## 1 Introduction  
Data-hiding has a long history and has been widely used for information security for centuries. Moreover, a good review of the recent works appeared in [1], which classified modern data-hiding techniques and cryptography as two distinguishable domains for information security and explained comprehensive analysis of information-hiding techniques. An interactive buyer-seller watermarking protocol has been proposed to not allow invisible watermarked copy [2], and a secure comparison protocol in the encrypted domain was also proposed [3]. New techniques to hide information in encrypted images have drawn much interest of researchers who have extended its applications. Data-hiding in encrypted images involves several different problems since it is impossible to directly use the contents of encrypted data. Therefore, applications of data hiding in the encrypted images have been explained in a number of papers [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. The usage of an encrypted image as the cover medium for concealing data is not only a relatively new research subject, but an interesting subject due to simultaneous consideration of both cryptography and steganography technology, which have been independently developed. An encrypted image can be protected when it is transmitted through a public channel, increasing the difficulty of analyzing the embedded data. Among data-hiding technologies, reversibility is one of the major research topics since an encrypted image should be recovered to the original image after extracting the hidden data. Reversible data-hiding (RDH) systems are designed to assign embedded data to a cover media and to allow recovery of the original image without distortion after extraction. Especially, for military and medical images, which are important applications of data-hiding, there should be no distortion in the recovered image due to data hiding. The embedding procedure of the RDH systems with flipping least significant bit (LSB) is first to divide an encrypted image into blocks and to use the LSBs of pixels in the blocks according to the embedded data. In the receiver, spatial correlation in the natural digital image is calculated to estimate the embedded data. [PERSON] (2015) first proposed RDH systems with flipping LSBs and suggested the fluctuation function by using the four neighboring pixels to measure the spatial correlation between pixels in each block. [PERSON]'s system [PERSON] (2015) separately calculated the horizontal and vertical fluctuations of the correlation. Moreover, [PERSON] (2015) proposed a side-match technique that the updated border of neighboring blocks be used for calculating the correlation in order to improve the spatial correlation, whereas the marginal pixels of the block are not used in [PERSON] (2015). Modified fluctuation functions were proposed according to the position of pixels in block Hong (2015), and the functions were composed of three different equations. Reversible data hiding systems were proposed by using a new embedding pattern and multiple judgments [PERSON] _et al._ (2016). These RDH systems perfectly recovered the original image if there was no error in the extracted data. The RDH about lossless data compression are based on data compression to make new space for the data hider to hide data. Research about lossless data compression was designed to develop efficient data compression to reduce image distortion. In [PERSON] _et al._ (2016), a separable RDH scheme for encrypted images was proposed by compressing the encrypted data using a source coding scheme with side information, making data extraction independent of encryption. An efficient data compression from low-density parity-check codes was proposed, and a new data extraction method using side information was discussed in [PERSON] _et al._ (2016) to enhance rate-distortion performance.