With the rapid development and massive deployment of the Internet of things (IoT) networks, security related issues in the IoT networks have been paid more and more attention to. Among all the security concerns, message authentication is critical in preventing the unauthorized messages from being transmitted in the IoT networks. Many message authentication schemes have been proposed based on the public-key cryptosystem, where the key management is simple and scalable. Identity based cryptosystem is a special type of public-key cryptosystem and can further ease the process of the key management since the public keys can be obtained easily. In this paper, we devise an efficient message authentication with enhanced privacy (IMAEP) scheme using the identity based signature. Our proposed scheme can provide both unconditional privacy as well as the enhanced privacy under full key exposure attack. Our proposed scheme can also provide existential unforgeability under the adaptive chosen-message-and-identity attack. Compared with the scheme that has the same level of anonymity and security, our proposed scheme has much lower computational overhead, and can provide extra unconditional privacy. Next we propose an extended IMAEP (EIMAEP) scheme for the general access structures where the message is signed by a group of users instead of one user. We also conduct comprehensive analysis and demonstrate that the EIMAEP scheme can achieve the same level of privacy and unforgeability as the IMAEP scheme.