This paper presents a factoringalgorithm for computing source-to-K terminal (SKT) reliability, the probability that a source s can send message to a specified set of terminals K, in acyclic directed networks (AD-networks) in which bothnodes and edges can fail. Based on Pivotal decomposition theorem, a newformula is derived for computing the SKT reliability of AD-networks. By establishing a topological property of AD-networks, it is shown that the SKT reliability of AD-networks can be computed by recursively applying this formula. Two new Reliability-Preserving Reductions are alsointroduced. The recursion tree generated by the presented algorithm hasat most 2(|V| - |K|- |C|) leaf nodes, where |V| and |K| are the numbers of nodes and terminals, respectively, while |C| is the number of the nodes satisfying some specified conditions. The computation complexity of the new algorithm is O (|E||V|2(|V| -|K| -|C|)) in the worst case, where |E| is the number of edges. Forsource-to-all-terminal (SAT) reliability, its computation complexity is O (|E|). Comparison of the new algorithm with the existing ones indicates that the new algorithm is more efficient for computing the SKT reliability of AD-networks.