Broadcast encryption provides an efficient way to encrypt a message for a large number of receivers. This paper investigates whether it is possible to further improve efficiency of an existing state-of-the-art broadcast encryption scheme by re-using some of the random choices among different encryptions, without compromising the security of the original scheme. We introduce two schemes: the first allows a transmitter to efficiently encrypt several messages to a set of users; the second scheme extends the first by allowing the transmitter to efficiently send independent messages to different groups at once. We illustrate two scenarios where our schemes provide significant advantages compared to existing solutions.