<p><b>Abstract</b>—<it>Independence</it> is a fundamental property needed to achieve security in fault-tolerant distributed computing. In practice, distributed communication networks are neither fully synchronous or fully asynchronous, but rather <it>loosely synchronized</it>. By this, we mean that in a communication protocol, messages at a given round may depend on messages from other players at the same round.</p><p>These possible dependencies among messages create problems if we need <tmath>$n$</tmath> players to announce independently chosen values. This task is called <it>simultaneous broadcast</it>. In this paper, we present the first <it>constant</it> round protocol for simultaneous broadcast in a reasonable computation model (which includes a common shared random string among the players). The protocol is provably secure under general cryptographic assumptions. In the process, we develop a new and stronger formal definition for this problem. Previously known protocols for this task required either <tmath>$O(\log n)$</tmath> or expected constant rounds to complete (depending on the computation model considered).</p>