Byzantine Agreement. Gábor Mészáros. CEU Budapest, Hungary

Transcription

1 CEU Budapest, Hungary

2 1453 AD, Byzantium

3 Distibuted Systems Communication System Model

4 Distibuted Systems Communication System Model G = (V, E) simple graph

5 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines)

6 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels

7 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism

8 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism Different Attributes - Different Fields of Interest

9 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism Different Attributes - Different Fields of Interest Cryptography

10 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism Different Attributes - Different Fields of Interest Cryptography Data Compression

11 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism Different Attributes - Different Fields of Interest Cryptography Data Compression Distributed Computing

12 Distibuted Systems Communication System Model G = (V, E) simple graph V : nodes - participants (finite state machines) E: edges - communication channels Description of the communication mechanism Different Attributes - Different Fields of Interest Cryptography Data Compression Distributed Computing...

13 Byzantine Generals Problem Attributes

14 Byzantine Generals Problem Attributes Synchronous

15 Byzantine Generals Problem Attributes Synchronous Reliable

16 Byzantine Generals Problem Attributes Synchronous Reliable Authenticated

17 Byzantine Generals Problem Attributes Synchronous Reliable Authenticated Point-to-Point

18 Byzantine Generals Problem Attributes Synchronous Reliable Authenticated Point-to-Point Presence of faulty participants ("traitors") which can behave arbitrarily ("Byzantine failures").

19 Byzantine Generals Problem Attributes Synchronous Goals Reliable Authenticated Point-to-Point Presence of faulty participants ("traitors") which can behave arbitrarily ("Byzantine failures"). Given the set of initial assessments x i {0, 1} of each G i L V (G) ("loyal generals") calculate decisions d i {0, 1} satisfying:

20 Byzantine Generals Problem Attributes Synchronous Goals Reliable Authenticated Point-to-Point Presence of faulty participants ("traitors") which can behave arbitrarily ("Byzantine failures"). Given the set of initial assessments x i {0, 1} of each G i L V (G) ("loyal generals") calculate decisions d i {0, 1} satisfying: Termination: each process terminates in finitely many steps

21 Byzantine Generals Problem Attributes Synchronous Goals Reliable Authenticated Point-to-Point Presence of faulty participants ("traitors") which can behave arbitrarily ("Byzantine failures"). Given the set of initial assessments x i {0, 1} of each G i L V (G) ("loyal generals") calculate decisions d i {0, 1} satisfying: Termination: each process terminates in finitely many steps Agreement: d i = d j G i, G j L (the set of "loyal generals")

22 Byzantine Generals Problem Attributes Synchronous Goals Reliable Authenticated Point-to-Point Presence of faulty participants ("traitors") which can behave arbitrarily ("Byzantine failures"). Given the set of initial assessments x i {0, 1} of each G i L V (G) ("loyal generals") calculate decisions d i {0, 1} satisfying: Termination: each process terminates in finitely many steps Agreement: d i = d j G i, G j L (the set of "loyal generals") Nontriviality: x i = c {0, 1} G i L d i = c

23 Byzantine Generals Problem Definition A protocol P is t-resilient if it tolerates byzantine failure of at most t faulty participants.

24 Byzantine Generals Problem Definition A protocol P is t-resilient if it tolerates byzantine failure of at most t faulty participants. Question How many byzantine failures can a network tolerate?

25 Byzantine Generals Problem Example The "Simple Majority" strategy is not 1-resilient.

26 Byzantine Generals Problem Theorem (Lamport, Pease, Shostak, 1980) There exists t-resilient protocol t < n 3.

27 Byzantine Generals Problem Theorem (Lamport, Pease, Shostak, 1980) There exists t-resilient protocol t < n 3. Lemma No 1-resilient protocol P exists on K 3.

28 No 1-resilient P in K 3 Proof

29 No 1-resilient P in K 3 Proof

30 No 1-resilient P in K 3 Proof

31 t-resilient t < n 3 Corollary of the Lemma - Reduction A t n 3 -resilient protocol is 1-resilient in K 3.

32 t-resilient t < n 3 Corollary of the Lemma - Reduction A t n 3 -resilient protocol is 1-resilient in K 3. Constructions for t < n 3 (sketch)

33 t-resilient t < n 3 Corollary of the Lemma - Reduction A t n 3 -resilient protocol is 1-resilient in K 3. Constructions for t < n 3 (sketch) 1 Exponential data trees - "x told me, that y told him, that..." - fill() and resolve() -not efficient

34 t-resilient t < n 3 Corollary of the Lemma - Reduction A t n 3 -resilient protocol is 1-resilient in K 3. Constructions for t < n 3 (sketch) 1 Exponential data trees - "x told me, that y told him, that..." - fill() and resolve() -not efficient 2 Efficient (polinomial) Broadcast- firefly effect, echoes...

35 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k

36 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes

37 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous

38 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous Reliable

39 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous Reliable Authenticated

40 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous Reliable Authenticated Not necessarily Point-to-Point (communication on edges only)

41 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous Reliable Authenticated Not necessarily Point-to-Point (communication on edges only) Presence of faulty participants

42 Generalized Byzatine Generals Problem I. - Graphs Communication Model G = (V, E) simple (not necessarily complete) graph with connectivity number k(g) := k Attributes Synchronous Reliable Authenticated Not necessarily Point-to-Point (communication on edges only) Presence of faulty participants Goal Unanimity between the non-faulty processors

43 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient t < n 3 and t < k 2.

44 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient t < n 3 and t < k 2. Theorem (Kumar,2002) Given S 2 V (G) set of corruptible subsets in G = (V, E) unanimity is attainable

45 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient t < n 3 and t < k 2. Theorem (Kumar,2002) Given S 2 V (G) set of corruptible subsets in G = (V, E) unanimity is attainable no union S 1 S 2 of any pair S 1, S 2 S contains a cut of G,

46 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient t < n 3 and t < k 2. Theorem (Kumar,2002) Given S 2 V (G) set of corruptible subsets in G = (V, E) unanimity is attainable no union S 1 S 2 of any pair S 1, S 2 S contains a cut of G, no union S 1 S 2 S 3 of any triple S 1, S 2, S 3 S covers V (G).

47 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient iff t < n 3 and t < k 2.

48 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient iff t < n 3 and t < k 2. Proof (" ")

49 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient iff t < n 3 and t < k 2. Proof (" ") 1 For each G i, G j V (G), (G i G j ) E(G) fix disjoint paths P 1, P 2,..., P k between the nodes ("delivery channels").

50 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient iff t < n 3 and t < k 2. Proof (" ") 1 For each G i, G j V (G), (G i G j ) E(G) fix disjoint paths P 1, P 2,..., P k between the nodes ("delivery channels"). 2 Send messages from G i to G j via P 1, P 2,..., P k and consider majority of the 0-1 messages. t < k 2 guaranties reliability.

51 Generalized Byzantine Generals Problem I. - Graphs Theorem (Dolev, 1982) G = (V, E) is t-resilient iff t < n 3 and t < k 2. Proof (" ") 1 For each G i, G j V (G), (G i G j ) E(G) fix disjoint paths P 1, P 2,..., P k between the nodes ("delivery channels"). 2 Send messages from G i to G j via P 1, P 2,..., P k and consider majority of the 0-1 messages. t < k 2 guaranties reliability. 3 Emulate the solution of the original BA problem.

52 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph.

53 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes

54 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes Synchronous

55 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes Synchronous Reliable

56 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes Synchronous Reliable Authenticated

57 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes Synchronous Reliable Authenticated Broadcast on the edges

58 Generalized Byzantine Generals Problem II. - Hypergraphs Communication Model H = (V, E) hypergraph. Attributes Synchronous Reliable Authenticated Broadcast on the edges Presence of faulty participants

59 Generalized Byzantine Generals Problem II. - Hypergraphs Theorem (Fitzi, Maurer, 2000) H = (V, E) 3-uniform complete hypergraph is t-resilible n 2 t + 1.

60 Other Possible Generalizations Variants

61 Other Possible Generalizations Variants Asynchronous communication

62 Other Possible Generalizations Variants Asynchronous communication General Hypergraphs

63 Other Possible Generalizations Variants Asynchronous communication General Hypergraphs Corruptible subsets

64 Other Possible Generalizations Variants Asynchronous communication General Hypergraphs Corruptible subsets Random processes

65 Other Possible Generalizations Variants Asynchronous communication General Hypergraphs Corruptible subsets Random processes...

66 THANK YOU!