A visually secure image encryption method based on integer wavelet transform and rhombus prediction

1.   Introduction

Encryption and steganography are the two information branches of the information security, they are used in a wide of the privacy protect and conceal communication of the digital media ^[1], such as text, image, audio video and so on. The image encryption focuses on preventing unauthorized accessing by encoding the secret information into an encrypted image similar to noise or texture ^[2,3]. Whereas steganography is committed to hiding the conceal communication by embedding the secret information into a carrier image and which will not change the visual appearance of the original image ^[4,5,6]. Although there are differences in principles and methods, they can protect information effectively.

The image encryption technology can be divided into two classes: the spatial domain method ^[7,8] and the frequency domain method ^[9]. The encryption method in spatial domain treats the image as two-dimensional data and works on the secret pixels directly to generate an encrypted image with random noise. Typical algorithms include Advanced Encryption Standard (AES) ^[10], chaotic systems ^[11,12], wave transmission ^[13], DNA coding ^[14], and some recently published symmetric encryption and asymmetric encryption methods ^{[15,16,17,18]}. For the frequency domain encryption, Fourier transform (FFT) or discrete cosine transform (DCT) are used to transform the image from the spatial domain to the frequency domain firstly, and then the frequency data will be encrypted ^[19]. Image encryption is an efficient security tool during the privacy protection usually converts a two-dimensional image into one-dimensional data ^[20]. Since these methods destroy the correlation within the image, attackers can be extremely hard to extract useful information from the encrypted images ^[21], so it can protect the privacy of the secret image during transmission and storage in some extent ^[22].

However, as we all know, the encrypted images are cryptographic images similar to noise or texture, which will make an obvious visual sign indicating that the encrypted image contains important information, as shown in Figure 1 (a), (b). Therefore, these encrypted images may cause suspicion and even analysis by the attackers when they are transmitted in public channels ^[23,24,25].

To overcome the disadvantage that the noise and texture features are easy to attract the attention of the encrypted images and improve the visually security, some researchers have tried to encrypt images into normal ones to prevent the content from being suspected by combining the encryption and steganography. The appearance of the encrypted image is almost the same as that of the ordinary images, which can effectively avoid malicious tamper and attack, thus improving the security of encrypted images ^[26,27,28], as shown in the Figure 1(c). For example, in 2015, Bao and Zhou ^[29] proposed a visually meaningful image encryption (VMEI) called BZ scheme, in which they first pre-encrypt the secret image by an encryption algorithm ^[30], then embed the pre-processed image into the sub-bands of another carrier image. Although the encrypted image has a visually security appearance with the normal image in this method, the image quality is not very well. Subsequently, to improve the quality and security of the BZ scheme, Kanso and Ghebleh ^[31] proposed an encryption scheme that can restore the secret image losslessly. In the embedding phase, they first use a two-dimensional lifting wavelet transform (LWT) to divide the carrier image into three high-frequency coefficients and one low-frequency. Later, Yang and Zhang ^[32] combined with the knowledge of quantum computing to improve the BZ scheme, which reduced the texture features of the secret images to a certain extent and had a better performance. In 2017, Chai et al. ^[33] trained a data dictionary by using the compressed sensing technology to combine the secret image with carrier image. However, it has a practical problem that the secret image could not be restored losslessly, which limits its development potential for the application of content sensitive.

Focusing on the image quality and integrity, in this paper, we use rhombus prediction and wavelet transform to embed the pre-encrypted secret image into another natural-looking carrier image, the final encrypted image is not only visually secure, but also of high quality. The contributions of the proposed method are:

1). The visual quality of the final carrier image, containing secret image, is improved because the secret pixels are replaced by the prediction errors during the embedding stage, which greatly reduces the modification to the carrier image.

2). The proposed algorithm is reversible. In the proposed method, the receiver can extract secret information from the final encrypted image and reconstruct the secret image.

The paragraph structure of this article is listed as follows: The second part shows the related work. The third section is the introduction of the proposed method. The fourth part gives the experimental results, and the final section summarizes the full text.

2.   The introduction of BZ scheme

The BZ scheme ^[29] mainly includes two steps: 1) pre-encryption phase, 2) Content Transformation Based on Discrete Wavelet Transform (CTDWT) ^[34]. During the pre-encryption phase, the secret image is encrypted to noise or texture. In the CTDWT stage, the CTDWT is used to divide the carrier image C into four parts: LL, LH, HL and HH, which are denoted as ${\mathrm{C}}_{\mathrm{A}}, {\mathrm{C}}_{\mathrm{H}}, {\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ respectively. Then divide the pre-encryption image into two arrays called ${\mathrm{P}}_{\mathrm{V}}$ and ${\mathrm{P}}_{\mathrm{D}}$ respectively ^[35], and embed the two arrays into the HL and HH sub-bands by value substitution. Finally, an encrypted image containing a secret image is obtained by using inverse CTDWT transform, the details are listed as follows.

Suppose S represents the secret image, ${\mathrm{K}}_{\mathrm{t}}$ denotes the encryption key. Then, S can be encrypted into an indistinguishable pre-encrypted image $\mathrm{P}$ by the Eq (1):

where $\mathrm{F}$ is the pre-encryption function, such as the Advanced Encryption Standard (AES). After pre-processing, the secret image becomes a noise or texture image.

Decompose the carrier image C to four coefficient matrices ${\mathrm{C}}_{\mathrm{A}}$, ${\mathrm{C}}_{\mathrm{H}}, {\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ by CTDWT, then divide the pre-encrypted image $\mathrm{P}$ into two parts ${\mathrm{P}}_{\mathrm{V}}$ and ${\mathrm{P}}_{\mathrm{D}}$, which can be calculated by Eqs (3) and (4):

finally, embed the ${\mathrm{P}}_{V}$ and ${\mathrm{P}}_{D}$ and parameter into ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$, respectively.

where $\mathrm{D}$ is the CTDWT function, the ${\mathrm{K}}_{\mathrm{P}}$ represents the parameter set of DWT, which defines the DWT wavelet transform filter, and E is the final encrypted image.

Take a pixel of the pre-encrypted image with a value of 125 for example, ${\mathrm{P}}_{\mathrm{V}} = \left\lfloor {125/10} \right\rfloor = 12$, ${\mathrm{P}}_{\mathrm{D}} = \left\lfloor {125 \ \ \mathrm{m}\mathrm{o}\mathrm{d} \ \ 10} \right\rfloor = 5$, then, ${\mathrm{C}}_{\mathrm{V}}$ holds 12 and ${\mathrm{C}}_{\mathrm{D}}$ holds 5.

In the receiver side, they firstly decompose the encrypted image into four sub-bands namely ${\mathrm{E}}_{\mathrm{A}}, {\mathrm{E}}_{\mathrm{H}}', {\mathrm{E}}_{\mathrm{V}}', {\mathrm{E}}_{\mathrm{D}}'.$ To reconstruct the pre-encrypted image, the following formula is defined:

where, ${\mathrm{P}}'$ is the restored pre-encrypted image, and ${\mathrm{P}}_{\mathrm{V}}'$, ${\mathrm{P}}_{\mathrm{D}}'$ corresponding to the ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ sub-bands of image E.

Overall, this encryption system is visually safe, but it is an unwise choice for replacing the coefficient values in the carrier image's ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ matrices directly, and the visual quality of the encrypted image will be degraded largely.

3.   The proposed method

As described in the Section 2, BZ scheme ^[29] completely replaces the HL and HH parts of the carrier image with the ${\mathrm{P}}_{\mathrm{V}}$ and ${\mathrm{P}}_{\mathrm{D}}$ parts of the pre-encrypted secret image. This operation will directly affect the quality of the final encrypted image, which is also an obvious disadvantage of the scheme. To improve the image quality, we proposed a new meaningful encryption scheme based on the BZ scheme ^[29], the flowchart of encryption process is shown in Figure 2.

3.1.   Encryption and embedding process

Suppose the size of the secret image $\mathrm{S}$ is $\mathrm{m}\times \mathrm{n}$ and the size of the carrier image $\mathrm{C}$ is $2\mathrm{m}\times 2\mathrm{n}$. Firstly, choose the rhombus predictor proposed by Sachnev et al. ^[36] to calculate the prediction value of $\mathrm{S}$, which is an effective prediction scheme, and the prediction mode is shown in Figure 3. The pixel value $\mathrm{U}\left(\mathrm{i}, \mathrm{j}\right)$ (denoted as Cross set) can be predicted by the four neighboring pixels $\mathrm{v}\left(\mathrm{i}, \mathrm{j}\right)$ (denoted as Dot set), and vice versa. The prediction value can be calculated using the Eq (6).

Then get the prediction error sequence ${\mathrm{P}}_{\mathrm{s}}$ by $\mathrm{S}\left(\mathrm{i}, \mathrm{j}\right)-\mathrm{U}(\mathrm{i}, \mathrm{j})$ and scan it by zigzag manner. After that, using AES algorithm to encrypt e with secret key ${\mathrm{K}}_{\mathrm{t}}$, the encryption process is calculated by the Eq (7):

where $\mathrm{F}$ is the encryption function and ${\mathrm{P}}_{\mathrm{e}}$ is the encrypted prediction error.

Secondly, divide the carrier image into $\mathrm{s}\times \mathrm{s}$ non-overlapping sub-blocks, then use IWT to decompose each image block into a low frequency sub-band and three high frequency sub-bands, which are denoted as ${\mathrm{C}}_{\mathrm{A}}$, ${\mathrm{C}}_{\mathrm{H}}$, ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ respectively, where ${\mathrm{C}}_{\mathrm{A}}$ is a low-frequency matrix, which concentrates most of the energy and features of the image, and ${\mathrm{C}}_{\mathrm{H}}$, ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$ represent the middle frequency matrix and high-frequency matrix, which contain the details of the image.

Then, the parameter set ${\mathrm{K}}_{\mathrm{P}}$, the encrypted prediction error sequence ${\mathrm{P}}_{\mathrm{e}}$, the basic pixel $\mathrm{B}$ used to restore original pixel are embedded into the coefficient matrixes of the carrier image by value replacement with Eq (8).

where, E is the final encrypted image similar to the carrier image.

To prevent the pixels from being outside the range of [0,255], we first adjust the pixels of the carrier image $\mathrm{C}$ to be between 10 and 245, and called the adjusted carrier image ${\mathrm{C}}'.$ Input the secret image $\mathrm{S}$, the encryption key ${\mathrm{K}}_{\mathrm{t}}$, output the final encrypted image, and the details of the embedding algorithm are listed as follows:

Step 1. Shrink the carrier image C with the formula ${\mathrm{C}}' = \left\lceil { {\rm{ \mathsf{ ρ} }}-({\rm{ \mathsf{ ω} }}-{\rm{ \mathsf{ ρ} }})/255\mathrm{C}} \right\rceil$ to prevent the overflow and underflow of the pixel value. Where ${\mathrm{C}}'$ is the modified carrier image, and ${\rm{ \mathsf{ ρ} }} = 10, \ \ {\rm{ \mathsf{ ω} }} = 245$.

Step 2. Divide the processed carrier image ${\mathrm{C}}'$ into $8\times 8$ blocks and then divide each block into four parts by IWT, which are represented by ${\mathrm{C}}_{\mathrm{A}}$, ${\mathrm{C}}_{\mathrm{H}}$, ${\mathrm{C}}_{\mathrm{V}}$ and ${\mathrm{C}}_{\mathrm{D}}$.

Step 3. Calculate the prediction value $\mathrm{U}(\mathrm{i}, \mathrm{j})$ of the secret image using the rhombus prediction scheme, and then calculate the prediction error ${\mathrm{P}}_{\mathrm{s}}(\mathrm{i}, \mathrm{j}) = \mathrm{S}(\mathrm{i}, \mathrm{j})-\mathrm{U}(\mathrm{i}, \mathrm{j})$, and the basic pixels are listed as $\mathrm{B}$.

Step 4. Scramble the prediction error ${\mathrm{P}}_{\mathrm{s}}\left(\mathrm{i}, \mathrm{j}\right)$ first and encrypt it into the sequence ${\mathrm{P}}_{\mathrm{e}}$ using AES algorithm with the secret key ${\mathrm{K}}_{\mathrm{t}}.$

Step 5. Segment the basic pixels into two parts ${\mathrm{B}}_{1}$ and ${\mathrm{B}}_{2}$, then divided ${\mathrm{B}}_{2}$ into ${\mathrm{B}}_{\mathrm{V}}$ and ${\mathrm{B}}_{\mathrm{D}}$ by the Eqs (2) and (3).

Step 6. Embed the prediction error into ${\mathrm{C}}_{\mathrm{H}}$, the basic pixel ${\mathrm{B}}_{1}$ into ${\mathrm{C}}_{\mathrm{V}}$, and ${\mathrm{B}}_{2}$ into ${\mathrm{C}}_{\mathrm{D}}$ by the value substitution. The modified sub-bands are denoted as ${\mathrm{C}}_{\mathrm{A}}, {\mathrm{C}}_{\mathrm{H}}', {\mathrm{C}}_{\mathrm{V}}' \ \ \mathrm{a}\mathrm{n}\mathrm{d} \ \ {\mathrm{C}}_{\mathrm{D}}'$.

Step 7. Apply inverse IWT to the modified sub-band of each block to get the final encrypted image E.

3.2.   Decryption and reconstruction process

The restoration of the secret image can be carried out according to the steps in the encryption phase, which is usually the reverse operation. According to the decryption framework in Figure 4, the decryption phase mainly includes two steps: information extraction and image reconstruction.

In the information extraction stage, we first use IWT to divide each block of the final encrypted image E into four parts called ${\mathrm{E}}_{\mathrm{A}}, {\mathrm{E}}_{\mathrm{H}}, {\mathrm{E}}_{\mathrm{V}} \ \ \mathrm{a}\mathrm{n}\mathrm{d} \ \ {\mathrm{E}}_{\mathrm{D}}$. Secondly, the encrypted prediction error sequence is extracted from ${ \ \ \mathrm{E}}_{\mathrm{H}}$, and two parts of the basic pixel are extracted from ${\mathrm{E}}_{\mathrm{V}}\ \ \mathrm{a}\mathrm{n}\mathrm{d} \ \ {\mathrm{E}}_{\mathrm{D}}$, i.e., ${{\mathrm{P}}'}_{\mathrm{e}} = {\mathrm{E}}_{\mathrm{H}}$, ${{\mathrm{B}}'}_{1} = {\mathrm{E}}_{\mathrm{V}}$. ${{\mathrm{B}}'}_{2}$ are restored from ${\mathrm{E}}_{\mathrm{D} \ \ }$ by the Eq (9).

where ${{\mathrm{B}}_{2}}'\left(\mathrm{i}, \mathrm{j}\right)$ is the extracted basic pixel, ${\mathrm{B}}_{\mathrm{V}}'\left(\mathrm{i}, \mathrm{j}\right)$ and ${\mathrm{B}}_{\mathrm{D}}'(\mathrm{i}, \mathrm{j})$ represent the pixels obtained by applying inverse IWT to the final encrypted image.

Finally, ${{\mathrm{P}}'}_{\mathrm{e}}$ is decrypted with the key, the original prediction error can be restored in the reverse order of zigzag, and the secret image is reconstructed according to the basic pixel and prediction error.

4.   Experiments and evaluations

All experiments are implemented in MatlabR2016a, and the configuration of the computer is i5-2450 M and 8 GB of memory. To compare fairly with the previous schemes, we use the same wavelet kernel function "db1" with the BZ scheme to decompose the carrier image. All images are downloaded from Bose base image library ^[37], and the secret image is one fourth size of the carrier image.

4.1.   The appearance of the encrypted image

The appearance of the final encrypted image will change with the selected carrier image. The proposed scheme can not only embed secret images into gray carrier images of different sizes, but also can be used for color image encryption. The three channels of the color image are separated, and each channel is encrypted according to the gray image processing method described above. Finally, combine the secret carriers of the three channels together to generate the final color image encryption. Figure 5 is an example of the proposed scheme, the first line shows the secret image, the carrier image, the final encrypted image, and the reconstructed secret image, respectively, and the second line is the histograms of the corresponding images. From the experimental results we can see that there is almost no difference in histograms between the carrier and the final encrypted images. That is, the two pictures are resemble in the human visual system and will not cause people's doubt.

As shown in Figure 6, we randomly selected five groups of the secret images (girl face, milk drop, boat, butterfly and hill) and the carrier images (pepper, sailing, boat, birds, and lake) to show the final encrypted images obtained by our algorithm and the BZ scheme ^[29]. The results of the two algorithms are slightly different, and our method has a better visual effect.

As Zeng et al. mentioned in ^[38], the gray histogram counts the frequency of all pixels in the image according to the gray value, if the histograms of the two images are close, they can be considered to be visually similar. To show the difference between the final encrypted image and the carrier image more clearly, we calculate the difference between their histograms, as shown in Figure 7, which indicates that the overall effect of the proposed algorithm is better than the BZ scheme ^[29]. This is because the value of prediction error is smaller than the pre-encrypted image, so the impact of the replacement on the carrier image is small, which can improve the visual quality of the encrypted image.

4.2.   Image quality evaluation

To evaluate the visual quality of the encrypted image, we mainly use two common image quality evaluation standards, Peak signal to noise ratio (PSNR) and structural similarity (SSIM) ^[39]. PSNR is an objective standard to evaluate the image. The larger the PSNR value is, the less distortion is. SSIM is an index to measure the similarity of two images in brightness, contrast and structure, and the closer the value of SSIM is to 1, the more similar the two images are. The calculation formulas are listed as follows

where the image's size is $\mathrm{M}\times \mathrm{N}$, $\mathrm{C}\left(\mathrm{i}, \mathrm{j}\right)$ and $\mathrm{E}(\mathrm{i}, \mathrm{j})$ are the pixel value of the carrier image and the final encrypted image, respectively. Then, the similarity between the secret image $\mathrm{S}$ and the reconstructed secret image R is calculated by SSIM ^[39], and it is based on luminance, contrast and structure, and here is the specific equation:

where, ${\mu }_{S}$ and ${\mu }_{R}$ are the mean of images S and R, ${\rm{ \mathsf{ σ} }}_{\mathrm{S}}$ and ${\rm{ \mathsf{ σ} }}_{\mathrm{R}}$ are the variance of images S and R, ${\sigma }_{SR}$ is the covariance of S and R, ${\mathrm{c}}_{1}$ and ${\mathrm{c}}_{2}$ are constants to avoid denominators becoming zero, ${\mathrm{c}}_{1} = {({\mathrm{k}}_{1}\times \mathrm{l})}^{2}$, ${\mathrm{c}}_{2} = {({\mathrm{k}}_{2}\times \mathrm{l})}^{2}$, usually, ${\mathrm{k}}_{1} = 0.01, {\mathrm{k}}_{2} = 0.03 \ \ \mathrm{a}\mathrm{n}\mathrm{d} \ \ \mathrm{l} = 255.$

In addition to the above two typical quality evaluation criteria, we introduce a new method called the Two Dimensional-Detrended fluctuation analysis (2D-DFA) to further analyze the image ^[43]. Suppose the size of an image I is $\mathrm{M}\times \mathrm{N}$, dvide I into ${\mathrm{M}}_{\mathrm{s}}\times {\mathrm{N}}_{\mathrm{s}}$ non-overlapping blocks of the same size $\mathrm{s}\times \mathrm{s}$, where, ${\mathrm{M}}_{\mathrm{s}} = \mathrm{M}/\mathrm{s}$, and ${\mathrm{N}}_{\mathrm{s}} = \mathrm{N}/\mathrm{s}$. The detrended fluctuation of the image I can be calculated as:

where, ${\mathrm{F}}^{2}(\mathrm{v}, \mathrm{w}, \mathrm{s})$ is the detrended fluctuation of each sub-block.

To evaluate the fractal scale characteristics of the image, ${\mathrm{F}}^{2}\left(\mathrm{s}\right)$ should be displayed as power law scale:

where $\alpha$ is a scaling exponent, and when the $\alpha$ of the final encrypted image is closer to that of the carrier image, the visual quality of final encrypted image will be better ^[43].

Figure 8 shows the PSNR and SSIM of 50 pairs of randomly selected carrier images and the corresponding final encrypted images ^[41]. As can be seen from Figure 8, the data obtained by our scheme is generally better than that of the BZ algorithm. The average values of these 50 groups of results are recorded in Table 1. The results show that compared with BZ scheme ^[29], the PSNR and SSIM of this scheme are improved by 3.421 db and 0.0269, respectively.

The quality of extracted secret image also evaluates the algorithm. We use the images in Table 2 to discuss the reconstruction ability of the proposed method. The PSNR and SSIM of the reconstructed secret image in Table 2 is 31.5874db and 0.9801, respectively, therefore, the reconstructed images are visually acceptable.

Although PSNR is considered as an objective index of encrypted images, there are some defects occasionally. To make up for this shortcoming, 2D-DFA is used to further analyze the encrypted image and compare it with the BZ scheme ^[29]. Table 3 shows that the proposed algorithm is superior to BZ scheme ^[29] under different image evaluation standards for the images in Table 2. Although fluctuation function ${\mathrm{F}}^{2}\left(\mathrm{s}\right)$ present similar values for the two schemes, the scaling exponent ${\rm{ \mathsf{ α} }}$ of the encrypted image obtained by our scheme is more similar than the BZ scheme ^[29].

4.3.   Security analysis

In this section, we will analyze the security of the proposed method in terms of secret key sensitivity, noise attack and anti-clipping attack.

4.3.1.   Key sensitivity analysis

Generally speaking, the key security mentioned in the image encryption system mainly refers to that in the process of decryption and image reconstruction, it has a significant impact on the recovery of secret image. In this paper, we use the AES algorithm to encrypt the prediction error, and choose the key = ([${\mathrm{K}}_{\mathrm{t}}, {\mathrm{K}}_{\mathrm{p}}$]). Where, ${\mathrm{K}}_{\mathrm{t}}$ is the encryption key of AES and ${\mathrm{K}}_{\mathrm{p}}$ is the parameter of integer wavelet. We set ${\mathrm{K}}_{\mathrm{t}}$ = [60,46,98,30,45,90,105,78], ${\mathrm{K}}_{\mathrm{p}}$ = db1. In the process of image reconstruction, we consider the influence of the subtle changes of keys to the experimental results. We first keep the ${\mathrm{K}}_{\mathrm{p}}$ unchanged, and use ${\mathrm{k}\mathrm{e}\mathrm{y}}_{1} = \left(\right[{\mathrm{K}}_{\mathrm{t}1}, {\mathrm{K}}_{\mathrm{p}}\left]\right)$ to decrypt the image, where ${\mathrm{K}}_{\mathrm{t}1}$ = [56,46,98,30,45,90,105,78]. Then, keep the ${\mathrm{K}}_{\mathrm{t}}$ unchanged, and use ${\mathrm{k}\mathrm{e}\mathrm{y}}_{2} = \left(\right[{\mathrm{K}}_{\mathrm{t}}, {\mathrm{K}}_{\mathrm{p}2}\left]\right)$ for decryption, where ${\mathrm{K}}_{\mathrm{p}2} = \mathrm{d}\mathrm{b}3$. The correct key can recover the secret image, but only the tampered image can be obtained by slightly modifying the key set, as shown in Figure 9.

4.3.2.   Noise attack

As is known to all, in the process of image transmission and processing, images are inevitably contaminated by various noises ^[40,42]. For this purpose, we add different degrees of Gaussian noise, salt and pepper noise and speckle noise into the final encrypted image, and then analyze whether the proposed method can recover the secret image. As can be seen from Figure 10, adding 0.001% noise, the reconstructed secret image has different degrees of visual damage, and Gaussian noise has the greatest impact on the image, but the outline of the secret image is still clear and visible. Table 4 shows the comparison of performance between the proposed scheme and BZ scheme ^[29] under different kinds of noise attacks. From Table 4, we can observe that even after the noise attack and the cutting attack, the image quality of the proposed method is better than that of the BZ scheme both in encryption process and decryption.

4.3.3.   Different degree of clipping attack

In addition, the image may be subject to cropping attacks during image transmission, we apply $64\times 64$ and $128\times 128$ cropping attacks to the secret carrier image, and then reconstruct the secret image. Figure 11 shows us the cropped encrypted images and the corresponding reconstructed images and Table 4 shows that the visual quality of the secret image is acceptable even though a large amount of data is cropped.

4.3.4.   Running time and computation complexity analysis

Running time and computation complexity are two important standards to measure the algorithm. In this part, we will compare the computation time between the BZ scheme ^[29] and our algorithm both in encryption and decryption process. For the fairness, we use the same pre-encryption algorithm in BZ scheme and this paper, i.e., AES encryption method, and change the key to do the test. Table 5 shows the experimental results. We can see that the difference of encryption time and decryption time between the two methods is 0.9547 and 0.0590 respectively. The time difference is acceptable, and the computation complexity of this paper is similar to that of BZ scheme.

5.   Conclusions

This paper proposes a visual secure image encryption algorithm, which not only improves the visual quality of the final encrypted image, but also restores the secret image. In the proposed method, the rhombus predictor is used to calculate the prediction value of the secret image. The prediction error value is far less than the secret variable value, so it improves the visual quality of the final encrypted image to a certain extent. Visually meaningful image encryption can well protect the image and the encrypted image can be transmitted safely in the channel without the suspicion of the attacker, moreover, the secret image extraction is independent of the carrier image, which increases the usability of the proposed scheme. In the future work, we are committed to the application of the method for privacy protection of the internet users.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgements

This work is supported by the National Key R & D Program of China under grant 2018YFB1003205; by the National Natural Science Foundation of China under grant U1836208, U1536206, U1836110, 61972207, 61602253, 61672294; by the Engineering Research Center of Digital Forensics, Ministry of Education; by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) fund; by the Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET) fund, China.