## 1 Introduction  
Over-the-air (OTA) technology refers to the remote management of systems through the air interface of mobile communication, which distributes updated software and updates configurations to devices using communication capabilities and secure encryption [1]. This technology can be classified as either software-over-the-air (SOTA) [2] and firmware-over-the-air (FOTA) [3]. SOTA is a technology that updates software applications on smart devices over a wireless network and is used for software systems. FOTA updates the firmware of smart devices to upgrade the core underlying hardware software. As industrial Internet and Internet of Things technologies evolve, industrial equipment will become more advanced, the amount of software for smart devices will become richer, and the addition of OTA configurations will become more important for service updates and rollouts. OTA technology can promote rapid industrial development by decoupling the software and hardware development of industrial equipment, remotely eliminating defects, and providing continuous maintenance and updates. Decoupling software and hardware development, wherein user data are uploaded to the cloud for iterative development, results in a new model of personalized and customized software and hardware development oriented according to user needs [4]. If there are defects or errors in the use of the equipment, the remote update mechanism can solve the machine errors. In industrial production, continuous OTA maintenance and updates can further improve production efficiency without interruptions [5]. As OTA technology is increasingly being applied in the industrial field, its high demand for service bandwidth and service quality will require a large amount of bandwidthresources, without which there may be upgrade interruptions. The 5G-U (5G on unlicensed band) network has a high data transmission rate and low delay communication capability, which can significantly improve wireless communication performance [6, 7]. For the centralized upgrade of various equipment in factories, the 5G-U network can meet strict requirements for the data transmission rate. However, transferring all firmware images to a large number of devices that need to be upgraded is inefficient. Sending one byte wirelessly may consume the same amount of energy as executing 1000 instructions [8]. The difference compression algorithm can record the changes between the two versions. The principle of this algorithm is to find the common string between the old and new version files and compress the data by replacing the substring with a reference copy to obtain the \"delta\" [9], namely the difference file between the two version files. The regression algorithm creates a new version of the file by combining the differential file (incremental) with the old file, as illustrated in Figure 1, which is typically much smaller than the entire upgraded file, thus reducing the upgrade time and mobile network consumption during the transmission and enabling rapid OTA software upgrades. For differential compression algorithms, much of the literature is based on byte processing techniques or solving optimization problems to reduce increments in the differential process. [PERSON] et al. [10] provided a preprocessing step for the in-place sliding window (IPSW) algorithm to align the overlapping parts of the source file and new file when compared with the new file, thus reducing the space required for encoding and decoding and improving the upgrade efficiency. [PERSON] et al. [11] proposed a linked list method to reduce the impact of different operational changes on differences, which improves the similarity of firmware to optimize performance. However, such methods can lead to program memory fragmentation in the event of a change in a link, which results in significant energy consumption if the functions are called in different parts of a flash. [PERSON] et al.