Exact resilience threshold for two-round Prefix Consensus

Determine the exact resilience threshold (the tight relation between n and f) under which a two-round Prefix Consensus protocol is possible in the asynchronous Byzantine setting, closing the gap between the current impossibility for n ≤ 4f and the existing two-round construction that requires n ≥ 5f+1.

Background

The paper introduces Prefix Consensus, a consensus primitive where parties propose input vectors and output two consistent vectors (low and high) that extend the maximum common prefix of honest inputs and satisfy a mutual upper-bound property. The authors present a two-round protocol for Prefix Consensus under the stronger resilience assumption n ≥ 5f+1.

They also establish a lower bound showing that a two-round solution is impossible when n ≤ 4f. This leaves a gap between the lower bound (impossibility for n ≤ 4f) and the upper bound (feasibility for n ≥ 5f+1). Closing this gap would determine the precise resilience threshold for achieving two rounds in the asynchronous Byzantine model and sharpen the understanding of the trade-off between latency and fault tolerance.

References

Note that there remains a gap between our upper and lower bounds for 2-round Prefix Consensus. On the one hand, our lower bound (\Cref{thm:lower-bound}) shows that 2-round Prefix Consensus is impossible when n≤4f. On the other hand, our 2-round construction requires the stronger condition n≥5f+1. Closing this gap is an interesting open problem, which we summarize in~\Cref{sec:conclusion}.

— Prefix Consensus For Censorship Resistant BFT  (2602.02892 - Xiang et al., 2 Feb 2026) in Section 6.1 (Discussion: 2-Round Prefix Consensus under n ≥ 5f+1)