TPE-H2MWD:an exact thumbnail preserving encryption scheme with hidden Markov model and weighted diffusion＊

Xiuli CHAI ,Xiuhui CHEN ,Yakun MA ,Fang ZUO ,Zhihua GAN ,Yushu ZHANG

1School of Artificial Intelligence,Henan Engineering Research Center for Industrial Internet of Things,Henan University,Zhengzhou 450046,China

2Henan Key Laboratory of Cyberspace Situation Awareness,Zhengzhou 450001,China

3School of Software,Intelligent Data Processing Engineering Research Center of Henan Province,Institute of Intelligent Network System,Henan University,Kaifeng 475004,China

4College of Computer Science and Technology,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China

Abstract: With the substantial increase in image transmission,the demand for image security is increasing.Noise-like images can be obtained by conventional encryption schemes,and although the security of the images can be guaranteed,the noise-like images cannot be directly previewed and retrieved.Based on the rank-then-encipher method,some researchers have designed a three-pixel exact thumbnail preserving encryption (TPE2) scheme,which can be applied to balance the security and availability of images,but this scheme has low encryption efficiency.In this paper,we introduce an efficient exact thumbnail preserving encryption scheme.First,blocking and bit-plane decomposition operations are performed on the plaintext image.The zigzag scrambling model is used to change the bit positions in the lower four bit planes.Subsequently,an operation is devised to permute the higher four bit planes,which is an extended application of the hidden Markov model.Finally,according to the difference in bit weights in each bit plane,a bit-level weighted diffusion rule is established to generate an encrypted image and still maintain the same sum of pixels within the block.Simulation results show that the proposed scheme improves the encryption efficiency and can guarantee the availability of images while protecting their privacy.

Key words: Hidden Markov model;Weighted diffusion;Balance between usability and privacy;Image encryption

1 Introduction

With the development of the Internet and cloud service platforms,people upload their pictures and videos to cloud service platforms (Beaver et al.,2010).With fewer restrictions on hardware and easier file sharing,it is possible for users to access uploaded files at will (He et al.,2010).However,images may also contain sensitive information such as identity,location,and circumstances (Fan,2019),so people are more inclined to prevent others from viewing their private images.

There are two main methods of privacy leakage:inside attacks and external attacks (Ashiq,2015).Inside attacks are the most easily ignored security issue.Inside attacks generally refer to malicious damage by insiders,collusion between insiders and outsiders,abuse of power by managers,and threats caused by natural disasters such as floods and earthquakes.Even righteous employees can inadvertently leak data due to weak security awareness or incorrect use of software.External attacks refer to the attacks on nodes inside the network by malicious nodes outside the network that have not been authenticated.With the explosion of information and the maturity of attack techniques,there are increasingly more incidents of illegal intrusion in the network,which brings numerous difficulties to network security protection.

Some scholars have proposed conventional image encryption schemes to generate noise-like images (Joshi et al.,2020;Zhu et al.,2020).This prevents attackers from stealing the content in the image,but the image owners cannot identify the original image’s content through encrypted images.To balance the availability and security of encrypted images in the cloud,Wright et al.(2015) introduced two thumbnail preserving encryption (TPE) schemes,where the plaintext images and the encrypted images have the same thumbnails.In the generated encrypted images,users with a priori knowledge can quickly find the image they want,while those without a priori knowledge can recognize only the general outline of the images and cannot obtain more detailed information.A priori knowledge refers to the impressions left in the user’s mind by viewing the image.

The TPE schemes presented by Wright et al.(2015) change only the position of the pixels within the block,and although the thumbnail of the plaintext image is accurately preserved,the attackers can obtain all the pixel values in the plaintext image.Some studies (Li et al.,2008;Jolfaei et al.,2016) have shown that only the scrambling operation in the encryption process cannot satisfy the security requirements.To improve the security of encrypted images,Marohn et al.(2017) proposed two approximate TPE schemes,where the thumbnails of encrypted images and those of plaintext images are similar but not identical.The first method works using dynamic range preserving encryption (DRPE),which retains only the minimum and maximum pixel values of the plaintext in each thumbnail block.The second approach is an extended application of the least significant bit (LSB) embedding technique in steganography,which reveals only the maximum pixel value and the average value of the plaintext in the blocks.However,the encrypted images still reveal more information than the plaintext image thumbnails and cannot be fully decrypted to the original images.Later,Zhang et al.(2022) proposed a high-fidelity TPE scheme (HF-TPE),which improves the perceptual quality of encrypted images.Decrypted images have a lower noise intensity and an upper limit of noise quantity.

Tajik et al.(2019) achieved the first exact TPE scheme with nonce-respecting (NR) security.To preserve accurate thumbnails of the plaintext images,the adopted strategy is to ensure that the sum of pixels within a single block of a single channel remains unchanged.They summarized and employed the rank-then-encipher method proposed by Bellare et al.(2009).That is,first calculate the sumSof a vector,compute the rank of the vector through the function,and then encrypt the rank.Finally,the encrypted rank is converted into an integer vector via the function.All the vectors with the same sumSare numbered,and the rank of the vector is the corresponding number.This scheme encrypts the image in groups of two pixels,but it uses Markov chains to demonstrate its security.The fewer the number of pixels in a group,the worse the chain state connectivity.Therefore,based on the rank-then-encipher approach,Zhao et al.(2021) proposed the TPE2 scheme for encrypting images,which extended the exact TPE scheme to a group of three pixels for the first time;however,the TPE2 scheme has the problem of low efficiency.

To solve the above problems,we propose an exact TPE algorithm based on a hidden Markov model and weighted diffusion (TPE-H2MWD).The encryption process takes the entire block as a unit while changing the position and value of the pixels and keeping the sum of the pixels constant.After multiple rounds of encryption,an encrypted image with the same thumbnail as the plaintext image is obtained,making it impossible for attackers to identify whether the encrypted image comes from the plaintext image or from another image with the same sum of pixels in the block (e.g.,a mosaic image of the plaintext) while improving the encryption efficiency.The main contributions of this paper are as follows:

1.A new exact TPE-H2MWD is presented.Bitlevel permutation and diffusion methods are used to realize the exact TPE directly,which improves the encryption efficiency.

2.A bit-level permutation method based on the hidden Markov model is presented.First,blocking and bit-plane decomposition operations are performed on the plaintext image to obtain eight bit planes,and then different scrambling operations are performed on different bit planes.This makes the permutation process more random while improving the security of the encryption scheme.

3.A weighted diffusion method is introduced under the condition that the sum of pixels is unchanged.This method modifies the bit values and is able to prevent the leakage of statistical information of the bits.

4.In the TPE-H2MWD,the thumbnails of the ciphertext images and those of the plaintext images are absolutely identical,and the encrypted images can be recovered completely by the decryption process.Simulation results show that the proposed scheme balances the security and usability of encrypted images.In addition,there is good retrieval performance for thumbnails of ciphertext images.

2 Fundamentals

In this section,the definition of TPE and the fundamental knowledge of the hidden Markov model are presented.

2.1 Thumbnail preserving encryption

TPE is a special type of format-preserving encryption,which is the encryption of plaintext images into ciphertext images with the same properties.The definition of TPE is provided below.

Definition 1In the TPE,a collection of images M is selected.An encryption scheme is claimed to beΦ-preserving within the plaintext image group M,if it satisfies the following criteria:

where the plaintext imageM∈M,with the keyK,nonceT,and the plaintext imageMas input and the ciphertext imageCas output in the encryption process,while with the keyK,nonceT,and the ciphertext imageCas input and the plaintext imageMas output in the decryption process,as well as the plaintext image and the corresponding encrypted image having the same dimensionality and thumbnail.

Definition 2Let FΦbe the set of functionsF:{0,1}*× M→M,whereFsatisfies theΦ-preserving property,andΦ(F(T,M))=Φ(M) is met for allTandM.If for any probabilistic polynomial time (PPT) oracle machineA,the following condition is satisfied (Bellare et al.,2009):

where negl(λ) is negligible inλ,the format-preserving encryption scheme is deemed to satisfy the pseudo random permutation (PRP) security.

Definition 3Let A be the PPT oracle machine with two parameters.A is denoted as NR if A never takes the same first parameter (i.e.,nonceT) to make two oracle calls.A format-preserving encryption scheme is considered NR-secure if it can satisfy Definition 2 under only the condition that Definition 3 is respected (Tajik et al.,2019).

2.2 Hidden Markov model

Hidden Markov models are the statistical models (Franzese and Iuliano,2019) that are used to describe Markov processes with hidden uncertain parameters.There are three elements:

1.Visible random sequence.This is used to describe the physical quantity of interest,which changes over time.

2.Hidden state sequence.Each physical quantity at each time point corresponds to a state quantity,and each state quantity cannot be directly observed.

3.Relationships among variables.The initial state quantity,the relationship between the current hidden state quantity and the next hidden state quantity,and the relationship between the current hidden state quantity and the visible random variable,all these three relationships or variables are described in probabilistic terms.

The hidden Markov model is a kind of statistical model with quite common applications.The theory was proposed by Baum and Petrie (1966) and subsequently became an important direction of signal processing over a decade of development.At present,the hidden Markov model has also started to be widely applied to the fields of speech recognition (Srivastava et al.,2022),smart home applications (Youngblood and Cook,2007),and behavior recognition (Hartmann et al.,2022;Xue and Liu,2022),among which the application of the hidden Markov model in image processing is also relatively successful and effective.

In our scheme,the hidden state chains and the visible state chains are generated based on one bit plane,and then another bit plane is scrambled based on the generated sequences.The data in each bit plane are different,making the permutation process more random and not easy to predict.

3 Proposed TPE-H2MWD

After performing bit-plane decomposition on the plaintext image,one may obtain eight bit planes,as shown in Fig.1.From this figure,we can see that the lower four bit planes contain less information about the plaintext image,and the higher four bit planes contain more information about the image.

Fig.1 Bit-plane decomposition:(a) 8th bit plane;(b) 7th bit plane;(c) 6th bit plane;(d) 5th bit plane;(e) 4th bit plane;(f) 3rd bit plane;(g) 2nd bit plane;(h) 1st bit plane

Inspired by this,an exact TPE-H2MWD is proposed,and the specific algorithm framework is illustrated in Fig.2.First,the plaintext imageP(M×N×3) is decomposed to obtain the red,green,and blue components,and then blocking and binary decompositions are performed on each component in turn to yield eight bit planes.The lower four bit planes are randomly zigzag scrambled,and then the scrambled lower four bit planes are used to generate two hidden state sequences and two visible state sequences.The higher four bit planes are scrambled with these sequences,and then,a weighted diffusion operation is performed between two adjacent bit planes.Finally,an encrypted image is obtained.

Fig.2 Flowchart of the proposed scheme (ZMP:zigzag model permutation;PMH2M:permutation method based on hidden Markov model;WDPCS:weighted diffusion preserving constant sum)

3.1 Zigzag model permutation (ZMP) for the lower four bit planes

The pixel blockBof sizeK×Kis binarily decomposed to obtain eight bit planes,denoted asB8,B7,B6,B5,B4,B3,B2,andB1.Zigzag scrambling operations are performed on the lower four bit planesB1,B2,B3,andB4to obtain four new bit planesB'1,B'2,B'3,andB'4,respectively.Generally,there are eight types of zigzag scrambling models,as shown in Fig.3,where the traversal rules of different models will be changed.

Fig.3 Eight types of zigzag models

In this scheme,to improve the randomness of confusion,different zigzag models and starting positions are selected for different bit planes,and accordingly,the data in this bit plane are scrambled.The selected zigzag model is determined by ((count1–count0) % 8),and the starting position can be computed by ((count0 %K)+1,(count1 %K)+1),where count1 indicates the total number of bits with value 1 in the current bit plane,and count0 indicates the total number of bits with value 0 in the current bit plane.

3.2 Permutation method based on hidden Markov model (PMH2M) for the higher four bit planes

Step 1:Based on the bit values in the bit plane,two hidden state sequences (Imp1,Imp2) and two visible state sequences (Vis1,Vis2) are created,and all of them have lengthK.The process of sequence generation is expressed by

wheretis an integer,t∈[1,K].

where Imp1tand Imp2tare thetthelements in the hidden state sequences Imp1 and Imp2,respectively.

where Vis1tand Vis2tare thetthelements in the visible state sequences Vis1 and Vis2,respectively.

To generate the sequence Imp1,the,andbits of the first row inB'iare taken,three bits are converted to a decimal number,and the remainder of the number divided by 4 is thetthdata Imp1t.Similarly,to generate the sequence Imp2,the,andbits of the first column inare selected,and then thetthdata Imp2tare obtained.

To generate the sequence Vis1,the total number of bits 1 in the,andcolumns ofis taken,and the remainder of the total number divided byKis thetthdata Vis1t.Similarly,to generate the sequence Vis2,the total number of bits 1 in the,androws ofis obtained,and the remainder of the total number divided byKis thetthdata Vis2t.

A complete example of generating sequences is shown in Fig.4.

Fig.4 Generating two hidden state sequences (Imp1,Imp2) and two visible state sequences (Vis1,Vis2)

Step 2:According to the data in Imp1 and Vis1,the bit populations and the moving distance of the bits will be determined.In bit planeBj,the first row of bits is used as a reference to change the position of the bits,and the new bit planeis obtained.

For bit populations in the minor and principal diagonal directions,circular downwards shift operations will be performed,and the moving distances are determined by the data in the corresponding visible state sequence.For bit populations in the row direction (as shown in Algorithm 1),bits in odd positions shift sequentially to the left,and bits in even positions shift sequentially to the right.For bit populations in the column direction,bits in odd positions shift successively upwards,and bits in even positions shift successively downwards.After moving the corresponding distance,if there are already data at the target location,continue moving one bit per time until a vacant location is found.

As shown in Fig.5,the first data point in Imp1 is 1,and then the set of bits in the column where the first bit in the first row in the planeBjis located is called the bit population.The first data point in the visible state sequence Vis1 is 1.When finding the target position in this bit population,it moves one bit first.The 1stbit is moved up cyclically by one bit,reaching the 5thposition of the first column inBj1;the 2ndbit is moved down cyclically by one bit,reaching the 3rdposition of the first column inBj1;the 3rdbit is moved up cyclically by one bit,reaching the 2ndposition of the first column inBj1;the 4thbit is moved down cyclically by one bit,reaching the 5thposition of the first column,but there are already data in the 5thposition,so it continues down one bit to the 1stposition in the first column inBj1;the 5thbit is moved up cyclically by one bit,reaching the 4thposition of the first column inBj1(the resulting bit population is outlined by the blue line inBj1).The other operations are similar.

Fig.5 Illustration of the shift of bits based on data in the hidden and visible state sequences (References to color refer to the online version of this figure)

Step 3:According to the data in Imp2 and Vis2,the bit populations and the moving distance of the bits will be determined.In bit plane,the first column of bits is used as a reference to change the position of the bits,and the new bit planeis obtained.

3.3 Weighted diffusion preserving constant sum (WDPCS)

A bit-level weighted diffusion scheme that maintains sum invariance is proposed using the different weights occupied by bits in various bit planes.

For eight bit planes,two adjacent bit planes are selected as a group,which can be divided into seven groups in total.Starting from the group containing the lowest bit plane,perform the following operations on each group successively:for two adjacent bit planesBiandBi+1with dimensionsK×K,the weights occupied by the bits inBi+1are twice the weights of the bits inBi.Each bit plane is divided into two equal parts on the left and right.IfKis odd,the firstK?1 columns are picked for the operation.Then,three bits are chosen fromBiandBi+1,one bit is from the left part ofBi+1,and two bits correspond to the same position in the left and right parts ofBi.If the three bits are 0,1,1 (or 1,0,0),the bits can be converted to 1,0,0 (or 0,1,1),and the sum of the values represented by all bits inBiandBi+1remains constant.Repeat this operation until all the bits in the left part ofBihave been selected.A concrete example is shown in Fig.6.

Fig.6 Variation of bits in two adjacent bit planes (References to color refer to the online version of this figure)

In Fig.6,the weight of each bit inBi+1is 2i,and the weight of each bit inBiis 2i?1.When the three selected bits are 1,0,0 (or 0,1,1),they are marked with the same color and modified to 0,1,1 (or 1,0,0).Before and after modification,the sum of the values represented by all bits inBiandBi+1is 17×2i.

4 Simulations

In this section,a Helen dataset (Zhao et al.,2021) containing 500 face images with a size of 512×512 and a Holiday dataset (Jegou et al.,2008) containing 1491 holiday photos with a size of 512×512 are used to assess the empirical evaluation of the proposed scheme.

4.1 Analysis of the encryption effect

The encrypted images are obtained by iterating the encryption process for 10 rounds.As shown in Fig.7,the block size can be adjusted to achieve a balance between the privacy and availability of the image.WhenK=0,the image indicates a plaintext image;whenKreaches 8 and 16,many details are shown in the encrypted images,which is not secure enough;when the block size is 32×32,the encrypted image retains the rough information of the plaintext image for availability,and it also has a certain level of security.

Fig.7 TPE-H2MWD encrypted images with different block sizes (original,8×8,16×16,32×32,and 64×64) and corresponding preserved thumbnails

4.2 Ciphertext image perception quality

The peak signal-to-noise ratio (PSNR) is an objective standard for evaluating image quality (Chai et al.,2022a).The larger the value is,the less distortion of the images is.Structural similarity (SSIM) (Wu et al.,2022) is another metric for measuring the similarity of two images in terms of brightness,contrast,and structure,and the closer its value is to 1,the more similar the two images are.

Five hundred images from the Helen dataset are used for testing.The block sizes are set to 8×8,16×16,and 32×32,corresponding to thumbnail sizes of 64×64,32×32,and 16×16,respectively.That is,the pixel average value within each block corresponds to a value in the thumbnail.From the data in Tables 1 and 2,it can be seen that the mean value of PSNR for the TPE-H2MWD and TPE2 schemes reached +∞ and the mean value of SSIM reached 1,indicating that encrypted images completely preserve the thumbnails of the corresponding plaintext images.

Table 1 Peak signal-to-noise ratio (PSNR) values between 500 plaintext and encrypted image thumbnails

Table 2 Structural similarity (SSIM) values between 500 plaintext and encrypted image thumbnails

4.3 Information entropy analysis

Information entropy is used to reflect the randomness of the information.Information entropy can be computed by (Mishra et al.,2021)

wherep(mi) denotes the probability of occurrence of the pixel valuemi,andnrefers to the bit depth of the image,which has 8-bit depth for grayscale images.

When the block size is 32×32,the plaintext image shown in Fig.7 is encrypted with different schemes,and the information entropies of the obtained ciphertext images are listed in Table 3.It is clear that among the approximate TPE schemes,the information entropy obtained by the TPE-LSB (1-bit) scheme is the largest;in the exact TPE schemes,the information entropy obtained by the TPE2 scheme is slightly larger than the result in TPE-H2MWD,indicating that the ciphertext image in the TPE2 scheme has stronger randomness.

Table 3 Information entropy of plaintext and ciphertext images

4.4 Histogram analysis

In image processing,the pixel values of specific channels of an image are used as thexaxis of the histogram,the number of occurrences of each value is used as theyaxis,and the resulting histogram is the image histogram.The histogram of the image reflects the distribution density of image pixels.The images in Fig.7 are selected for testing,and the histograms of the plaintext image and ciphertext images are shown in Fig.8.The ciphertext image histogram contains some features of the plaintext image thumbnail,thus allowing effective use of the encrypted image (Chai et al.,2022b).

Fig.8 Histogram in plaintext image and ciphertext images:(a)–(c) are histograms of the R component of the plaintext image and ciphertext images with K=8 and 32,respectively;(d)–(f) are histograms of the G component of the plaintext image and ciphertext images with K=8 and 32,respectively;(g)–(i) are histograms of the B component of the plaintext image and ciphertext images with K=8 and 32,respectively

4.5 Adjacent pixel correlation analysis

The adjacent pixel correlation reflects the degree of correlation between pixel values at adjacent locations in the image,including mainly the correlation among horizontal pixels,vertical pixels,and diagonal pixels in the image.Here,we randomly select 5000 pairs of neighboring pixels (horizontal,vertical,and diagonal directions) from the plaintext and ciphertext images and then calculate the correlation coefficient by (Chen et al.,2021)

wherexandyrefer to the values of two adjacent pixels in the image,Nrefers to the pair of pixels selected from the image,andE(x) andD(x) refer to the mathematical expectation and variance ofx,respectively,where the larger the correlation coefficient is,the greater the adjacent pixel correlation is,and vice versa.

The plaintext image shown in Fig.7 is encrypted when the block size is 32×32,and the results are shown in Fig.9.In addition,the adjacent pixel correlations of the encrypted images corresponding to the different encryption schemes are listed in Table 4.It can be concluded that compared to the approximate TPE scheme (TPE-LSB),the ciphertext image obtained by the TPEH2MWD scheme has the lower adjacent pixel correlation;furthermore,compared to the exact TPE scheme (TPE2),our data still have significant advantages.

Table 4 Correlation coefficients of plaintext and ciphertext images

Fig.9 Correlation analysis of the R channel in the image:(a)?(c) are the results of correlation of plaintext image in the horizontal,vertical,and diagonal directions,respectively;(d)?(f) are the results of correlation of encrypted image with K=32 in the horizontal,vertical,and diagonal directions,respectively

4.6 Running time

The time complexity of the TPE2 scheme isO(mnd3),wheremandnare the dimensions of the plaintext image anddis the maximum value of pixels in the plaintext image,and the time complexity of the TPE-H2MWD scheme isO(mn).Here,we intercept a subblock of a single channel in the color image for encryption,where the block size is 32×32.The data in Table 5 show that the encryption time of the exact TPE scheme (TPE2) grows rapidly with increasing subblock size and is longer than the running time in TPE-H2MWD.The approximate TPE scheme (TPE-LSB) has the shortest encryption time.

Table 5 Running time

Table 6 Mean average precision (mAP) values of image retrieval on encrypted image thumbnails based on the color histogram features

Table 7 Mean average precision (mAP) values of image retrieval on encrypted image thumbnails based on VGG16 depth features

4.7 Image retrieval effect test

At present,content-based image retrieval is a mainstream retrieval method.Encrypted image thumbnails are obtained using in-block averaging,and then the color histogram feature and the depth feature based on VGG16 are extracted from the ciphertext image thumbnails.The cosine distance between feature vectors is used to calculate the similarity between images,and finally,the image retrieval accuracy is obtained.Encrypted image thumbnails from the Holiday dataset are tested,and the mean average precision (mAP) results are listed in Tables 6 and 7.The larger the mAP value is,the higher the retrieval accuracy is.Plaintext images have the highest retrieval accuracy.In addition,when the block size reaches 16×16,32×32,and 64×64,the highest retrieval accuracy can be obtained if the thumbnail size is 32×32;when the block size reaches 8×8,the highest retrieval accuracy can be obtained if the thumbnail size is 64×64.Thus,users can choose the block size and thumbnail size according to their specific requirements.

4.8 Security analysis

Most likely,the biggest threat to TPE algorithms is super resolution (SR) technology,which reconstru?cts low-resolution images into corresponding highresolution images by specific algorithms.Fig.10a shows an image with a resolution of 128×128,Fig.10b corresponds to a thumbnail with a resolution of 32×32,and the results obtained by reconstructing the thumbnail with advanced SR algorithms FSRCNN (Dong et al.,2016) and DASR (Wang et al.,2021) are shown in Figs.10c and 10d,respectively.While the SR technique can enhance the visual quality of thumbnail images,the details of the reconstructed images and the original images differ,suggesting that the SR scheme restores only the approximate images but not the real ones.

Fig.10 Super resolution simulation results:(a) image with a resolution of 128×128;(b) a thumbnail with a resolution of 32×32;(c) results obtained by reconstructing the thumbnail with FSRCNN;(d) results obtained by reconstructing the thumbnail with DASR

4.9 Size expansion rate

In this subsection,we use the PNG images obtained from the Helen dataset to evaluate the expansion rate of the encrypted images.Suppose that the file size of the encrypted image isxand the file size of the plaintext image isy.Then,the encrypted image expansion rate is equal tox/y.For images in lossless PNG format,the redundancy of image information is used to accomplish lossless image compression.The redundancy of the plaintext image tends to be more than the redundancy of the encrypted image,resulting in a lower compression rate of the encrypted image than the plaintext image.As shown in Fig.11,the average encrypted image expansion rates in different methods are demonstrated.In the TPE-H2MWD scheme,the cipher image expansion rate expands with increasing block size because the larger the block size is,the greater the number of available bit random shift positions,the more severely the adjacent pixel correlation is broken,and the lower the compression efficiency.

Fig.11 Average size expansion rate of encrypted images

5 Conclusions

In this paper,an exact TPE is implemented with hidden Markov model and weighted diffusion,and the specific encryption process and simulation results are given.The proposed TPE-H2MWD consists of two stages:bit-level permutation and weighted diffusion.In the scrambling phase,the zigzag models are applied in the lower four bit planes,and the hidden Markov model is used in the higher four bit planes.In the diffusion phase,the sum of the pixel values is kept constant.Simulation results show that the TPEH2MWD can effectively balance the security and availability of images,where illegal users cannot obtain detailed information about the plaintext images from the encrypted images,while image owners can achieve image usability based on the relevant information about the plaintext images retained in the encrypted images.

Contributors

Xiuhui CHEN processed the data.Yakun MA validated the study.Fang ZUO supervised the study.Xiuli CHAI drafted the paper.Zhihua GAN helped organize the paper.Yushu ZHANG revised and finalized the paper.

Compliance with ethics guidelines

Xiuli CHAI,Xiuhui CHEN,Yakun MA,Fang ZUO,Zhihua GAN,and Yushu ZHANG declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.