Impossibility of Synchronous Byzantine Agreement

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Say we have processes \(A, B, C\) and \(B\) is Byzantine.
• \(B\) tells \(A\) that \(C\) is byzantine and \(C\) that \(A\) is byzantine
• From an honest \(C\)'s point of view, both \(A,B\) are telling the same
• Disallows consensus

• Indistinguishability - processes cannot tell the difference in two worlds 
• Hybridization - build intermediate worlds between the two contradicting worlds

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof. Consider world \(\text{I}\):

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof. Consider world \(\text{I}\):

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof. Consider world \(\text{I}\):

1. \(C\) frames \(A\) as if \(A\) starts from \(0\)
2. Similarly, \(C\) frames \(B\)
3. Since validity must hold, \(A,B\) proceed to decide \(1\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{II}\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{II}\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{II}\)

1. \(A\) is byzantine
2. \(A\) frames \(B,C\) similar to the world \(\text{I}\)
3. \(B,C\) proceed to decide \(0\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{III}\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{III}\)

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Consider world \(\text{III}\)

1. \(B\) is byzantine,
2. \(A,B,C\) starts with \(1,1,0\) respectively
3. \(A,C\) output \(1\) and \(0\) respectively. Why?

Impossibility of Synchronous BA

• There is no consensus algorithm for \(n=3\) processes out of which \(f=1\) process is byzantine in a synchronous setting.

• Proof.  Compare with world \(\text{I}\)

• Validity in worlds \(\text{I,II} \ \implies\) agreement violation in world \(\text{III}\) 

Closing Notes

• Proof for Byzantine Broadcast for \(f < \frac{n}{3}\) in synchronous systems is similar in flavour
• In fact, if we can achieve broadcast and \(f < \frac{n}{2}\), then we can achieve Byzantine agreement
• Other way round is also true for any \(f < n\)
• To sum up, \(\text{BB} \Longleftrightarrow \text{BA}\) (to be discussed in detail in the next talk)

1. M. Fischer, N. Lynch, and M. Merritt. 1985. "Easy impossibility proofs for distributed consensus problems". In PODC '85. ACM, NY, USA, 59–70. DOI:https://doi.org/10.1145/323596.323602
2. K. Nayak and I. Abraham. 2019. "Byzantine Agreement is Impossible for \(n\le 3f\) if the Adversary can Simulate", Decentralized Thoughts Blog, Online: link