Summary: We discuss aspects of secure quantum communication by proposing and analyzing a quantum analog of the Vernam cipher (one-time-pad). The quantum Vernam cipher uses entanglement as the key to encrypt quantum information sent through an insecure quantum channel. First, in sharp contrast with the classical Vernam cipher, the quantum key can be recycled securely. We show that key recycling is intrinsic to the quantum cipher-text, rather than using entanglement as the key. Second, the scheme detects and corrects for arbitrary transmission errors, and it does so using only local operations and classical communication (LOCC) between the sender and the receiver. The application to quantum message authentication is discussed. Quantum secret sharing schemes with similar properties are characterized. We also discuss two general issues, the relation between secret communication and secret sharing, the classification of secure communication protocols.