## 1 Introduction  
With the rapid development of cloud services and computing, users' behavior has drastically altered. A massive amount of data has been stored and processed on cloud servers, and users can download required resources at any time through wired or wireless networks. However, the security vulnerability of the cloud is a critical issue, and user information disclosure events on the cloud emerge frequently. The data encryption technique [1], which can encrypt the plaintext into ciphertext, is one of the solutions proposed to protect the content security of users. However, cloud administrators usually need some functions, such as authentication, access control, and multimedia management. The additional data should be embedded into the multimedia files on the cloud. Therefore, the reversible data hiding technique [2; 3; 4] is introduced, and the reversible data hiding in the encrypted domain becomes a potential solution that has flourished in recent years. In the cloud, there are three roles in such applications: the content owner encrypts the multimedia files and uploads them into the cloud; the cloud server embeds some additional data into the encrypted files and stores them in the cloud; and an authorized third-party can acquire the embedded encrypted files and extract the embedded data or recover the original files according to different keys. Most researchers focus on the digital image domain, a series of reversible data hiding schemes in encrypted images that have been proposed and can be classified into two categories: vacate the spare room after image encryption (VRAE) and reserve the spare room before image encryption (RRBE). In the conventional VRAE schemes, the content owner needs only to encrypt the image directly and the spare room is vacated on the fully encrypted image by the LSB flipping technique [5; 6; 7], LSB compression technique [8; 9], prediction error detection [10], and pixel rotation [11]. However, because the fully encrypted image lacks redundancy, most of these schemes cannot totally extract the accurate embedded data, and all of them are unable to recover the original image without loss. Therefore, recent schemes have proposed some special image encryption methods, such as homomorphic encryption [12; 13; 14] and block based encryption [15; 16; 17; 18], to recover the original image and further improve the embedding capacity. With the enhancement of the capabilities of personal computing devices, some RRBE schemes [19; 20; 21; 22; 23; 24; 25] have been proposed to preprocess the original image before image encryption to reserve redundancy by compressing the original image. As the original image has sufficient redundancy, the embedding capacities of RRBE schemes are higher than those in VRAE schemes. Reversible data hiding in encrypted 3D mesh models finds practical applications across diverse fields to enhance their utility and security. In healthcare, patients' information can be embedded into encrypted MRI scans to ensure patient privacy during transmission and storage to facilitate secure remote consultations and diagnoses. In the manufacturing industry, CAD models of proprietary designs can incorporate watermarking to safeguard intellectual property rights and prevent unauthorized replication or modification of critical components. Moreover, 3D models of characters and environments can embed copyright information to ensure the protection of creative assets in virtual worlds and gaming environments in the field of digital entertainment. In the realm of augmented reality (AR) and virtual reality (VR), encrypted 3D models enriched with additional data can enable immersive training simulations for aviation and defense, where sensitive information needs to be securely integrated into training modules. Therefore, introducing reversible data hiding technique into encrypted 3D mesh models has broad application scenarios. As the 3D mesh model is an emerging file format, there are only a few schemes that have explored reversible data hiding in encrypted 3D mesh models.