Secure Network Function Computation

In this paper, we put forward the model of secure network function computation. In this model, a target function, of which the inputs are generated at multiple source nodes, is required to be computed with zero error at a sink node over a network while being protected from a wiretapper who can access any one but not more than one wiretap set in a given collection of wiretap sets. The secure computing rate of a secure network code is the average number of times the target function can be securely computed with zero error for one use of the network. However, characterizing this secure capacity with this general setup is overwhelmingly difficult. In the paper, we only consider this secure model for linear functions with the wiretapper being able to eavesdrop any subset of edges in the network up to a certain size, referred to as the security level. We prove a non-trivial upper bound on the secure computing capacity. Also, we discover the surprising fact that for some models, there is no penalty on the secure computing capacity compared with the computing capacity without security consideration. We further present a lower bound by designing a secure network coding scheme. Both the upper and lower bounds are tight for some cases but not tight in general. Finally, by comparing the upper and lower bounds thus obtained, we can exactly characterize the secure computing capacity when network topology satisfies a certain condition.