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Linear network coding is a promising technology that can maximize the throughput capacity of communication network. Despite this salient feature, there are still many challenges to be addressed, and security is clearly one of the most important challenges. In this paper, we will address the design of secure linear network coding. Specifically, we will investigate the network coding design that can both satisfy the weakly secure requirements and maximize the transmission data rate of multiple unicast streams between the same source and destination pair, which has not been addressed in the literature. In our study, we first prove that the secure unicast routing problem is equivalent to a constrained link-disjoint path problem. We then develop efficient algorithm that can find the optimal unicast topology in a polynomial amount of time. Based on the topology, we design deterministic linear network code that is weakly secure and can be constructed at the source node. And finally, we investigate the potential of random linear code for weakly secure unicast and prove the low bound of the probability that a random linear code is weakly secure.