High capacity reversible data hiding in encrypted images based on adaptive quadtree partitioning and MSB prediction＊

Kaili QI,Minqing ZHANG,Fuqiang DI,Yongjun KONG

Key Laboratory of Network ＆ Information Security under Chinese People Armed Police Force,Engineering University of PAP,Xi’an 710086,China

Abstract: To improve the embedding capacity of reversible data hiding in encrypted images (RDH-EI),a new RDH-EI scheme is proposed based on adaptive quadtree partitioning and most significant bit (MSB) prediction.First,according to the smoothness of the image,the image is partitioned into blocks based on adaptive quadtree partitioning,and then blocks of different sizes are encrypted and scrambled at the block level to resist the analysis of the encrypted images.In the data embedding stage,the adaptive MSB prediction method proposed by Wang and He (2022) is improved by taking the upper-left pixel in the block as the target pixel,to predict other pixels to free up more embedding space.To the best of our knowledge,quadtree partitioning is first applied to RDH-EI.Simulation results show that the proposed method is reversible and separable,and that its average embedding capacity is improved.For gray images with a size of 512×512,the average embedding capacity is increased by 25 565 bits.For all smooth images with improved embedding capacity,the average embedding capacity is increased by about 35 530 bits.

Key words: Adaptive quadtree partitioning;Adaptive most significant bit (MSB) prediction;Reversible data hiding in encrypted images (RDH-EI);High embedding capacity

1 Introduction

With the rapid development of cloud computing,an increasing number of users choose to store data (such as images) in the cloud.However,while users enjoy convenience,their privacy security is under unprecedented threat.To protect the data privacy of image owners while performing necessary operations,information processing on encrypted images has attracted increasing attention (Zhang et al.,2014;Ke et al.,2020;Du et al.,2022;Yu et al.,2022).In many applications,for example,in cloud storage,to protect the images,the owners of images encrypt the images before sending them to the cloud server.After receiving the encrypted images,the cloud server embeds the secret data and thus can protect the privacy of the image owner.Moreover,information processing on encrypted images is conducted for image storage management,labeling,etc.Reversible data hiding in encrypted images (RDH-EI) has thus received much attention (Puech et al.,2008;Zhang,2011;Liao and Shu,2015).At present,RDH-EI schemes can be roughly divided into two categories:vacating room after encryption (VRAE) (Zhang,2011;Zhou et al.,2016) and reserving room before encryption (RRBE) (Yi and Zhou,2017;Puteaux and Puech,2018).

In recent years,VRAE methods have included the original image encryption scheme using homomorphic encryption proposed by Xiong and Dong (2019) and Ke et al.(2020),and a general RDH-EI method based on most significant bit (MSB) prediction and error embedding proposed by Gao et al.(2023).Xiong and Dong (2019) used homomorphic encryption to encrypt the original image.The data hider embeds the secret data using prediction error expansion based on the redundancy of pixels in each block in the encrypted image.It was shown that the proposed method can recover the original image well only when the embedding rate was low;when the embedding rate was high,the quality of the reconstructed image was not satisfactory.The embedding rate was basically no more than 0.45 bits per pixel (bpp).Ke et al.(2020) used homomorphic encryption to encrypt the original image and designed a least significant bit (LSB) data hiding method based on key exchange,by which data could be directly extracted from the encrypted image without a private key.It was shown that the embedding rate of the VRAE method was relatively high,but it was also no more than 0.5 bpp.Gao et al.(2023) proposed an RDH-EI method based on MSB prediction and error embedding.Flags,errors,and message blocks were used to mark the location of prediction errors and data that could be embedded.This method can reconstruct the original image better and improve the embedding rate,but the embedding rate is below 1 bpp.The above reflects an important problem in current VRAE methods:data are hidden after image encryption,and the embedding capacity is limited because the redundancy between pixels is not fully considered;meanwhile,it is difficult to achieve lossless image reconstruction.

To ensure a high embedding rate and good image quality after data extraction and image restoration,Wang and He (2022) used the correlation between pixels and proposed a reversible information hiding scheme with a large capacity.To further improve the embedding capacity,in this paper we propose an improved scheme based on Wang and He (2022)’s method,and adopt adaptive quadtree partitioning according to the smoothness of the image to adaptively generate a number of blocks with different sizes.For blocks greater than 2×2,we use the upper-left pixels to predict other pixels,so that it can free up more space for embedding.As far as we know,this is the first time that adaptive quadtree partitioning has been applied to RDH-EI.The proposed algorithm combines adaptive MSB prediction with adaptive quadtree partitioning,and can implement adaptive quadtree partitioning according to the image texture and free up the space of embedded information.The proposed algorithm can increase the average embedding capacity based on improved adaptive MSB prediction,significantly improving the embedding rate of smooth images.For gray images with a size of 512×512,the average embedding capacity is increased by 25 565 bits.For all smooth images with improved embedding capacity,the average embedding capacity is increased by about 35 530 bits.What is more,the embedding capacity can be adjusted according to the thresholdT,which enables our algorithm to have good practicability.Also,the proposed algorithm has reversibility and separability.

2 Related works

2.1 Analysis of adaptive MSB prediction

After the data hider receives the encrypted image,by adaptive MSB prediction the upper-left pixel is used as the target pixel for each pixel block,to predict the MSB shared among the remaining pixels,so that the remaining pixels can adaptively retain the remaining unshared bits to free up the embedding space.

In the encryption process of Wang and He (2022)’s method,after 2×2 block partitioning,block-level encryption and scrambling are conducted to resist the analysis of encrypted images.Fig.1 shows the structure of each pixel block.Each pixel block is composed of four pixelsP,C1,C2,andC3.PixelPis used to predict the three other pixels,and here it is the target pixel.For each pixel block,all the four pixels are first decomposed into binary 8 bits,and then the three variablesD1,D2, andD3are obtained using the following formula:

Fig.1 Structure demonstration and pixel bit reconstruction of a 2×2 block

where dif(a,b) returns the maximum number of MSBs thataandbshare.For example,because 127=(01111111)2,98=(01100010)2,we can obtain dif(127,98)=3.This means that the number of MSB values shared by 127 and 98’s binary 8 bits is at most three;that is,the first three bits are the same.According to variablesD1,D2, andD3,the minimum bit value of MSB shared byPwithC1,C2,andC3can be calculated,and the variable md can be calculated as follows:

It can be seen from Fig.1 that after pixel block reconstruction,Pis first arranged,followed by the minimum number of shared MSBs (MD).Then,arrange the remaining binary value after removing the minimum number of shared MSBs of other pixels.Finally,the remaining bits can be used to embed additional information,which we define as nc.nc represents (3md?2) bits in quantity,indicating that each sub-block can free up (3md?2) bits.

Similarly,when the block is a sub-block larger than 2×2,the pixel structure demonstration and pixel bit reconstruction of a 4×4 block can be expressed as Fig.2,using the improved prediction method proposed by Wang and He (2022).

Fig.2 Structure demonstration and pixel bit reconstruction of a 4×4 block

PixelPis used to predict the other 15 pixels,which are the target pixels.For each reconstructed pixel block,pixelPis first arranged,followed by the size of the pixel block Dim,which is represented by two bits.Then,the minimum number of shared MSBs is arranged.Since the minimum number of shared MSBs md is within [1,8],there are a total of eight possibilities.To express md uniformly with three bits,md?1 (MD) is used to represent the minimum number of shared MSBs,and the value range of MD here is [0,7].Next,the remaining binary values of the remaining pixels are arranged after removing the shared MSB values,and finally the binary values used to embed the data are arranged.

2.2 Quadtree partitioning

Quadtree is a typical tree-like data structure.The quadtree structure is designed to provide an efficient data organization structure.The quadtree block is implemented based on the quadtree structure.Because of its high efficiency in multiresolution partitioning of spatial data,quadtree partitioning has been widely used in image processing and other fields.The basic idea of quadtree segmentation is to iteratively and adaptively divide the square image (if the carrier image is not square,it can be extended by supplementing the background) according to the preset rules until all the image sub-blocks meet the rules.The whole quadtree structure can be represented by a binary group (I,rule).In each round,the current image block is divided into four sub-blocks:upper left,upper right,lower left,and lower right.Each sub-block corresponds to a node of the quadtree structure.Sub-blocks that do not meet the preset rule will be further divided into the next level’s nodes,and sub-blocks that meet the rule will not be further divided and become leaf nodes.The quadtree partitioning rules can be set according to the actual situation (Di et al.,2019).

In the following we take a 16×16 image block as an example to briefly introduce the basic quadtree partitioning principle.Suppose that the rule is set as follows:the difference between the maximum and minimum pixel values of the image block is less than a thresholdT.If the rule is met,the current node is set as a leaf node,represented by binary bit 1.If the rule is not met,the current node is divided into four child nodes in the quadtree,represented by 0b1b2b3b4,whereb1,b2,b3,andb4correspond to the northwest,northeast,southwest,and southeast sub-blocks,respectively.Then,determine whether the child nodes meet the rule.If the rule is not met,continue to divide the child nodes (Di et al.,2019).

3 The proposed method

The overall structure of our algorithm is shown in Fig.3,divided mainly into three stages:adaptive quadtree partitioning and encryption,data hiding,and data extraction and image recovery.In the partitioning and encryption stage,first the original image is divided into several 64×64 sub-blocks,and then each sub-block is adaptively divided into blocks of different sizes using the quadtree partitioning method according to thresholdT.The image owner uses the encryption key to encrypt the image and then sends it to the cloud server.In the data hiding stage,adaptive quadtree partitioning is used to divide the image,and for each block,improved adaptive MSB prediction is used to free up more space for the embedded data.Then,the encrypted secret data are embedded in the image,and the image is transmitted to the receiver.In the stage of data extraction and image recovery,the receiver uses the data hiding key to extract data directly and the decryption key to recover the original image.

Fig.3 Algorithm framework and process

3.1 Adaptive quadtree partitioning

The core problem of the quadtree coding algorithm is to determine the preset rule of quadtree partitioning and related parameters,that is,to determine when the image block meets the conditions for the next-level partitioning.For a specificT,the quadtree partitioning threshold is defined as the standard value which the minimum number of MSBs shared by the upper-left corner pixels and other pixels should be no less than.The owner adopts adaptive quadtree partitioning to the original image processing before image encryption.Initially,theM×Nimage is divided into several 64×64 non-overlapping sub-blocks.For each 64×64 sub-block,the owner should judge whether it meets the preset rule.If it meets the condition of partitioning,it will no longer be divided into four average blocks until a block of size 2×2 can no longer be partitioned.As shown in Fig.4b,whenTis set to 3,the 8×8 block at the upper-left corner of the Lena image shown in Fig.4a has been partitioned into quadtree blocks.

Fig.4 Pixel values of the selected block (a) and the partitioning result (b)

As shown in Fig.4b,the image can be divided into sub-blocks of different sizes according to thresholdT,which is the minimum value of md.Through such a threshold setting,blocks of different sizes can be divided according to the different smoothness of the image.Compared with the case in which the image is simply divided into 2×2 sub-blocks in Wang and He (2022),more embedding space can be freed up according to the different texture distributions of each image.

3.2 Image encryption

To protect the privacy of the image owner,the owner needs to encrypt the image before the original image transmission.For the first encryption,all the pixels in the block are decomposed into eight binary pixels.Then,encryption keys and a classic stream cipher are used to perform block-level encryption,thus maintaining the relevance between the block pixels to some extent.The encryption formula of thekthbitof themthoriginal pixel in sub-block (i,j) is encrypted intoas follows:

whererk(i,j) represents thekthvalue of the element at (i,j) in the pseudorandom matrix.

For image encryption,a number of 64×64 subblocks are first scrambled.Next,in each stage of quadtree partitioning scrambling,four same-level blocks of each 64×64 sub-block are scrambled according to the key of scrambling,to resist the analysis of the encrypted image,improve the security of the encrypted image,and satisfy the image owner’s demand on protecting the privacy of the image.

3.3 Adaptive quadtree MSB prediction

After receiving the encrypted image,the data hider adopts first adaptive MSB prediction (Wang and He,2022) to free up a large amount of embedding space for data hiding and then adaptive quadtree partitioning.When the block sizea×ais greater than 2×2,pixels of the block are reconstructed,first to arrange pixelPwith eight bits,using the first three bits to arrange the minimum number of shared MSBs (MD),and then two bits are used to represent the size of the block (Dim).For each 64×64 block,after quadtree partitioning by settingT,the blocks have four sizes of 16×16,8×8,4×4,and 2×2,which are represented by 00,01,10,and 11,respectively.Next,arrange the binary values of the remaining pixels after removing the shared MSB value.Finally,arrange the binary value nc of embedding data,which is md(a2?1)?5.

As shown in Fig.5,when the pixel bits of 4×4 blocks are reconstructed toP=11111111,the minimum number of shared MSBs is md=6 (that is,MD=5=(101)2),the type of block size is 4×4 (that is,Dim=10),then arrange the corresponding remaining binary values of other pixels,which aree1=(11)2,e2=(11)2,e3=(11)2,e4=(10)2,e5=(10)2,e6=(11)2,e7=(11)2,e8=(11)2,e9=(01)2,e10=(11)2,e11=(10)2,e12=(11)2,e13=(11)2,e14=(11)2,ande15=(01)2,and finally arrange the secret data,represented by nc.The secret data that can be embedded in this example are 85 bits.

Fig.5 Pixel structure demonstration and pixel bit reconstruction of a 4×4 block

When the block size is 2×2,the block cannot be further divided,so there will be a kind of block that cannot be embedded in the data.When the block is available,the pixel bit reconstruction is as shown in Fig.6a,first to arrange the top-left corner pixelP,then to arrange 11 to represent the size of the block,followed by “1” indicating an available block.Then,MD and the remaining binary values of the three other pixels are arranged,and finally the secret data to be embedded are arranged.

Fig.6 Demonstration and pixel bit reconstruction of a 2×2 available (a) or unavailable (b) block

When the block is unavailable,the reconstruction of pixels is as shown in Fig.6b,first to arrange the top-left corner pixelP,then to arrange pixel block types Dim for 11,followed by 0 indicating that the block is unavailable.Next,MD and the remaining binary value of the three other pixels are arranged.Finally,the embedding secret data are arranged.

Since 2×2 blocks cannot be further divided,according to the characteristics of quadtree blocks,2×2 blocks always appear as four consecutive blocks,so when representing 2×2 blocks,we need only to arrange the size of the block (Dim) in the pixel bit reconstruction of the first 2×2 blocks in the four consecutive blocks.For the remaining three blocks,we need only to directly arrange the value indicating an available or unavailable block after pixelPin the upper-left corner and then the remaining binary value of other pixels and secret data (nc) to be embedded.Each 2×2 block can thus be embedded with (3md?4.5) bits of data on average.

3.4 Data hiding

The data hider first divides the received encrypted image into several 64×64 sub-blocks and then performs adaptive quadtree partitioning for each subblock according toT.In the image encryption stage,a number of 64×64 scrambled sub-blocks are first decrypted.After scrambling,four same-level blocks of each 64×64 sub-block are scrambled according to the key of encryption in each stage of quadtree partitioning scrambling.Therefore,in the data hiding stage,when the data hider performs adaptive quadtree partitioning for each 64×64 sub-block according toT,the partitioning result obtained by the data hider is the same as that sent to the data hider by the image owner after scrambling.

After adaptive quadtree partitioning,the data hider performs adaptive MSB prediction for each size of the block to free up space for embedding the secret data.

To prevent malicious attacks from reading the embedded data,classical symmetric cryptographic algorithms such as RC4 and AES are used to encrypt the secret data using the data hiding key.

Finally,the data hider embeds the encrypted secret data into each available block and side information such as the secret key,thresholdT,and hidden key into the end of the image.The labeled encrypted image with embedded secret data is sent to the receiver.

3.5 Data extraction and image restoration

In the scheme designed in this study,data extraction and image restoration can be truly separated.There are three types of receivers based on the type of key they hold,and receivers with different types of keys will recover different contents.

When the receiver has only the decryption key,the original image can be reversibly restored.First,the labeled encrypted image is divided into several nonoverlapping sub-blocks,and the pixels in each block are decomposed to obtain the corresponding bit sequence.Then,additional information is retrieved from the bit sequence of the last few blocks until the end tag is extracted.Then,adaptive MSB prediction is used for retrieval.First,Pand Dim are retrieved from the bit sequence,followed by recognition of three bits to retrieve MD,and theneiand hidden data.If Dim is retrieved as 00,01,and 10,Ciis restored as

where Trunc(a,b) indicates thatbMSBs inaare truncated,and “+” indicates that left join operation is performed.For example,in Fig.4bC1is identified asC1=(11111111,6)+11=11111111,and the blocks in the first part are processed successively until Dim is identified as 11,indicating that the retrieved blocks and the next three blocks in a row are all 2×2 ones.At this time,one bit is further retrieved backwards.If the bit is 1,it indicates that the blocks are available,and then MD is identified.If the bit is 0,it indicates that the block is unavailable,and then the search for this block is stopped but the search for the next block is continued until the end tag is recognized.After the pixel values of all blocks are identified,the decryption key is used for block-level decryption,and the order of all blocks is recovered;that is,the original image is obtained.

When the receiver has only the data hiding key,the original encrypted image and the encrypted secret data are obtained similarly,as described in the previous case.Next,the embedded secret data are decrypted using the data hiding key,and eventually only the secret data are reversibly recovered.

When the receiver possesses not only the decryption key but also the data hiding key,the receiver can perfectly recover the original image while obtaining the secret data.

4 Simulations

Compared with Wang and He (2022)’s method,the algorithm proposed in this study does not simply divide the image into blocks,but uses the adaptive quadtree partitioning algorithm to divide the image into blocks of different sizes according to the smoothness of the image,and then improves the adaptive MSB algorithm by adding tagged bits to predict the pixels.The improved algorithm fully considers the redundancy between pixels and can release more embedding space in most cases.At the same time,the algorithm has the characteristic of separability and can reversibly extract data and restore the image.

To compare the performances of our proposed method and Wang and He (2022)’s method,simulations are carried out for embedding rate comparison and embedding capacity comparison.The simulations are conducted in a computer with an Intel i7-10875H 2.30 GHz CPU,32 GB RAM,and MATLAB R2018a.To more accurately analyze the effectiveness of the proposed scheme in improving the embedding capacity,eight standard test gray images with different texture features,obtained from the USC-SIPI image database (http://sipi.usc.edu/database/),are used as the test images.

4.1 Embedding capacity

The embedding performance about which data hiders are concerned is the number of data bits that can be embedded in the received encrypted images.Compared with Wang and He (2022)’s method,our method is more effective in improving the embedding capacity.Since MSB prediction is improved using adaptive quadtree partitioning,the redundancy between pixels is more fully considered than simply 2×2 blocks,and more space can be released for embedding.

As shown in Fig.7,for the eight standard test images,the embedding capacity of the proposed method is greatly improved compared with that of Wang and He (2022)’s method.

Fig.7 Comparison of the embedding capacity between the two schemes

To better verify the embedding ability of our method,100 grayscale images are randomly selected from the BOW-2 database (http://dde.binghamton.edu/download/),which has 10 000 images with strong statistical characteristics.It can be seen from Fig.8 that the maximum embedding rate can reach about 4.30 bpp,and that the embedding rates of most images are above 1 bpp.Generally half of the images have an embedding rate higher than 2 bpp.

Fig.8 Embedding rate of the proposed method on 100 images randomly selected from the BOW-2 image database

To better illustrate the embedding performance,the two schemes are compared on 100 images randomly selected from the BOW-2 database,and the results are shown in Table 1.The average embedding capacity of our method is increased by 25 565 bits,and the average embedding rate is increased by 0.0975 bpp.This proves the effectiveness of our proposed method in improving the embedding performance.

Table 1 Comparison of the embedded performance between the two schemes on 100 images randomly selected from the BOW-2 database

To obtain a comprehensive understanding of the embedding performance of our algorithm,the average embedding capacity and average embedding rate of 10 000 images are calculated.Simulation results show that for 10 000 images in the BOW-2 database,the average embedding capacity is 512 863 bits,and the average embedding rate is 1.9564 bpp,indicating that our algorithm has good embedding performance.

Among the 100 images randomly selected,the embedding capacity of 76 images with smooth texture has been improved.As shown in Table 2,EC1and ER1represent the embedding capacity and embedding rate of 76 images with improved embedding performance,respectively,and EC2and ER2represent the embedding capacity and embedding rate of 24 images without improved embedding performance,respectively.Among all images with improved embedding capacity,the average embedding capacity reaches 518 062 bits,and the average embedding rate reaches 2.1118 bpp.The average embedding capacity of our method is increased by 35 530 bits,and the average embedding rate is increased by 0.1355 bpp.However,for the remaining 24 images with complex texture,the average embedding capacity is decreased by 9682 bits,and the average embedding rate is decreased by 0.0369 bpp.Thus,the proposed method can adaptively partition the image according to the image smoothness,so that smooth texture images can have a great increase in embedding capacity.However,for the images with complex texture,more side information is required for quadtree partitioning;therefore,the embedding capacity is not improved.

Table 2 Comparison of the embedding performance on 100 images randomly seclected from the BOW-2 database between the two schemes

For blocks greater than 2×2,although more than two tagged bits are used for each block,compared to Wang and He (2022)’s method,in the proposed method one block spares more space than two bits.Moreover,the proposed method does not need a location map.Therefore,for an image with smooth texture,the proposed method can obtain more blocks greater than 2×2,and the embedding capacity is larger than that of Wang and He (2022)’s method.For the 2×2 block,however,the number of tagged bits of each block is 1.5 more than that in Wang and He (2022)’s method.Although their method needs side information,through simulation discovery,for an image with more complex texture,there are more 2×2 blocks.Therefore,for images with complex texture,the embedding capacity of our method is lower.For the 2×2 blocks,using our method the increased number of tagged bits leads to less embedding space.

To further verify the effectiveness of the proposed method in increasing the embedding capacity,the proposed method is compared with the existing methods (Khanam et al.,2016;Nguyen et al.,2016;Puteaux and Puech,2018;Di et al.,2019;Bhardwaj and Aggarwal,2020;Wang and He,2022).As shown in Table 3,the embedding capacity of the proposed method is higher than those of the existing methods.

Table 3 Comparison of embedding capacity among our method and state-of-the-art methods

To better show that the embedding performance of our method is superior to that of the latest reversible information hiding method after encryption using MSB prediction,six standard test images with a size of 512×512,Airplane,Lena,Peppers,Baboon,Man,and Crowd,are tested using our method and Gao et al.(2023)’s method.Simulations show that the average embedding rate of Gao et al.(2023)’s method for six standard test images is 0.9385 bpp,while that of our method is 1.5776 bpp.Moreover,there have been some other recent VRAE methods,such as those proposed by Xiong and Dong (2019) and Ke et al.(2020).In Xiong and Dong (2019),the embedding capacity is almost no more than 0.45 bpp,and if the embedding capacity is high,the quality of the reconstructed image is not satisfactory.In Ke et al.(2020),the embedding rate is relatively high,but it is also no more than 0.5 bpp,and the average peak signal-to-noise ratio (PSNR) of the reconstructed image can reach 50 dB.However,the average embedding rate of our method reaches 1.8773 bpp and the average embedding rate for all smooth images reaches 2.1118 bpp,higher than those of the methods proposed by Ke et al.(2020) and Gao et al.(2023).Meanwhile,our method can achieve lossless image reconstruction.

4.2 Selection of parameter T

When the quadtree is partitioned,if the minimum number of shared MSBs of pixels in a block is smaller thanT,the block is partitioned;otherwise,improved adaptive MSB prediction is performed.The proposed method enables adaptive quadtree partitioning according to the smoothness of the image by adjustingT.We can adjustTto make full use of the smoothness of the image,or choose a proper intensity to perform adaptive quadtree partitioning.In Table 4,T=t(t=1,2,…,8) indicates that when the maximum number of shared MSBs of pixels in a block is smaller thant,adaptive quadtree partitioning is performed.For the eight standard test images,adjusting the value ofTwill produce different embedding capacities,and each image has an optimal value,which enables the best use of the smoothness of the image to adaptively partition the quadtree to maximize the embedding capacity.For example,for Lena,whenT=3,the embedding capacity reaches a maximum of 478 891 bits.

Table 4 Comparison of image embedding capacity with different T’s

To analyze the relationship between the optimalTand the image texture in more detail,we perform a statistical analysis of the optimalTcorresponding to the 100 images randomly selected from the BOW-2 database.As shown in Fig.9,the optimalTis distributed mainly in three values 3,4,and 5.Among them,T=3 accounts for 57%,andT=4 andT=5 account for 23% and 20%,respectively.By combination with image texture analysis,the image texture is smooth whenTis 4 or 5,and most of the selectedT’s are 3 when the image texture is complex.

Fig.9 Distribution diagram of the optimal parameter T in 100 randomly selected images

4.3 Reversibility

Fig.10 shows the line charts of embedding rate vs.PSNR of the existing methods (Cao et al.,2016;Nguyen et al.,2016;Chen and Chang,2019;Xiong and Dong,2019;Ke et al.,2020) and the proposed method on the Lena image.For different embedding rates,the PSNR of previous methods is no more than 70 dB;in contrast,the PSNR of the proposed method is approaching +∞ for any given embedding rate.In other words,the proposed method achieves true reversibility to restore the original image with a given embedding rate.

Fig.10 Embedding rate vs.PSNR for our method and previous methods on image Lena

4.4 Security

To ensure the security of the encrypted images,the proposed method first adopts block-level encryption in the encryption stage.Fig.11b shows the results of image complexity analysis after block-level encryption.It can be seen that the image can be revealed by complexity analysis to a certain extent.Fig.11c shows the result of complexity analysis of the encrypted image after scrambling.The image cannot be revealed;that is,the proposed encryption method is resistant to the analysis of the encrypted image and ensures the privacy of the image owner.

Fig.11 Image demonstration of several key stages of the proposed method:(a) original image;(b) results of complexity analysis of the block-level encrypted image;(c) final encrypted image complexity analysis results;(d) marked encrypted image;(e) extracted secret data;(f) recovered image

Fig.12 Scatter maps of different images:(a) original image of Lena;(b) original image of F16;(c) encrypted image of Lena;(d) encrypted image of F16;(e) marked encrypted image of Lena;(f) marked encrypted image of F16

To achieve more accurate and objective analysis,taking Lena and F16 as examples,we obtain the scatter diagrams of the original images,the encrypted images obtained by our method,and the marked encrypted images.As we can see from Figs.12a and 12b,pixels are unevenly distributed,the information of the images can be easily analyzed,and the correlation between pixels is strong.As can be easily seen in Figs.12c–12f,it is difficult to analyze the related information of the images,the correlation between pixels is weak,and the security of the image is strong.It can be concluded that the pixel correlation between the encrypted image obtained by the proposed method and the marked encrypted image is weak;that is,the proposed scheme can resist statistical analysis attacks.

4.5 Separability

To verify the separability of our method,we select the Lena image for image restoration and data extraction.When the receiver holds only the decryption key,as shown in Fig.11f,it fully recovers the original image.When the receiver holds only the data hiding key,as shown in Fig.11e,the embedding secret data are fully restored.When the receiver has both the decryption key and data hiding key,as shown in Figs.11e and 11f,the receiver can fully recover the original image and the embedded secret data.That is,our method has the ability to extract the secret data and restore the original images.

5 Conclusions

We have proposed high capacity reversible data hiding in encrypted images based on adaptive quadtree partitioning and MSB prediction.This method is developed based on Wang and He (2022)’s method.It adopts adaptive quadtree partitioning according to the thresholdTand the smoothness of the image while ensuring reversibility to restore the original image.It can adaptively change the embedding capacity by adjusting thresholdT.For images with smooth textures,the embedding capacity of our method is greatly improved compared with that of Wang and He (2022)’s method.In addition,our method has separability and reversibility,and thus is more practical than previous methods.

In the future,we will further improve the universality and practicability of our method in increasing the image embedding capacity.

Contributors

Kaili QI designed the research.Kaili QI and Fuqiang DI processed the data.Kaili QI drafted the paper.Yongjun KONG helped organize the paper.Kaili QI and Minqing ZHANG revised and finalized the paper.

Compliance with ethics guidelines

Kaili QI,Minqing ZHANG,Fuqiang DI,and Yongjun KONG declare that they have no conflict of interest.

Data availability

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