Secure high capacity tetris-based scheme for data hiding

: 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 L Q × L Q 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.


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][2][3][4] or the error resilience of multimedia communications [5][6][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.
In this paper, another Tetris-based DH scheme is proposed to further enhance both the EC and image quality of puzzle gamebased 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 L Q × L Q 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 L Q , 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. 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.

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][21][22][23][24][25][26][27][28][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][21][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 Sudokubased 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].

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 (p i ) and a y-axis (p i +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. 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 (s j , s j+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(s j , s j+1 ); (ii) its projected value in RM should be equal to the value of LT(s j , s j +1 ). Taking a specific case as an example, assume that we want the pixel pair (5,3) to carry secret messages (s j , s j+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 predetermined 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.

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.

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.

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 L Q × L Q . Obviously, there are L 2 Q empty grid-squares in the Q. Although the selection of L Q has various possibilities, it should meet this rule, i.e. the L 2 Q must be divisible by 4 (Any one of tetrominoes has 4 grid-squares), making those L 2 Q /4 tetrominoes can just appropriately fill out the corresponding square Q without gaps. As a result, values of L Q can be determined, and are ranged in {2, 4, 6, 8, 10, …, 256}. Next, for simplicity, we mainly discuss the cases when L Q 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 L Q is set to 4 and 6, respectively. For simplicity, we name the spliced Q in Figs. 4a and 5a as Q 4 and Q 6 , respectively.
Concretely, Fig. 4a shows that the Q 4 consists of 'T's with four different orientations, namely, 'T 1 ', 'T 2 ', 'T 3 ', and 'T 4 ', as shown in Fig. 4b. Fig. 5a demonstrates that the Q 6 is made up of nine different shapes of tetrominoes, including 'T 1 ', 'T 2 ', 'T 3 ', 'T 4 ', 'SQ', 'RS 1 ', 'RS 2 ', 'LS 1 ', and 'LS 2 '. In our approach, each different shape of tetrominoes will be used to represent a string of  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 MT 4 and MT 6 , which are derived from Q 4 and Q 6 , respectively. Fig. 6 demonstrates the following features: (i) In MT 4

, in any 4
× 4 square, such as the white dotted rectangles shown, four different shapes of tetrominoes, i.e. 'T 1 ', 'T 2 ', 'T 3 ', and 'T 4 ' can always be found. (ii) Similarly, nine shapes of tetrominoes, i.e. 'T 1 ', 'T 2 ', 'T 3 ', 'T 4 ', 'SQ', 'RS 1 ', 'RS 2 ', 'LS 1 ', and 'LS 2 ' can always be found for any given 6 × 6 square in MT 6 . Such unique feature guarantees that given a coordinate in MT 4 , 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 MT 6 , 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.

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 (L 2 Q/4-1). Finally, the LUT can be derived, such as LUT 4 and LUT 6 as shown in Fig. 7. As Fig. 7a shows, LUT 4 is designed for MT 4 to carry a 4-bit of secret message ranging from ('00 00') to ('11 11'). In LUT 4 , 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 LUT 4 . 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 LUT 4 . For instance, the (called an LUT-shape) located at (2,2) in LUT 4 means the following: (i) this LUT-shape belongs to 'T 3 ' (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 LUT 6 to carry ((log 2 9) + 2) bits of secret messages. In LUT 6 , 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 LUT 6 . 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 LUT 6 , 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 (p i , p i+1 ). First, the corresponding LUT-shape must be located by projecting the secret messages into the LUT; then, the pixel pair (p i , p i+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 (p i , p i+1 ) is chosen. The distortion d is measured by: 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.  6 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 = {(s t , s t+1 )| t = 1, 2, 3, …, L B /(log 2 9 + 2) } for case 2, and the major difference from case (1) is that s t is a base-9 digit while s t+1 is a base-4 digit in case 2.

Secret message embedding phase
Next, in order to embed secret digits (s t , s t+1 ) into an original pixel pair (p i , p i+1 ), we can process it by the following rules. Firstly, an LUT-shape in LUT is determined according to the to-beembedded secret digits (s t , s t+1 ), where the s t and s t+1 are treated as the row and column index values in LUT, respectively. Secondly, the pixel pair (p i , p i+1 ) is projected into the selected MT and denoted as MT(p i , p i+1 ), where the p i is row index value and the p i +1 is the column index value. By the properties of MT introduced above, it is certain that in the adjacent areas of MT(p i , p i+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 (p i , p i+1 ) is selected as the final pixel pair of the stegoimage. In other words, the pair p′ i , p′ i + 1 is changed into pair p′ i , p′ i + 1 to carry the secret digits (s t , s t+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 MT 4 and MT 6  column index value; thus, the corresponding LUT-shape is determined. Then, we do a search in MT 4 to find multiple candidate pixel pairs whose tetromino has the same shape ('T 2 ') as the determined LUT-shape, and its projected digit is equal to s t+1 = 0. The white rhombuses shown in Fig. 8 are some candidate pixel pairs, and MT 4 (2,9) is ultimately selected as the pixel pair of stego-image because it is the one with the shortest Euclidean distance to the original pixel pair (0, 8). Finally, the pixel pair (0, 8) is changed to a stego-pixel pair (2,9) to carry the secret messages '01 00'. Case 2: Assume there are secret messages '10 11' whose corresponding converted 4-ary secret digits are (s t , s t+1 ) = (2, 3). Also, assume the current pixel pair is (6,4). Similarly, we first find the LUT-shape in LUT 4 , where its row index value is 2, and its column index value is 3. Then, according to the determined LUTshape, the pixel pair (7, 5) is chosen because its shape and value are the same as that of the determined LUT-shape, while it also has the minimum distance with the pixel pair (6,4). Finally, the original pixel pair (6, 4) is modified to a stego-pixel pair (7,5), and the secret messages '10 11' were carried.
Examples for MT 6 : Case 1: Assume that the (p i , p i+1 ) is (1, 1) and a segment of secret messages are '100 10'. In this case, the first three secret messages '100' are converted into a 9-ary digit s t and the remaining two secret messages are transferred into a 4-ary digit s t+1 , and hence it results as (s t , s t+1 ) = (4, 2). To begin with, we think of the s t as a row index value and s t+1 as a column index value and then project them in LUT 6 . Finally, an LUT-shape located at LUT 6 (4, 2) is found. Afterwards, a series of candidates, e.g. MT 6 (3, 3), MT 6 (3,9) and MT 6 (9, 3), as red rhombuses shown in Fig. 9, are looked for. Then, the distance between (p i , p i+1 ) and the candidates is measured to select a candidate that causes the least distortion. Obviously, the MT 6 (3, 3) is the closest one of these, so the original pixel pair (1, 1) is modified to stego-pixel pair p′ i , p′ i + 1 = (3, 3). Case 2: Take another specific case for example. Suppose the secret messages are '1000 00' and its corresponding converted secret digits (s t , s t+1 ) are (8, 0). They should be concealed into the pixel pair (6,7). Certainly, we use (s t , s t+1 ) = (8, 0) to find an LUT-shape in LUT 6 in the same way, i.e.

Secret message extraction phase
When the recipient obtains the stego-image SI and the spliced Q, the LUT can be reconstructed in the same way mentioned in Section 3.2, and then the process of data extraction can be performed. Firstly, s/he constructs the reference matrix MT using the spliced Q. Secondly, the stego-image SI is divided into nonoverlapping pixel pairs in the same way and is denoted as SI = { p′ i , p′ i + 1 | i = 1, 3, 5, 7, …, (H ⋅ W−1)}. To extract the secret messages, the recipient projects each stego-pixel pair p′ i , p′ i + 1 into the MT, i.e. MT p′ i , p′ i + 1 . Then, an LUT-shape can be determined by combining the shape of the tetromino the MT p′ i , p′ i + 1 locates on and the value of MT p′ i , p′ i + 1 . According to the aforementioned determined LUT-shape, we can find its corresponding coordinate (s t , s t+1 ) in LUT, those found coordinate shall be the secret digits. Finally, those secret digits are converted into binary codes. After concentrating all derived binary codes, the original secret messages are generated. The same operations are performed until all pixel pairs are processed and the secret messages can be extracted.
Examples: For simplicity, we only take two examples for MT 6 and LUT 6 to explain the process of secret message extraction.
) is locked, and the secret messages '100 10' are extracted. Case 2: Take the pixel pair (6,8) as the second example. We project it into MT 6 as the red hexagon inside the dotted rectangle shown in Fig. 9. It is apparent that MT 6 (6, 8) is located on the tetromino 'LS 2 ' and its shape (i.e. ) matches that in the eighth row in LUT 6 , so that the first 4 bits of secret messages are extracted from its row index value s t , i.e. '1000'. Meanwhile, we can derive the following 2-bit secret messages from the value of MT 6 (6, 8) = 0, and results in a binary code of '00'. Finally, the joint secret messages '1000 00' are obtained by concatenating '1000' and '00'. IET Image Process., 2020, Vol. 14 Iss. 17 Fig. 10, were served as test images. Besides, in our experiments, the binary secret messages were randomly generated by using a seed, except for the discussion of the specific sequence of secret messages as shown in Table 3.
In our experiments, several statistical metrics are computed for the performance evaluation. Firstly, the PSNR [31] is utilised to evaluate the visual quality of the stego-image and is defined by: where the I i, j and SI i, j represent the pixel values of the original image I and the stego-image SI, respectively. Generally, the higher the PSNR is, the better the quality of the image is. As long as PSNR is larger than 30 dB, the human vision system is hard to distinguish the stego-image from the original image. Secondly, both the EC and ER are used to measure the ability of the stego-image to carry secret messages and is defined as (3) where L B (or EC) represents the length of embedded secret messages. Thirdly, we employ the SSIM (structural similarity) [32] to evaluate the change of the structure of stego-image. It is a measure that can estimate the similarity between two images with respect to the structure and is ranged in [0, 1]. SSIM considers image degradation as a perceived change in structural information and can be defined as the following: where μ I and μ SI are the mean of the images I and SI, respectively; σ I 2 and σ SI 2 are the variance of the images I and SI, respectively.
σ I&SI is the covariance of I and SI. c 1 = (k 1 ⋅ L) 2 , c 2 = (k 2 ⋅ L) 2 are two variables to stabilise the division with the weak denominator, where L is the dynamic range of the pixel values and k i (i = 1, 2) is a constant much <1. The closer to 1 the SSIM index, the higher the similarity is. Finally, NCC (normal cross correlation) [33], as another method for measuring the similarity of two data sequences, is also measured in our experiments. It ranges from 0 to 1 and is defined by: If NCC = 1, it indicates two images are the same. Fig. 11 shows the stego-images when the maximum ER of 2.56 bpp was achieved. It is clear that the visual quality of the stegoimage remains good even if the embedded secret messages are up to 669,992 bits. So, it is hard for the human eye to detect any distortion caused in the images. Tables 1 and 2 list the performances of our approach when L Q is set to 4 and 6, respectively. Note that the sequence of secret messages is randomly generated. When L Q = 4, the average ER was 2.0 bpp, and the average PSNR of the stego-images was 46.38 dB. We also can see that the corresponding SSIM and NCC were 0.9977 and 0.9997, respectively, both of which imply that the similarity between the original image and the stego-images is very high. When L Q = 6, our proposed scheme had a higher ER of 2.56 bpp, and the stegoimage's PSNR was around 43.12 dB. It is also obvious to see that the SSIM and NCC of this scheme remains a high value, 0.9960 and 0.9994, respectively. This indicates that the stego-image's structure remains good even when the ER is up to 2.56 bpp. Furthermore, the last column of Tables 1 and 2 show the BER (bit error rate) of the extracted secret messages, with values of 0, indicating that the proposed scheme can extract the secret messages with error-free. Additionally, we also experimented the proposed scheme on four specific sequences of secret messages, which are quite different from the aforementioned random secret messages. Those four specific sequences of secret messages are: (i) all 0's; (ii) all 1's; (iii) a sequence of '01010101…'; (iv) a sequence of '10101010…'. The results are demonstrated in Table 3. As we can see, no matter ER is set to 2.0 or 2.56 bpp, the performances tested on specific secret messages is almost the same as that tested on random secret messages, with respects to the PSNR, SSIM, and NCC. Hence, it is concluded that the visual quality and structure of stego-images also keep well with our proposed scheme based on our experimental data. Besides, those specific secret messages also can be extracted losslessly, thus, resulting in BER = 0.

Security analysis
While achieving high EC, our approach also provides secure covert communication. To prove this point, more detailed, security analyses like PVD (pixel value difference) histogram [34], RS steganalysis [35] and relative entropy (RE) [36] are employed to analyse the stego-images with an ER of 2.56 bpp. In the security

PVD histogram:
The PVD histogram is a method for examining the degree of differences in the neighbouring pixels between the original image and the stego-image. According to the conclusions of the PVD histogram analysis in [34], the abnormal behaviour of the PVD histogram reveals the presence of the hidden messages. It is also possible to estimate the size of the embedded secret messages. Fig. 12 shows that the PVD histogram curves of the 'Baboon' and 'Peppers' images embedded with secret messages were well preserved, which is quite similar to those of the original images without carrying secret messages.

RS steganalysis:
We also analyse the performance of our approach in terms of resisting the RS steganalysis. The RS steganalysis is based on the following functions: a discrimination function DF, a flipping function F + and a shifting function F − . The divided image blocks are separated into three groups, including R (Regular), S (Singular) and U (Unchanged), by combining those    Fig. 13 shows the graphs of the RS steganalysis for the stegoimages that are produced for two test images 'Lena' and 'Peppers' when the ER varies from 0.2 to 2.56 bpp. As we can see from the graphs, the gaps between R M and R −M , and between S M and S −M curves are very close to each other. Obviously, (6) is satisfied in the stego-images 'Lena' and 'Peppers' with the proposed scheme. This implies that our approach is robust against the RS steganalysis.

Relative entropy:
Furthermore, the statistics of entropy and relative entropy were introduced to evaluate the divergence of the stego-image (SI) from the original image (I). Relative entropy is an indicator that can be used to measure the difference between probability distributions of pixel values for two images, i.e. I(p) and SI(p), can be defined by: When RE = 0, two images are a coincidence and the system is perfectly secure. Table 4 lists the experimental results of entropy and RE for all test images. We can observe that the entropy values for the image I and the stego-image SI are extremely close to each other, with an average difference of 0.0103, while the corresponding value of RE between them is about 0.004, which is also close to zero. It is concluded that our approach is quite secure. To sum up, three security analyses confirm that the difference between the original image and stego-images are quite small, and no clue can be found with our proposed scheme.

Comparisons in performance:
To further demonstrate the superior performance of our proposed Tetris-based scheme, we compared the results provided by the proposed scheme with the results provided by other previous works [14, 16-21, 26-29, 37-39].
Firstly, a performance comparison of our proposed scheme, a magic cube-based DH scheme [19], and five representative TSbased DH schemes [14,[16][17][18]39] is conducted for six test images and is shown in Table 5. From Table 5, we can see that the average ER (or EC) of our approach is the same as that of schemes [16,18,39] and is higher than schemes [14,17,19] with differences of 0.5,  Table 4 Entropy and Relative entropy between I and SI Images Entropy (I) Entropy (SI) RE(I, SI) 0.15, and 0.25 bpp, respectively. Moreover, our approach provides a better stego-image visual quality than that of [16,18,39] under the same ER, with the differences of 1.35, 1.50, and 2.21 dB, respectively. Secondly, the proposed scheme is also compared to the Sudokubased schemes [20,21,[26][27][28][29], with respects to the EC, ER and image quality of the four test images. Those schemes are all designed on the concept of a puzzle game. The results are presented in Table 6. It is apparent that the proposed Tetris-based scheme has the highest PSNR value when compared to schemes [20,[26][27][28][29], where ER reaches 1.5 or 2.0 bpp. When EC is set to 414,188 bits, the PSNR of our approach is slightly less than that of Hong et al.'s scheme [21]. However, based on the reports presented in [21], their EC is limited to the ER of 1.58 bpp. By contrast, our approach promotes an ER up to 2.56 bpp while maintaining the average PSNR of 43.12 dB. Therefore, our proposed scheme is an excellent one that obtains both a higher EC and a higher PSNR.
Finally, we further provide a summary and comparison of PSNR and ER between the proposed Tetris-based scheme and other recent works [14, 16, 18, 20, 26-28, 37, 38], as shown in Table 7. Herein, '-' represents unavailable or unspecific. In Table 7, it is easy to find that the PSNR of our approach maintains a considerable level when ER is 1.5 and 2.0 bpp, that is, 48.72 and 46.38 dB, respectively. It is also clear that given the larger ER of 2.5 or 3.0 bpp, the visual quality of our proposed scheme's stegoimage reaches 43.22 and 40.72 dB, respectively. The Sudoku-based DH schemes, like [20], have a weakness of the limitation of EC. For the TS-based schemes, such as Leng [37], their payload and image quality depend on the width and height of both the octagon and the corner of the octagon. As a result, the PSNR seriously decreases as the octagon is extended, e.g. a PSNR of 39.33 dB is lower than that of our approach when the ER reaches 3.0 bpp. Similarly, Xie et al. [38] presented a revised version based on the EMD-2 scheme by constructing an extended squared magic matrix. Although their scheme can achieve a larger ER high up to 3.15 bpp, it is relatively poor in the visual quality of the stego-image, with an average PSNR of 39.89 dB (ER = 3.0 bpp). Only our approach, Li et al.'s scheme [18] and Chen et al.'s scheme [28] have both a larger ER up to 3.0 bpp and a satisfactory PSNR not lower than 40.00 dB. Among which, the PSNR of our approach is ranked second after that of Li et al.'s scheme [18] when ER reaches 3.0 bpp. That is because, theoretically, the maximum distortion of 128 in our approach is a little greater than that of 78 in the scheme [18] at a time. However, compared to our approach, Li et al.'s scheme [18] and Chen et al.'s scheme [28] have a weakness of visual quality when the ER was set as 1.5, 2.0 and 2.5 bpp. In Table 5 Comparisons of EC (bits), ER (bpp) and PSNR (dB) among the proposed scheme, the magic cube based DH scheme [19] and TS-based DH schemes Images Liu et al. [14] Lee et al. [19] Liu et al. [17] Lee et al. [ Table 6 Comparisons of EC (bits), ER (bpp) and PSNR (dB) between the proposed scheme and six Sudoku-based schemes Images Chang et al. [20] Hong et al. [21] Lyu et al. [26] He et al. [ summary, it is enough to see that the proposed Tetris-based DH scheme significantly outperforms other state-of-the-art schemes [14, 16, 18, 20, 26-28, 37, 38].
• Compare with Sudoku or mini-Sudoku-based DH schemes [20][21][22][26][27][28][29]. Firstly, similar to our approach, the pre-shared knowledge is also required in those schemes. Besides, it is undeniable that the Sudoku-based schemes provide relatively reliable security because of the hundreds of millions of fundamental possible solutions of Sudoku. On the aspects of ER and PSNR, both our approach and scheme [28] obtain the maximum ER of 3.0 bpp, however, the PSNR of stego-image provided by our approach is 0.71 dB which is higher than that of scheme [28]. • Compare with Jigsaw-and Tetris-based schemes [24,25].
Although there is no requirement in sharing knowledge in advance, the ER and PSNR provided by schemes [24,25] are far lower than those of our approach. Concretely, they hide secret messages with the use of shapes of Jigsaw or Tetris, thus, resulting in a relatively low ER. Meanwhile, scheme [24] provided an unsatisfactory visual quality of stego-image, with a PSNR of 34.00 dB.

Computational cost analysis:
In our approach, the required computational cost mainly lies in five aspects: construct the reference matrix, construct the LUT, divide pixel pairs, embed secret messages and extract secret messages. Table 9 lists a comparison of the computational cost, and details are analysed as below: the computational complexity in constructing reference  [26] 47.51 ---Liu et al. [14] 49.95 45.55 --He et al. [27] 48.06 46.37 --Liu et al. [16] 52  Considering the L Q is usually far smaller than H, thus, the overall computational complexity of our approach can be represented by O((H¬W/2)¬(4¬L 2 Q /4)) = O(H¬W¬L 2 Q /2). By the same way, the computational complexity of scheme [14] is analysed as O((H¬W/2)¬(4¬4)) = O(8¬H¬W) because of the required processing of embedding secret messages with the help of a 4 × 4 location table. Similarly, the computational complexity of scheme [18] is summarised at O((H¬W/2)¬(4¬16)) = O(32¬H¬W), where the LUT is sized 4 × 16. For scheme [28], duo to the threelayer mini-Sudoku is designed to serve for secret messages embedding or extraction, its computational complexity is analysed as O((H¬W/2)¬(4¬4¬4)) = O(32¬H¬W). Additionally, it is also worth note that, the temporary storage cost of reference matrix for those four schemes are all sized of 256 × 256, whereas the storage cost of the LUT is sized of 4 × 4, 4 × 16, 4 × 4 × 4 and 4 × (L 2 Q /4), respectively. More specifically, we also compared the execution time of both secret message embedding and extraction. The results are shown in Table 10. It is apparent that most of schemes can implement experiments in <1 s, which implies that they are suitable to be used in real-time applications. It is also seen that our approach achieved a good performance in the lightweight computation, with an overall execution time of 0.591 s.

Potential failure analysis:
Based on empirical results, it is noted that the proposed scheme has excellent performance, exceeding the performance of some state-of-the-art schemes in both EC and visual quality. Nevertheless, some potential failures our approach may occur when the following extreme cases were encountered.
• The length of the secret message (i.e. L B ) is too long. The longer the L B is, the larger the L Q should be. Doing so that, the stegoimage will be seriously distorted. It means that the balance between the EC and stego-image quality is broken. • L Q is set to larger. When L Q ≥ 9, it requires 9 2 /4 ≈ 20 different shapes of tetrominoes to make an entire Q filled without gaps.
However, in fact, there are only 19 different tetrominoes at most in our system. This may result in a failure in constructing the workable reference matrix and loop-up-table. • Without owning the prior knowledge of the usable spliced Q or the rule of constructing LUT. In this case, it is difficult to extract the secret messages with error-free.
However, the above three cases can be prevented in advance by either setting rules for determining the maximinimal size of the length of secret message and L Q or presharing the knowledge of spliced Q or the rule of constructing LUT.

Conclusions
In this paper, we proposed a novel Tetris-based DH scheme to achieve a larger EC along with better image quality. A tetrominobased falling puzzle game was played first within an L Q × L Q square lattice Q. Then, according to the decided Q, the reference matrix and corresponding LUT are constructed for DH. The maximum EC can be flexibly selected according to the L Q . The experimental results showed that our proposed scheme had a more excellent performance than other state-of-the-art schemes in both EC and the image quality. In addition, the security analysis proved our approach is secure covert communication. In the future, we will explore the novel strategy to solve the potential failures in our system. Also, we will try to investigate more applications by using the proposed Tetris-based DH mechanism, such as watermarking, image authentication, and hiding information in encrypted images.

Acknowledgments
This work was supported in part by the Natural