## 1 Introduction  
The Internet of Things (IoT) emerged when everyday objects began connecting to the Internet and interacting with each other autonomously without human intervention. IoT devices follow the IEEE 802.15.4 standard [1] to enable network connectivity for resource-constrained devices using short-range, lightweight communication protocols [1]. Due to the limited availability of device resources in addition to the bandwidth problem, the IoT ecosystem prefers multicast communication instead of unicast messages to send updates and patch-ups [2; 3]. To send multicast messages securely, a common secret key need to be shared between the devices in the multicast group. To efficiently distribute the keys to all group devices, a vast amount of group key management techniques have been proposed in the literature. Group key management enables the group to operate with integrity and confidentiality [4]. There are three methods for managing group keys. The Key Management Server (KMS), a trustworthy third party, is utilized for key distribution in the centralized key management technique [5; 6]. In decentralized key management, both the server and group member devices contribute to key management [7; 8; 9]. In distributed key management, there is no centralized trust, but every member participates in key management [10; 11; 12; 13]. Among all three methods, centralized key management offers less communication and computation overhead on the member devices with simpler functions. Hence, the proposed work focuses on centralized key management techniques, considering them to be most suitable for resource-constrained IoT devices. Centralized key management relies on a dependable third-party server for key management and key distribution. The proposed group management server uses an SDN controller to obtain centralized control over the heterogeneous network. The controller provides opinions on whether to forward network traffic, while the routers just obey the controller [14]. The advantages of SDN in comparison with traditional networks [15; 16; 17] are,  
* Easy patching and upgradation
* Knowledge of the sleep/wake cycle of IoT devices
* Supports security services by routing traffic through virtualized service functions known as Virtual Service Functions (VSF)  
The distribution of a common group key or key materials needed for group key generation occurs during the rekeying procedure. Rekeying ensures the group's confidentiality by putting forward and backward secrecy into action. When a group member leaves, the departing member should not have any access to any key materials that could be used to access any unapproved group data after the member exit event. This is termed forward secrecy. Similarly, when a new member enters an existing group, the joining member ought not to receive any key materials to obtain any unauthorized group data preceding the member join event. This is termed backward secrecy. A collusion attack is when two or more members join to obtain key material to access the group's data that are unauthorized for the colluding members. An efficient key management technique should ensure both forward secrecy and backward secrecy, as well as resistance to collusion attacks. Chinese Remainder Theorem (CRT)-based methods [18] offer the lowest communication costs out of all the centralized group key management techniques that have been proposed in the literature. However, because of its limited scalability, the method cannot be applied to dynamic groups. A group key management strategy called OFT [19] lowers the communication cost for rekeying by a factor of log=n. However, a collusion attack is possible on this approach. There are many proposed collusion-resistant OFT-based methods; however, they come at a higher communication cost. In the proposed MOFT, the binary tree's neighboring nodes share a second key, which is termed adjacent secret.