Probabilistic Byzantine Fault Tolerance (Extended Version)

TL;DR

The paper addresses Byzantine fault-tolerant consensus under practical, less pessimistic adversaries in partially synchronous permissioned systems. It introduces ProBFT, a leader-based probabilistic protocol that uses verifiable random functions to form probabilistic quorums of size $q= l\sqrt{n}$, achieving $O(n\sqrt{n})$ message complexity with a best-case latency of three rounds. The authors provide a protocol specification, correctness proofs, and numerical evaluation, showing safety with probability $1-\exp(-\Theta(\sqrt{n}))$ and termination with probability $1$ under GST, while maintaining significantly lower messaging than traditional PBFT. The work demonstrates scalable and practical Byzantine resilience, with potential applications to state machine replication and blockchain-style consensus; future work includes broader adoption in scalable distributed systems. $$\text{ProBFT combines probabilistic quorum systems with VRFs to balance safety, liveness, and efficiency in real-world adversarial settings.}$$

Abstract

Consensus is a fundamental building block for constructing reliable and fault-tolerant distributed services. Many Byzantine fault-tolerant consensus protocols designed for partially synchronous systems adopt a pessimistic approach when dealing with adversaries, ensuring safety in a deterministic way even under the worst-case scenarios that adversaries can create. Following this approach typically results in either an increase in the message complexity (e.g., PBFT) or an increase in the number of communication steps (e.g., HotStuff). In practice, however, adversaries are not as powerful as the ones assumed by these protocols. Furthermore, it might suffice to ensure safety and liveness properties with high probability. In order to accommodate more realistic and optimistic adversaries and improve the scalability of the BFT consensus, we propose ProBFT (Probabilistic Byzantine Fault Tolerance). ProBFT is a leader-based probabilistic consensus protocol with a message complexity of $O(n\sqrt{n})$ and an optimal number of communication steps that tolerates Byzantine faults in permissioned partially synchronous systems. It is built on top of well-known primitives, such as probabilistic Byzantine quorums and verifiable random functions. ProBFT guarantees safety and liveness with high probabilities even with faulty leaders, as long as a supermajority of replicas is correct, and using only a fraction of messages employed in PBFT (e.g., $20\%$). We provide a detailed description of ProBFT's protocol and its analysis.

Figures (5)

• Figure 1: Comparing the normal case of three consensus protocols --- PBFT, ProBFT, and HotStuff --- regarding the number of communication steps and message complexity.
• Figure 2: Overview of PBFT. Each correct replica broadcasts its Prepare and Commit messages. The size of any quorum is $|Q| = \lceil{(n+f+1)/2}\rceil$. The set of replicas $I$ in the intersection of two quorums contains at least one correct replica.
• Figure 3: Overview of ProBFT. The size of any probabilistic quorum is $q = |Q| = O(\sqrt{n})$. Each correct replica multicasts its Prepare and Commit messages to random samples of sizes $o\times q$, where $o$ is a constant. The set of replicas $\bar{I}$ in the intersection of two probabilistic quorums contains at least one correct replica with high probability.
• Figure 4: Different scenarios in which a Byzantine leader can cause disagreements in a view.
• Figure 5: ProBFT agreement and termination probability analysis with $q=2\sqrt n$. For $f/n=0.2$, the top-left figure depicts the probability of ensuring agreement with faulty leaders in every view, while the top-right figure shows the probability of terminating in a view after GST when the leader is correct. For $n=100$, the bottom-left figure depicts the probability of ensuring agreement with faulty leaders in every view, while the bottom-right figure depicts the probability of terminating in a view after GST when the leader is correct.

Theorems & Definitions (29)

• Theorem 1: Informal main result
• Theorem 2
• Theorem 3
• Theorem 4: Main liveness result
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Corollary 1: Main safety result
• Definition 1: Negative association concentration