###### Abstract  
Reversible data hiding in encrypted images (RDHEI) is a technology which embeds secret data into encrypted images in a reversible way. In this paper, we proposed a novel high-capacity RDHEI method which is based on the compression of prediction errors. Before image encryption, an adaptive linear regression predictor is trained from the original image. Then, the predictor is used to obtain the prediction errors of the pixels in the original image, and the prediction errors are compressed by Huffman coding. The compressed prediction errors are used to vacate additional room with no loss. After image encryption, the vacated room is reserved for data embedding. The receiver can extract the secret data and recover the image with no errors. Compared with existing approaches, the proposed method efficiently improves the embedding capacity. reversible data hiding; image encryption; linear regression; Huffman coding 2021 9 2166 10.3390/math9172166 10.  
from encrypted images or decrypted images. In [11], some pixels were selected from the original image, and they were estimated by their surrounding pixels to generate estimation errors. Histogram shift was used to modify these estimation errors for embedding data. In [12], an interpolation technique in [13] was used to generate interpolation error of pixels, and a designed histogram shift scheme was used for embedding data. In [14], the image was divided into patches, and each patch was encoded by the patches-level sparse coding to vacate additional room. In [15], the most significant bit (MSB) planes of the image were divided into non-overlapping blocks, and each block was encoded by a designed sparse matrix coding scheme to vacate room. In [16], an MSB predictor was designed to vacate room in the MSB plane of the original image. In [17], an MSB plane rearrangement scheme and extended run-length coding were used jointly to compress the MSB planes for additional room. In [18], the median edge detector (MED) was used to predict multi-MSBs of each pixel, and a set of Huffman coding-based labels was used to label the prediction result of each pixel for additional room. In [19], the image was divided into non-overlapping blocks, and an adaptive reversible integer transformation was used for each block to vacate embedding room. In the CRBE methods, embedding room was created based on the designed image encryption scheme. At present, the idea that is used most extensively is that the image encryption scheme partially keeps the redundancy of the original image in the encrypted image. In [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31], different block-level image encryption schemes were used to maintain the spatial correlation inside the blocks of the encrypted image. In [20], the image was encrypted by a stream encryption scheme that consisted of block-level permutation and block-level bit-XOR. The proposed histogram shifting-based RDH methods can be used directly on the encrypted image. In [21], a cross division image encryption scheme was used to keep spatial correlation in cross blocks, and a difference histogram shifting scheme was used for hiding data. In [22], a pixel value ordering scheme was designed to embed data into the 2 \(\times\) 2 encrypted blocks. In [23], a block-level prediction-error expansion scheme was used to embed data into the 2 \(\times\) 2 encrypted blocks. In [24], a run-length coding compression and a matrix compression were used to compress 2 \(\times\) 2 encrypted blocks to create room for embedding data. In [25], a multi-level histogram shifting scheme was used for each encrypted block to embed data. In [26; 27; 28], different encoding schemes were used to compress the MSB planes of each encrypted block. In [29; 30], for each encrypted block, one pixel was used to label the other pixels, so that additional bits could be spared for embedding data in the other pixels. In [31], a block histogram modification scheme was used to embed data into the LSB planes of each encrypted block.