## 1 Introduction  
The Internet of Things (IoT) plays an important role in many smart city applications, such as traffic management, public safety, environmental monitoring, smart parking, smart waste management, disaster management, and more. The IoT is crucial in many use cases in industrial IoT, such as asset management, inventory management, remote monitoring and control, supply chain management, smart manufacturing, and others [1]. Security is a significant challenge in the successful implementation of these IoT applications. The various security concerns in the IoT environment include identification, authentication, data integrity, trust, data confidentiality, access control, data privacy, and data availability. Although the generic IoT architecture addresses security concerns of perception, network, middleware, and application, trust and identity management are not given much consideration [2]. Most of the proposed security mechanisms are for IoT devices controlled by a single trusted authority (TA). However, in many use cases, the IoT device of one TA may need to communicate with the IoT device of another TA to carry out a coordinated task. A review study carried out by researchers [3] highlighted key parameters for a good authentication protocol. An authentication protocol for an IoT network should be lightweight, robust, have low overhead network traffic, have low computation cost, support the heterogeneity of the network, be scalable, and should include hardware security using physical unclonable functions (PUFs).  
Keeping the suggested properties of an authentication protocol in mind, in this research, a group authentication framework to authenticate a group of IoT devices controlled by different TAs is presented. In the past, the Chebyshev polynomial has been used for public key cryptosystems and exhibits an important property of a _semigroup_. This property of the Chebyshev polynomial is utilized in the proposed authentication framework to authenticate the group of IoT devices. The smart contract of Blockchain technology is used for secure communication among different TAs in a distributed environment to facilitate Chebyshev polynomial-based group authentication. The presented group authentication framework provides a platform for the secure collaboration of different TAs and can help in developing future use cases of smart cities and Industrial IoT. In comparison with existing authentication protocols, the contributions of the proposed authentication scheme are as follows:  
* Most authentication protocols are developed for individual device authentication and the same protocol is used for group authentication. The proposed authentication scheme is specifically proposed for group authentication.
* In the proposed scheme, the Chebyshev polynomial is used to create a shared secret so that the group can be authenticated without creating large overhead in network traffic. The Chebyshev polynomial has not been used earlier for group authentication for IoT devices.
* Blockchain technology is used to securely exchange short quick messages between various IoT devices of different entities located in different geographic locations.
* In the proposed scheme, a hardware-based identification PUF is employed.  
The rest of the paper is organized as follows. A high-level overview of our system is presented in this Section 1. Section 2 discusses the work conducted by researchers in the related field. Section 3 explains the Chebyshev polynomial and its application in cryptography. In Section 4, the proposed group authentication framework is explained. In Section 5, a security analysis of the proposed authentication framework is carried out using informal security analysis methods. Section 6 discusses the analysis of the results. This section also presents results obtained by the deployment of smart contracts on Ethereum's _Goerli testnet_. The paper concludes with some final remarks and future works in Section 7.