Round Complexity of Authenticated Broadcast with a Dishonest Majority

Broadcast among n parties in the presence of t \geqslant n/3 malicious parties is possible only with some additional setup. The most common setup considered is the existence of a PKI and secure digital signatures, where so-called authenticated broadcast is achievable for any t \le n. It is known that t + 1 rounds are necessary and sufficient for deterministic protocols achieving authenticated broadcast. Recently, however, randomized protocols running in expected constant rounds have been shown for the case of t \le n/2. It has remained open whether randomization can improve the round complexity when an honest majority is not present. We address this question and show upper/ lower bounds on how much randomization can help: For t \leqslant n/2 + k, we show a randomized broadcast protocol that runs in expected O(k^2 ) rounds. In particular, we obtain expected constant-round protocols for t = n/2 + O(1). On the negative side, we show that even randomized protocols require \Omega (2n/(n - t)) rounds. This in particular rules out expected constant-round protocols when the fraction of honest parties is sub-constant.