## 1 Introduction  
Data-hiding encompasses steganography and watermarking techniques, primarily leveraging the information redundancy of carrier data, to embed additional data. This embedding serves to fulfill security functions, including privacy protection, covert communication, digital signatures, and copyright authentication. Among modern data-hiding technologies, reversible data-hiding in the encrypted domain (RDH-ED) stands out as a prominent branch. Its defining characteristic lies in the encryption of the carrier used for data embedding. This approach mandates that the embedded data is not only precisely extracted from the ciphertext but also allows for the reversible restoration of the original plaintext [1]. RDH-ED addresses the dual security concerns of content privacy and secure communication, emerging as a crucial encrypted domain signal-processing technique. Its applications are wide-ranging, spanning telemedicine diagnosis, cloud services, and data security management in judicial or military contexts [2]. Many RDH-ED methods have been proposed [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25], depending on the different sources of redundancy. The prevailing RDH-ED technologies can be categorized into three primary types: vacating room after encryption (VRAE), vacating room before encryption (VRBE), and vacating redundancy in encryption (VRIE) [23; 24]. In 2011, ref. [4] introduced an RDH-ED approach based on stream cipher encryption. During the calculation, the fluctuations within each image block are calculated, but the pixels along the four boundaries are ignored. To address this limitation, a novel and enhanced function was proposed in [5], to assess the complexity of each image block more accurately, thereby effectively reducing the average extraction error rate. Ref. [15] first proposed the RDH-ED method of VRBE. Before encryption, the owner segments the cover image into two distinct sets, \(S\) and \(T\). They embed the least significant bits (LSBs) from set \(S\) into set \(T\), creating space for additional data embedding, followed by the encryption of the cover image. This approach enables the data hider to embed additional data into the encrypted image, ensuring error-free data extraction and image restoration, thus achieving genuine reversibility. Subsequently, ref. [17] introduced a novel method termed high-capacity separable reversible data-hiding in encrypted images (HC-SRDHEI). Compared to [PERSON]'s methodology, it offers superior embedding capacity and improved decryption image quality. In 2018, ref. [18] introduced the most significant bit (MSB) prediction technique as a substitute for the conventional LSB-replacement embedding approach. Their method involves predicting pixels before image encryption and presenting two techniques: correcting prediction error (CPE) and embedding prediction error (EPE). While these effectively enhance embedding capacity, they fall short of fully reversibly restoring the original image. Later that year, ref. [20] refined the EPE method through a recursive approach, achieving complete reversible recovery of the original image and boosting embedding capacity. Following this, ref. [13] put forth a novel reversible data-hiding in encrypted images (RDH-EI) method based on adaptive MSB prediction. This advanced technique takes full advantage of pixel correlations within blocks, thereby achieving an optimal capacity increase while ensuring reversibility and separability. To further refine the method proposed [13], ref. [14] introduced a new approach that combines adaptive quadtree partitioning (AQP) with MSB prediction. In this method, regions characterized by smooth textures exhibit superior embedding performance, whereas regions with more intricate textures demonstrate poorer embedding capabilities.