Toward Optimal-Complexity Hash-Based Asynchronous MVBA with Optimal Resilience

Multi-valued validated Byzantine agreement (MVBA), a fundamental primitive of distributed computing, allows $n$ processes to agree on a valid $\ell$-bit value, despite $t$ faulty processes behaving maliciously. Among hash-based solutions for the asynchronous setting with adaptive faults, the state-of-the-art HMVBA protocol achieves optimal $O(n^2)$ message complexity, (near-)optimal $O(n\ell+n^2 λ\log n)$ bit complexity, and optimal $O(1)$ time complexity. However, it only tolerates up to $t < \frac15 n$ adaptive failures. In contrast, the best known optimally resilient protocol, FIN-MVBA, exchanges $O(n^3)$ messages and $O(n^2\ell + n^3λ)$ bits. This highlights a fundamental question: can a hash-based protocol be designed for the asynchronous setting with adaptive faults that simultaneously achieves both optimal complexity and optimal resilience? In this paper, we take a significant step toward answering the question. Namely, we introduce Reducer, an MVBA protocol that retains HMVBA's complexity while improving its resilience to $t<\frac14 n$. Like HMVBA and FIN-MVBA, Reducer relies exclusively on collision-resistant hash functions. A key innovation in Reducer's design is its internal use of strong multi-valued Byzantine agreement (SMBA), a variant of strong consensus we introduce and construct, which ensures agreement on a correct process's proposal. To further advance resilience toward the optimal one-third bound, we then propose Reducer++, an MVBA protocol that tolerates up to $t < (\frac13-ε)n$ adaptive failures, for any fixed constant $ε> 0$. Unlike Reducer, Reducer++ does not rely on SMBA. Instead, it employs a novel approach involving hash functions modeled as random oracles to ensure termination. Reducer++ maintains constant time complexity, quadratic message complexity, and quasi-quadratic bit complexity, with constants dependent on $ε$.