Network function computation with vector linear target function and security function

Abstract: In this paper, we study the problem of securely computing a function over a network, where both the target function and the security function are vector linear. The network is modeled as a directed acyclic graph. A sink node wishes to compute a function of messages generated by multiple distributed sources, while an eavesdropper can access exactly one wiretap set from a given collection. The eavesdropper must be prevented from obtaining any information about a specified security function of the source messages. The secure computing capacity is the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. We establish two upper bounds on this capacity, which hold for arbitrary network topologies and for any vector linear target and security functions. These bounds generalize existing results and also lead to a new upper bound when the target function is the sum over a finite field. For the lower bound, when the target function is the sum, we extend an existing method, which transforms a non-secure network code into a secure one, to the case where the security function is vector linear. Furthermore, for a particular class of networks and a vector linear target function, we characterize the required properties of the global encoding matrix to construct a secure vector linear network code.