Optimal Design of Linear Network Coding for information theoretically secure unicast

In this paper, we study the optimal design of linear network coding (LNC) for secure unicast against passive attacks, under the requirement of information theoretical security (ITS). The objectives of our optimal LNC design include (1) satisfying the ITS requirement, (2) maximizing the transmission rate of a unicast stream, and (3) minimizing the number of additional random symbols. We first formulate the problem that maximizes the secure transmission rate under the requirement of ITS, which is then transformed to a constrained maximum network flow problem.We devise an efficient algorithm that can find the optimal transmission topology. Based on the transmission topology, we then design a deterministic LNC which satisfies the aforementioned objectives and provide a constructive upper bound of the size of the finite field. In addition, we also study the potential of random LNC and derive the low bound of the probability that a random LNC is information theoretically secure.