## 1 Introduction  
Advancements in cloud service technology and mobile communication terminals have led to an increased upload of multimedia data to cloud servers for storage via the Industrial Internet of Things (IIoT) [1]. This trend amplifies security risks concerning users' private data. In response to this, the technology of reversible data hiding in encrypted images (RDH-EI) [2, 3] has been developed to enhance data security in cloud environments, and facilitate the authentication and management of encrypted data. This technique allows for the embedding of secret data, such as check digits, into encrypted images while maintaining the privacy of the embedded secrets. Moreover, it enables receivers to fully recover the image and extract the secret data accurately. The transmission of information in cloud service environments is intricate and diverse. Current RDH-EI algorithms are typically categorized based on their suitability for single-user or multi-user scenarios. Single-user-oriented algorithms fall under three main classes: vacating room after encryption (VRAE) [4, 5], vacating room before encryption (VRBE) [6, 7], and vacating room in encryption (VRIE) [8, 9]. While most VRAE methods [10] employ lightweight ciphers to preserve pixel correlations when encrypting the carrier image, it has been shown that these ciphers, such as group substitution, are vulnerable under ciphertext-only attacks, as highlighted by [PERSON] et al. [11]. To mitigate these vulnerabilities, some embedding algorithms based on homomorphic encryption [12; 13] have been proposed. However, VRAE algorithms often face limitations in embedding capacity due to the high entropy of ciphertext state data. On the other hand, VRBE algorithms [14; 15] directly utilize correlations between plaintext pixels to embed additional redundancy space. For instance, [PERSON] et al. [16] employed Huffman coding to compress the most significant bits (MSBs) of image pixels, creating embeddable space and maintaining high-quality image recovery [17]. [PERSON] et al. 's algorithm [18] enhances embedding capacity through weighted prediction techniques, although it remains constrained by image texture features. Breaking through these limitations, VRIE algorithms, like that of [PERSON] et al. [9], use random number substitution to embed secret data during the encryption process. While these algorithms enable covert communication, they are typically limited to single data hiders, falling short in meeting the multi-user data storage demands and thereby limiting their practicality in IIoT scenarios. In 2020, [PERSON] et al. introduced a hiding model based on secret sharing and multi-party embedding [19], distributing secret shares among various data hiders to achieve independent embedding. [PERSON] et al. 's algorithm adeptly conceals communication activities during multi-party data exchanges [20]. [PERSON] et al. [21] elaborated on the application of the RDH-EI algorithm in the context of the double-blind Inter Partes Review (IPR) scenario, as depicted in Figure 1a. The applicant entrusts the review committee with an encrypted image representing the patent file, which subsequently forwards the patent file and authorization data to each examiner. If the reviewer approves the patent, authorization data will be embedded into encrypted images to produce marked encrypted images. When the administrator extracts sufficient authorization data from the collected marked encrypted images, the patent passes the review. In 2022, [PERSON] et al. [22] proposed an RDH-EI algorithm based on feedback secret sharing, enhancing the security of existing algorithms. [PERSON] et al. [23] also proposed an RDH-EI scheme based on matrix-based secret sharing (MSS), enabling the complete restoration of the carrier image. In 2023, [PERSON] et al. [24] proposed an RDH-EI scheme based on ciphertext sharing and hybrid coding to enhance embedding capacity. [PERSON] et al. [25] also introduced a high-capacity RDH-EI scheme based on preprocessing-free secret sharing.