Byzantine broadcast is a distributed primitive that allows a specific party to consistently distribute a message among n parties in the presence of potential misbehavior of up to t of the parties. All known protocols implementing broadcast of an ℓ-bit message from point-to-point channels tolerating any t < n Byzantine corruptions have communication complexity at least Ω(ℓn ²). In this paper we give cryptographically secure and information-theoretically secure protocols for t < n that communicate bits when ℓ is sufficiently large. This matches the optimal communication complexity bound for any protocol allowing to broadcast ℓ-bit messages. While broadcast protocols with the optimal communication complexity exist for t < n/2, this paper is the first to present such protocols for t < n.