Security communication and information sharing among mobile devices are important application technologies of the intelligent information system (IIS). Because IIS is vulnerable to attacks, so the security of information sharing among mobile devices is seriously threatened. Thence, it is necessary to establish a secure channel for communication among mobile devices of IIS over an opening network. Group key agreement (GKA) can establish a secure channel among mobile devices of IIS by encryption technology. Due to the resource-constraints of mobile devices, such as weak computing power, small storage capacity, and limited communication range. To address these issues, an asymmetric GKA protocol among terminals of IIS for mobile edge computing (GKA–IIS–MEC) network is proposed in this paper. Adopted asymmetric GKA to achieve the group secure communication mechanisms that message sender unfettered in this protocol; the protocol also uses edge computing environment to migrate the computation and communication loads of mobile devices of IIS to edge nodes, thereby ensuring that mobile devices have lightweight computation and communication loads; and the members participating in the GKA can verify whether the group session keys they calculated are correctness. Under the hardness assumption of bilinear inverse Diffie–Hellman problem, the proposed protocol is proven that it can resist negative attacks. After evaluating the performance of the protocol, GKA–IIS–MEC has higher efficiency than the referred works in terms of time cost, communication consumption, and computation consumption.