Current advancements in mobility of devices and also Internet of Things (IoT) have replaced the central networks by distributed infrastructure. The more a network is distributed, the more the security of infrastructure and the communication is getting complex. The members in a distributed network create different groups according to their coverage area or their requirements. Mobility nature of the members brings a problem called hand-over of members between groups. Current authentication methods are not applicable due to the lack of resources in the devices.A lightweight authentication method and an easy and fast hand-over process are the current need for the distributed networks. Shamir Secret Sharing algorithm is used for the authentication process in the studies before, but still secure group authentication algorithm and hand-over process are challenges in the group authentication. In this study, a new method is proposed to provide a secure group authentication and hand-over process between groups based on Lagrange's Interpolation.