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This paper presents a method to cope with reliable broadcasting in faulty hypercubes using local safety information. A new definition, broadcast subcube, is introduced, with which various techniques are proposed to improve performance of the broadcast algorithm. Local safety information is well used in the fault-tolerant broadcast algorithm by considering only safety of the broadcast subcube. An unsafe hypercube can be split into a set of maximal safe subcubes. If these maximal safe subcubes meet certain requirements (listed in this paper), then broadcasting can still be done successfully and, in some cases, optimal broadcast is still possible. The sufficient condition for optimal broadcast of a message is presented in an unsafe hypercube. Extensive simulation results show that the proposed method outperforms previous methods, in all cases.