Dining-cryptographers networks (DCN) can achieve information-theoretical privacy. Unfortunately, they are not well suited for peer-to-peer networks as they are used in blockchain applications to disseminate transactions and blocks among participants. In previous but preliminary work, we proposed a threephase approach with an initial phase based on a DCN with a group size of k while later phases take care of the actual broadcast within a peer-to-peer network. This paper describes our DCN protocol in detail and adds a performance evaluation powered by our proof-of-concept implementation. Our contributions are (i) an extension of the DCN protocol by von Ahn for fair delivery of arbitrarily long messages sent by potentially multiple senders, (ii) a privacy and security analysis of this extension, (iii) various performance optimisation especially for best-case operation, and (iv) a performance evaluation. The latter uses a latency of 100 ms and a bandwidth limit of 50 Mbit/s between participants. The interquartile range of the largest test of the highly secured version took 35s+-1.25s for a full run. All tests of the optimized common-case mode show the dissemination of a message within 0.5s+-0.1s. These results compare favourably to previously established protocols for k-anonymous transmission of fixed size messages, outperforming the original protocol for messages as small as 2 KiB.