International Association
for Cryptologic Research

Byzantine Agreement (BA) allows a set of $n$ parties to agree on a value even when up to $t$ of the parties involved are corrupted. While previous works have shown that, for $\ell$-bit inputs, BA can be achieved with the optimal communication complexity $\mathcal{O}(\ell n)$ for sufficiently large $\ell$, BA only ensures that honest parties agree on a meaningful output when they hold the same input, rendering the primitive inadequate for many real-world applications.

This gave rise to the notion of Convex Agreement (CA), introduced by Vaidya and Garg [PODC'13], which requires the honest parties' outputs to be in the convex hull of the honest inputs. Unfortunately, all existing CA protocols incur a communication complexity of at least $\Omega(\ell n^2)$. In this work, we introduce the first CA protocol with the optimal communication of $\mathcal{O}(\ell n)$ bits for inputs in $\mathbb{Z}$ of size $\ell = \Omega(\kappa \cdot n^2 \log n)$, where $\kappa$ is the security parameter.