Bit-efficient consensus

1
Bit-efficient consensus

• Michal Strojnowski

2
The Byzantine Generals Problem

• n entities
• Each has its value
• They communicate using only point-to-point
  messages
• The goal is to choose one of this values, and
  make it known to everyone

3
The Byzantine Generals Problem

• n entities
• Each has its value
• They communicate using only point-to-point
  messages
• The goal is to choose one of this values, and
  make it known to everyone
• Simplest solution one leader sends its value to
  everyone

4
Traitors

• In the Byzantine Generals Problem, we assumed any
  malicious behavior of traitors
• In this case, no protocol is resilient to n/3
  such aberrations

5
Traitors

• In the Byzantine Generals Problem, we assumed any
  malicious behavior of traitors
• In this case, no protocol is resilient to n/3
  such aberrations

6
Traitors

• In the Byzantine Generals Problem, we assumed any
  malicious behavior of traitors
• In this case, no protocol is resilient to n/3
  such aberrations

7
Crashes

• Assume that only possible violation of protocol
  is a crash (i.e. death of general)
• Consensus is possible for any number of crashes,
  but still not easy to achieve
• t crashes require t1 rounds of communication
• (in asynchronous case problem is unsolvable)
• Most protocols use big number of lengthy messages

8
Lowering the number of messages

• Define a communication graph, and send messages
  through it edges.
• Ensure that there exists a big, strongly
  connected subset of correct nodes
• Devise a protocol in which the final value is
  easy to obtain

9
Communicators

• Degree of every node is d
• In each subset of 4n/d nodes, there exists a
  subset in which each node has k neighbors
• Each such subset has at least n/4d vertices
• Each two sets of n/4d vertices are connected

10
Family of communicators
degree d2 diameter (log n)/2
degree d diameter log n

degree n diameter 1
11
Preliminaries

• 1 is a dominant value. Any node that receives 1,
  change its own value to 1.
• Depending on value, we will call node
• either 1-node or 0-node.
• We will be using only 1-bit messages.

12
Algorithm (case tltn/2)

• Each 1-node sends its value to its neighbors. If
  less than k nodes reply, it switches to the next
  communicator.
• After log n rounds of communication without
  switching, node makes its decision. After that,
  it only answers to received messages.
• 0-nodes start the same procedure t2log n rounds
  later

13
Analysis

• Why wait t rounds?

14
Analysis

• Why wait t rounds?

15
Analysis

• Why wait t rounds?

16
Analysis

• Why wait t rounds?

17
Analysis

• Why wait t rounds?

18
Analysis

• Why wait t rounds?

19
Analysis

• If all 1-nodes die out, the final decision is 0
• If not, in any round without failure,
  neighborhood of
• 1-nodes expands k/2 times, until it reaches size
  n/2
• Any node eventually communicates with the set of
• 1-nodes.
• Every node sends O(log2n) messages before making
  its decision.
• For tgtn/2, we use sequence of log n epoches.

20
Conclusions

• It is possible to achieve consensus in tO(log2n)
  rounds, using messages of total length O(n log3n)
  bits.