Quantum steganography based on reflected gray code for color images

Abstract

Digital steganography is the art and science of hiding information in covert channels, so as to conceal the information and prevent the detection of hidden messages. On the classic computer, the principle and method of digital steganography has been widely and deeply studied, and has been initially extended to the field of quantum computing. Quantum image steganography is a relatively active branch of quantum image processing, and the main strategy currently used is to modify the LSB of the cover image pixels. For the existing LSB-based quantum image steganography schemes, the embedding capacity is no more than 3 bits per pixel. Therefore, it is meaningful to study how to improve the embedding capacity of quantum image steganography. This work presents a novel steganography using reflected Gray code for color quantum images, and the embedding capacity of this scheme is up to 6 bits per pixel. In proposed scheme, the secret qubit sequence is considered as a sequence of 6-bit segments. For 6 bits in each segment, the first 3 bits are embedded into the second LSB of RGB channels of the cover image, and the remaining 3 bits are embedded into the LSB of RGB channels of the cover image using reflected-Gray code to determine the embedded bit from secret information. Following the transforming rule, the LSBs of stego-image are not always same as the secret bits and the differences are up to almost 50%. Experimental results confirm that the proposed scheme shows good performance and outperforms the previous ones currently found in the literature in terms of embedding capacity.

Keywords

Quantum image processing quantum image steganography gray code least-significant-bit embedding capacity

1. Introduction

Steganography is an early important branch of information hiding, which is a technique that embeds secret information into seemingly common other carrier information to prevent third parties from detecting secret information. The main requirement of steganography is undetectability, which, loosely defined, means that no algorithm exists that can determine whether a work contains a hidden message. Steganography and watermarking are both forms of data hiding and share some common foundations. Nevertheless, it is worth reiterating the goals of these two data-hiding applications in order to highlight the key differences. In steganography, the cover image is a mere decoy and has no relationship to the secret message. In contrast, a watermark usually carries supplemental information about the cover image or some other data related to the cover, such as labels identifying the sender or the receiver. The second and perhaps even more important difference between steganography and watermarking is the issue of the existence of a secret message in an image. While in steganography it is of utmost importance to make sure the image does not exhibit any traces of hidden data, the presence of a watermark is often advertised to deter illegal activity, such as unauthorized copying or redistribution. Additionally, steganography is a mode of communication and as such needs to allow sending large amounts of data. On the contrary, even a very short digital watermark can be quite useful. For example, the presence of a watermark (a one-bit payload) may testify about the image’s ownership. These very different requirements imposed on these two applications make their design and analysis quite different [1].

Since Feynman proposed a novel computation model (called quantum computers) [2] based on principles of quantum physics that seemed to be more powerful than classical ones in 1982, quantum computing, as a new computing method, has been paid more and more attention because of its high parallelism. In the near future, quantum computers are expected to replace the classic computer. Quantum Image Processing (QImP) is an emerging sub-discipline that focuses on extending conventional image processing tasks and operations to the quantum computing framework.

Research in the field of QImP started with proposals on quantum image representations. In 2003, Venegas-Andraca and Bose proposed the Qubit Lattice representation to encode quantum images [3]. This was closely followed by Latorre’s Real Ket representation for quantum images [4]. Years later, Le et al.’s FRQI (flexible representation for quantum images) was proposed in 2010 [5] and later reviewed in 2011 [6]. These three model are collectively regarded as the pioneers of the sub-discipline of quantum image processing [7]. A variety of models proposed later are a modification or extension of these three models. For example, similar to Qubit Lattice representation, radiation energy of objects is transformed into a quantum state $|I\rangle$ , which is named SQR model [8], through a converter $C$ , i.e., $C$ is capable of detecting an recording infrared radiation energy and then producing a quantum state $|I\rangle$ as output; By extending the gray-scale information in FRQI to color representation, a multichannel representation for quantum images, MCQI, was proposed in [9, 10], which uses R, G, and B channel to represent the different color information of image and also keeps the normalized state. In 2013, [11] proposed the NEQR (novel enhanced quantum representation). The model stores images with $(2n+q)$ qubits, where $2n$ qubits represent the pixel position and $q$ qubits represent the gray-scale intensity. For a concise review of quantum image representations, we kindly refer the reader to [7], which gives the most comprehensive review of the principles, features and relationships of all the existing image representation models. After establishing the quantum representation model of classical image, designing the specific methods of quantum image processing become the main task of this sub-discipline. However, the research on image processing for quantum computers has just started. At present, compared with the classical image processing, the quantum image processing is still in infancy, and the related theories and techniques are far from mature.

It is worth pointing out that researchers have focused on data-hiding methods in QImP as a recent field of study, which includes quantum image steganography and quantum image watermarking. In the quantum image watermarking, the work in [12] by Iliyasu et al. is credited as being the first quantum image security protocol. In that study, a scheme called WaQI was proposed based on restricted geometric transformations on the images. Following the WaQI scheme, its gray-scale version, the gray WaQI [13] (or gWaQI) was proposed. In [14], the quantum Fourier transform (QFT) was used to embed and extract the watermarking image without having to know what it looks like. In addition, two dynamic watermarking schemes, i.e. quantum wavelet transform (QWT)-based watermarking [15] and Hadamard transform-based watermarking [16] were proposed. Using the multichannel extension of the FRQI representation, i.e. the MCQI representation, [17] presented an MCQI extension of the WaQI scheme (the MC-WaQI). In extending quantum watermarking to gray-scale images, [18] proposed the use of simple and small-scale quantum circuits to embed a scrambled image onto the carrier image using CNOT gates. More recently, in the study presented in [19], a new watermarking strategy was investigated, which the carrier and the watermark images are stored in the $\theta$ and $\phi$ phases of the same qubit, respectively, and the visual imperceptibility is guaranteed irrespective of the size of image being embedded.

In the quantum image steganography, the cover image can be a gray-scale image or a color image. In general, the embedding capacity of color cover images is greater than that of gray-scale cover images. The following briefly reviews several existing scenarios of quantum image steganography.

In terms of the steganography of gray-scale cover images, in 2015, a novel strategy for quantum image based on Moire Pattern was proposed for the first time in [20]. The embedding process consisted of three steps: (1) Arbitrarily select an original image, which is also called a “moire grating”, as a carrier; (2) The original carrier image is processed using a deformation operation to obtain a moire pattern; (3) Using the denoising operation, the moire pattern is transformed into a carrier image containing hidden information. The embedding capacity of this scheme is 1 bit per pixel. Shortly thereafter, a Least significant qubit (LSQb) based quantum image steganography was proposed in [21]. Firstly, the QFT is applied to the carrier image, and then the LSB of the cover image is matched with the secret bit by the quantum comparator. Finally, the inverse QFT is carried out on the carrier image to complete the embedding process. The extraction process is relatively simple, and the LSBs of the corresponding pixels in the stego-image are secret information. The embedding capacity of this scheme is also 1 bit per pixel. In 2016, [22] proposed two blind LSB steganography algorithms, where one algorithm is plain LSB which uses the message bits to substitute for the pixels’ LSB directly, and the other is block LSB which embeds a message bit into a number of pixels that belong to one image block. The embedding capacity of the two methods is at most 1 bit per pixel. In 2017, a new secure quantum steganography scheme was investigated in [23]. In this scheme, in addition to using the LSB, the most significant bit (MSB) is also employed. In fact, in this method, only half of the secret information is embedded in the cover image, and the remaining half is treated as a key, and its embedding capacity is also 1 bit per pixel. Soon after, a novel steganography scheme based on Arnold scrambling and LSB is proposed in [24]. The scheme first expand a secret image with $2^{n-1}\times 2^{n-1}$ size and 8-bit gray-scale to an image with $2^{n}\times 2^{n}$ size and 2-bit one, and then scramble the expanded secret image to a meaningless image using Arnold scrambling. Finally, the scrambled secret image is embedded in the cover image. This scheme has better security, and the embedding capacity has reached 2 bits per pixel.

Figure 1.

A $2\times 2$ color image and its NEAR representation.

In terms of the steganography of color cover images, in 2016, by employing a novel quantum representation for color digital images, [25] investigated the feasibility of the classical image LSB information hiding algorithm on quantum computer, and proposed a LSB information hiding algorithm of quantum image. In this method, the secret information is embedded by replacing the LSB of RGB three-channel of the cover image directly as the message bits. Although the embedding capacity of this method reaches 3 bits per pixel, the security of this method is very low, because the embedded secret information can be obtained by directly extracting the LSB of the three channel of the cover image. In 2017, [26] investigated three quantum color images steganography algorithms based on LSB. In the first two methods, the message bits are embedded in the R channel or B channel of the cover image, and the embedding capacity is 1 bit per pixel. The third method embeds the message bits into the RG channel or RB channel, and the embedding capacity is 2 bit per pixel. Although the three methods proposed in [26] have better security, the embedding capacity is low. Not long after, [27] proposed a novel quantum LSB-based steganography method using the Gray code for colored quantum images. This method uses the Gray code to accommodate two secret qubits in 3 LSBs of each pixel simultaneously according to reference tables. Considering that four reference tables are needed in the process of embedding and extracting, this method is relatively complex. In addition, in this method, the advantage of Gray code has not been fully reflected. In other words, in [27], using ordinary binary code instead of the Gray code is exactly the same. Although this method has many advantages (e.g., high security, robustness, etc.), the embedding capacity is only 2 bits per pixel. In [28], a new blind quantum copyright protection method based on owner’s signature in RGB images is proposed, which utilizes one of RGB channels as indicator and two remained channels are used for embedding information about the owner. In this scheme, the owner’s signature is considered as a text. Unlike LSB steganography, the scheme embeds each message bit into the fourth bit of the three channel of the cover image, and the embedding capacity is only 1 bit per pixel.

The rest of this paper is organized as follows. In Section 2, some preliminary knowledge related steganography scheme proposed in this paper are briefly reviewed. Then we introduce the proposed information hiding and extracting schemes in Section 3, which is the main contribution of this paper. Section 4 gives the complexity analysis. Section 5 is devoted to the experimental results with steganalysis and performance comparisons with several existing quantum color image steganography schemes. Finally, the conclusions are presented in Section 6.

2. Preliminary knowledge

2.1 NEQR representation of color image

For a monochrome image, NEQR encodes 8 bits of binary gray-scale value into 8 qubits. For a color images, NEQR encodes the RGB color values into 24 qubits. In NEQR, a color image of size $2^{n}\times 2^{n}$ can be represented by the following equation.

$\displaystyle|Q|={\displaystyle\frac{1}{2^{n}}}\sum\limits_{y=0}^{2^{n}-1}\sum% \limits_{x=0}^{2^{n}-1}|f(y,x)||yx|={\displaystyle\frac{1}{2^{n}}}\sum\limits_% {y=0}^{2^{n}-1}\sum\limits_{x=0}^{2^{n}-1}\bigotimes\limits_{k=0}^{7}|r_{yx}^{% k}|\bigotimes\limits_{k=0}^{7}|g_{yx}^{k}|\bigotimes\limits_{k=0}^{7}|b_{yx}^{% k}||yx|=\frac{1}{2^{n}}\sum\limits_{y=0}^{2^{n}-1}\sum\limits_{x=0}^{2^{n}-1}|% c_{yx}^{23}c_{yx}^{22}\ldots c_{yx}^{0}||yx|,$ (1)

where $f(y,x)=r_{yx}^{7}\ldots r_{yx}^{0}g_{yx}^{7}\ldots g_{yx}^{0}b_{yx}^{7}\ldots b% _{yx}^{0}$ , $r_{yx}^{k}$ , $g_{yx}^{k}$ , $b_{yx}^{k}\in\{0,1\}$ , and $f(y,x)\in\{0,1,\ldots,2^{24}-1\}$ .

It is clear from [11] that NEQR needs $24+2n$ qubits to represent a $2^{n}\times 2^{n}$ color image with color range $(2^{8},2^{8},2^{8})$ . An example of a $2\times 2$ NEQR color image and its corresponding representation are shown in Fig. 1.

2.2 Hilbert scrambling of quantum images

In 1891, Hilbert discovered a traversal curve in a two-dimensional space named the Hilbert curve and described by the Hilbert matrix. Along the Hilbert curve, the image can be scrambled, and the scrambling effect is better. For a $2^{n}\times 2^{n}$ matrix $S_{n}$ , by numbering its elements from top to bottom and from left to right, we obtain the following results.

$\displaystyle S_{n}=\begin{bmatrix}1&2&\ldots&2^{n}\\ 2^{n}+1&2^{n}+2&\ldots&2^{n+1}\\ \vdots&\vdots&\ddots&\vdots\\ 2^{2n-1}+1&2^{2n-1}+2&\ldots&2^{2n}\\ \end{bmatrix}$ (2)

The Hilbert matrix $H_{n}$ is an arrangement of the initial matrix $S_{n}$ , for example

$\displaystyle H_{0}=[1],H_{1}=\begin{bmatrix}1&2\\ 4&3\end{bmatrix},H_{2}=\begin{bmatrix}1&2&15&16\\ 4&3&14&13\\ 5&8&9&12\\ 6&7&10&11\\ \end{bmatrix}$

In $H_{n}$ , the Hilbert curve can be obtained by sequentially connecting the element $1\sim 2^{2n}$ . When $n=$ 3, 4, 5, 6, the corresponding Hilbert curves are shown in Fig. 2.

Figure 2.

The hilbert curve at $n=3,4,5,6$ . (a) $n=3$ . (b) $n=4$ . (c) $n=5$ . (d) $n=6$ .

Obviously, for a $2^{n}\times 2^{n}$ sized image $S_{n}$ , the image can be scrambled by repositioning the pixels in $S_{n}$ along the Hilbert curve $H(n)$ .

For the Hilbert scrambling of quantum images, the specific implementation method is given in [31]. In their method, first of all, three circuit modules (i.e. PARTITION $(k)$ , $O(k)$ , $E(k)$ ) are designed, and then three basic circuits are built with these three modules: (1) Initialization, (2) Odd $(k)$ , (3) Even $(k)$ . Finally, using these three basic circuits, a complete Hilbert scrambling circuits are constructed, as shown in Fig. 3. Considering this complexity of the circuits, the functions of each sub-module are not mentioned, and to acquire more information, readers can refer [31].

Because the quantum logic gates are unitary, the whole quantum Hilbert scrambling circuits are unitary. The quantum inverse Hilbert scrambling circuits can be obtained only by arranging all logic gates in the opposite order.

Figure 3.

Complete hilbert scrambling circuits (figure adapted from [31]).

2.3 Gray code and its application in LSB-based steganography

Gray codes are named after Frank Gray who first patented the idea for use in shaft encoders in 1953 [29]. In a group of codes, if any two adjacent codes have only one different binary number, it is called Gray Code. In addition, since there is only one digit difference between the maximum number and the minimum number, that is, “end to end”, it is also called cyclic code or reflection code. One possible way to produce a reflection Gray code with length $n$ is the one that starts with all bits zero and successively flip the right-most bit at odd order number or flip the left bit of the right-most 1 digit at even order number [30]. The reflection Gray codes from 1 bit $G_{1}$ to 4 bits $G_{4}$ are illustrated as follows:

$\displaystyle\begin{array}[]{l}G_{1}=\begin{cases}0=0\\ 1=1\end{cases}\\ G_{2}=\begin{cases}00=0\\ 01=1\\ 11=2\\ 10=3\end{cases}\end{array}G_{3}=\begin{cases}000=0\\ 001=1\\ 011=2\\ 010=3\\ 110=4\\ 111=5\\ 101=6\\ 100=7\end{cases}$ $\displaystyle G_{4}=\begin{cases}0000=0,1100=8\\ 0001=1,1101=9\\ 0011=2,1111=10\\ 0010=3,1110=11\\ 0110=4,1010=12\\ 0111=5,1011=13\\ 0101=6,1001=14\\ 0100=7,1000=15\end{cases}$

In order to convert a number in binary code $b_{n}b_{n-1}$ $\ldots$ $b_{1}$ to its reflected Gray code $g_{n}g_{n-1}\ldots g_{1}$ , the following iterative equation can be used [30].

$\displaystyle g_{k}=\begin{cases}b_{k},&k=n,\\ b_{k+1}\oplus b_{k},&1\leqslant k\leqslant n-1,\end{cases}$ (3)

and the conversion of a reflected Gray code $g_{n}g_{n-1}\ldots$ $g_{1}$ to its corresponding binary code $b_{n}b_{n-1}\ldots b_{1}$ can be achieved by the following equation [30].

$\displaystyle b_{k}=\begin{cases}g_{k},&k=n,\\ b_{k+1}\oplus g_{k},&1\leqslant k\leqslant n-1.\end{cases}$ (4)

Taking 4 bits Gray code $G_{4}$ as an example (other cases are similar), the Fig. 4 illustrats the quantum circuits that implements the conversion between binary code and Gray code.

Figure 4.

The quantum circuits that implements the conversion between binary code and gray code. (a) Quantum circuits of converting a number in binary code to its Gray code. (b) Quantum circuits of converting a Gray code to its corresponding binary code.

Figure 5.

Quantum comparator (figure adapted from [32]).

In 2008, Chen and Chang [30] investigates the application of Gray code in LSB-based image steganography. In their proposed method, gray-scale images are used as cover images. The message is embedded according to the rules reflected-Gray code.

For the embedding process, let the Gray code function Gray ( $g$ ) be the corresponding Gray code value of $g$ , where $g$ is the last 4 bits of pixel value. If the message bit is 0, the LSB of the cover pixel is modified so that the last 4 bits corresponds to the even Gray code that is closest to $g$ ; otherwise, modify the LSB of the cover pixel so that its last 4 bits correspond to the odd Gray code closest to $g$ . Taking the Gray code $G_{4}$ as an example, if the pixel value of the cover image is $197_{10}=11000101_{2}$ , then Gray $(0101_{2})=6_{\text{Gray}}$ . If the message bit is 1, the nearest odd number of Gray code is $5_{\text{Gray}}=0111_{2}$ or $7_{\text{Gray}}=0100_{2}$ . In view of the fact that $0111_{2}$ may bring greater distortion to the image, we choose $0100_{2}$ as the best solution, and by modifying the LSB, the stego-image pixel value $197_{10}=11000101_{2}$ is rewritten to be $196_{10}=11000100_{2}$ . If the message bit is 0, we do not change the pixel value, because Gray $(0101_{2})=6_{\text{Gray}}$ is an even number.

The extraction process is relatively simple. When the last 4-bit Gray code of the stego-image pixel is odd, the embedded message bit is 1, and when it is even, the embedded message bit is 0. For example, if the value of the stego-image pixel is $212_{10}=11010100_{2}$ , then, it is clear from G $(0100_{2})=7_{\text{Gray}}$ that the embedded message bit is 1.

The outstanding advantage of this scheme is that, following the transforming rule, the LSBs of stego-image are not always equal to the secret bits and the experiment shows that the differences are up to almost 50%. However, its embedded capacity is only 1 bit per pixel.

2.4 Quantum comparator

The quantum comparator occupies an important position in our proposed steganographic scheme. Here, we use the quantum comparator designed in [32], and its quantum circuit is shown in Fig. 5.

The quantum comparator compares $x$ and $y$ , where $x=x_{n-1}x_{n-2}\ldots x_{0}$ , $y=y_{n-1}y_{n-2}\ldots y_{0}$ , $x_{i},y_{i}\in\{0,1\},i=n-1,n-2,\ldots,0$ . Qubits $e_{1}$ and $e_{0}$ are outputs. If $e_{1}e_{0}=10$ then $x>y$ ; if $e_{1}e_{0}=01$ then $x<y$ ; and if $e_{1}e_{0}=00$ then $x=y$ .

3. Quantum steganography using reflected gray code for color images

It is first pointed out that in this method, a $2^{n}\times 2^{n}$ sized RGB image $|CI\rangle$ is used as the cover image, and a $2^{u}\times 2^{v}$ sized gray-scale image $|SI\rangle$ is used as a secret image, where $u+v\leqslant 2n-1$ . According to [11], the NEQR of the secret image can be expressed as

$\displaystyle|SI|={\displaystyle\frac{1}{\sqrt{2^{u+v}}}}\sum\limits_{y=0}^{2^% {u}-1}\sum\limits_{x=0}^{2^{v}-1}|f(y,x)||yx|={\displaystyle\frac{1}{\sqrt{2^{% u+v}}}}\sum\limits_{y=0}^{2^{u}-1}\sum\limits_{x=0}^{2^{v}-1}\bigotimes\limits% _{k=0}^{7}|s_{yx}^{k}||yx|={\displaystyle\frac{1}{\sqrt{2^{u+v}}}}\sum\limits_% {y=0}^{2^{u}-1}\sum\limits_{x=0}^{2^{v}-1}|s_{yx}^{7}s_{yx}^{6}\ldots s_{yx}^{% 0}||yx|,$ (5)

where $f(y,x)=s_{yx}^{7}s_{yx}^{6}\ldots s_{yx}^{0},s_{yx}^{k}\in\{0,1\}$ , and $f(y,x)\in\{0,1,\ldots,2^{8}-1\}$ .

The steganographic scheme proposed in this paper consists of four processes: scrambling, embedding, extracting, and inverse scrambling, as shown in Fig. 6. Among them, the scrambling scheme uses the quantum Hilbert scrambling proposed in [31], and the inverse scrambling also adopts the corresponding method in [31]. Next, we introduce the embedding and extracting process of secret information using the reflected-Gray code in detail, which is the main contribution of this paper.

Figure 6.

The outline of the steganographic scheme presented in this paper.

Figure 7.

The quantum circuits that implements the secret image conversion.

3.1 Quantum embedding process of secret image

As previously shown, in this paper, the secret information (i.e., messages that need to be embedded into the cover color image $|\rm CI\rangle$ ) refers to a $2^{u}\times 2^{v}$ sized gray-scale image $|\rm SI\rangle$ with gray range $2^{8}$ .

3.1.1 Pixel decomposition of secret image

The basic idea of embedding message into cover image is: first, the secret qubit sequence in $|SI\rangle$ is consider as a sequence of 6-bit segments, and each 6-bit segment is embedded into three channels of the pixel of the cover pixel, where the LSB and the second LSB of three-channel accommodate 3 bits, respectively. If the last 6-bit segment is less than 6 bits, it is padded with ‘0 s. Therefore, to embed $|SI\rangle$ into $2^{n}\times 2^{n}$ sized $|CI\rangle$ , we first convert $|SI\rangle$ to a $2^{u+v+1-n}\times 2^{n}$ sized image with gray range $2^{6}$ , where $u+v+1\leqslant 2n$ . Taking the $n=2,u=2,v=1$ as an example, the quantum circuits that implements the secret image conversion are shown in Fig. 7.

In Fig. 7, the upper part is an original secret image (with gray range $2^{8}$ ), and the lower half is a decomposed secret image (with gray range $2^{6}$ ). The $H$ denotes the Hadamard gate with the matrix form $H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}$ , and its role is to make the position qubits in the equilibrium superposition state. The $I=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$ represents identity matrix.

From the decomposition process of the secret image, it can be seen that the decomposed secret image has the same number of columns as the cover image, and the number of rows is less than or equal to the number of rows of the cover image. Taking the secret image represented by Eq. (3.1.1) as an example, the decomposed secret image can be expressed as

$\displaystyle|S|=$ (6) $\displaystyle{\displaystyle\frac{1}{\sqrt{2^{u+v+1}}}}\sum\limits_{y=0}^{2^{u+% v+1-n}-1}\sum\limits_{x=0}^{2^{n}-1}|s_{yx}^{5}s_{yx}^{4}s_{yx}^{3}s_{yx}^{2}s% _{yx}^{1}s_{yx}^{0}||yx|.$

Note: after the pixel decomposition, the secret image usually contains some redundant pixels. In this paper, the gray values of these redundant pixels are set to $|0\rangle$ .

Suppose the size of the cover image is $4\times 4$ , and the size of the original secret image is $4\times 2$ . At this point, the secret image contains $4\times 2\times 8=64$ message bits. These secret bits can be seen as 11 6-bit segments, where the last 6-bit segment has only 4 bits and is padded with two ‘0 s. The 11 6-bit segments can be regarded as a $3\times 4$ sized image with gray range $2^{6}$ . Obviously, it is necessary to employ 4 qubits to represent the position of the 11 6-bit pixels, resulting in five redundant pixels, as shown in Fig. 8.

Figure 8.

The hilbert curve at $n=3,4,5,6$ . (a) the original secret image. (b) the decomposed secret image.

3.1.2 Quantum embedding of secret image

The basic idea of embedding secret image into cover image can be summarized as follows. The secret qubit $s_{ij}^{5},s_{ij}^{4},s_{ij}^{3}$ are embedded into the cover qubit $r_{ij}^{1},g_{ij}^{1},b_{ij}^{1}$ while the $s_{ij}^{2},s_{ij}^{1},s_{ij}^{0}$ are embedded into the cover qubits $r_{ij}^{0},g_{ij}^{0},b_{ij}^{0}$ using the reflected-Gray code, respectively. Figure 9 illustrates the specific embedding scheme.

Next, taking a single pixel of secret image as an example, we give the concrete algorithm of embedding it into cover image.

Input: A cover image pixel $P_{ij}$ ( $r_{ij}^{7}$ $\ldots r_{ij}^{1}r_{ij}^{0},g_{ij}^{7}$ $\ldots$ $g_{ij}^{1}$ $g_{ij}^{0},b_{ij}^{7}$ $\ldots b_{ij}^{1}b_{ij}^{0}$ ), a Gray code length $n$ of the $G_{n}$ system and 6 embedded bits $s_{ij}^{5},s_{ij}^{4},s_{ij}^{3},s_{ij}^{2},s_{ij}^{1},s_{ij}^{0}$ .Output: A stego-image pixel $\hat{P}_{ij}$ ( $r_{ij}^{7}\ldots\hat{r}_{ij}^{1}\hat{r}_{ij}^{0},g_{ij}^{7}\ldots$ $\hat{g}_{ij}^{1}\hat{g}_{ij}^{0}$ , $b_{ij}^{7}\ldots\hat{b}_{ij}^{1}\hat{b}_{ij}^{0}$ ).Step 1: Let $g_{R}=r_{ij}^{4}r_{ij}^{3}r_{ij}^{2}r_{ij}^{1},g_{G}=g_{ij}^{4}∼{}g_{ij}^{3}∼{% }g_{ij}^{2}∼{}g_{ij}^{1}$ , $g_{B}=b_{ij}^{4}b_{ij}^{3}b_{ij}^{2}b_{ij}^{1}$ .Step 2: If (( $s_{ij}^{5}$ is 1) and ( ${\rm Gray}(g_{R})$ is even)) or (( $s_{ij}^{5}$ is 0) and ( ${\rm Gray}(g_{R})$ is odd)) $\hat{r}_{ij}^{1}=1-r_{ij}^{1}$ . else $\hat{r}_{ij}^{1}=r_{ij}^{1}$ .If (( $s_{ij}^{4}$ is 1) and ( ${\rm Gray}(g_{G})$ is even)) or (( $s_{ij}^{4}$ is 0) and ( ${\rm Gray}(g_{G})$ is odd)) $\hat{g}_{ij}^{1}=1-g_{ij}^{1}$ . else $\hat{g}_{ij}^{1}=g_{ij}^{1}$ .If (( $s_{ij}^{3}$ is 1) and ( ${\rm Gray}(g_{B})$ is even)) or (( $s_{ij}^{3}$ is 0) and ( ${\rm Gray}(g_{B})$ is odd)) $\hat{b}_{ij}^{1}=1-b_{ij}^{1}$ . else $\hat{b}_{ij}^{1}=b_{ij}^{1}$ .Step 3: Let $g_{R}=r_{ij}^{3}r_{ij}^{2}\hat{r}_{ij}^{1}r_{ij}^{0},g_{G}=g_{ij}^{3}∼{}g_{ij}% ^{2}\hat{g}_{ij}^{1}∼{}g_{ij}^{0}$ , $g_{B}=b_{ij}^{3}b_{ij}^{2}\hat{b}_{ij}^{1}b_{ij}^{0}$ .Step 4: If (( $s_{ij}^{2}$ is 1) and ( ${\rm Gray}(g_{R})$ is even)) or (( $s_{ij}^{2}$ is 0) and ( ${\rm Gray}(g_{R})$ is odd)) $\hat{r}_{ij}^{0}=1-r_{ij}^{0}$ . else $\hat{r}_{ij}^{0}=r_{ij}^{0}$ .If (( $s_{ij}^{1}$ is 1) and ( ${\rm Gray}(g_{G})$ is even)) or (( $s_{ij}^{1}$ is 0) and ( ${\rm Gray}(g_{G})$ is odd)) $\hat{g}_{ij}^{0}=1-g_{ij}^{0}$ . else $\hat{g}_{ij}^{0}=g_{ij}^{0}$ .If (( $s_{ij}^{0}$ is 1) and ( ${\rm Gray}(g_{B})$ is even)) or (( $s_{ij}^{0}$ is 0) and ( ${\rm Gray}(g_{B})$ is odd)) $\hat{b}_{ij}^{0}=1-b_{ij}^{0}$ . else $\hat{b}_{ij}^{0}=b_{ij}^{0}$ .

The quantum circuits of implementing the above algorithm are shown in Fig. 10. At the input, there are 18 auxiliary qubits with initial values of $|0\rangle^{\otimes 18}$ , of which 12 auxiliary qubits are reset to $|0\rangle^{\otimes 12}$ at the output by quantum circuits in the two dashed boxes.

Figure 9.

The outline of the embedding scheme presented in this paper.

Figure 10.

Quantum circuits for embedding a single secret pixel into cover image.

The quantum circuits of embedding a whole secret image into the cover image are shown in Fig. 11, in which, the $|r_{yx}^{k}\rangle,|g_{yx}^{k}\rangle,|b_{yx}^{k}\rangle,k=0,1,\ldots,7$ are the color qubits of the pixel in the cover image, and $|y\rangle$ and $|x\rangle$ are the corresponding position qubits. The $|s_{y^{\prime}x^{\prime}}^{5}∼{}s_{y^{\prime}x^{\prime}}^{4}∼{}s_{y^{\prime}x^% {\prime}}^{3}∼{}s_{y^{\prime}x^{\prime}}^{2}∼{}s_{y^{\prime}x^{\prime}}^{1}∼{}% s_{y^{\prime}x^{\prime}}^{0}\rangle$ denote the gray-scale value of pixel in secret image, and $|y^{\prime}x^{\prime}\rangle$ denotes the corresponding positions. Firstly, two quantum comparators are used to compare the pixel position between the cover image and the secret image. If the two positions are equal, the embedding operation is performed. At the output of the circuits, the stego-image embedded with the secret image can be obtained.

Figure 11.

Quantum circuits for embedding a whole secret image into the cover image.

3.2 Quantum extracting process of secret image

The extraction of secret images is the inverse process of embedding. Figure 12 illustrates the specific extracting scheme. Taking the pixel at the position $(i,j)$ in the secret image as an example, and letting its gray-scale value be $|s_{ij}^{5}∼{}s_{ij}^{4}∼{}s_{ij}^{3}∼{}s_{ij}^{2}∼{}s_{ij}^{1}∼{}s_{ij}^{0}\rangle$ , next, we give a brief description of extracting it from stego-image.

Figure 12.

The outline of the extracting scheme presented in this paper.

In stego-image, let $g_{R}=r_{ij}^{4}r_{ij}^{3}r_{ij}^{2}\hat{r}_{ij}^{1}$ , $g_{G}=g_{ij}^{4}∼{}g_{ij}^{3}$ $g_{ij}^{2}\hat{g}_{ij}^{1}$ , $g_{B}=b_{ij}^{4}b_{ij}^{3}b_{ij}^{2}\hat{b}_{ij}^{1}$ , $g^{\prime}_{R}=r_{ij}^{3}r_{ij}^{2}\hat{r}_{ij}^{1}\hat{r}_{ij}^{0}$ , $g^{\prime}_{G}=g_{ij}^{3}$ $g_{ij}^{2}$ $\hat{g}_{ij}^{1}\hat{g}_{ij}^{0}$ , $g^{\prime}_{B}=b_{ij}^{3}b_{ij}^{2}\hat{b}_{ij}^{1}\hat{b}_{ij}^{0}$ . From the embedding process we can see that in the stego-image, the parity of Gray code $g_{R},g_{G},g_{B}$ is consistent with the parity of $s_{ij}^{5},s_{ij}^{4},s_{ij}^{3}$ , and the parity of Gray code $g^{\prime}_{R},g^{\prime}_{G},g^{\prime}_{B}$ is consistent with the parity of $s_{ij}^{2},s_{ij}^{1},s_{ij}^{0},$ respectively. Therefore, $s_{ij}^{5}$ , $s_{ij}^{4}$ , and $s_{ij}^{3}$ can be obtained directly according to the parity of the Gray code $g_{R}$ , $g_{G}$ , and $g_{B}$ , and $s_{ij}^{2}$ , $s_{ij}^{1}$ , and $s_{ij}^{0}$ can be obtained directly according to the parity of the Gray code $g^{\prime}_{R}$ , $g^{\prime}_{G}$ , and $g^{\prime}_{B}$ . The quantum circuits implementing the above process are shown in Fig. 13, where the quantum circuits in the two dashed boxes are used to reset the 12 auxiliary qubits to the state $|0|^{\otimes 12}$ at the output. The quantum circuits of extracting a whole secret image from the stego-image are shown in Fig. 14.

In Fig. 14, the $|r_{yx}^{7}\rangle\sim|\hat{r}_{yx}^{0}\rangle,|g_{yx}^{7}\rangle\sim|\hat{g}_% {yx}^{0}\rangle,|b_{yx}^{7}\rangle\sim|\hat{b}_{yx}^{0}\rangle$ are the color qubits of the pixel in the stego-image, and $|y\rangle$ and $|x\rangle$ are the corresponding position qubits. The $|s_{y^{\prime}x^{\prime}}^{5}∼{}s_{y^{\prime}x^{\prime}}^{4}∼{}s_{y^{\prime}x^% {\prime}}^{3}∼{}s_{y^{\prime}x^{\prime}}^{2}∼{}s_{y^{\prime}x^{\prime}}^{1}∼{}% s_{y^{\prime}x^{\prime}}^{0}\rangle$ with initial value $|0\rangle^{\otimes 6}$ denote the gray-scale value of pixel in extracted secret image, and $|y^{\prime}x^{\prime}\rangle$ denotes the corresponding positions.

It is worth pointing out that the secret image extracted directly from the stego-image is a $2^{u+v+1-n}\times 2^{n}$ sized image with 6-bit gray-scale value, and it needs to be further restored to the $2^{u}\times 2^{v}$ sized image with 8-bit gray-scale value. Taking the $n=2,u=2,v=1$ as an example, the quantum circuits that implements this process are shown in Fig. 15. This circuits are similar to that in Fig. 7. At the input of the circuits, the upper part is a $2^{u}\times 2^{v}$ empty image, and it is converted to the original secret image at the output.

Figure 13.

Quantum circuits for extracting a single secret pixel from stego-image.

Figure 14.

Quantum circuits for extracting a whole secret image from the stego-image.

In order to help readers understand the method proposed in this paper including the embedding and extracting clearly, an example is given here, which depicts how 6 secret qubits are embedded into a cover pixel and extracted form it. Assume that the color value of the cover pixel $(i,j)$ is $P=(|r_{ij}|,|g_{ij}|,|b_{ij}|)=|01100111||10110101||00111011|$ and 6 secret qubits to be embedded is $|s_{ij}^{5}∼{}s_{ij}^{4}$ $s_{ij}^{3}∼{}s_{ij}^{2}∼{}s_{ij}^{1}∼{}s_{ij}^{0}|=|110101|$ . The embedding and extracting process are as follows.

(1) Embedding

Let $g_{R}=0011,g_{G}=1010,g_{B}=1101$ , and then ${\rm Gray}(g_{R})=2$ , ${\rm Gray}(g_{G})=12$ , ${\rm Gray}(g_{B})=9$ .

From $s_{ij}^{5}=1$ is odd and ${\rm Gray}(g_{R})=2$ is even, $\hat{r}_{ij}^{1}=1-r_{ij}^{1}=1-1=0$ .

From $s_{ij}^{4}=1$ is odd and ${\rm Gray}(g_{G})=12$ is even, $\hat{g}_{ij}^{1}=1-g_{ij}^{1}=1-0=1$ .

From $s_{ij}^{3}=0$ is even and ${\rm Gray}(g_{B})=9$ is odd, $\hat{b}_{ij}^{1}=1-b_{ij}^{1}=1-1=0$ .

At this time, the cover pixel $P$ becomes $P^{\prime}=|01100101\rangle|10110111\rangle|00111001\rangle$ .

Let $g^{\prime}_{R}=0101,g^{\prime}_{G}=0111,g^{\prime}_{B}=1001$ , and then ${\rm Gray}(g^{\prime}_{R})=6$ , ${\rm Gray}(g^{\prime}_{G})=5$ , ${\rm Gray}(g^{\prime}_{B})=14$ .

From $s_{ij}^{2}=1$ is odd and ${\rm Gray}(g^{\prime}_{R})=6$ is even, $\hat{r}_{ij}^{0}=1-r_{ij}^{0}=1-1=0$ .

From $s_{ij}^{1}=0$ is even and ${\rm Gray}(g^{\prime}_{G})=5$ is odd, $\hat{g}_{ij}^{0}=1-g_{ij}^{0}=1-1=0$ .

From $s_{ij}^{0}=1$ is odd and ${\rm Gray}(^{\prime}g_{B})=14$ is even, $\hat{b}_{ij}^{0}=1-b_{ij}^{0}=1-1=0$ .

Therefore the corresponding stego-pixel is $\hat{P}=|01100100\rangle|10110110\rangle|00111000\rangle$ .

(2) Extracting

For the extracting process, let $g_{R}=0010,g_{G}=1011,g_{B}=1100$ , $g^{\prime}_{R}=0100,g^{\prime}_{G}=0110,g^{\prime}_{B}=1000$ , and then ${\rm Gray}(g_{R})=3$ , ${\rm Gray}(g_{G})=13$ , ${\rm Gray}(g_{B})=8$ , ${\rm Gray}(g^{\prime}_{R})=7$ , ${\rm Gray}(g^{\prime}_{G})=4$ , ${\rm Gray}(g^{\prime}_{B})=15$ .

From ${\rm Gray}(g_{R})=3$ is odd, $s_{ij}^{5}$ should be odd, and so, $s_{ij}^{5}=1$ .

From ${\rm Gray}(g_{G})=13$ is odd, $s_{ij}^{4}$ should be odd, and so, $s_{ij}^{4}=1$ .

From ${\rm Gray}(g_{B})=8$ is even, $s_{ij}^{3}$ should be even, and so, $s_{ij}^{3}=0$ .

From ${\rm Gray}(g^{\prime}_{R})=7$ is odd, $s_{ij}^{2}$ should be odd, and so, $s_{ij}^{2}=1$ .

From ${\rm Gray}(g^{\prime}_{G})=4$ is even, $s_{ij}^{1}$ should be even, and so, $s_{ij}^{1}=0$ .

From ${\rm Gray}(g^{\prime}_{B})=15$ is odd, $s_{ij}^{0}$ should be odd, and so, $s_{ij}^{0}=1$ .

At this point, the extracted secret bits are 110101, which is exactly the same as embedded information.

Figure 15.

Quantum circuits for secret image restoration.

Figure 16.

Six cover images used in the experiments. (a) 512 $\times$ 512. (b) 512 $\times$ 512. (c) 512 $\times$ 512. (d) 512 $\times$ 512. (e) 512 $\times$ 512. (f) 512 $\times$ 512.

4. Complexity analysis

The circuit network complexity depends on the number of elementary gates used in the quantum image processing. The complexity of elementary quantum gates is taken to be unity. This includes the NOT-gate, Hadamard gate, Control-Not gate, and any $2\times 2$ unitary operator [33]. Below, we start by analyzing the complexity of each sub-module and gradually establish the complexity of the entire steganograpgy circuits.

•
Some auxiliary modules.

In the steganographic scheme proposed in this paper, the auxiliary modules used include: Hilbert scrambling, quantum comparator, Gray code conversion. From [31], it is clear that the complexity of Hilbert scrambling is $O(n^{2})$ . In [34], quantum comparator is used in quantum image translation. From the analysis presented in [34], the complexity of this model is no more than $O(n^{2})$ . According to Fig. 4, the complexity of Gray code conversion is $O(3)$ .
•
Pixel decomposition of secret images.

According to the pixel decomposition method of the secret image introduced in Section 3.1.1, the quantum circuits consists of $2^{u+v+3}$ $(2u+2v+2)$ -Control-NOT gates. According to [33], only $(4k-8)$ 2-Control-Unitary gates were needed to construct 1 $k$ -Control-Unitary gate, as well as some assistant qubits. Therefore, 1 $(2u+2v+2)$ -Control-NOT gate can be constructed by $(8u+8v)$ 2-Control-NOT gate. From $u+v+1\leqslant 2n$ , the complexity of quantum circuits is no more than $O(n2^{2n+2})$ .
•
Embedding a single secret pixel into cover image.

From Fig. 10, this quantum circuits consists 12 Gray code conversions, 60 Control-Not gates, and 12 3-Control-NOT gates ( $O(4)$ for a 3-Control-NOT gates from [33]). Therefore, the complexity of this circuits is $O(12\times 3+60\times 1+12\times 4)=O(144)$ .
•
Embedding a whole image into the cover image.

From Fig. 11, this quantum circuits consists 2 quantum comparators, 1 Single pixel embedding module. Obviously, the complexity of this circuits is $O(2n^{2}+144)$ .
•
Extracting a single secret pixel from stego-image.

From Fig. 13, this quantum circuits consists 12 Gray code conversions, 54 Control-Not gates. Therefore, the complexity of this circuits is $O(12\times 3+54\times 1)=O(90)$ .
•
Extracting a whole secret image into the cover image.

From Fig. 14, this quantum circuits consists 2 quantum comparators, 1 Single pixel extracting module. Obviously, the complexity of this circuits is $O(2n^{2}+90)$ .
•
Secret image restoration.

From Fig. 15, the structure of this circuits is similar to that of Fig. 7, their complexity is the same, and so, the complexity of this circuits is also $O(n2^{2n+2})$ .

Table 1
The PSNR the four steganographic schemes for the various combinations of cover images and secret images

Scheme Secret image Cover image

(a) (b) (c) (d) (e) (f)

Scheme in [26] Fig. 17a 52.9010 52.9022 52.8982 52.9010 52.9226 52.9099

Fig. 17b 52.8957 52.9009 52.8991 52.9108 52.9042 52.9018

Fig. 17c 52.9013 52.9017 52.9138 52.9083 52.9198 52.9077

Scheme in [27] Fig. 17a 50.3606 50.3382 50.3265 50.3169 50.2848 50.3324

Fig. 17b 50.2368 50.2334 50.1826 50.1860 50.1511 50.2267

Fig. 17c 50.2999 50.2855 50.2814 50.2631 50.2116 50.2952

Scheme in [25] Fig. 17a 52.8954 52.8939 52.9102 52.9060 52.9105 52.9078

Fig. 17b 52.9007 52.9012 52.9003 52.9121 52.9037 52.9074

Fig. 17c 52.8978 52.8809 52.9091 52.9028 52.9072 52.9030

Fig. 18a 51.1333 51.1349 51.1491 51.1481 51.1546 51.1480

Fig. 18b 51.1407 51.1427 51.1321 51.1457 51.1376 51.1456

Fig. 18c 51.1404 51.1245 51.1471 51.1445 51.1486 51.1433

Scheme in this paper Fig. 17a 48.9121 48.9178 48.9060 48.8659 48.8876 48.9802

Fig. 17b 48.9215 48.9176 48.8976 48.8849 48.8812 48.9807

Fig. 17c 48.9061 48.9174 48.9014 48.8678 48.8927 48.9366

Fig. 19a 44.1371 44.1394 44.1307 44.0969 44.0855 44.1316

Fig. 19b 44.1331 44.1413 44.1175 44.1150 44.0639 44.1406

Fig. 19c 44.1347 44.1382 44.1238 44.1104 44.0779 44.1233

In summary, the complexity of the steganographic scheme is $O(n^{2}+2n^{2}+144+2n^{2}+90+n^{2})=O(6n^{2}+243)\approx O(n^{2})$ , regardless of the pixel decomposition and restoration process of the secret image, but considering the scrambling and inverse scrambling process. If all processes are considered, the complexity is $O(n2^{2n+3}+n^{2})$ .
5. Simulation and analysis on classical computer

Scheme	Secret image	Cover image
Scheme in [26]	Fig. 17a	52.9010	52.9022	52.8982	52.9010	52.9226	52.9099
	Fig. 17b	52.8957	52.9009	52.8991	52.9108	52.9042	52.9018
	Fig. 17c	52.9013	52.9017	52.9138	52.9083	52.9198	52.9077
Scheme in [27]	Fig. 17a	50.3606	50.3382	50.3265	50.3169	50.2848	50.3324
	Fig. 17b	50.2368	50.2334	50.1826	50.1860	50.1511	50.2267
	Fig. 17c	50.2999	50.2855	50.2814	50.2631	50.2116	50.2952
Scheme in [25]	Fig. 17a	52.8954	52.8939	52.9102	52.9060	52.9105	52.9078
	Fig. 17b	52.9007	52.9012	52.9003	52.9121	52.9037	52.9074
	Fig. 17c	52.8978	52.8809	52.9091	52.9028	52.9072	52.9030
	Fig. 18a	51.1333	51.1349	51.1491	51.1481	51.1546	51.1480
	Fig. 18b	51.1407	51.1427	51.1321	51.1457	51.1376	51.1456
	Fig. 18c	51.1404	51.1245	51.1471	51.1445	51.1486	51.1433
Scheme in this paper	Fig. 17a	48.9121	48.9178	48.9060	48.8659	48.8876	48.9802
	Fig. 17b	48.9215	48.9176	48.8976	48.8849	48.8812	48.9807
	Fig. 17c	48.9061	48.9174	48.9014	48.8678	48.8927	48.9366
	Fig. 19a	44.1371	44.1394	44.1307	44.0969	44.0855	44.1316
	Fig. 19b	44.1331	44.1413	44.1175	44.1150	44.0639	44.1406
	Fig. 19c	44.1347	44.1382	44.1238	44.1104	44.0779	44.1233

In this Section, we make the simulations of steganoraphy scheme for several color images on a classical computer due to the condition that the physical quantum computer is not in our grasp right now. The simulations are demonstrated with a classical computer with Intel (R) Core (TM) i5-3470 CPU @ 3.20 GHz 4.00 GB RAM and 32-bit operating system. The simulations are based on linear algebra with complex vectors as quantum states and unitary matrices as unitary transforms using Matlab 7.8.0 (R2009a).

In order to reflect the advantages of steganoraphy scheme proposed in this paper, this scheme will be compared with other steganoraphy schemes in [25, 26, 27] under two cases of the same embedding amount and different embedding amount. Specific contrast items include: the peak signal-to-noise ratio of stego-image, robust performance in noisy environment, histogram and so on. The six $512\times 512$ sized color images (as cover images) used in the simulation are taken from the web [35], as shown in Fig. 16.

For a $512\times 512$ sized cover image, since the embedding capacity of the steganographic scheme in [26, 27] is 2 bits per pixel, that is, it can embed at most $512\times 512\times 2=256\times 256\times 8$ secret bits, so the maximum size of embedded gray-scale image is $256\times 256$ . Similarly, for the steganographic scheme in [25], the embedding capacity is 3 bits per pixel, it can embed at most $512\times 512\times 3=384\times 256\times 8$ secret bits, so the maximum size of embedded secret image is $384\times 256$ . For the steganographic scheme proposed in this paper, the embedding capacity is 6 bits per pixel, it can embed at most $512\times 512\times 6=768\times 256\times 8$ secret bits, so the maximum size of the secret image can reach $768\times 256$ . Three $256\times 256$ sized gray-scale level images (as secret images) used in the simulation are shown in Fig. 17a–c, Similarly, three $384\times 256$ and three $768\times 256$ sized gray-scale level images are shown Fig. 18a–c and Fig. 19a–c, respectively.

Figure 17.

Three $256\times 256$ sized secret images used in the experiments. (a) 256 $\times$ 256. (b) 256 $\times$ 256. (c) 256 $\times$ 256.

5.1 Imperceptibility evaluation

Invisibility is a fundamental requirement for the performance of steganographic schemes, and is usually measured using two metrics: peak-signal-to-noise ratio (PSNR) and histogram analysis.

5.1.1 Comparison of PSNR of four steganographic schemes

Let $C$ be a $2^{n}\times 2^{n}$ sized color cover image, and $S$ be its stego-version. The PSNR of the stego-image $S$ is defined as

$\displaystyle\text{PSNR}=$ (7) $\displaystyle 20\log_{10}\left(\frac{255\sqrt{3\times 2^{2n}}}{\displaystyle% \sqrt{\sum\limits_{i=0}^{2^{n}-1}\sum\limits_{j=0}^{2^{n}-1}\sum\limits_{k=0}^% {3}[C(i,j,k)-S(i,j,k)]^{2}}}\right)$

With the nine gray-scale images in Figs 17–19 as the secret images, the PSNRs of the four steganographic schemes are shown in Table 1. The visual effects of the proposed scheme in which the secret image in Fig. 19c is accommodated in the depicted cover images is illustrated in Fig. 20.

Figure 18.

Three 384 $\times$ 256 sized secret images used in the experiments. (a) 384 $\times$ 256. (b) 384 $\times$ 256. (c) 384 $\times$ 256.

Figure 19.

Three $768\times 256$ sized secret images used in the experiments. (a) 768 $\times$ 256. (b) 768 $\times$ 256. (c) 768 $\times$ 256.

Figure 20.

Six cover images used in the experiments. (a) 512 $\times$ 512. (b) 512 $\times$ 512. (c) 512 $\times$ 512. (d) 512 $\times$ 512. (e) 512 $\times$ 512. (f) 512 $\times$ 512.

By considering Table 1, after embedding the secret image with the same size, compared with other schemes, the PSNR of proposed scheme is only reduced by 2–4 dB. When the maximum embedding capacity is achieved, compared with schemes in [26, 27], although the PSNR is reduced by 6–8 dB, the embedding capacity is increased by two times, and compared with schemes in [25], although the PSNR is reduced by about 7 dB, the embedding capacity is doubled. In addition, for the proposed scheme, the PSNR values above 44 dB are also acceptable after reaching the maximum embedding capacity. Moreover, it can be seen from Fig. 20 that the original cover images and the corresponding stego-image are indistinguishable from the naked eye alone, which also verifies the security of this scheme in a certain extent.

5.1.2 The histogram analysis

Figure 21 shows the histogram of the three original cover images and the corresponding stego-versions where Fig. 19b is considered as a secret image. It is clear from Fig. 21 that, compared with the histogram of the original cover image, although the histogram of the stego-image has a slight jitter, they still maintain a high degree of consistency.

Figure 21.

The histogram diagram of the images including the cover images’ histogram depicted in second column and the stego-images’ histogram depicted in third column where the gray-scale image in Fig. 19b is considered as the secret image. (a) Stego. (b) Histogram of cover Histogram of stego. (c) Stego. (d) Histogram of cover Histogram of stego. (e) Stego. (f) Histogram of cover Histogram of stego.

5.2 Safety evaluation

At present, many steganographic programs are designed by modifying the LSB of the cover image. In this section, we examine the security of the steganographic scheme by directly extracting the LSB of the stego-image. The specific extraction method is as follows.

For scheme in [26], the LSB of the R channel is first extracted as the first bit of the 2-bit segment, and then the LSB of the G or B channel is extracted as the second bit of the 2-bit segment. The extraction scheme in [27] is based on two previously obtained secret keys. For scheme in [25], because the secret information is directly used as the LSB of the RGB three-channel in the cover image, the LSB of RGB three-channel is directly extracted as the 3-bit segment of secret information. For scheme proposed in this paper, first, the second LSB of RGB three-channel is extracted as the first three bits of the 6-bit segment, and then three-channel’s LSB is extracted as the last three bits of the 6-bit segment. Figure 22 shows the results of the four steganographic schemes obtained by directly extracting the LSB of the stego-image, where the color image in Fig. 16a is considered as the cover image, and the gray-scale image in Fig. 17a is considered as the secret image.

As can be seen from Fig. 22, the scheme proposed in this paper and the scheme in [27] are safer, the security of the scheme in [26] is slightly less, and the scheme in [25] has no security at all.

Figure 22.

The secret image obtained by directly extracting the LSB of the RGB three channels in the stego-image for four schemes. (a) Ref. [26]. (b) Ref. [27]. (c) Ref. [25]. (d) Ours.

Unlike the scheme in [25], in the scheme proposed in this paper, the secret bits are not directly embedded in the cover image. According to the reflected Gray-code rule, about 50% of the embedded secret bits has flipped in the stego-image. Taking the three secret images in Fig. 19 as an example, the specific numerical results are shown in Table 2.

Table 2

Quantity comparison of hiding secret bits and LSBs of cover image

Coverimage inFig. 16	Secret image (a) in Fig. 19				Secret image (b) in Fig. 19				Secret image (c) in Fig. 19
Coverimage inFig. 16	Secret Bit 0		Secret Bit 1		Secret Bit 0		Secret Bit 1		Secret Bit 0		Secret Bit 1
	(837345)		(735519)		(807168)		(765696)		(820977)		(751887)
	Flip to 0	Flip to 1	Flip to 0	Flip to 1	Flip to 0	Flip to 1	Flip to 0	Flip to 1	Flip to 0	Flip to 1	Flip to 0	Flip to 1
(a)	419543	417802	368190	367329	403261	403907	383109	382587	409704	411273	375839	376048
(b)	418961	418384	366986	368533	405535	401633	382979	382717	411017	409960	374322	377565
(c)	422467	414878	363862	371657	408059	399109	379511	386185	414663	406314	372776	379111
(d)	419671	417674	364592	370927	404564	402604	383290	382406	411335	409642	374936	376951
(e)	423515	413830	364630	370889	407242	399926	377388	388308	415720	405257	371783	380104
(f)	419974	417371	365942	369577	404633	402535	383232	382464	411414	409563	375034	376853

Table 2 gives the comparison results of hiding secret bits and LSBs of cover image. For instance, taking the cover image in Fig. 16a and the secret image in Fig. 19a as an example, the total secret bits are $837345+735519=1572864$ , the LSBs of cover image are 837345 bits, where 419543 bits are changed to 0 s and 417802 bits are changed to 1 s when the secret bits are 0 s, and the LSBs of cover image are 735519 bits while 368190 bits are changed to 0 s and 367329 bits are changed to 1 s when the secret bits are 1 s. In a average, the ratio of cover image’s LSBs changed to 0 s or 1 s is almost up to 50%. In this paper, the LSBs of stego-image are modified according to the rule of reflected Gray-code instead of the simple LSB substitution scheme which replaces the secret bit into LSB of stego-image directly, which helps to improve the security of the steganographic scheme.

Figure 23.

Three secret images extracted from stego-images superimposed with different noises for proposed scheme, in which G represents Gauss, SP represents salt and pepper, and Q represents qubit flipping. (a) G noise. (b) SP noise. (c) Q noise.

Table 3

The comparison of the correlation coefficients of the four schemes under Gaussian noise

Scheme	Secret image	Cover image
		(a)	(b)	(c)	(d)	(e)	(f)	Avg.
Scheme in [27]	Fig. 17a	0.0009	0.0038	$-$ 0.0034	0.0047	$-$ 0.0004	0.0014	0.0012
	Fig. 17b	0.0004	$-$ 0.0025	0.0026	0.0010	0.0032	0.0030	0.0013
	Fig. 17c	0.0010	$-$ 0.0011	$-$ 0.0060	$-$ 0.0027	$-$ 0.0000	0.0017	$-$ 0.0012
Scheme in [26]	Fig. 17a	0.0052	0.0021	$-$ 0.0051	0.0086	0.0058	$-$ 0.0048	0.0020
	Fig. 17b	0.0042	$-$ 0.0026	0.0055	$-$ 0.0056	0.0035	0.0044	0.0016
	Fig. 17c	0.0039	0.0073	0.0016	$-$ 0.0010	$-$ 0.0035	0.0019	0.0017
Scheme in [25]	Fig. 17a	$-$ 0.0113	0.0033	$-$ 0.0055	0.0040	0.0034	0.0075	0.0003
	Fig. 17b	0.0015	0.0052	0.0013	0.0059	0.0088	$-$ 0.0001	0.0038
	Fig. 17c	$-$ 0.0041	0.0027	0.0033	$-$ 0.0029	0.0078	0.0047	0.0019
	Fig. 18a	$-$ 0.0045	0.0019	$-$ 0.0005	0.0004	0.0014	0.0053	0.0006
	Fig. 18b	$-$ 0.0029	0.0017	0.0026	0.0010	0.0035	$-$ 0.0009	0.0008
	Fig. 18c	$-$ 0.0056	0.0052	0.0021	$-$ 0.0024	0.0044	0.0024	0.0010
Scheme in	Fig. 17a	$-$ 0.0010	$-$ 0.0069	0.0029	0.0044	0.0005	$-$ 0.0112	$-$ 0.0019
this paper	Fig. 17b	0.0019	$-$ 0.0015	0.0116	0.0090	0.0032	$-$ 0.0030	0.0035
	Fig. 17c	$-$ 0.0028	$-$ 0.0017	$-$ 0.0026	$-$ 0.0020	0.0021	$-$ 0.0032	$-$ 0.0017
	Fig. 19a	0.0023	$-$ 0.0000	0.0003	0.0037	0.0012	$-$ 0.0009	0.0011
	Fig.19b	$-$ 0.0026	$-$ 0.0021	$-$ 0.0038	0.0031	0.0066	0.0013	0.0004
	Fig.19c	$-$ 0.0024	$-$ 0.0013	$-$ 0.0020	$-$ 0.0026	0.0018	$-$ 0.0044	$-$ 0.0018

Figure 24.

Three secret images extracted from stego-images superimposed with different noises for schemes in [25, 26, 27], in which G represents Gauss, SP represents salt and pepper, and Q represents qubit flipping. (a) G noise. (b) SP noise. (c) Q noise. (d) G noise. (e) SP noise. (f) Q noise. (g) G noise. (h) SP noise. (i) Q noise.

5.3 Robustness performance under noise

In order to evaluate the robustness of the proposed steganographic scheme in the presence of channel noise, first, we mixed in with each stego-image one of three noise patterns: Gaussian noise with mean 0 and variance of 0.01, Salt-and-Pepper noise with a density of 0.1, and Qubit flip noise with a flipping probability of 0.01. We then extract the secret image from each noise-imposed stego-image. Finally, we investigate the correlation between the extracted secret image and the original secret image. To further verify the effectiveness of the proposed method, we also compared the performance of our method with that of [25, 26, 27].

Before beginning, we first explain what Qubit flip noise is. Qubit flip noise is essentially a channel noise that is caused when information is transmitted in a quantum channel. It is consistent with classic channel noise, i.e., “1” is received when “0” is sent, or “0” is received when “1” is sent. For a quantum image transmitted in a channel, the qubit flip noise transitions the state of the qubit from $|0|$ to $|1|$ (or vice versa) with probability $1-p$ , and this action can be described by operators such as $E_{0}=\sqrt{p}I=\sqrt{p}\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$ and $E_{1}=\sqrt{1-p}X=\sqrt{1-p}\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$ . With respect to the qubit flip noise used in this experiment, the flip probability $1-p$ is equal to 0.01.

Taking the secret image in Fig. 19a as an example, for the stego-image (where Fig. 16a is considered as a cover image) with superimposed noise, the extraction effect of the proposed scheme is shown in Fig. 23. Similarly, taking the secret image in Fig. 17a as an example, for the stego-image with superimposed noise, the extraction effect of the scheme in [27] is shown in Fig. 24a–c, and the extraction effect of the scheme in [26] is shown in Fig. 24d–f. Taking the secret image in Fig. 18a as an example, the extraction effect of the scheme in [25] is shown in Fig. 24g–i.

The simulation results show that the robust performance of several schemes is similar, that is, it has good adaptability to Salt and Pepper noise and Qubit flip noise, and is very fragile for Gaussian noise. To obtain a quantitative comparison of the robustness of the four schemes, we first define the correlation coefficient between the two images as follows.

$\displaystyle R(x,y)={\displaystyle\frac{E((x-E(x))(y-E(y)))}{\sqrt{D(x)D(y)}}},$ (8)

where $E(x)$ and $D(x)$ denote the expectation and variance, respectively, of the pixel gray-scale values of the image $x$ .

Using the upper equation and taking $x$ as the original secret image, $y$ is regarded as the secret image extracted from the stego-image with superimposed noise, the comparison of the correlation coefficients of the four schemes is shown in Tables 3–5.

To make the contrasts clearer and more intuitive, we show the last column in Tables 3–5 as a bar graph, as shown in Fig. 25.

Figure 25.

The comparison of average correlation coefficients of the four schemes under three different noise. (a) The comparison of average correlation coefficients of the four schemes under Gaussian noise. (b) The comparison of average correlation coefficients of the four schemes under Salt-and-Pepper noise. (c) The comparison of average correlation coefficients of the four schemes under Qubit flip noise.

The experimental results show that the robustness of the four schemes is quite poor for the Gaussian noise. For the Salt-and-Pepper noise, the robustness of proposed scheme is roughly equivalent to that of scheme in [25, 26], and is better than that of the scheme in [27]. For the Qubit flip noise, the proposed scheme is roughly equivalent to the scheme in [27], but is slightly weaker than that in [25, 26].

Table 4

The comparison of the correlation coefficients of the four schemes under Salt-and-Pepper noise

Scheme	Secret image	Cover image
		(a)	(b)	(c)	(d)	(e)	(f)	Avg.
Scheme in [27]	Fig. 17a	0.7317	0.7317	0.7325	0.7348	0.7372	0.7342	0.7337
	Fig. 17b	0.6595	0.6632	0.6563	0.6585	0.6592	0.6666	0.6606
	Fig. 17c	0.6797	0.6861	0.6829	0.6814	0.6787	0.6844	0.6822
Scheme in [26]	Fig. 17a	0.8648	0.8646	0.8645	0.8660	0.8645	0.8680	0.8654
	Fig. 17b	0.7807	0.7799	0.7784	0.7774	0.7796	0.7798	0.7793
	Fig. 17c	0.7981	0.7961	0.7986	0.8001	0.7987	0.8004	0.7987
Scheme in [25]	Fig. 17a	0.8644	0.8695	0.8658	0.8629	0.8635	0.8663	0.8654
	Fig. 17b	0.7776	0.7798	0.7731	0.7782	0.7823	0.7784	0.7782
	Fig. 17c	0.7980	0.7965	0.8029	0.8040	0.8001	0.7996	0.8002
	Fig. 18a	0.8665	0.8698	0.8662	0.8644	0.8648	0.8678	0.8666
	Fig. 18b	0.7603	0.7617	0.7573	0.7614	0.7619	0.7611	0.7606
	Fig. 18c	0.7906	0.7885	0.7927	0.7935	0.7901	0.7920	0.7913
Scheme in	Fig. 17a	0.8619	0.8624	0.8621	0.8615	0.8615	0.8638	0.8622
this paper	Fig. 17b	0.7850	0.7871	0.7823	0.7822	0.7847	0.7847	0.7844
	Fig. 17c	0.8184	0.8241	0.8166	0.8219	0.8167	0.8237	0.8202
	Fig. 19a	0.8639	0.8624	0.8618	0.8624	0.8630	0.8603	0.8623
	Fig. 19b	0.7826	0.7844	0.7859	0.7837	0.7850	0.7845	0.7843
	Fig. 19c	0.8208	0.8198	0.8195	0.8209	0.8194	0.8216	0.8203

Table 5

The comparison of the correlation coefficients of the four schemes under Qubit flip noise

Scheme	Secret image	Cover image
		(a)	(b)	(c)	(d)	(e)	(f)	Avg.
Scheme in [27]	Fig. 17a	0.9024	0.9029	0.9022	0.9013	0.9030	0.9034	0.9025
	Fig. 17b	0.8006	0.8037	0.8018	0.7998	0.7995	0.8016	0.8012
	Fig. 17c	0.8293	0.8325	0.8310	0.8325	0.8322	0.8309	0.8314
Scheme in [26]	Fig. 17a	0.9705	0.9717	0.9717	0.9722	0.9709	0.9721	0.9715
	Fig. 17b	0.9473	0.9475	0.9443	0.9489	0.9474	0.9485	0.9473
	Fig. 17c	0.9556	0.9561	0.9546	0.9532	0.9526	0.9534	0.9542
Scheme in [25]	Fig. 17a	0.9719	0.9716	0.9728	0.9710	0.9734	0.9717	0.9721
	Fig. 17b	0.9440	0.9460	0.9474	0.9489	0.9461	0.9478	0.9467
	Fig. 17c	0.9516	0.9540	0.9534	0.9540	0.9555	0.9554	0.9540
	Fig. 18a	0.9721	0.9715	0.9718	0.9713	0.9730	0.9718	0.9719
	Fig. 18b	0.9401	0.9413	0.9426	0.9432	0.9412	0.9424	0.9418
	Fig. 18c	0.9493	0.9519	0.9504	0.9507	0.9513	0.9523	0.9510
Scheme in	Fig. 17a	0.8941	0.8963	0.8963	0.8965	0.8953	0.8950	0.8956
this paper	Fig. 17b	0.8209	0.8182	0.8225	0.8209	0.8204	0.8269	0.8216
	Fig. 17c	0.8421	0.8373	0.8398	0.8393	0.8366	0.8342	0.8382
	Fig. 19a	0.8950	0.8926	0.8953	0.8947	0.8951	0.8941	0.8945
	Fig. 19b	0.8218	0.8207	0.8221	0.8227	0.8232	0.8236	0.8223
	Fig. 19c	0.8390	0.8393	0.8399	0.8413	0.8391	0.8400	0.8398

5.4 Embedding capacity

On the embedding capacity of the steganography scheme, [20] defines it as the ratio of the number of secret qubits and the number of cover pixels.

$\displaystyle C={\displaystyle\frac{{\text{the number of secret qubits}}}{% \text{the number of cover image's pixel}}}.$ (9)

According to this definition, the implicit capacity of the proposed scheme is

$\displaystyle C=\frac{6\times 2^{n}\times 2^{n}}{2^{n}\times 2^{n}}=6\text{(% bit/pixel)}.$

As a comparison, the embedding capacity of the scheme in [25] is 3 bit/pixel, while in [26, 27], the embedding capacity of two schemes is only 2 bit/pixel.

Based on the above results, the proposed steganography scheme not only has the acceptable PSNR, high security and robustness, the most outstanding advantage is that it has a higher embedding capacity. At the same time embedding process shows that the scheme is not only no secret key, but it is blind.

6. Conclusions

Computer technology has made it a lot easier to hide messages and more difficult to discover them. Digital steganography is the science of hiding secret messages within digital media, such as digital images, audio files, or video files. There are many different methods for digital steganography, including least-significant-bit (LSB) substitution, message scattering, masking and filtering, and image-processing functions. These existing methods have been deeply studied on classical computers. However, similar studies on quantum computers are rare. This paper investigates the realization of color image steganography based on the idea of LSB substitution on quantum computer. The proposed scheme does not replace the LSBs of the cover image directly with the secret bits, but embeds the secret information indirectly into the LSBs by reflecting the Gray code rule, which enhances the security of the stego-image. Compared with several existing quantum steganography schemes for color images, this scheme has the advantages of acceptable PSNR and better security, and the most prominent advantage is that it has a larger embedded capacity of up to 6 bits per pixel. In addition, the proposed scheme is very sensitive to Gauss noise, and so, how to improve the robust performance under Gauss noise is the next step to be carried out.

Footnotes

Acknowledgments

This work was supported by the Youth Science Foundation of Northeast Petroleum University (Grant No. 2018QNL-08), and the Guiding Innovation Fund of Northeast Petroleum University (Grant No. 2018YDL-20).

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