this behavior is used to specify the behavior of faulty processors in executions of A in the triangle. Set 10: Consensus with Byzantine Failures CSCE 668 Getting a Contradiction 5 Let 0 be this execution: p1 act like p3 to p4 in p0 B p1 and p2 must decide 0 0 0 0 p2 C act like p0 to p5 in Set 10: Consensus with Byzantine Failures CSCE 668 Getting a Contradiction 6 Let 1 be this execution: p0 and p1 must decide 1 A p1 B act like p2 to p1 in 1 p0 1 1 p2 act like p5 to p0 in Set 10: Consensus with Byzantine Failures CSCE 668 The Contradiction 7 Let be this execution: What do p0 and p2 decide? act like p1 to p0 in A p0 1 p1 ? act like p4 to p5 in 0 p2 C view of p0 in = view of p0 in = view of p0 in 1 p0 decides 1 view of p2 in = view of p2 in = view of p2 in 0 p2 decides 0 Set 10: Consensus with Byzantine Failures CSCE 668 Views 8 β: A p1 p2 1 p1 ? 1 0 p3 p0 1 0 act like p1 to p0 in p0 C 1 : A B 0 p2 0 p5 p4 C B act like p4 to p5 in C A 1: A p0 p1 B act like p2 to p1 in 1 1 Set 10: Consensus with Byzantine Failures 1 p2 act like p5 to p0 in CSCE 668 Processor Lower Bound for Any f 9 Theorem: Any consensus algorithm for f Byzantine failures must have at least 3f+1 processors. Proof: Use a reduction to the 3:1 case. Suppose in contradiction there is an algorithm A for f > 1 failures and n = 3f total processors. Use A as a subroutine to construct an algorithm for 1 failure and 3 processors, a contradiction to theorem just proved. Set 10: Consensus with Byzantine Failures CSCE 668 The Reduction 10 Partition the n ≤ 3f processors into three sets, P0, P1, and P2, each of size at most f. In the n = 3 case, let p0 simulate P0 p1 simulate P1 p2 simulate P2 If one processor is faulty in the n = 3 system, then at most f processors are faulty in the simulated system. Thus the simulated system is correct. Let the processors in the n = 3 system decide the same as the simulated processors, and their decisions will also be correct. Set 10: Consensus with Byzantine Failures CSCE 668 Exponential Tree Algorithm 11 This algorithm uses f + 1 rounds (optimal) n = 3f + 1 processors (optimal) exponential size messages (sub-optimal) Each processor keeps a tree data structure in its local state Values are filled in the tree during the f + 1 rounds At the end, the values in the tree are used to calculate the decision. Set 10: Consensus with Byzantine Failures CSCE 668 Local Tree Data Structure 12 Each tree node is labeled with a sequence of unique processor indices. Root's label is empty sequence ; root has level 0 Root has n children, labeled 0 through n - 1 Child node labeled i has n - 1 children, labeled i : 0 through i : n-1 (skipping i : i) Node at level d labeled v has n - d children, labeled v : 0 through v : n-1 (skipping any index appearing in v) Nodes at level f + 1 are leaves. Set 10: Consensus with Byzantine Failures CSCE 668 Example of Local Tree 13 The tree when n = 4 and f = 1 : Set 10: Consensus with Byzantine Failures CSCE 668 Filling in the Tree Nodes 14 Initially store your input in the root (level 0) Round 1: send level 0 of your tree to all store value x received from each pj in tree node labeled j (level 1); use a default if necessary "pj told me that pj 's input was x" Round 2: send level 1 of your tree to all store value x received from each pj for each tree node k in tree node labeled k : j (level 2); use a default if necessary "pj told me that pk told pj that pk's input was x" Continue for f + 1 rounds Set 10: Consensus with Byzantine Failures CSCE 668 Calculating the Decision 15 In round f + 1, each processor uses the values in its tree to compute its decision. Recursively compute the "resolved" value for the root of the tree, resolve(), based on the "resolved" values for the other tree nodes: value in tree node labeled if it is a leaf resolve() = majority{resolve( ') : ' is a child of } otherwise (use a default if tied) Set 10: Consensus with Byzantine Failures CSCE 668 Example of Resolving Values 16 The tree when n = 4 and f = 1 : 0 0 0 0 (assuming 0 is the default) 0 1 0 0 1 0 1 1 1 1 Set 10: Consensus with Byzantine Failures 1 1 CSCE 668 0 Resolved Values are Consistent 17 Lemma (5.9): If pi and pj are nonfaulty, then pi 's resolved value for its tree node labeled ' j (what pj tells pi for node ') equals what pj stores in its node '. ' ' 'j original value = v resolved value = v part of pi's tree part of pj's tree Set 10: Consensus with Byzantine Failures CSCE 668 Resolved Values are Consistent 18 Proof Ideas: By induction on the height of the tree node. Uses inductive hypothesis to know that resolved values for children of the tree node corresponding to nonfaulty procs are consistent. Uses fact that n > 3f and fact that each tree node has at least n - f children to know that majority of children are nonfaulty. Set 10: Consensus with Byzantine Failures CSCE 668 Validity 19 Suppose all inputs are v. Nonfaulty proc. pi decides resolve(), which is the majority among resolve(j), 0 ≤ j ≤ n-1, based on pi 's tree. Since resolved values are consistent, resolve(j) (at pi) is value stored at the root of pj 's tree, which is pj 's input value if pj is nonfaulty. Since there are a majority of nonfaulty processors, pi decides v. Set 10: Consensus with Byzantine Failures CSCE 668 Common Nodes 20 A tree node is common if all nonfaulty procs. compute the same value of resolve(). same resolved value part of pi's tree part of pj's tree Set 10: Consensus with Byzantine Failures CSCE 668 Common Frontiers 21 A tree node has a common frontier if every path from to a leaf contains a common node. pink means common Set 10: Consensus with Byzantine Failures CSCE 668 Common Nodes and Frontiers 22 Lemma (5.10): If has a common frontier, then is common. Proof Ideas: By induction on height of . Uses fact that resolve is defined using majority. Set 10: Consensus with Byzantine Failures CSCE 668 Agreement 23 The nodes on each path from a child of the root to a leaf correspond to f + 1 different procs. Since there are at most f faulty processors, at least one such node corresponds to a nonfaulty processor. This node is common (by the lemma about the consistency of resolved values). Thus the root has a common frontier. Thus the root is common (by preceding lemma). Set 10: Consensus with Byzantine Failures CSCE 668 Complexity 24 Exponential tree algorithm uses n > 3f processors f + 1 rounds exponential size messages: each msg in round r contains n(n-1)(n-2)…(n-(r-2)) values When r = f + 1, this is exponential if f is more than constant relative to n Set 10: Consensus with Byzantine Failures CSCE 668 A Polynomial Algorithm 25 We can reduce the message size to polynomial with a simple algorithm The number of processors increases to n > 4f The number of rounds increases to 2(f + 1) Uses f + 1 phases, each taking two rounds Set 10: Consensus with Byzantine Failures CSCE 668 Phase King Algorithm 26 Code for each processor pi: pref := my input first round of phase k, 1 ≤ k ≤ f+1: send pref to all receive prefs of others let maj be value that occurs > n/2 times (0 if none) let mult be number of times maj occurs second round of phase k: if i = k then send maj to all // I am the phase king receive tie-breaker from pk (0 if none) if mult > n/2 + f then pref := maj else pref := tie-breaker if k = f + 1 then decide pref Set 10: Consensus with Byzantine Failures CSCE 668 Unanimous Phase Lemma 27 Lemma (5.12): If all nonfaulty processors prefer v at start of phase k, then all do at end of phase k. Proof: Each nonfaulty proc. receives at least n - f preferences for v in first round of phase k Since n > 4f, it follows that n - f > n/2 + f So each nonfaulty proc. still prefers v. Set 10: Consensus with Byzantine Failures CSCE 668 Phase King Validity 28 Unanimous phase lemma implies validity: Suppose all procs have input v. Then at start of phase 1, all nf procs prefer v. So at end of phase 1, all nf procs prefer v. So at start of phase 2, all nf procs prefer v. So at end of phase 2, all nf procs prefer v. … At end of phase f + 1, all nf procs prefer v and decide v. Set 10: Consensus with Byzantine Failures CSCE 668 Nonfaulty King Lemma 29 Lemma (5.13): If king of phase k is nonfaulty, then all nonfaulty procs have same preference at end of phase k. Proof: Let pi and pj be nonfaulty. Case 1: pi and pj both use pk 's tie-breaker. Since pk is nonfaulty, they both have same preference. Set 10: Consensus with Byzantine Failures CSCE 668 Nonfaulty King Lemma 30 Case 2: pi uses its majority value v and pj uses king's tie-breaker. Then pi receives more than n/2 + f preferences for v So pk receives more than n/2 preferences for v So pk 's tie-breaker is v Set 10: Consensus with Byzantine Failures CSCE 668 Nonfaulty King Lemma 31 Case 3: pi and pj both use their own majority values. Suppose pi 's majority value is v Then pi receives more than n/2 + f preferences for v So pj receives more than n/2 preferences for v So pj 's majority value is also v Set 10: Consensus with Byzantine Failures CSCE 668 Phase King Agreement 32 Use previous two lemmas to prove agreement: Since there are f + 1 phases, at least one has a nonfaulty king. Nonfaulty King Lemma implies at the end of that phase, all nonfaulty processors have same preference Unanimous Phase Lemma implies that from that phase onward, all nonfaulty processors have same preference Thus all nonfaulty decisions are same. Set 10: Consensus with Byzantine Failures CSCE 668 Complexities of Phase King 33 number of processors n > 4f 2(f + 1) rounds O(n2f) messages, each of size log|V| Set 10: Consensus with Byzantine Failures CSCE 668