An Improved Capacity Bound for Secure Network Function Computation

The problem of secure network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function f, of which the inputs are generated at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, obtains no information about the source inputs. The secure computing rate of a secure function-computing network code is the average number of times the target function can be securely computed for one use of the network. In the paper, we are interested in securely computing linear functions with the wiretapper who can eavesdrop any subset of edges up to a certain size r, referred to as the security level. We obtain an improved upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. When the security level r is equal to 0, our improved upper bound reduces to the computing capacity without security consideration. Furthermore, by applying the improved upper bound, we obtain a non-trivial upper bound on the maximum security level such that the function can be securely computed with a positive rate. We also present a lower bound on the secure computing capacity and give some sufficient conditions in terms of the network topology on the tightness of the lower bound. Together with the improved upper bound, the secure computing capacity for a class of security models can be fully characterized.