Wireless Sensor Network (WSN) and Internet of Things (IoT) allow sensor devices to collect information about various critical sectors through wireless networks. However, when the WSNs are connected to a public network, the security of the WSN is vulnerable. Besides, WSN needs a key distribution scheme to secure data among other sensor devices. Furthermore, IoT devices have low computing, energy, and memory storage capabilities. Thus, designing a lightweight, efficient, and secure protocol communication for WSN is always a challenge due to the resource constraint of sensor devices. The existing schemes result in the number of keys stored by sensor devices depending on group size. When the group size increases, the number of the stored key by the sensor also increases. Other research proposes key establishment based on polynomial multiplicative and causes high computational capability. This paper proposed a key distribution scheme based on (p, q)-Lucas polynomial and XOR to achieve lightweight, memory overhead efficiency and security. The proposed method is evaluated in several parameters: memory overhead, communication overhead, energy consumption, computational complexity analysis, and security. The results indicate that our scheme outperforms the existing approaches regarding memory overhead, computation efficiency, and support security.