Data hiding technology is an essential technology in the field of data security. At the early stage of development, it can only guarantee the security of the secret data. Still, it cannot fully recover the cover image at the receiver, such as the least significant bit (LSB) algorithm [1]. Research focuses on reversible data hiding (RDH) at the second stage of technology development to achieve lossless recovery of the cover image. It has been widely applied in military, medical, and information security applications. In the last two decades, RDH technology has flourished [2]. General RDH algorithms are based on three types of techniques: lossless compression ^[3][4][5][3,4,5], extended transformations ^{[6][7][8][9][10][11]}[6,7,8,9,10,11], and histogram shifting (HS) ^[12][13][12,13]. With these techniques, many algorithms can achieve better performance. For example, in ^[12][13][12,13], a general framework for constructing HS-based RDH is used to effectively achieve high capacity and low distortion by employing specific shifting and embedding functions.

The above traditional RDH methods are used in plaintext cover images. During data transmission, the visual quality of the transmission image is close to that of the original cover image because of the technology demand, so it is not easy for the secret data to be discovered. However, in this case, the content of the cover image is constantly exposed. Data security has become increasingly important in recent years with the continuous development of data privacy protection technology. On the one hand, the data sender does not trust the transferred server because the data are easily stolen. On the other hand, the plaintext cover image is more likely to be cracked. Then, RDH in encrypted images (RDH-EIs) was developed to meet the needs of privacy and security. RDH-EI technology embeds data in encrypted images so that the contents of secret data and the cover image are not visible during transmission. This benefit has significant implications for military communications, private medical data, trade secret data, and other data security areas. Therefore, it has been a research hotspot in recent years.

The existing RDH-EI schemes are mainly divided into two main categories: vacating room after encryption (VRAE) ^{[14][15][16][17][18][19][20][21][22]}[14,15,16,17,18,19,20,21,22] and reserving room before encryption (RRBE) ^{[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]}[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. They are based on the order in which the hidden room is vacated and image encryption. In general, the vacating room and reserving room mean analyzing the current image through certain techniques to create some space for other data hiding. The space can be empty bits, and data will be placed directly in empty positions. It can also be locations with data, but the data can be modified to reflect embedded information and can be completely restored.

Under the VRAE framework, the sender first encrypts the cover image, then uses some technologies to make space in the encrypted image, and finally uses the data hiding key to embed data in the vacated room. Zhang [14] realized data hiding through flipping the low 3 bits LSB of pixels, but there is error data extraction. Hong et al. ^[15][16][15,16] improved the wave function and used the correlation of pixels to reduce the error rate. To vacate the room, the LSBs of encrypted pixels were compressed with a special matrix [17]. Kim [18] used Hamming code to hide data in compressed and encrypted image blocks and introduced quantization to adjust the embedding capacity. Block encryption ^{[19][20][21][22]}[19,20,21,22] was used to preserve the correlation in the encrypted image, and then traditional RDH technologies such as HS and difference expansion were designed to embed data.

In the above schemes, some methods may have error codes in data extraction or image recovery, which are not completely reversible. Some methods are completely reversible but limit the embedding capacity. Therefore, the RRBE framework was developed to improve the situation. Compared with the previous VRAE schemes, RRBE schemes are more convenient for the high embedding capacity, reversibility, and security.

Under the RRBE framework, the sender pre-processes the cover image before encryption to reserve room or generate a new image format conducive to hiding. In previous studies, classifying image blocks and pixels by using some rules is a common technical means. Ma et al. [23] classified image blocks by smoothing function calculation to reserve the LSBs of pixels in blocks. Qin et al. [24] divided image blocks according to the pre-designed threshold and reserved room by compressing some blocks’ LSB. Wu et al. [25] classified the cover image into blocks with different scales. Then, the lowest two bits of each pixel were used to vacate room. The pixels were sorted according to the predefined fixed classification mode in ^[26][27][26,27]. For the above methods, the adaptive classification method ^[23][24][25][23,24,25] has the advantage of mining the hidden space as much as possible but requires uncertain auxiliary information. The fixed classification methods ^[26][27][26,27] have fixed costs but limit the amount of hidden space. Prediction technology is also applied through high performance ^[28][29][30][28,29,30]. Most significant bit (MSB) prediction was proposed in ^{[31][32][33][34]}[31,32,33,34] based on the high correlation of the adjacent pixel values of the cover image, which significantly improves the embedding capacity and the image recovery quality. Interpolation techniques ^[26][35][36][26,35,36], compression ^[24][37][38][24,37,38], and HS [39] have also been further studied and applied to encrypted images.

These schemes all use a single hider and maintain the amount of data that is always kept consistent with the size of the cover image during transmission. In addition, in recent years, some scholars have proposed multiple hiders schemes ^[40][41][42][40,41,42] based on secret sharing, which has also promoted the development of this field.

The above technologies have improved recovery quality and embedding rates. However, there may be problems with reserving room and how data embedding can be achieved after encryption.

It can be found that some problems exist in previous studies, such as the fact that the secret data cannot be extracted completely and accurately in some algorithms ^[14][15][16][14,15,16], so complete reversibility cannot be achieved. The embedding capacity needs to be improved ^{[19][20][23][33]}[19,20,23,33], but the image recovery quality will be affected. The joint schemes at the receiver ^{[14][26][27][37]}[14,26,27,37] and others need to process the data in order for the operation to be simple but low in flexibility; the separable schemes such as ^{[17][18][19][20][21][22][23][24][25][33][36]}[17,18,19,20,21,22,23,24,25,33,36] are crucial to achieving independence in data extraction and image recovery but also increase the complexity. Therefore, RDH-EI is more challenging than traditional RDH. Balancing the embedding rate, the accuracy of secret data extraction, the quality of image recovery, and separability is a technical challenge.