The locally twisted cube is a newly topological structure of network. The authors we design a unicast fault-tolerant routing algorithm on the locally twisted cube by utilizing safety levels and the feature of the network. The performance of the proposed algorithm is evaluated through simulation experiments. When the number of faulty nodes reaches or exceeds half of the total, it can still achieve a quite high percentage of successful routing. An additional advantage of the routing algorithm is that it is highly probable that the selected route be a shortest route between the associated nodes.