Impossibility for Byzantine Generals Problem where $n \leq 3m$

Question

Wiki: https://en.wikipedia.org/wiki/Byzantine_fault_tolerance

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In the paper "Reaching Agreement in the Presence of Faults", M. Pease et al. proved that there is no protocol (of some kind) to solve the problem for $n \leq 3m$, where $n$ stands for the number of generals and $m$ stands for the number of traitors. The key of their proof this the impossibility of the case $n=3,m=1$. However, the method they used does not look like an information theoretic proof. Thus, it seems like that their result is not "impossibility of arbitrary protocol".

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My question: Is there a infomation-theory-based proof for the case $n=3,m=1$? More formally, is there a proof or counterexample for the proposition "there does not exist any kind of protocol, which solves Byzantine generals problem where $n=3,m=1$"?

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Note: The typical protocol $\mathrm{SM}(m)$ (it works for arbitrary $n,m$) suggested by L. Lamport et al. is NOT a suitable counterexample, because it needs a signature mechanism, which is NOT perfect reliable in the sense of information theory, if we assume that traitors have infinity computing resources.

Answer 1

In the synchronous model of communication, there are $n$ agents which share a clock. In each round of communication, each agent sends an arbitrary message to each other agent, and then receives the message sent her by every other agent.

A protocol for byzantine agreement on $n$ agents supporting $m$ byzantine agents is a communication protocol for the agents satisfying the following properties:

• Each agent receives an input bit.
• The agents all start talking at time 0.
• There are at most $m$ byzantine agents, whose behavior is arbitrary.
• The other agents follow the protocol.
• The protocol always terminates (this means that the non-byzantine agents always reach a special "terminate" state of the protocol, and then stop talking forever), with a return value which is also a bit.
• The return values must all agree (note that this only applies to the non-byzantine agents).
• If all input bits are the same, then the return values have to be the same bit.

The impossibility result states that such a protocol exists if and only if $n > 3m$.

There is a different model in which an agent can sign a message, and this signature cannot be tampered with. In this model (which I will not specify formally) the problem can be solved for any $n,m$.

One of the difficulties in the area of distributed systems is the complicated nature of the computation model. If you want to understand the meaning of impossibility results, you have to familiarize yourself with these models in full detail (even more detail than the rather informal treatment in this answer).

Answer 2

I have thought the same issue as yours for a long time(I mean from the view of information theory). Maybe there is some fundemental things behind the consensus in the paper, such as entropy (uncertainty) in the whole distributed system of all the nodes, and the purpose of a consensus is to eliminate the entropy or you can say to figure out something that will make the uncertainty disappear. And maybe identifying the false nodes(byzantine nodes) in the system is a way to remove uncertainty, what do you think of this idea? (But I don't know how to quantify the entropy) oh, there is a solid proof in the paper "Impossibility of Distributed Consensus with One Faulty Process", but it is focused on the CFT scenario, not BFT.