Network Function Computation With Different Secure Conditions

In this paper, we investigate function computation problems under different secure conditions over a network with multiple source nodes and a single sink node which desires a function of all source messages without error. A wiretapper has access to some edges of the network. Based on different practical requirements, we consider two secure conditions named as secure and user secure respectively. The main parameter concerned here is the computing rate, which is the average times of the target function that can be computed securely or user securely without error for one use of the network. In the secure case, a new upper bound which is tighter than the previous one is provided for arithmetic sum functions and arbitrary networks. Moreover, we show that the improved upper bound is strictly tight for tree-like networks. In the user secure case, we give a sufficient and necessary condition for the existence of user secure network codes and obtain an upper bound for the computation capacity.