COOL Is Optimal in Error-Free Asynchronous Byzantine Agreement

COOL (Chen'21) is an error-free, information-theoretically secure Byzantine agreement (BA) protocol proven to achieve BA consensus in the synchronous setting for an $\ell$-bit message, with a total communication complexity of $O(\max\{n\ell, nt \log q\})$ bits, four communication rounds in the worst case, and a single invocation of a binary BA, under the optimal resilience assumption $n \geq 3t + 1$ in a network of $n$ nodes, where up to $t$ nodes may behave dishonestly. Here, $q$ denotes the alphabet size of the error correction code used in the protocol. In this work, we present an adaptive variant of COOL, called OciorACOOL, which achieves error-free, information-theoretically secure BA consensus in the asynchronous setting with total $O(\max\{n\ell, n t \log q\})$ communication bits, $O(1)$ rounds, and a single invocation of an asynchronous binary BA protocol, still under the optimal resilience assumption $n \geq 3t + 1$. Moreover, OciorACOOL retains the same low-complexity, traditional $(n, k)$ error-correction encoding and decoding as COOL, with $k=t/3$.