Wireless sensor networks (WSNs) exhibit their potential capacity in the next generation of mobile communication networks and wireless systems (5G). Collected data in WSNs are different from most data transmitted in digital communication applications. Most collected data in WSNs contain only few bits of information. Conventional protocols are not suitable for WSNs since this environment needs more flexible and lightweight protocols for secure group communications. Hence, how to realize the mutual secure and lightweight communication is a big challenge for WSNs. User authentication and key establishment are two fundamental security services in secure communications for WSNs. In this paper, we propose a novel design which embeds the function of membership authentication and group key establishment in WSNs. By using an asymmetric bivariate polynomial, membership authentication and pairwise shared keys distribution are realized. Then, each member mixes his/her input with pairwise shared keys with other members and releases the encrypted value in a broadcast channel. After collecting all released values, each member can compute the group key efficiently. Our proposal is noninteractive and lightweight. As it enjoys low computation and communication costs compared with the state-of-the-art cryptographic solutions, this design is more suitable for efficient membership authentication and group key establishment in WSNs.