Look into verifiable shuffles, private information retrieval, and dining cryptographer networks. By combining such techniques, it's possible to be much much be

Look into verifiable shuffles, private information retrieval, and dining cryptographer networks. By combining such techniques, it's possible to be much much better than naive broadcast, and also possible to use a client/server architecture so that low bandwidth endpoints can participate. Have a look at systems such as Dissent and Riffle.