Guest post by Zhuolun Xiang

## State Machine Replication and Broadcast

Many existing permission blockchains are built using Byzantine fault-tolerant state machine replication (BFT SMR), which ensures all honest replicas agree on the same sequence of client inputs. Most of the practical solutions for BFT SMR are based on the Primary-Backup paradigm. In this approach, in each view, there is a leader in charge to drive decisions efficiently, until replaced by the next leader. The Primary-Backup approach for SMR exposes deep connections to broadcast. Each view in BFT SMR is similar to an instance of broadcast where the leader taking on a similar role as the broadcaster, and hence an efficient broadcast protocol can be converted to an SMR protocol with similar efficiency guarantees.

## Good-case Latency

Practical SMR solutions care about the good case performance measured as the latency to commit when the Primary is honest. For many applications, latency is crucial. In a talk from 2000, Barbara Liskov, the author of PBFT, commented on PBFT needing 3 rounds in the good case:

I don’t know about a minimality proof that would show you require three phases, though I certainly haven’t been able to think of a way of doing it with fewer.

Therefore, it is natural and important to ask

What is the best latency a BFT SMR can achieve to commit decisions in the good case?

We refer to the above latency notion as good-case latency. For broadcast, we similarly define the good-case latency to be the latency to commit when the broadcaster is honest.

Somehow surprisingly, the above question has not been formally answered yet. Although a sequence of efforts improves the good-case latency of BFT SMR, there lacks a complete and rigorous characterization of the whole picture. Before we present our results, let’s take a quick look at the existing best solutions for BFT SMR on the good-case latency. For the synchrony model where the network delay is bounded by $\Delta$, Sync HotStuff commits in $2\Delta$ under $n\geq 2f+1$. For the partial synchrony model where the network delay is bounded only after a Global Stable Time (GST), PBFT commits in 3 rounds after GST under $n\geq 3f+1$, and FaB commits in 2 rounds after GST under $n\geq 5f+1$. We show that all these protocols can be improved!

## Results Overview

### Theory

Our good-case latency paper gives a complete categorization of the good-case latency for broadcast, under synchrony, partial synchrony and asynchrony. As mentioned, the protocols for broadcast can be converted to BFT SMR with similar good-case latency guarantees. The lower bound results also shed light on what is the limitation of good-case latency for BFT SMR. All of our bounds are tight except for just one case, as summarized in the table below (new and non-trivial results are marked bold).

• For asynchrony, Byzantine broadcast is impossible and the standard broadcast formulation is Byzantine reliable broadcast (BRB), which has a tight bound of 2 rounds for the good-case latency.
• For partial synchrony, we propose a new broadcast formulation called partially synchronous Byzantine broadcast (Psync-BB) that captures a single-shot of BFT SMR protocols like PBFT. We show that $n\geq 5f-1$ is the tight resilience bound for solving psync-BB with good-case latency of 2 rounds. Since psync-BB solves a single-shot of BFT SMR, our results directly refute the claim made in FaB saying that $n=5f+1$ is the best possible resilience for $2$-round BFT SMR protocols.
• For synchrony, we reveal a surprisingly rich structure of the good-case latency for Byzantine broadcast (BB). For a more accurate characterization, we adopt the separation of assumed network delay $\Delta$ and the actual (unknown) network delay $\delta$. For instance, 1 round in the bounds for asynchrony and partial synchrony above equals $\delta$, as the protocols can proceed with the network speed. We also distinguish two assumptions about the clock synchronization – the synchronized start case where all parties can start the protocol and local clock at the same time, and the unsynchronized start case where all parties start the protocol and local clock within $\Delta$ time of each other. To strengthen the results, all lower bounds assume sync start and all upper bounds assume unsynchronized start, except the case when $n/3<f<n/2$, which is especially interesting as the tight bounds depend on the clock synchronization assumption, and for unsynchronized start the tight bound is $\Delta+1.5\delta$, not even an integer multiple of the delay!

### Practice

For the practical side, the investigation on good-case latency leads to better BFT SMR protocols over PBFT, FaB, and Sync HotStuff in terms of good-case latency.

• For partial synchrony, we obtain a 2-round BFT SMR protocol that only requires $n\geq 5f-1$. Our protocol refutes the claim made in FaB saying that $n=5f+1$ is the best possible resilience for $2$-round BFT SMR protocols. Interestingly, for the canonical example with $n=4$ and $f=1$, we can have a 2-round PBFT protocol with the optimal resilience!
• For synchrony, we obtain a $1\Delta$-SMR protocol that commits in $\Delta+2\delta$ with $n\geq 2f+1$, reducing the commit latency (which is $2\Delta$) of Sync HotStuff by almost half when $\delta\ll\Delta$.

## Protocols and Impossibility Proofs

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