Asynchronous Byazantine Broadcast

#2.1

Recall

Asynchronous Byazantine Broadcast

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Unbounded delay in processing and delivering messages
• Undetectable faulty process

• Agreement
• Probabilistic termination
• Validity

• Byzantine failures

• \(\texttt{send}(e), \ e=(p,m)\)

Asynchronous Byazantine Broadcast

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Unbounded delay in processing and delivering messages
• Undetectable faulty process

• Agreement
• Probabilistic termination
• Validity

• Byzantine failures

• \(\texttt{send}(e), \ e=(p,m)\)

• \(\texttt{receive}(p)\)

Asynchronous Byazantine Broadcast

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Unbounded delay in processing and delivering messages
• Undetectable faulty process

• Agreement
• Probabilistic termination
• Validity

• Byzantine failures

• \(\texttt{send}(e), \ e=(p,m)\)

• \(\texttt{receive}(p)\)

Asynchronous Byazantine Broadcast

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Unbounded delay in processing and delivering messages
• Undetectable faulty process

• Agreement
• Probabilistic termination
• Validity

• Byzantine failures

• \(\texttt{send}(e), \ e=(p,m)\)

• \(\texttt{receive}(p)\)

Asynchronous Byazantine Broadcast

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Unbounded delay in processing and delivering messages
• Undetectable faulty process

• Agreement
• Probabilistic termination
• Validity

• Byzantine failures

• \(\texttt{send}(e), \ e=(p,m)\)

• \(\texttt{receive}(p)\)

Consensus in Async. BB

• Desginate one of the processes as being the transmitter that sends
  a value to all the rest of the processes.

• Consensus is defines as:

1. If the transmitter is correct, all the correct processes decide on its value
2. If the transmitter is malicious, then either no correct process will decide or they will all decide on the same value

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Proof. Consider the case \(n=10,f=4\) for simplicity

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Groups of at most \(f\) processes in each

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Case \(\text{I}\): All processes in \(C\) are malicious

• Honest \(T\) sends \(0\) to all the processes

• No process in \(C\) responds
• Processes in \(A,B\) decide \(0\) in time \(t\)

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Case \(\text{II}\): \(T\) is malicious
• \(T\) sends \(0\) to \(C\) and \(1\) to the rest

• Suppose processes in \(C\) delayed by \(t\)
• \(A,B\)'s view same as Case \(\text{I} \implies\) decide \(1\)

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Case \(\text{III}\): Again \(T\) is malicious
• \(T\) sends \(0\) to \(B\) and 1 to the rest

• Suppose messages in \(B\) delayed by \(t'\)
• \(A,C\) will proceed to decide \(1\)

The Proof

• It is impossible to achieve asynchronous Byzantine Broadcast with
  \(f \ge \frac{n}{3},\) where out of n processes, at most f are malicious processes.

• Case \(\text{IV}\): All processes in \(A\) malicious
• \(T\) sends \(1,0\) to \(C,B\) resp.

• Messages from \(C,B\) delayed by \(\text{max}\{t,t'\}\) 
• \(C,B\) individually decide \(1,0\).

Coming Soon...

• In the upcoming talk, we will talk about lower bound for synchronous Byzantine Agreement
• We will also look at the similarities/differences between BA and BB which will drive our way forward
• Next would be Authenticated Byzantine Agreement...

1. G. Bracha and S. Toueg, “Asynchronous consensus and broadcast protocols,” J. ACM, vol. 32, no. 4, pp. 824–840, Oct. 1985. [Online]. Available: https://doi.org/10.1145/4221.214134
2. L. Lamport, R. Shostak, and M. Pease, “The byzantine generals problem,” ACM Trans. Program. Lang. Syst., vol. 4, no. 3, pp. 382–401, Jul. 1982. [Online]. Available: https://doi.org/10.1145/357172.357176