Secure high capacity tetris‐based scheme for data hiding

IET Image ProcessingVolume 14, Issue 17 p. 4633-4645 Research ArticleOpen Access Secure high capacity tetris-based scheme for data hiding Correction(s) for this article Erratum: Secure high capacity tetris-based scheme for data hiding Volume 15Issue 12IET Image Processing pages: 3020-3020 First Published online: June 18, 2021 Guo-Dong Su, Guo-Dong Su Engineering Research Center for ICH Digitalization and Multi-source Information Fusion (Fujian Polytechnic of Fujian Normal University), Fujian Province University, Fuzhou, 350500 People's Republic of China Department of Information Engineering and Computer Science, Feng Chia University, Taichung, 40724 TaiwanSearch for more papers by this authorChin-Chen Chang, Chin-Chen Chang Department of Information Engineering and Computer Science, Feng Chia University, Taichung, 40724 TaiwanSearch for more papers by this authorChia-Chen Lin, Corresponding Author Chia-Chen Lin ally.cclin@ncut.edu.tw Department of Computer Science and Information Engineering, National Chin-Yi University of Technology, Taichung, 41170 Taiwan Department of Computer Science and Information Management, Providence University, Taichung, 40724 TaiwanSearch for more papers by this authorZhi-Qiang Yao, Zhi-Qiang Yao College of Mathematics and Informatics, Fujian Normal University, Fuzhou, Fujian, 350117 People's Republic of ChinaSearch for more papers by this author Guo-Dong Su, Guo-Dong Su Engineering Research Center for ICH Digitalization and Multi-source Information Fusion (Fujian Polytechnic of Fujian Normal University), Fujian Province University, Fuzhou, 350500 People's Republic of China Department of Information Engineering and Computer Science, Feng Chia University, Taichung, 40724 TaiwanSearch for more papers by this authorChin-Chen Chang, Chin-Chen Chang Department of Information Engineering and Computer Science, Feng Chia University, Taichung, 40724 TaiwanSearch for more papers by this authorChia-Chen Lin, Corresponding Author Chia-Chen Lin ally.cclin@ncut.edu.tw Department of Computer Science and Information Engineering, National Chin-Yi University of Technology, Taichung, 41170 Taiwan Department of Computer Science and Information Management, Providence University, Taichung, 40724 TaiwanSearch for more papers by this authorZhi-Qiang Yao, Zhi-Qiang Yao College of Mathematics and Informatics, Fujian Normal University, Fuzhou, Fujian, 350117 People's Republic of ChinaSearch for more papers by this author First published: 11 March 2021 https://doi.org/10.1049/iet-ipr.2019.1694AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Information hiding is a technique that conceals private information in a trustable carrier, making it imperceptible to unauthorised people. This technique has been used extensively for secure transmissions of multimedia, such as videos, animations, and images. This study proposes a novel Tetris-based data hiding scheme to flexibly hide more secret messages while ensuring message security. First, an LQ × LQ square lattice Q is selected to determine the maximum embedding capacity, and then it is filled without gaps through rotating and sliding tetrominoes while making the shape of each tetromino different. Secondly, according to the decided Q, the reference matrix and corresponding look-up table are constructed and then used for secret messages embedding and extraction. In the authors approach, each pixel pair of the original image can be processed to conceal 4- or 6-bit secret messages. The experimental results show that their proposed Tetris-based scheme has excellent performance, exceeding the performance of some state-of-the-art schemes in both embedding capacity and visual quality. The proposed scheme also provides secure covert communication. 1 Introduction Data hiding (DH) is one technique that has been designed to hide secret messages using various cover media, i.e. images, videos, and audios files, for secret and secure communication [1-4] or the error resilience of multimedia communications [5-7]. The key point of DH lies on the imperceptible modifications made on cover media so that the human eyes have difficulty detecting any distortion caused by DH. In the past decade, many studies in terms of DH have been presented and have achieved quite good visual quality and a promising high embedding capacity (EC) of stego-images. Many literature works have indicated that there still exists a trade-off between the EC and image quality of the stego-image. However, just because of this challenge, this technique keeps attracting attention from many scholars. In 2006, an improved LSB (least significant bit) substitution scheme [8], namely the LSB matching revisited scheme, was proposed by Mielikainen [9]. His scheme concealed 2-bit secret messages into a pixel pair, where the LSB of the first pixel carries one bit and a function of those two-pixel values carries another bit. In the same year, Zhang and Wang [10] observed that there was room for improvement in fully exploiting the directions of modification in Mielikainen's scheme [9]. Thus, they proposed the EMD (exploiting modification direction) scheme, where n cover pixels as a unit can carry a (2n + 1)-ary secret digit, and only one pixel is added or subtracted by 1 at most. EMD's advantage is that it successfully controlled the distortion while its hiding capacity still has not been enhanced. In 2010, the schemes of Kim et al. [11], also called EMD-2 and 2-EMD, were proposed to promote the EC. Experimental results confirmed that Kim et al.’s embedding rate (ER) is higher than the original EMD scheme while maintaining a similar stego-image quality. Meanwhile, by extension, i.e. EMD-k and k-EMD, their scheme is also easy to be conducted. The following year, Kieu and Chang [12] proposed a new DH scheme by exploiting eight modification directions to hide several secret messages into a pixel pair at a time. In this way, it provides a high EC up to 4.5 bpp (bits per pixel), and the corresponding image quality is around 31.70 dB. Recently, a series of DH schemes [13-19] based on the concept of a special structure, i.e. turtle shell (TS), were also developed to obtain a better balance between EC and stego-image quality. To enhance the level in terms of security, some researches [20-29] have explored using the idea of a puzzle game to hide messages. The review of them [13-29] will be presented in Section 2. However, those schemes might still have difficulty achieving a win-win in both EC and image quality. Now, this scenario presents a requirement for the continuous innovations of the DH technique. In this paper, another Tetris-based DH scheme is proposed to further enhance both the EC and image quality of puzzle game-based DH scheme. Tetris is a tetromino-based falling a puzzle game whose main point is to fill a gameboard by rotating and sliding seven tetrominoes. In our proposed Tetris-based scheme, first, a square lattice Q with the size of LQ × LQ is filled without gaps through rotating and sliding these tetrominoes. Secondly, a reference matrix sized 256 × 256 is generated by replicating this decided Q and then the corresponding look-up table (LUT) is constructed. Afterwards, according to this reference matrix and LUT, each pixel pair can be processed to embed flexible secret messages. The larger the LQ, the higher the EC has. The experimental results show that the proposed scheme achieves a considerable EC along with satisfactory image quality than that of other puzzle game-based DH schemes. The rest of this paper is organised as follows. Section 2 gives the related works. Details of the proposed Tetris-based DH scheme are discussed in Section 3. Experimental results are presented in Section 4, and Section 5 presents our conclusions. 2 Related works To provide sufficient background knowledge, several TS, Sudoku and Tetris-based DH schemes will be briefly reviewed in Section 2.1. Next, we introduce a TS-based scheme [14] in Section 2.2 because their innovative in designing location table inspires the idea of our designed Tetris-based scheme. Finally, we state the basic knowledge of Tetris in Section 2.3. 2.1 Review of the related DH schemes In 2014, Chang et al. [13] proposed a novel DH scheme based on the concept of a special structure, i.e. TS. In their scheme, an 8-ary secret digit was concealed into a pixel pair of cover image with the guidance of TS. Their scheme provides a good visual quality, i.e. 49.40 dB, of stego-image on average along with a high ER of 1.5 bpp. To further enhance the EC for Chang et al.’s TS-based scheme, a DH scheme combing the reference matrix and a location table was presented by Liu et al. [14] in 2016. In Liu et al.'s scheme, each pixel pair conceals 4-bit secret messages, and the average image quality of the stego-images is about 45.55 dB. In 2017, Jin et al. [15] used a particle swarm optimisation algorithm to reduce the distortion from the original image. Compared to schemes [13, 14], the average improvement of peak signal-to-noise ratios (PSNRs) of Jin et al.’s scheme was 0.03 and 0.01 dB, respectively. At the same year, Liu et al. [16] extended the structure of TS-based reference matrix into various models to meet different ECs and the needs of image quality. With the guidance of an extended reference matrix, an N-ary secret digit can be embedded into a pixel pair. Their scheme provides a high EC with the ER of 2.5 bpp and good image quality with an average PSNR of 41.87 dB. Afterwards, in order to overcome the problem of the drastic decrease in stego-image quality as N increases in [16], in 2019, Liu et al. [17] further proposed a DH scheme based on a multi-matrix structure of TS to balance the requirements of EC and image quality. For different smooth-component image blocks, the different TS-based reference matrices were employed to implement the DH to ensure as few distortions as possible. Their scheme achieves an ER of 2.35 bpp, while the image quality is around 44.31 dB. At the same year, Li et al. [18] defined an upgraded reference matrix based on TS by adding 8 to partial digits to ensure digits of the three neighbouring TSs ranged from 0 to 15. Meanwhile, with the help of the self-defined numbering rule and LUT, each pixel pair can carry 6-bit secret messages, while maintaining an average PSNR of 40.93 dB. Also, in 2020, a DH scheme based on a 3D magic cube was developed by Lee et al. [19], where they achieved an ER of 2.25 bpp and average PSNR of 44 dB. In addition, some research studies [20-29] have explored using the idea of a puzzle game to hide messages. In 2008, Chang et al. [20] proposed a DH scheme based on a number placement puzzle game, i.e. Sudoku, in which a pixel pair of the original image was utilised to carry a 9-ary secret digit with the help of a 256 × 256 reference matrix that can be constructed by a determined solution of Sudoku. The average ER provided by scheme in [20] is about 1.58 bpp, and the stego-image visual quality is around 44.90 dB. In the same year, Hong et al. [21] observed that there was space for improvement in searching for candidate elements in Chang et al.’s scheme [20]. That is, in some cases, a couple more suitable candidate pixel pairs in the reference matrix could be found for DH, with fewer distortions. As a result, scheme [21] achieves better image quality in the stego-image than that of Chang et al.’s scheme [20], with an average improvement of 2.60 dB. Later, Hong et al. [22] again provided another suggestion, which was to measure the difference between the original pixel pair and the candidate pixel pair by using the Euclidean distance instead of the Manhattan distance. In this scheme, the stego-image quality is 0.60 dB higher than their previous work proposed in [21]. Also, Chang et al. [23] presented an improved version of the ones present in [20-22] using the greedy method, where PSNR reaches 48.14 dB. In 2009, Farn and Chen [24] proposed a new DH scheme using jigsaw puzzle images. In their scheme, the image was first divided into non-overlapping blocks. For each block, a semicircle was drawn and attached to the right and bottom sides. Then, the secret messages were embedded through adjusting the attached positions and the orientations of semicircles. The EC of a 1024 × 1024 image offered by their scheme is 992 bytes. In 2014, a steganographic method based on the Tetris game on a practical scenario was proposed by Ou and Chen [25]. The secret messages were carried by using a generated tetromino sequence. Their scheme is undetectable and has a weakness in EC. In 2018, Lyu et al. [26] presented a DH scheme based on kinds of irregularly shaped Sudoku. A benefit of the reshaped Sudoku is that their scheme provides more secure than that of the original Sudoku-based scheme. Furthermore, their scheme also achieves a PSNR of 47.48 dB when the ER is set to 1.5 bpp. To increase the EC of Sudoku-based scheme, the single-layer and multi-layer mini-Sudoku matrix-based high capacity schemes were individually presented by He et al. [27] and Chen et al. [28]. Those two schemes inherit the idea of coordinate mapping used in scheme [14]. Benefit by this, the former scheme achieved the ER of 2.0 bpp and the PSNR of 46.37 dB on hiding capacity and image quality, respectively, and the latter one reached the ER of 3.0 bpp and the PSNR of 40.01 dB on hiding capacity and image quality, respectively. Additionally, Horng et al. [29] developed the multidimensional mini-Sudoku-based DH scheme in 2020. It achieves good image quality and EC, which are like the results in [27]. 2.2 Liu et al.'s TS-based DH scheme In 2016, a high capacity TS-based DH scheme was proposed by Liu et al. [14], where each pixel pair is processed to conceal a 4-bit secret message. In their scheme, a reference matrix RM is constructed in advance, as shown in Fig. 1, and then shows the renewal strategy during data embedding. To reduce the caused distortion and minify the search area, RM must possess the following features: (i) The RM has an x-axis (pi) and a y-axis (pi+1). Each of them ranges from 0 to 255 and maps to the pixel value of a given pixel pair in a cover image. (ii) The difference of two adjacent pixel values in the same row of RM always remains 1, but the differences between two adjacent pixel values in the same column are 2 and 3 in turn. (iii) RM consists of a series of interconnected TSs, and each TS covers eight different distinct digits that all range in [0, 7]. Meanwhile, the digits located inside the TS are named as back elements, such as the red digits shown in Fig. 1; conversely, other digits located on the edge of the TS are called edge elements, as the blue digits shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Reference matrix RM In Liu et al.’s scheme, there are 16 different cases in total that have been determined according to four pre-defined locations and different digits to enhance the EC. Based on those cases, a location table LT as shown in Fig. 2 is constructed and used for DH. For simplicity, each element in LT is called as an LT-shape, and its corresponding digit d represents the value of LT-shape. To carry secret messages (sj, sj+1), a pixel pair is changed to a renewed pixel pair with the principle of minimum Euclidean distance. The renewed pixel pair should satisfy two properties: (i) its projected location in RM should be the same as the LT-shape LT(sj, sj+1); (ii) its projected value in RM should be equal to the value of LT(sj, sj+1). Fig. 2Open in figure viewerPowerPoint Location table LT Taking a specific case as an example, assume that we want the pixel pair (5, 3) to carry secret messages (sj, sj+1) = (01 00). First, the LT-shape LT(01, 00) (see the red circle in Fig. 2) is determined in LT, and it is found that the corresponding value is 6. Then, some valid candidate pixel pairs whose shape is the same as the pre-determined LT-shape are searched, e.g. RM(7, 3), RM(4, 1) and RM(2, 5). Finally, the RM(7, 3) is chosen because it has the shortest distance to RM(5, 3). In other words, the original pixel pair (5, 3) is changed to (7, 3) in the stego-image and two secret messages (01 00) have been hidden. 2.3 Pazhitnov's Tetris puzzle game Tetris [25, 30] was invented by Alexey Pazhitnov in 1985, and it is one of most popular computer games. Tetris is a tetromino-based falling puzzle game whose objective is to form some solid horizontal lines without gaps by rotating and sliding tetrominoes within an M × N gameboard, and then the filled line disappears. Once a line is filled and disappears, the player gains points. A tetromino is a geometric shape composed of four grid-squares connected orthogonally. In general, there are seven types of tetrominoes in total in a Tetris game, including ‘T’ (T), ‘Square’ (SQ), ‘Right Snake’ (RS), ‘Left Snake’ (LS), ‘Right Gun’ (RG), ‘Left Gun’ (LG), and ‘I’ (I), as shown in Fig. 3a. The allowed rotation angles θ include four types: { 0 ° , 90 ° , 180 ° , 270 ° }. Take tetromino ‘T’ for example. Assume that the first shape ‘T’ shown in Fig. 3b is a base tetromino with the orientation of θ = 0 °, and three other shapes of derivatives with three orientations of 90 °, 180 °, and 270 °, are depicted on its right side, respectively. To sum up, Tetris includes seven types of tetrominoes in total, and each tetromino can be derived into different shapes with each shape also consisting of four grid-squares. Concretely, in our approach, a series of combinations of the tetrominoes and their derivatives will be conducted and then utilised to fill an M × N gameboard without gaps to generate a meaningful matrix, which will be adopted as the fundamental framework of the reference matrix in our proposed DH scheme. Fig. 3Open in figure viewerPowerPoint Definition of tetrominoes (a) Seven tetrominoes, (b) Four derivative tetrominoes of ‘T’ with four orientations θ s 3 Proposed scheme Inspired by Liu et al.'s location-based DH scheme [14], in this paper, we present a novel DH scheme based on Tetris to further enhance the image quality of stego-image and EC. The proposed scheme mainly consists of two phases: (i) secret messages embedding phase; and (ii) secret messages extraction phase. Before that, we shall introduce how to construct the reference matrix based on Tetris, called MT, and the corresponding LUT in Sections 3.1 and 3.2, respectively, for later use in the data embedding phase. 3.1 Reference matrix construction Similar to Sudoku-based DH scheme, in order to effectively embed secret messages, a Tetris-based reference matrix, called MT, with a size of 256 × 256, should be first constructed. In our proposed scheme, the structure of the MT is quite novel, and the construction of MT is like to play a jigsaw puzzle game. We take a square lattice, Q, with the size of LQ × LQ. Obviously, there are L2Q empty grid-squares in the Q. Although the selection of LQ has various possibilities, it should meet this rule, i.e. the L2Q must be divisible by 4 (Any one of tetrominoes has 4 grid-squares), making those L2Q/4 tetrominoes can just appropriately fill out the corresponding square Q without gaps. As a result, values of LQ can be determined, and are ranged in {2, 4, 6, 8, 10, …, 256}. Next, for simplicity, we mainly discuss the cases when LQ is 4 and 6, which can lead to a considerable EC, in this paper. After a Q has been prepared, a jigsaw puzzle game is conducted in an entire Q and its objective is to make all grid-squares are filled without gaps by rotating and sliding some tetrominoes while making each tetromino has different shape. It is certain that various splicing solutions of the Q will be derived, and Figs. 4a and 5a illustrate one of solutions when LQ is set to 4 and 6, respectively. For simplicity, we name the spliced Q in Figs. 4a and 5a as Q4 and Q6, respectively. Fig. 4Open in figure viewerPowerPoint Constructing Q4 (a) Example of spliced Q when LQ is 4, namely, Q4, (b) The corresponding four different shapes for tetrominoes, namely, ‘T1’, ‘T2’, ‘T3’, and ‘T4’ Fig. 5Open in figure viewerPowerPoint Constructing Q6 (a) Example of spliced Q when LQ is 6, namely, Q6, (b) The corresponding nine different shapes for tetrominoes, namely, ‘T1’, ‘T2’, ‘T3’, ‘T4’, ‘SQ’, ‘RS1’, ‘RS2’, ‘LS1’, and ‘LS2’ Concretely, Fig. 4a shows that the Q4 consists of ‘T's with four different orientations, namely, ‘T1’, ‘T2’, ‘T3’, and ‘T4’, as shown in Fig. 4b. Fig. 5a demonstrates that the Q6 is made up of nine different shapes of tetrominoes, including ‘T1’, ‘T2’, ‘T3’, ‘T4’, ‘SQ’, ‘RS1’, ‘RS2’, ‘LS1’, and ‘LS2’. In our approach, each different shape of tetrominoes will be used to represent a string of secret messages. In addition, each shape also has four grid squares and is always filled in four distinct digits ranging in [0, 3], as shown in Figs. 4b and 5b. Similarly, each digit can represent a two-bit secret message later. After the spliced Q has been generated, the MT with a size of 256 × 256 is constructed by replicating this pre-decided Q. Fig. 6 shows examples of the corresponding reference matrices MT4 and MT6, which are derived from Q4 and Q6, respectively. Fig. 6Open in figure viewerPowerPoint Examples of reference matrices MT4 and MT6 Fig. 6 demonstrates the following features: (i) In MT4, in any 4 × 4 square, such as the white dotted rectangles shown, four different shapes of tetrominoes, i.e. ‘T1’, ‘T2’, ‘T3’, and ‘T4’ can always be found. (ii) Similarly, nine shapes of tetrominoes, i.e. ‘T1’, ‘T2’, ‘T3’, ‘T4’, ‘SQ’, ‘RS1’, ‘RS2’, ‘LS1’, and ‘LS2’ can always be found for any given 6 × 6 square in MT6. Such unique feature guarantees that given a coordinate in MT4, a 4 × 4 square can be determined; then, four different shapes of tetrominoes within the square can always be found. A similar feature also exists in MT6, but the type of distinct tetromino shapes is changed to nine. Based on this feature, we will construct a mapping relation between tetrominoes’ shapes and secret digits (see Fig. 7) and then use it to serve for concealing secret messages. Fig. 7Open in figure viewerPowerPoint Look-up table (a) LUT4, (b) LUT6 3.2 LUT construction After Q and MT have been prepared, the corresponding LUT based on tetrominoes shall be constructed for data embedding. The construction of LUT can be processed by the rules as follows: (i) Scan all shapes of used tetrominoes within Q. (ii) Sort those tetrominoes first by the order in which they appear in Fig. 3a. (iii) For the same kind of different shapes of tetrominoes, sort them by the ascending order of the rotation angle. (iv) Number this sorted sequence using the shape number from 0 to (L2Q/4–1). Finally, the LUT can be derived, such as LUT4 and LUT6 as shown in Fig. 7. As Fig. 7a shows, LUT4 is designed for MT4 to carry a 4-bit of secret message ranging from (‘00 00’) to (‘11 11’). In LUT4, each column includes four distinct shapes of tetrominoes, which can represent four distinct secret digits ranging from 0 to 3 and their corresponding binary representations are mapped to [‘00’, ‘01’, ‘10’, ‘11’], which were listed in the left of LUT4. Meanwhile, each row has the same tetromino, where its four grid-squares separately cover four different secret digits ranging from 0 (‘00’) to 3 (‘11’), which were listed in the bottom of LUT4. For instance, the (called an LUT-shape) located at (2, 2) in LUT4 means the following: (i) this LUT-shape belongs to ‘T3’ (see Fig. 4b), which represents a binary code ‘10’ in terms of its shape number; (ii) this LUT-shape contains a value of 2, which also maps to a binary code ‘10’. Based on above two facts, it is concluded that the LUT-shape can be used to carry a 4-bit binary stream (‘10 10’). Similarly, we also designed the LUT6 to carry (( log 2 ⁡ 9) + 2) bits of secret messages. In LUT6, each column includes nine different shapes of tetrominoes, which can be used to conceal log 2 9 -bit of secret messages, and its corresponding digits and binary codes were listed in the left of LUT6. Also, each row covers four different digits separately filled in different grid-squares of a tetromino, which is able to carry 2-bit of secret messages. In other words, for example, when the LUT-shape is encountered, we can look for it in LUT6, and then its location is finally determined as (6, 2). Therefore, it can be used to carry secret bits (‘110 10’), which its binary code maps to. When the MT and LUT have been prepared, the data embedding and extraction can be processed. Assume that secret messages need to be embedded into a pixel pair (pi, pi+1). First, the corresponding LUT-shape must be located by projecting the secret messages into the LUT; then, the pixel pair (pi, pi+1) is changed to the other pixel pair so that the new shape and shape's value are the same as the determined LUT-shape. During the process of concealing secret messages into a pixel pair, there are multiple candidates of pixel pairs that can be found by combining LUT and MT. Finally, a candidate pixel pair p ′ i , p ′ i + 1 which causes the least distortion with the original pixel pair (pi, pi+1) is chosen. The distortion d is measured by: d = p i − p ′ i 2 + p i + 1 − p ′ i + 1 2 .(1) In addition, it is worth noting that the usable spliced Q should be shared between the sender and the recipient in advance for the aim of extracting the secret messages correctly. With the shared spliced Q, the reference matrix MT and LUT can be correctly derived. 3.3 Secret message embedding phase For a greyscale image I with the size of H × W, all pixels are paired by a pairing technique. For simplicity, the paired results are denoted as I = {(pi, pi+1)| i = 1, 3, 5, 7, …, (H ⋅ W−1)}, and both pi and pi+1 are ranged in [0, 255]. At the same time, assume that the secret messages are B and its length is LB. According to the LB, the selection of MT is classified into two cases. (i) If L B ≤ 2 ⋅ H ⋅ W, the MT4 is chosen to serve for DH to guarantee the visual quality of stego-image as high as possible. In this case, the secret messages are converted into the secret digits in the base-4 numeral system. The converted secret digits are further paired sequentially and denoted as S = {(st, st+1)| t = 1, 2, 3, …, LB/4}, where both the st and st+1 are the base-4 digits. (ii) If 2 ⋅ H ⋅ W < L B ≤ ( log 2 ⁡ 9 + 2 ) ⋅ H ⋅ W / 2, the MT6 is selected to achieve higher EC along with satisfactory stego-image quality. Correspondingly, we convert the secret messages into secret digits in the base-9 and base-4 numeral systems in turn. Concretely, the first several (3 or 4) bits of secret messages are converted into a base-9 digit firstly and then the following 2 bits are converted into a base-4 digit secondly. This process is repeated until all secret messages have been handled. For convenience, the converted secret digits are denoted as S = {(st, st+1)| t = 1, 2, 3, …, L B / ( l o g 2 9 + 2 ) } for case 2, and the major difference from case (1) is that st is a base-9 digit while st+1 is a base-4 digit in case 2. Next, in order to embed secret digits (st, st+1) into an original pixel pair (pi, pi+1), we can process it by the following rules. Firstly, an LUT-shape in LUT is determined according to the to-be-embedded secret digits (st, st+1), where the st and st+1 are treated as the row and column index values in LUT, respectively. Secondly, the pixel pair (pi, pi+1) is projected into the selected MT and denoted as MT(pi, pi+1), where the pi is row index value and the pi+1 is the column index value. By the properties of MT introduced above, it is certain that in the adjacent areas of MT(pi, pi+1), we can find multiple candidates MT p ′ i , p ′ i + 1 whose tetromino location is with the same shape as the determined LUT-shape and its projected digit is equal to the value of the determined LUT-shape. Finally, an p ′ i , p ′ i + 1 that has the least distortion with the original pixel pair (pi, pi+1) is selected as the final pixel pair of the stego-image. In other words, the pair p ′ i , p ′ i + 1 is changed into pair p ′ i , p ′ i + 1 to carry the secret digits (st, st+1). After the embedding procedure, the stego-image SI is sent to the recipient. In order to demonstrate the embedding procedure more clearly, two examples regarding secret digits embedding using MT4 and MT6 are illustrated in Figs. 8 and 9, respectively. Fig. 8Open in figure viewerPowerPoint Example of secret messages embedding using MT4 Fig. 9Open in figure viewerPowerPoint Example of secret messages embedding using MT6 Examples for MT4: Case 1: Assume an original pixel pair is (0, 8), the secret messages are ‘01 00’ and its 4-ary secret digits are (st, st+1) = (1, 0). Using the st = 1 as the row index value of LUT4 and the st+1 = 0 as the column index value; thus, the corresponding LUT-shape is determined. Then, we do a search in MT4 to find multiple candidate pixel pairs whose tetromino has the same shape (‘T2’) as the determined LUT-shape, and its projected digit is equal to st+1 = 0. The white rhombuses shown in Fig. 8 are some candidate pixel pairs, and MT4(2, 9) is ultimately selected as the pixel pair of stego-image because it is the one with the shortest Euclidean di