In this article, certain Hermite-Jensen-Mercer type inequalities are proved via Caputo fractional derivatives. We established some new inequalities involving Caputo fractional derivatives, such as Hermite-Jensen-Mercer type inequalities, for differentiable mapping h((n)) whose derivatives in the absolute values are convex.

Industrial internet security is a critical component of cyberspace safety. Furthermore, the encryption protocol is a critical component of cyberspace security. Due to the rapid development of industrial internet and edge computing, increasingly more devices are outsourcing their data to cloud servers to save costs. Edge devices should have a secure session key to reduce communication costs and share information. However, most key generation and storage are completed by a centralized third-party organization, which carries some security risks. In this context, this paper will propose a lightweight multi-dimensional virtual iteration of the group key agreement protocol. Group key agreement protocol allows for one-at-a-time encryption and timely key updates without the involvement of a trusted third party, and each device in the network can agreement a large number of keys. According to the analysis of this protocol, it has high security, rapid computation speed, and little storage space.

With the rapid development of edge computing in the Industrial Internet, data sharing schemes among edge users require reliable cross-domain authentication and key agreement mechanisms to guarantee the security and reliability of inter-device communication. To tackle the deficiencies of existing group key agreement schemes, including dependence on trusted third parties, high computational overhead, and the difficulty of achieving both privacy preservation and attack resistance, this paper presents a cross-domain authenticated key agreement protocol designed for edge computing environments. This protocol supports anonymous identity authentication between cross-domain users, and innovatively constructs a multi-dimensional virtual iterative cyberspace model to generate massive secure keys via the collaborative iteration of multi-user key sequences. The proposed protocol is decentralized, lightweight, and resistant to replay attacks and man-in-the-middle attacks, while satisfying forward and backward secrecy. Security analysis and performance comparison experiments illustrate that the protocol significantly reduces computational and communication overhead, matches the resource-constrained characteristics of edge devices, and can be widely deployed in large-scale data encryption and sharing scenarios under edge computing environments.