Epidemic protocols have two fundamental assumptions. One is the availability of a mechanism that provides each node with a set of log(N) (fanout) nodes to gossip with at each cycle. The other is that the network size N is known to all member nodes. While it may be trivial to support these assumptions in small systems, it is a challenge to realize them in large open dynamic systems, such as peer-to-peer (P2P) systems. Technically, since the most fundamental parameter of epidemic protocols is log(N), without knowing the system size, the protocols will be limited. Further, since the network churn, frequently observed in P2P systems, causes rapid membership changes, providing a different set of log(N) at each cycle is a difficult problem. In order to support the assumptions, the fanout nodes should be selected randomly and uniformly from the entire membership. This paper investigates one possible solution which addresses both problems; providing at each cycle a different set of log(N) nodes selected randomly and uniformly from the entire network under churn, and estimating the dynamic network size in the number of nodes. This solution improves the previously developed distributed algorithm called Shuffle to deal with churn, and utilizes the Shuffle infrastructure to estimate the dynamic network size. The effectiveness of the proposed solution is evaluated by simulation. According to the simulation results, the proposed algorithms successfully handle network churn in providing random log(N0 fanout nodes, and practically and accurately estimate the network size. Overall, this work provides insights in designing epidemic protocols for large scale open dynamic systems, where the protocols behave autonomically.