Modeling and Optimal Design of Linear Network Coding for Secure Unicast with Multiple Streams

In this paper, we will address the modeling and optimal design of linear network coding (LNC) for secure unicast with multiple streams between the same source and destination pair. The objectives include 1) satisfying the weakly secure requirements, 2) maximizing the transmission data rate, and 3) minimizing the size of the finite field. To fulfill the first two objectives, we formulate a secure unicast routing problem and prove that it is equivalent to a constrained link-disjoint path problem. Based on this fact, we develop an efficient algorithm that can find the optimal unicast topology in a polynomial amount of time. With the given topology, we investigate the design of both weakly secure deterministic LNC and weakly secure random LNC. In the designs of deterministic LNC and random LNC, we prove that the required size of the finite field decreases with the decrease of the number of intermediate nodes in the topology. Therefore, to meet the third objective, we formulate a problem to minimize the number of intermediate nodes. We prove that this problem is NP-Complete and develop an approximation algorithm to solve it. Finally, extensive simulation experiments have been conducted, and the results demonstrate the effectiveness of the proposed algorithms.