## I Introduction

In the past decades, with the increasing demand for information security, reversible data hiding(RDH) technology [1, 22, 23, 4]have been widely studied. Reversible data hiding can restore the embedded information and the original carrier without loss. That is to say, the reversible data hiding technology can not only embed and extract the additional information, but also restored the carrier to the original state. At present, many reversible data hiding methods have been proposed, which can be divided into three categories according to different technologies: lossless compression [7, 35, 19]. difference expansion [25, 11, 12]. histogram shifting [14, 24, 15, 29, 34, 26]. As the demand for privacy and data security continuously strengthen, the wide application of encryption algorithm makes the signal processing of encrypted domain generalised. This application provides a good platform for reversible data hiding in the encrypted domain, at the same time, puts forward a new challenge: carrier of encrypted domain lost the expressly structure redundancy, cause traditional reversible data hiding algorithm failed to apply for encrypted domain.

To meet this challenge, reversible data hiding in encrypted images (RDHEI) [37, 18, 10, 6] are proposed and developed in recent years. RDHEI method is gradually divided into two different types: vacating room after encryption (VRAE) [38, 8, 39, 33], reserving room before encryption (RRBE) [13, 30, 17, 31, 27]. The VRAE method vacated room for embedding after encryption. Zhang proposed a block-splitting method for the encrypted image in [38]. By flipping three Least Significant Bit(LSB) of pixels in the block, additional information can be embedded and extracted. Although this method can extract the information and restore the image, it may produce errors in some regions. In order to improve this method, Hong in [9] proposed a spatial correlation and edge matching mechanism between adjacent blocks to reduce the error rate in the recovery stage. Recently, an adaptive coding method has been proposed in [8] and applied to RDHEI. The adaptive coding method is used to compress the Most Significant Bit(MSB) of the encrypted image to vacate the room for embedding information. These methods make use of the correlation between the pixels of the encrypted image to vacate room for embedding information. Although the information can be extracted and the original image can be restored, insufficiencies such as low embedding capacity and bit error rate in the information extraction or image recovery stage also existed.

Embedding capacity is an important indicator to measure the performance of RDHEI. To further improved the embedding capacity, Ma in [13] put forward a RDHEI method of reserving room before encryption for the first time. The RRBE method combined the traditional method of RDH and encryption methods well. Using the traditional RDH method to reserve room before encryption, so that the reserved room can be used for embedding information. This method can not only protect the original image information on being leaked, but also realize the real reversibility. Then, many methods are proposed to improve the performance. Zhang proposed a RDHEI method combining the prediction error histogram with encryption in [36], which embedded information in the prediction error histogram. In [16], the MSB of the pixel is predicted to mark, so as to reserve the room where the information can be embedded. This method uses the MSB which is easier to prediction instead of LSB for a RDHEI, and obtains a high capacity. Recently, Qiu proposed a RDHEI method in [20] that utilizes reversible integer transformation to generate data redundancy, embedding information in the encrypted redundancy position. These methods greatly improve the embedding capacity and provide more embedding space for additional information.

In recent years, some high-capacity RDHEI methods use predicted value are proposed. Yi in [31] proposed a method of parameter binary tree label, pixels with different range prediction error are marked in the image respectively, and then information hiding is carried out in the reserved room of marker pixels, so as to obtain a high embedding capacity . In [32], compare the binary form of the original pixel and the predicted value, Then, record the number of the same bits from MSB to LSB and encoding it with Huffman coding. Finally, the reserved room in MSB is substituted by additional information. In [27] Wu divided images into LSB images and MSB images, and embedded LSB images into MSB images by using histogram translation of prediction error, so as to embed information in LSB.

Among the aforementioned methods, the embedding capacity is improved by exploiting the spatial redundancy of the original image. This indicates that finding a suitable carrier with higher redundancy is very important to improve the performance of RDHEI method. Of course, it is also important to make the most of these redundancies. Based on this, this paper puts forward a RDHEI method based on bit plane compression of prediction error. It not only selects the prediction error with high redundancy as the compression carrier, but also proposes a joint compression algorithm, this algorithm can reserve more room to improve the embedding capacity of RDHEI method.

In this paper, firstly, the prediction error bit planes is lossless compressed, and the compressed bit stream is connected to the uncompressed bit stream of bit plane to reserve room in LSB planes. Then the compressed image is encrypted using a stream cipher to prevent the information from leaking. Next, the information hiding device can embed the additional information into the reserved room of the encrypted image. Finally, the image receiver can extract the additional information and recover the original image. The experimental results show that this method greatly improves the embedding rate of RDHEI.

This paper has the following two contributions:

A joint compression algorithm on bit plane compression is proposed in this paper. This compression algorithm combine the Huffman coding and run-length coding well, with this method, the bit plane can be compressed efficiently.

A RDHEI method based on bit plane compression of prediction error is proposed. This method make well use of the correlation between adjacent pixels, obtains a higher capacity compared with state-of-the-art RDHEI methods.

The rest of this paper will be introduced from the following aspects: Section ii@ introduced the joint compression algorithm for bit plane reconstruction. The detailed description of the RDHEI method based on the bit plane compression of prediction error is shown in Section iii@. Section iv@ analyzed the experimental results. Finally, Section v@ summarized the whole paper.

## Ii Bit plane compression algorithm

To take advantage of the spatial redundancy of the image, this paper proposes a joint bit plane compression algorithm. For the carrier image with high redundancy, the adjacent bits of the bit plane are often the same, compress the bit stream of bit plane can reduce the room occupied by the original carrier image information, thus providing room for additional information.

The carrier image can be divided into eight bit planes, and each bit plane has its corresponding bit stream. In this paper, a compression algorithm combining Huffman coding and run-length coding is adopted to compress these bit streams. The concrete compression algorithm is introduced in Section ii@-A. Section ii@-B introduced the special treatment of compression algorithm.

### Ii-a Joint compression algorithm

The bit plane bit stream of the carrier image with high redundancy contains many bit strings of the same bit. By compressing the bit stream, more room is reserved for embedding information. We define the length of bit string as , compare with a predefined parameter named , if , treat the current bit string as a long bit string, otherwise, regard the current bit string as a short bit string. In the bit stream compression algorithm, the short bit string are compressed by Huffman coding, while the long bit string are encoded by run-length encoding. The compression algorithm is described as follows:

Case1: When , it is judged as a short bit string, and the Huffman coding is adopted. The encoding consists of two parts: the prefix and the bits starting from the head of current short bit string. is the flag bit, and means that the current bit string is a short code word, compress it with Huffman coding. As for the

bits, record the occurrence probability of each bit string, and encoding these bit strings with Huffman coding. The Huffman coding proposed in Section ii@-A is used to compress.

Case2: When , it is determined to be a long bit string, and the run-length coding is adopted. The run-length coding consists of three parts: the prefix , the middle part and the suffix . is the flag bit, equal to means it is a long code word. is represented bits binary form of , is a predefined positive integer. If and , then ; The suffix is represented by or to represent its repeating bit.

Coding in sequence until traversing the entire bit stream. Taking a part of bit stream as an example, the bit stream with the length of 23 is ’00000000000000000101010’. When and , the corresponding Huffman coding Table is shown in Table i@. Combined with the Huffman coding table in Table i@, the specific compression of bit stream is as Algorithm 1.

Bit string | 001 | 010 | 011 | 100 | 101 | 110 |
---|---|---|---|---|---|---|

Huffman coding | 011 | 010 | 101 | 00 | 11 | 100 |

After the compression algorithm, the bit stream is compressed into ’01000101111010’, whose length is 14. Obviously, the length of compressed bit stream is less than the length of the original bit stream, so compress the bit string can reserve room for embedding information.

### Ii-B Special treatment on compression algorithms

A joint bit plane compression method is obtained by special processing of Huffman and run length coding. As Table ii@ is shown, when compress Lena with Huffman coding, Run-length coding and joint compression algorithm respectively. The original bit planes size of Lena is 2097152 bits. After to compress Lena with different compression algorithms, the size of compressed image is 1957560 bits, 3393345 bits and 1271438 bits. The joint compression algorithm can obtain minimum compressed image. In other words, the joint compression algorithm is more effective than the single compression algorithm.

compression algorithm | original image | Huffman coding | Run-length coging | Joint compression algorithm |
---|---|---|---|---|

compressed image size/bit | 2097152 | 1957560 | 3393345 | 1271438 |

#### Ii-B1 Huffman coding

Huffman coding constructs the code word with the shortest average length according to the probability of character occurrence. When constructing the Huffman encoding table, the probability of each character’s occurrence should be firstly calculated. The Huffman tree is then constructed from the two characters of the lowest probability (or minimum weight), and the sum of the probabilities of the two characters is deemed as the probability of new character, then compare the new probability of the probability of the remaining character. Repeat the above steps until traversed all the characters to form a complete Huffman tree. Finally, mark each edge of the generated Huffman tree corresponds to and (left side corresponds to , right side corresponds to ), so that the Huffman coding corresponding to each character is obtained.

As shown in Fig.1, character A,B,C and D can generate a Huffman tree according to the probability of occurrence, and then encode characters according to the corresponding value of the edge to obtain the Huffman coding table. Table iii@ shows the Huffman code corresponding to the Huffman tree in Fig.1. In this way, the code with the shortest average length can be obtained.

character | probability | Huffman coding |

A | 0.1 | 010 |

B | 0.2 | 011 |

C | 0.2 | 00 |

D | 0.5 | 1 |

To apply Huffman coding to the bit plane compression algorithm proposed in this paper, some special treatments are needed. First, a flag bit named is adopted to distinguish the current compression coding. if current bit string is compressed by Huffman coding, the flag bit . Then, due to the bit stream of bit plane includes a lot of short bit strings, coding all of these short bit strings with Huffman coding doesn’t work very well. Therefore, we select the fixed-length short bit strings for Huffman coding, so that better compression effect can be obtained, this fixed-length is depending on the predefined parameter .

#### Ii-B2 Run-length coding

To obtain better bit plane compression effect, this paper not only uses Huffman coding, but also uses run-length coding to obtain a joint bit plane compression algorithm with higher compression rate.

The run-length coding consists of three parts, respectively is a prefix , a infix and a suffix . The prefix is used to represent the type of current bit string, represents the current bit string is a long bit string and adopt the run-length coding to compress. The infix represents the binary form of the bit string length L, a total of bits. a suffix that represents the current repetitive bits. when the length of compressible bit string is , adopt the run-length coding to compress the bit string. The prefix . The infix , the number of bits is represented by the predefined parameter . The suffix represents the current duplicate bit, that is .

## Iii RDHEI method based on bit plane compression of prediction error

In Section ii@, we mentioned that the compression algorithm is more suitable for carriers with high spatial redundancy. Therefore, combined with the compression algorithm proposed to Section ii@, this paper proposes a reversible data hiding method in encrypted images based on bit plane compression of prediction error. The proposed compression algorithm is used to compress each bit plane of the carrier, then the compressed bit plane is reconstructed, and the reconstructed carrier with more reserved room can be obtained.

As shown in Fig.2, the proposed method is divided into three stages: The content owner preprocesses the image to reserve room and encrypts the image; Information hiding device embeds additional information in encrypted images; The image receiver performs information extraction and image recovery. In the first stage, in order to reserve more room, the processed prediction error of the original image is used for bit plane rearrangement and compression. The details are described in Section iii@-A and iii@-B. The image encryption is then performed in Section iii@-C. In the second phase, Section iii@-D describes the embedding of additional information. For the receiver, the additional information can be extracted correctly and the image can be recovered losslessly, Section iii@-E describes this process.

### Iii-a Processing of prediction error

For any pixel in the original image with the size of , where , . The first row and first column pixels are recorded as reference pixels without any operations. As shown in Fig.3, the unprocessed predicted value of is calculated according to the median edge predictor(MED), and the formula is as follows:

(1) |

Next, according to the pixel value and its predicted value of the original image, the prediction error is calculated as follows:

(2) |

After the above steps, the unprocessed prediction error is obtained. As is shown in Fig.4, only pixels with prediction error of , named as available pixels are used, and the pixel value beyond the prediction error range is denoted as overflow pixels, without any modification. Then, convert the prediction error between into an eight-bit binary number. The highest bit represents the symbol marker bit. When the prediction error is negative, the highest bit replaced by , if not, replace the highest bit with . The lower seven bits represent the binary form of its absolute value. After above operations, the processed prediction error which can be compressed is obtained. For example, if , convert it with above description and can be obtained, where denotes as the pixel value of the processed prediction error. The bit plane calculation formula of the processed prediction error is as follows:

(3) |

represents the value of prediction error, and represents the bit binary value of . After traversing all the values in processed prediction error, eight bit plane of the processed prediction error is obtained.

### Iii-B Bit plane rearrangement of prediction error

To make use of the correlation between adjacent pixels of an image, Chen proposed a bit stream rearrangement method in [5]. As shown in Fig.5, the bit plane is first divided into non-overlapping blocks of the same size , and then generated four kinds of bit plane rearrangement methods according to different rearrangement modes within and between blocks. The bit plane rearrangement type is composed of two bits. The first bit represents the rearrangement within the block. When it is , it represents the arrangement within the block row by row, and when it is , it represents the arrangement within the block column by column. The second bit represents the rearrangement between the blocks, and represent the same meaning as above. After the bit plane rearrangement, each bit plane corresponds to four different bit streams.

### Iii-C Compression of bit plane and image reconstruction

Since the adjacent pixels of the original image are correlated, the adjacent bits of the corresponding prediction error bit plane are often the same. For the processed prediction error, the bit plane could be compressed to reserve room for embedding information. Then the compressed bit plane can be reconstruct into a image. Experiments show that this algorithm can effectively compress the bit plane of processed prediction error. The compression has been introduced in Section ii@. The description of prediction error bit plane compression is shown in Section iii@-C-1. The reconstruction of compressed bit plane is described in Section iii@-C-2.

#### Iii-C1 Compression of prediction error bit plane

The complete bit plane compression algorithm of processed prediction error is introduced next. It mainly includes the following steps:

Eight bit planes of processed prediction error were obtained by using the prediction error calculation method mentioned in Section iii@-A.

The bit plane rearrangement method proposed by Chen in [39] was used to rearrange the eight bit planes of prediction error. The detailed process has been described in Section iii@-B.

After , each prediction error bit plane generated four different bit streams according to different rearrangement types. In order to obtain efficient Huffman coding table with less time, the generation of Huffman coding is determined according to the compression effect of long bit strings. It is assumed that only the compression of long bit strings (When ) is considered, according to the bit stream compression algorithm described in Section ii@, each bit plane will generate four kinds of bit streams that only compress long bit strings. Then the shortest bit stream after compression is selected and its rearrangement type is , is the bit plane index. The subsequent rearrangement of the bit plane only uses . For example, if the shortest bit plane rearrangement type of bit plane is , after that, the bit plane all adopts the as bit plane rearrangement mode. With the eight rearrangement type ,…, , eight bit streams are generated. Then according to the occurrence probability of each short bit string within the eight bit streams, a complete Huffman coding table is generated.

With the compression algorithm introduced in Section ii@, compress the rearrangement bit plane of prediction error obtained in step3 orderly.

After the above compression operations, the bit plane of prediction error has been compressed. After each bit plane is compressed, the corresponding compressed bit plane information is obtained. Compared the compressed bit plane information with the original bit plane. If the compressed bit plane information is greater than the original bit plane, no compression is performed. If less than, compress and mark the current bit plane. The compressed bit plane information is part of auxiliary information, include compression marker bit, bit plane rearrangement type, length of bit stream after compression, and bit stream of compressed bit plane. The compression marker bit is used to judge whether the current bit plane is compressed or not. When the flag bit is , it means the current bit plane can be compressed, and a flag bit of indicates that the bit plane will not be compressed. Then the image reconstruction is carried out.

#### Iii-C2 Image reconstruction

Before reconstruct the image, the auxiliary information should be introduced. The auxiliary information consists of the following parts:

The predefined parameters block size n, and ;

Huffman coding rules and information of overflow pixels;

The compressed bit plane information;

The net compressed space size, it means the net reserved room.

To losslessly recover the compressed bit stream, As shown in Fig.6, while reconstructing the compressed image, the auxiliary information except the net compressed space size should be stored in the MSB planes, the net compression space size is recorded in the LSB. For the blank region of compressed bit plane which means the net reserved room, fill it with . Finally, the reconstructed compressed image is obtained. The calculation formula of the reconstructed image is as follows:

(4) |

Where represents the value of reconstructed image , and represents the bit binary value of .

### Iii-D Image encryption

In the image encryption stage, firstly, an

pseudo-random matrix

is generated by the encryption key . The pixel value of the reconstruct image and the corresponding value in the pseudo-random matrix are first converted into an eight-bit binary number. The formula is as follows:(5) |

Where represents bit of the binary form of and , . Then implement each bit with xor operation to achieve encryption. Noticed the bits in the LSB that store the net compression space size are not encrypted. The encryption calculation formula is as follows:

(6) |

Where is the bit of the current binary number, represents the bit value of the binary pixel value of the encrypted image, represents exclusive or operation. Finally, translate the encrypted binary value into the decimal, that is, the encrypted reconstruct image .

### Iii-E Embedding additional information into the encrypted image

In this section, while embedding the additional information into the reserved room, the net compression space size recorded on the LSB is extracted firstly and the location of the reserved room is obtained based on it. Then, the additional information is encrypted with the information hiding key . Finally, the encrypted additional information is embedded into the reserved room by bit substitution. In the end, generate a marked encrypted image .

### Iii-F Information extraction and image recovery

In the stage of information extraction and image recovery, the image receiver first extracts all information of eight bit planes, then extracts the net compression space size in the LSB to locate the position of the embedded additional information, so the bit plane bit stream can be divided into two parts: bit stream of auxiliary information bit stream of embedded additional information. The above operation can be carried out without any key. According to the key held by the image receiver, it can be divided into the following three cases:

#### Iii-F1 Only information hiding key

When the image receiver has only the information hiding key , he can only extract the embedded additional information. Firstly, obtain the bit stream of embedded additional information from the bit stream of eight bit planes. Then decrypt the bit stream of the embedded additional information using the information hiding key , the decryption formula is shown in formula (6). In this way, additional information can be correctly obtained.

#### Iii-F2 Only the encryption key

When the image receiver has only the encryption key , he can only recover the original image. The information of the compressed bit plane is extracted first and then decrypted using the encryption key , next extract the auxiliary information from the decrypted bit stream and then recover the prediction error according to the auxiliary information. Finally, restore the original image according to formula (2).

#### Iii-F3 Both the encryption key and the information hiding key

The embedded information can be extracted correctly by the key and the original image can be recovered by the key .

## Iv Experimental results and analysis

In order to verify the feasibility of the proposed method, a lot of experiments are carried out in this section. Experimental results show that the proposed method has a higher embedding rate than the current method. Firstly, we analyze the reversibility of the proposed method in Section iii@-A. Next,to get better performance, the parameters involved are optimized. The details are described in Section iii@-B. Then, compare the performance of this RDHEI method with three state-of-the-art methods [16],[32],[28] in Section iii@-C. As shown in Fig.7, six gray scale images were selected as the test images, respectively Lena, Baboon, Tiffany, Peppers, Man, Lake. Three data sets: UCID [21], BOSSbase [3] and BOWS2OrigEp3 [2] are used. The experimental results and analyses are introduced as follows.

### Iv-a Reversibility analysis

Fig.8 shows the Lena image in different stages. Fig.8 is a standard gray scale image Lena with the size of , Fig.8 shows the encrypted Lena image, Fig.8 is the encrypted image after the additional information is embedded, while Fig.8 is the recovered Lena image. It can be seen that the restored image is exactly the same as the original image.

To intuitively prove the invertibility of the proposed method, mean square deviation (MSE) is introduced here. The calculation formula of mean square error (MSE) is:

(7) |

Where and represent the recovered image pixel and original image pixel respectively.

When MSE is , it means the two images are the same. Therefore, to prove the reversibility of this method, calculate the MSE of the recovered image and the original image. As shown in Table 3, MSE of the test images are all . This means that the recovered image is exactly the same as the original image. In other words, the proposed method can be truly reversible.

Image | Lena | Baboon | Tiffany | Peppers | Man | Lake |
---|---|---|---|---|---|---|

MSE | 0 | 0 | 0 | 0 | 0 | 0 |

### Iv-B Parameter optimization

In the proposed method, there are mainly three parameters, block size , which is used to judge the type of code word, and the length of extend run-length code . According to the description in Section ii@-C, one should choose the compression algorithm that can reserved more room after compression, and the choice of block size, and will affect the compression effect. To optimize the parameters, two of the three parameters are kept unchanged, the remaining parameters are changed, and the parameter with the best performance is selected as the optimized parameter. The subsequent experiments were carried out with optimized parameters.

Lena | Baboon | Tiffany | Peppers | Man | Lake | ||

3 | 3 | 3.1084 | 1.409 | 3.1831 | 2.6918 | 2.6771 | 2.2428 |

4 | 3.1117 | 1.398 | 3.1864 | 2.6958 | 2.6832 | 2.2511 | |

5 | 3.072 | 1.3325 | 3.165 | 2.6626 | 2.6397 | 2.2383 | |

6 | 3.0364 | 1.2633 | 3.1292 | 2.6081 | 2.574 | 2.2094 | |

4 | 3 | 3.1144 | 1.4512 | 3.1893 | 2.7072 | 2.6887 | 2.2454 |

4 | 3.1296 | 1.4697 | 3.2014 | 2.7304 | 2.715 | 2.2722 | |

5 | 3.1243 | 1.4393 | 3.1951 | 2.7215 | 2.697 | 2.2523 | |

6 | 3.1055 | 1.3929 | 3.1796 | 2.6877 | 2.6452 | 2.2372 | |

5 | 3 | 3.1094 | 1.4664 | 3.1863 | 2.7024 | 2.7115 | 2.2444 |

4 | 3.1312 | 1.4989 | 3.216 | 2.7346 | 2.7478 | 2.2837 | |

5 | 3.131 | 1.4865 | 3.2058 | 2.7402 | 2.7428 | 2.273 | |

6 | 3.1221 | 1.4594 | 3.1928 | 2.7149 | 2.7143 | 2.2542 | |

6 | 3 | 3.1203 | 1.4882 | 3.2053 | 2.7119 | 2.7182 | 2.2775 |

4 | 3.1458 | 1.5263 | 3.243 | 2.7494 | 2.7594 | 2.3241 | |

5 | 3.1498 | 1.5245 | 3.2531 | 2.7589 | 2.7625 | 2.323 | |

6 | 3.1443 | 1.5066 | 3.2283 | 2.7427 | 2.7436 | 2.3047 |

, | ||||||
---|---|---|---|---|---|---|

Block-size | Lena | Baboon | Tiffany | Peppers | Man | Lake |

3.1249 | 1.511 | 3.1993 | 2.733 | 2.74 | 2.3174 | |

3.1423 | 1.5256 | 3.2372 | 2.7571 | 2.7571 | 2.3236 | |

3.1498 | 1.5245 | 3.2531 | 2.7589 | 2.7625 | 2.323 | |

3.1428 | 1.5146 | 3.2274 | 2.7526 | 2.7516 | 2.3174 |

Image | bit plane | bit plane | bit plane | bit plane | bit plane | bit plane |

Lena | 1 | 67.096 | 15.943 | 4.915 | 1.768 | 1 |

Baboon | 1 | 7.021 | 2.021 | 1.982 | 1 | 1 |

Tiffany | 1 | 70.148 | 14.694 | 4.853 | 2.066 | 1.030 |

Peppers | 1 | 93.723 | 15.464 | 3.325 | 1.161 | 1 |

Man | 1 | 65.397 | 12.693 | 3.277 | 1.198 | 1 |

Lake | 1 | 69.905 | 5.689 | 1.874 | 1.053 | 1 |

BOSSBase | BOWS2 | UCID | ||
---|---|---|---|---|

3 | 3 | 2.2307 | 3.6738 | 2.4650 |

4 | 2.3648 | 3.6512 | 2.4536 | |

5 | 3.3854 | 3.5932 | 2.4061 | |

6 | 3.3174 | 3.5235 | 2.3446 | |

4 | 3 | 3.5155 | 3.7183 | 2.5049 |

4 | 3.5278 | 3.7304 | 2.5206 | |

5 | 3.4987 | 3.7049 | 2.4982 | |

6 | 3.4542 | 3.6624 | 2.4584 | |

5 | 3 | 3.5329 | 3.7346 | 2.5204 |

4 | 3.5615 | 3.7640 | 2.5504 | |

5 | 3.5533 | 3.7586 | 2.5454 | |

6 | 3.5276 | 3.7339 | 2.5196 | |

6 | 3 | 3.5502 | 3.7512 | 2.5364 |

4 | 3.5885 | 3.7894 | 2.5750 | |

5 | 3.5910 | 3.7938 | 2.5791 | |

6 | 3.5756 | 3.7802 | 2.5635 |

Table v@ describes the embedding rate of the corresponding test image when n is 4, is from 3 to 6, and is from 3 to 6. It can be seen from Table v@ that when and , a high embedding rate can be obtained. Beyond this boundary, the embedding rate shows a decreasing trend. To explore the influence of block size on embedding rate, Table vi@ describes the image embedding rate when and , corresponding block size is , , and . It can be seen from Table vi@ that when the block size is , a high embedding rate can be achieved. As the block size continues to increase, the embedding rate will decrease. To prove that the selection of parameters is not related to the texture complexity of the image, this paper also carries out experiments on the image set. Similarly, keep two of the three parameters unchanged, select the parameters which can obtain best average performance. Just as Table viii@, when the block size , is from 3 to 6, and is from 3 to 6, the average performance of three image set is recorded. By analyzing the embedding capacity corresponding to different parameters, , and was selected as the parameters of the subsequent experiment.

### Iv-C Preference analyses

To directly analyze the compression effect of the proposed method, the compression rate of the test image is first calculated. The image compression rate is the space occupied by the image bit plane before compression divided by the actual space occupied by the image bit plane. As shown in table Table vii@, with the same compression method, the high-order bit plane of prediction error image can obtain a higher compression rate. The compression rate=1 means that the compressed bit plane is greater than the original bit plane. As for the bit plane which represents the symbol bit of the prediction error image, its compression effect is poor.

In order to further explore the performance of the proposed method, some experiments have been done to demonstrate the improvement of embedding capacity. Test images are encrypted and the additional information is embedded into the encrypted test images, experimental results show that the embedded additional information can be extracted completely, and the recovered image is consistent with the original image.

In order to better verify the performance of the proposed method, we compare the embedding capacity of the proposed method with three state-of-the-art RDHEI methods [16],[32],[28]. In this experiment, images of Fig.7 are also selected as the test images, and the maximum additional information that can be embedded in each method is selected as its embedding capacity. As shown in Fig.9, we first compare the embedding rate of the test images. It can be seen that the proposed method has a highest embedding rate, that is, it can provide more embedding room for additional information.

Compared with other methods, the RDHEI method in this paper has a great improvement in embedding rate. Although some invalid pixels cannot participate in the reserved room process in the method proposed in this paper, the number of such invalid pixels is very small and the effect is weeny. In order to further prove the universality of such performance improvement, this paper also conducts contrast experiment on image sets such as UCID, BOSSBase and BOWS2OrigEp3. The effectiveness of the proposed method is illustrated by comparing the average embedding rate of all images in the image set. As shown in Fig.10, it is a comparison graph of the average embedding rate of different image sets. Similarly, the method proposed in this paper can also obtain the best average embedding performance in the image set. As can be seen from the figure, the average embedding rate obtained by the method of this paper is significantly higher than that of the other three methods. Compared with methods [16],[32],[28], the average embedding rate in UCID of proposed method is increased by 1.1153 , 0.7793 and 0.3596 , respectively.

## V Summary

In this paper, a reversible data hiding method in encrypted images based on bit plane compression of prediction error image is proposed. The method of bit plane rearrangement is adopted for the prediction error image, which not only makes well use of the correlation between adjacent pixels, but also makes use of the correlation between the pixel value of the whole image. Experimental results show that this method can extract information and restore images separately, and its embedding rate is higher than the state-of-the-art methods. In the future, more effective predictors can be found to generate prediction errors with smaller fluctuating values, so as to further improve the embedding capacity. A more efficient bit stream compression method can also be explored to further improve the performance of bit plane compression.

## Acknowledgement

This research work is partly supported by National Natural Science Foundation of China (61872003, U1636206).

## References

- [1] (1997) Method and apparatus for embedding authentication information within digital data. Note: US Patent 5,646,997 Cited by: §I.
- [2] (2017) Image database of bows-2. Accessed: Jun 20. Cited by: §IV.
- [3] (2011) ” Break our steganographic system”: the ins and outs of organizing boss. pp. 59–70. Cited by: §IV.
- [4] (2010) Reversible watermarking techniques: an overview and a classification. EURASIP Journal on Information Security 2010 (1), pp. 134546. Cited by: §I.
- [5] (2019) High-capacity reversible data hiding in encrypted images based on extended run-length coding and block-based msb plane rearrangement. Journal of Visual Communication and Image Representation 58, pp. 334–344. Cited by: §III-B.
- [6] (2019) A new reversible data hiding in encrypted image based on multi-secret sharing and lightweight cryptographic algorithms. IEEE Transactions on Information Forensics and Security 14 (12), pp. 3332–3343. Cited by: §I.
- [7] (2001) Invertible authentication. 4314, pp. 197–208. Cited by: §I.
- [8] (2019) Effective reversible data hiding in encrypted image with adaptive encoding strategy. Information Sciences 494, pp. 21–36. Cited by: §I.
- [9] (2012) An improved reversible data hiding in encrypted images using side match. IEEE Signal Processing Letters 19 (4), pp. 199–202. Cited by: §I.
- [10] (2016) New framework for reversible data hiding in encrypted domain. IEEE transactions on information forensics and security 11 (12), pp. 2777–2789. Cited by: §I.
- [11] (2005) Reversible data embedding into images using wavelet techniques and sorting. IEEE transactions on image processing 14 (12), pp. 2082–2090. Cited by: §I.
- [12] (2008) A novel difference expansion transform for reversible data embedding. IEEE Transactions on Information Forensics and Security 3 (3), pp. 456–465. Cited by: §I.
- [13] (2013) Reversible data hiding in encrypted images by reserving room before encryption. IEEE Transactions on information forensics and security 8 (3), pp. 553–562. Cited by: §I, §I.
- [14] (2006) Reversible data hiding. IEEE Transactions on circuits and systems for video technology 16 (3), pp. 354–362. Cited by: §I.
- [15] (2013) Pairwise prediction-error expansion for efficient reversible data hiding. IEEE Transactions on Image Processing 22 (12), pp. 5010–5021. Cited by: §I.
- [16] (2018) An efficient msb prediction-based method for high-capacity reversible data hiding in encrypted images. IEEE transactions on information forensics and security 13 (7), pp. 1670–1681. Cited by: §I, Fig. 7, Fig. 8, §IV-C, §IV-C, §IV.
- [17] (2018) Reversible data hiding in encrypted images with two-msb prediction. pp. 1–7. Cited by: §I.
- [18] (2015) Reversible data hiding in encrypted images with distributed source encoding. IEEE Transactions on Circuits and Systems for Video Technology 26 (4), pp. 636–646. Cited by: §I.
- [19] (2016) Reversible data hiding in vq index table with lossless coding and adaptive switching mechanism. Signal Processing 129, pp. 48–55. Cited by: §I.
- [20] (2020) Reversible data hiding in encrypted images using adaptive reversible integer transformation. Signal Processing 167, pp. 107288. Cited by: §I.
- [21] (2003) UCID: an uncompressed color image database. 5307, pp. 472–480. Cited by: §IV.
- [22] (2004) Lossless data hiding: fundamentals, algorithms and applications. 2004 IEEE international symposium on circuits and systems 2, pp. II–33–II–36. Cited by: §I.
- [23] (2004) Reversible data hiding. International workshop on digital watermarking, pp. 1–12. Cited by: §I.
- [24] (2007) Expansion embedding techniques for reversible watermarking. IEEE transactions on image processing 16 (3), pp. 721–730. Cited by: §I.
- [25] (2003) Reversible data embedding using a difference expansion. IEEE transactions on circuits and systems for video technology 13 (8), pp. 890–896. Cited by: §I.
- [26] (2019) Multiple histograms based reversible data hiding: framework and realization. IEEE Transactions on Circuits and Systems for Video Technology. Cited by: §I.
- [27] (2019) High-capacity reversible data hiding in encrypted images by bit plane partition and msb prediction. IEEE Access 7, pp. 62361–62371. Cited by: §I, §I.
- [28] (2019) An improved reversible data hiding in encrypted images using parametric binary tree labeling. IEEE Transactions on Multimedia. Cited by: Fig. 7, Fig. 8, §IV-C, §IV-C, §IV.
- [29] (2019) Reversible data hiding based on pairwise embedding and optimal expansion path. Signal Processing 158, pp. 210–218. Cited by: §I.
- [30] (2017) Binary-block embedding for reversible data hiding in encrypted images. Signal Processing 133, pp. 40–51. Cited by: §I.
- [31] (2018) Separable and reversible data hiding in encrypted images using parametric binary tree labeling. IEEE Transactions on Multimedia 21 (1), pp. 51–64. Cited by: §I, §I.
- [32] (2019) Reversible data hiding in encrypted images based on multi-msb prediction and huffman coding. IEEE Transactions on Multimedia 22 (4), pp. 874–884. Cited by: §I, Fig. 7, Fig. 8, §IV-C, §IV-C, §IV.
- [33] (2019) A high capacity reversible data hiding scheme for encrypted covers based on histogram shifting. Journal of Information Security and Applications 47, pp. 199–207. Cited by: §I.
- [34] (2020) Location-based pvo and adaptive pairwise modification for efficient reversible data hiding. IEEE Transactions on Information Forensics and Security 15, pp. 2306–2319. Cited by: §I.
- [35] (2013) Recursive histogram modification: establishing equivalency between reversible data hiding and lossless data compression. IEEE transactions on image processing 22 (7), pp. 2775–2785. Cited by: §I.
- [36] (2014) Reversibility improved data hiding in encrypted images. Signal Processing 94, pp. 118–127. Cited by: §I.
- [37] (2015) Lossless and reversible data hiding in encrypted images with public-key cryptography. IEEE Transactions on Circuits and Systems for Video Technology 26 (9), pp. 1622–1631. Cited by: §I.
- [38] (2011) Reversible data hiding in encrypted image. IEEE signal processing letters 18 (4), pp. 255–258. Cited by: §I.
- [39] (2011) Separable reversible data hiding in encrypted image. IEEE transactions on information forensics and security 7 (2), pp. 826–832. Cited by: §I.