Most voting schemes rely on a number of authorities. If too many of these authorities are dishonest then voter privacy may be violated. To give stronger guarantees of voter privacy Kiayias and Yung [1] introduced the concept of elections with perfect ballot secrecy. In this type of election scheme it is guaranteed that the only thing revealed about voters' choices is the result of the election, no matter how many parties are corrupt. Our first contribution is to suggest a simple voting scheme with perfect ballot secrecy that is more efficient than [1]. Considering the question of achieving maximal privacy in other protocols, we look at anonymous broadcast. We suggest the notion of perfect message secrecy; meaning that nothing is revealed about who sent which message, no matter how many parties are corrupt. Our second contribution is an anonymous broadcast channel with perfect message secrecy built on top of a broadcast channel.