## 1 Introduction  
Image data hiding involves concealing information within a publicly shared image in a manner that remains imperceptible to casual observers. The concealed data can be extracted by the intended recipient and serves various purposes, such as covert communication and copyright protection. Many techniques have been used to design data-hiding schemes, such as least significant bit substitution [1] and matrix coding [2]. However, traditional data-hiding methods often result in significant distortions to the cover image. To tackle this challenge, reversible data hiding (RDH) [3] has been introduced. RDH not only enables the extraction of hidden data but also ensures the recovery of the original image. This technology finds applications across various fields, including social media, medical data, and financial information. Moreover, RDH facilitates covert data transmission and storage, enhancing data transmission security and making it a preferred option in cloud computing and the Internet of Things (IoT). Additionally, RDH plays a vital role in safeguarding digital copyrights by embedding copyright information to enable data traceability and content protection. There are many RDH schemes, such as difference expansion [4], histogram shifting [5], prediction error expansion [6], and lossless compression-based schemes [7]. These schemes can be effectively used in scenarios that require high reversibility. Recently, numerous studies have significantly enhanced the performance and efficiency of RDH by integrating deep learning [8; 9], blockchain technology [10; 11], and IoT technology [12] into algorithm optimization. However, in certain situations, such as in cloud computing services, safeguarding the privacy of carrier images is paramount. Consequently, reversible data hiding in encrypted domain (RDH-ED) [13] has been proposed to address this need. RDH-ED refers to the embedding of additional information in an encrypted carrier, which is characterized by the fact that the carrier used for embedding is encrypted. This requires that the embedded information can be accurately extracted and that the original image can be recovered without any loss. The key to RDH-ED is to find and utilize redundant space in the carrier. Based on the method of generating redundancy, embedding algorithms can be classified into three categories: vacating room after encryption (VRAE) [14; 15], vacating room before encryption (VRBE) [16], and vacating redundancy in encryption (VRIE) [17; 18]. The VRIE scheme seamlessly integrates data hiding with cryptography, leading to enhancements in security, reversibility, and embedding rate (ER). However, the current cryptographic algorithms applicable to the VRIE schemes are difficult, so the research is limited. While these schemes offer complete reversibility, it is crucial to acknowledge that if the tagged encrypted image is lost due to a single point of failure, the embedded secret data and images become irretrievable. Therefore, distributed storage is needed. Moreover, with the rapid development and application of cloud computing, distributed environments represented by distributed cloud services [19], edge computing [20], and federated learning [21] continue to emerge. However, traditional RDH-ED schemes introduce only a single participant and cannot meet the security requirements of distributed systems [22; 23]. In recent years, several RDH-ED schemes based on secret sharing (SS-RDHED) [24] have been proposed. The threshold effect of secret sharing makes it more disaster-tolerant and suitable for distributed scenarios. In 2018, [PERSON] et al. [25] first proposed a VRAE SS-RDHED scheme, which embeds additional data into the shared image after secret sharing using difference expansion and prediction error histogram panning, with low computational complexity and high embedding capacity. [PERSON] et al.