Effectively combined multi-party quantum secret sharing and secure direct communication

In this paper using quantum secret sharing based on quantum secure direct communication, four-bit secret message is transmitted, that any of the receivers cannot achieve any rate of secret message alone. This study contains two parts: in the first part there are two receivers and in the second, there are an arbitrary number of receivers. If the receivers collaborate, they can recover the message. For encoding the secret message, by EPR (Einstein–Podolsky–Rosen) pairs, the sender uses Pauli operators. This scheme is more efficient and has better performance than other works which have presented in this field, because in the proposed protocol more bits of secret message are transmitted using less qubits. Besides, this protocol is secure against probable attacks of eavesdropper, because it uses the technique of inserted decoy single qubits.