1 Introduction

With the rapid development of quantum computation and quantum information, the research of quantum image processing is springing up. It can be divided into two directions: representations of quantum images and quantum image processing algorithms. The first direction, representations of quantum images, is the foundation of the quantum image processing. The task in this direction is the construction of a pattern to store the images on quantum computers. In recent years, lots of quantum image representations were studied, i.e., Qubit Lattice [1], Real Ket [2], Entangled Image [3], a flexible representation for quantum images (FRQI) [4], multi-channel representation of quantum image (MCRQI) [5], a normal arbitrary quantum superposition state (NAQSS) [6], and a novel enhanced quantum representation (NEQR) [7]. Of these quantum representation models, two of them are most commonly used: one is FRQI using an angle to encode color, and the other is NEQR using a qubit sequence to store the color value of every pixel.

Many algorithms have been proposed in the second direction as well, such as Geometric transformation [8, 9], Image scaling [10, 28], Color transformation [7], Image scrambling [11, 12], Feature extraction [13], Quantum image encryption [14,15,16,17,18], Quantum image information hiding [19,20,21,22,23,24,25,26,27], and so on.

The quantum image information hiding includes quantum watermarking and quantum steganography of which have many similarities. For example, the goal of both methods is to hide the information into the carrier. Moreover, they both include embedding and extracting procedures.

Due to quantum information hiding is important for hiding information into other images and protecting the image copyright. The quantum image information hiding has a wide range of uses. In recent years, researchers had been committed to study in it. In 2012 A.M. Iliyasu, et al. proposed a quantum image watermarking algorithm based on restricted geometric transformations [19]. Later on, in 2013, watermark image was embedded into the Fourier coefficients of the quantum carrier image in the protocol proposed by W. W. Zhang et al., in which the carrier images would not be visually affected by the watermark images [20]. Then X.H. Song et al. proposed a dynamic watermarking scheme for quantum images based on Hadamard transform [21]. Recently, a blind LSB steganography in the form of quantum circuits had been proposed by N. Jiang et al. [22]. In 2016, S. Heidari, et al. proposed a novel LSB based quantum watermarking protocol [23]. Before this, S. Miyake proposed a quantum watermarking scheme using simple and small-scale quantum circuits [24]. The proposed scheme took a gray scale image as the watermark image by extending a n × n sized image with eight bits of gray scale to a 2n × 2n sized image with two bits of gray scale. In 2017, B. Abd-El-Atty et al. proposed the New Quantum Image Steganography Scheme with Hadamard Transformation [26], in which the quantum text message was hided into the carrier image.

In order to enhance the security of the information image, a novel quantum steganography scheme based on LSB is proposed in this paper. The outline of this paper is structured as follows. Section 2 introduces the preliminary knowledge of quantum image model NEQR, LSB scheme, Bit-plane scrambling and Arnold scrambling methods. The proposed scheme is explained in Section 3 in detail. Section 4 presents the simulation and analysis of the scheme. Finally, a brief conclusion is drawn in Section 5.

2 Background

2.1 Novel Enhanced Quantum Representation for Quantum Images

A Novel Enhanced Quantum Representation of digital images (NEQR) was proposed in 2013 [7]. The expression form of a quantum image described by the NEQR model is:

$$ \vert I\rangle =\frac{{1}}{{2}^{\mathrm{n}}}\sum\limits_{Y = 0}^{2^{n}-1} {\sum\limits_{X = 0}^{2^{n}-1} {\vert C_{YX} \rangle \vert Y\rangle \vert X\rangle =}} \frac{{1}}{{2}^{\mathrm{n}}}\sum\limits_{YX= 0}^{2^{2n}-1} {\mathop \otimes\limits_{i = 0}^{q-1} C_{YX}^{i}} \otimes \vert YX\rangle $$
(1)

where |YX〉 stands for position information and \(\vert C_{YX}^{i} \rangle \) is the color information.

$$\begin{array}{@{}rcl@{}} \vert YX\rangle =\vert Y\rangle \vert X\rangle &=&\vert y_{0} y_{1} ...y_{n-1} \rangle \vert x_{0} x_{1} ...x_{n-1} \rangle ,y_{i} ,x_{i} \in \left\{ {0,1} \right\},i = 0,1,...,n-1\\ \vert \mathrm{C}_{YX} \rangle &=&\left\vert C_{YX}^{0} C_{YX}^{1} ...C_{YX}^{q-1} \right\rangle ,C_{YX}^{i} \in \left\{ {0,1} \right\},i = 0,1,...,q-1 \end{array} $$
(2)

Thus, there are q + 2n qubits to represent a quantum image of size 2n × 2n with the gray range of 2q. Figure 1 shows an example of an image of 2 × 2 and the corresponding NEQR representation is on the right of the image.

Fig. 1
figure 1

A 2 × 2 NEQR image representation

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2.2 The Least Significant Bit (LSB)

The least significant bit is one of the pixel bits shown in Fig. 2. If the control bit is “1”, we only need to change the least significant bit from “1” to “0”, i.e., color value 227 is changed into 226. Simply, the change of pixel value is only one gray level, so the visual effect is not affected basically.

Fig. 2
figure 2

An example of LSB

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2.3 Bit-Plane Scrambling

The pixel values in the image are represented by its corresponding binary values. And then, every single bit of all the pixels will constitute a binary image, it is called bit-plane. For example, if the gray range is between 0 and 28, the image will be separated into 8 bit-planes. Figure 3 shows the result of bit-planes of Lena image of size 28 × 28.

Fig. 3
figure 3

Bit-planes of Lena

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From above, it can be known that different bit-planes carry different visual information. If these bit-planes are measured, the information will become disordered. By scrambling the bit-plane, the image will be scrambled.

2.4 Arnold Scrambling

Arnold scrambling is used to change the pixel horizontal and vertical coordinates to disrupt the original image information. The Arnold scrambling for a digital image can be illustrated specifically as following.

Supposing that there is an original image with size of 2n × 2n, the pixel coordinates of the original image and the scrambled image are (X, Y ) and (X A , Y A ), respectively. Two-dimensional Arnold scrambling process is expressed by (3) and its inverse operation is shown in (4).

$$\begin{array}{@{}rcl@{}} \left[ {{\begin{array}{*{20}c} {X_{A}} \hfill \\ {Y_{A}} \hfill \end{array}} } \right]&=&\left[ {{\begin{array}{*{20}c} 1 \hfill & 1 \hfill \\ 1 \hfill & 2 \hfill \end{array}} } \right]\;\left[ {{\begin{array}{*{20}c} X \hfill \\ Y \hfill \end{array}} } \right]\;\bmod 2^{n}\\ X_{A} &=&(X+Y)\bmod 2^{n}\;,Y_{A} =(X + 2Y)\bmod 2^{n} \end{array} $$
(3)

The inverse operation is:

$$\begin{array}{@{}rcl@{}} \left[ {{\begin{array}{*{20}c} X \hfill \\ Y \hfill \end{array}} } \right]&=&\left[ {{\begin{array}{*{20}c} 1 \hfill & 1 \hfill \\ 1 \hfill & 2 \hfill \end{array}} } \right]^{-1}\;\left[ {{\begin{array}{*{20}c} {X_{A}} \hfill \\ {Y_{A}} \hfill \end{array}} } \right]\bmod 2^{n}=\;\left[ {{\begin{array}{*{20}c} 2 \hfill & {-1} \hfill \\ {-1} \hfill & 1 \hfill \end{array}} } \right]\;\left[ {{\begin{array}{*{20}c} {X_{A}} \hfill \\ {Y_{A}} \hfill \end{array}} } \right]\;\bmod 2^{n}\\ X&=&(2X_{A} -Y_{A} )\;\bmod \;2^{n}\;,Y=(Y_{A} -X_{A} )\;\bmod \;2^{n} \end{array} $$
(4)

According to literature [11], the circuits of Arnold scrambling and inverse-scrambling are shown in Figs. 4 and 5.

Fig. 4
figure 4

The quantum circuit of Arnold image scrambling: a|x A 〉 circuit b|y A 〉 circuit

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Fig. 5
figure 5

The quantum circuit of inverse Arnold image scrambling: a|x〉 circuit b|y〉 circuit

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3 The Proposed Quantum Image Steganography Scheme

The proposed quantum image steganography scheme hides a gray scale image (secret information image) into the other gray scale image (cover image), and the size of two images is n × n and 4n × 4n, respectively. The embedding and extracting procedures are shown in Fig. 6, where Fig. 6a is embedding procedure and Fig. 6b is extracting procedure. The specific scheme is described as follows.

Fig. 6
figure 6

Quantum image steganography scheme: a embedding procedure b extracting procedure

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3.1 Preparation Procedure

In this section the original information image |I〉 should be scrambled by the bit-plane scrambling. Figure 7 shows an example of the circuit that is used for processing an image with gray scale of 28. Firstly, there are four SWAP gates and four CNOT gates utilized in this circuit. The lowest bit |C7〉 swaps with the highest bit |C0〉, the bit |C6〉 swaps with the bit |C1〉, the bit |C5〉 swaps with the bit |C2〉 and the bit |C4〉 swaps with the bit |C3〉. Then, four low bits will be the control bits and apply the results to four high bits. Therefore, the scrambled information image |I1〉 is prepared.

Fig. 7
figure 7

Bit-plane scrambling for a NEQR quantum image with gray scale of 28

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3.2 Embedding Procedure

The embedding procedure consists of four steps that illustrate as follows:

  1. Step 1

    The aim of this step expands the scrambled image to the expanded image which has same size with the cover image. An example, as shown in Fig. 8a, both of the key image |K〉 that only known to the operator and the gray scale image |I1〉 are size of 1 × 1, in which |K〉 is a binary image and |I1〉 will be expanded to a 4 × 4 binary image \(\vert I_{1}^{\prime } \rangle \). Figure 8b gives the concrete quantum circuit realization of expanding the scrambled image to the expanded image which has same size with the cover image, where the key image |K〉 and the gray scale image |I1〉 both are size of 2n × 2n, respectively. If the color value |Ki〉 of key |K〉 is equal to 1, an 8 bits string is divided into 8 single bits whose positions are (4i,4j + 1), (4i,4j + 2), (4i + 1,4j), (4i + 1,4j + 1), (4i + 1,4j + 2), (4i + 2,4j), (4i + 2,4j + 1) and (4i + 2,4j + 2), respectively. The value of the position (4i,4j) is equal to 1 as well. If the key |Ki〉 is equal to 0, an 8 bits string will be divided into 8 single bits by reverse permutation. Similarly, the value of the position (4i,4j) is equal to 0, and the rest of the expanded image is covered by the value 1.

  2. Step 2

    The Arnold scrambling is used for scrambling the expanded image \(\vert I_{1}^{\prime } \rangle \) to an unordered image |I2〉.

  3. Step 3

    The unordered information image |I2〉, the key |K1〉 and the cover image |C〉 have been prepared. Their NEQR representations are:

    $$ \vert I_{2} \rangle =\frac{{1}}{{2}^{\mathrm{n}}}\sum\limits_{j = 0}^{2^{2n}-1} {\vert {I_{2}^{j}} \rangle} \otimes \vert YX\rangle ,{I_{2}^{j}} \in \left\{ {0,1} \right\} $$
    (5)
    $$ \vert K_{1} \rangle =\frac{{1}}{{2}^{\mathrm{n}}}\sum\limits_{j = 0}^{2^{2n}-1} {\vert {K_{1}^{j}} \rangle} \otimes \vert YX\rangle ,{K_{1}^{j}} \in \left\{ {0,1} \right\} $$
    (6)
    $$ \vert C\rangle =\frac{{1}}{{2}^{\mathrm{n}}}\sum\limits_{YX= 0}^{2^{2n}-1} {\mathop \otimes\limits_{i = 0}^{q-1} C_{YX}^{i}} \otimes \vert YX\rangle ,\vert C_{YX} \rangle =\vert C_{YX}^{0} C_{YX}^{1} \mathellipsis C_{YX}^{q-1} \rangle ,C_{YX}^{i} \in \left\{ {0,1} \right\},i = 0,1,...,q-1 $$
    (7)

    where |YX〉 = |yn− 1yn− 2...y0〉|xn− 1xn− 2...x0〉, y i , x i ∈ {0,1}, i = 0,1,..., n − 1

Fig. 8
figure 8

a An example of 1 × 1 image expands to 4 × 4 image b The expanding circuit for a 2n × 2n NEQR image

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Each position in the unordered information image |I2〉 will be compared with the position in the cover image |C〉. Literature [25] proposed a quantum equal circuit presented in Fig. 9. The comparator compares |YX〉 and |AB〉, where |YX〉 = |Y 〉|X〉 = |yn− 1...y0〉|xn− 1...x0〉 and |AB〉 = |A〉|B〉 = |an− 1...a0〉|bn− 1...b0〉, y i , x i , a i , b i ∈ 0,1, i = n − 1,...,0. Qubit |c〉 is output. If |c〉 = |1〉, |YX〉 = |AB〉, otherwise, |YX〉≠|AB〉.

Fig. 9
figure 9

Quantum Equal circuit (QE)

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When the unordered information image is embedded, the steganography image and the binary image key |K2〉 are obtained. The quantum steganography embedding circuit is shown in Fig. 10. If the position is the same, then implement the embedding procedure as follows:

$$\begin{array}{@{}rcl@{}} && \text{For}~~i = 0~~\text{to~}~2^{2n}-1\\ &&\text{If}~(\vert {I_{2}^{i}} \rangle ==\vert 0\rangle )~\text{then}\\ &&\text{If}\; (\vert {K_{1}^{i}} \rangle =\,=\vert 1\rangle )~\text{then}~\vert C{I_{i}^{0}} \rangle =\vert {C_{i}^{0}} \rangle \oplus \vert 1\rangle \\ &&\text{Else if}~(\vert {K_{1}^{i}} \rangle ==\vert 0\rangle)~\text{then}~\vert {K_{2}^{i}} \rangle ==\vert 1\rangle \\ &&\text{End} \end{array} $$
Fig. 10
figure 10

Circuit realization of quantum steganography embedding

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3.3 Extracting Procedure

Considering the security of the unordered information image, the extracting procedure is more complex than embedding procedure.

The first step extracts the information from the steganography image |CI〉 and the key |K2〉. It is necessary to have the cover image |C〉, the steganography image |CI〉 and two keys |K1〉 and |K2〉 extracted from the unordered information image. The quantum steganography extracting circuit is shown in Fig. 11.

Fig. 11
figure 11

Circuit realization of quantum steganography extracting

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If the position is the same, the corresponding color value will be compared. The extracting procedure can be done in the following steps:

$$\begin{array}{@{}rcl@{}} && \text{For}~~i = 0~~\text{to}~~2^{2n}-1\\ &&\text{If}~\vert {K_{1}^{i}} \rangle =\,=\vert 1\rangle~~\text{then}\\ &&\text{If}~~\vert {C_{i}^{0}} \rangle \oplus \vert C{I_{i}^{0}} \rangle ==\vert 1\rangle~\text{then}~~\vert {I_{2}^{i}} \rangle ==\vert 0\rangle \\ &&\text{Else if}~\vert {K_{1}^{i}} \rangle =\,=\vert 0\rangle~\text{~then}\\ &&\text{If}~~\vert {K_{2}^{i}} \rangle =\,=\vert 1\rangle ~\text{then}~\vert {I_{2}^{i}} \rangle ==\vert 0\rangle \\ &&\text{End} \end{array} $$

The second step of extracting procedure is recovering the unordered information image |I2〉 to the expanded image \(\vert {I}^{\prime }_{1} \rangle \) by the inverse operation of the Arnold scrambling.

The third step is restoring the expanded image \(\vert {I}^{\prime }_{1} \rangle \) to the scrambled information image |I1〉. To judge the value of the position (i, j) in key |K〉, where i and j are less than n, if the value equals to 1, the 8 bits string of the gray scale of the pixel (i, j) in image |I1〉 equals to the value of these positions [(4i,4j + 1), (4i,4j + 2), (4i + 1,4j), (4i + 1,4j + 1), (4i + 1,4j + 2), (4i + 2,4j), (4i + 2,4j + 1), (4i + 2,4j + 2)] in image \(\vert {I}^{\prime }_{1} \rangle \); if the value equals to 0, the 8 bits string is in the opposite order. The restoring circuit is shown in Fig. 12.

Fig. 12
figure 12

The restoring circuit

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The final step is the inverse bit-plane scrambling operation to the scrambled information image that is importing the color value in the right side of the circuit like Fig. 7.

4 Simulation and Analysis

Since there is no enough physical quantum hardware to implement the proposed scheme, the simulation of quantum operations is performed with MATLAB on the classical computer. Six familiar images (“Mandrill”, “Cameraman”, “Airplane”, “Pepper”, “Rice” and “Lena”) are used. And the sizes of cover image and information image are 256 × 256 and 64 × 64, respectively. The information images are shown in Fig. 13.

Fig. 13
figure 13

Six information images with size 64 × 64

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4.1 Visual Quality

There are quantities of techniques to measure the difference of pixels between the steganography image and the cover image, where the peak-signal-to-noise ratio (PSNR) is one of the most employed. It is defined as the mean squared error (MSE) for two m×n images P and Q.

$$ MSE=\frac{1}{mn}\sum\limits_{i = 0}^{m-1} {\sum\limits_{j = 0}^{n-1} {[(P(i,j)-Q(i,j))^{2}]}} $$
(8)

PSNR is defined as:

$$ PSNR= 20\log_{10} \left( {\frac{MAX_{P}} {\sqrt {MSE}} } \right) $$
(9)

where MAX P is the maximum pixel value of the cover image C. In our scheme, P is associated with the steganography image and Q is associated with the cover image.

Obviously, human eyes can’t identify the difference between the cover image and the steganography image, which is shown in Fig. 14. Compared Table 1 with Table 2, it can be obtained that the values of PSNR of our proposed scheme are higher than literature [24] which is also use the gray scale image as the information image.

Fig. 14
figure 14

a the cover images, b their corresponding histograms c the steganography images (which embeds information images of Mandrill, Pepper, Cameraman, Lena, Airplane, Rice, respectively). d their corresponding histograms

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Table 1 PSNR values in proposed scheme
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Table 2 PSNR values in [24]
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4.2 Capacity

The steganography capacity can be stated as the ratio between the number of embedded bits of information image and the number of pixels of cover image. The proposed scheme’s capacity is given as follows:

$$ C=\frac{the\;num.~of~information~bits}{the~num.~of~cover~image~pixels}=\frac{8\times 2^{n-2}\times 2^{n-2}}{2^{2n}}=\frac{1}{2}(bit/pixel) $$
(10)

In order to enhance the confidentiality, half of the pixels are not to carry information.

4.3 Security Analysis

In our proposed scheme, we put the security in a significant status. First, the original image is scrambled to a disordered image. And then the meaningless image consists of half of the information pixels and half of blank pixels will be hiding in the cover image. Furthermore, the extracting operation needs the cover image, the steganography image and two keys, which are all indispensable. In all, the security of the proposed scheme is excellent.

4.4 Circuit Complexity

In this section, the circuit complexity of the proposed scheme is discussed. We assume that the sizes of cover image and original information image are 2n × 2n and 2n− 2 × 2n− 2, respectively. Literature [10] pointed out that circuit complexity of a n-CNOT gate (\(n\geqslant \)3) is about 12n−9 and each swap gate can be realized by three CNOT gates.

  • a. The Time Complexity of Embedding Procedure

The embedding procedure is shown in Fig. 6a and the bit-plane scrambling circuit is shown in Fig. 7 of which complexity is (4 × 3 + 4). According to [11], the complexities of Arnold scrambling circuit |x A 〉 and |y A 〉 are 28n and 28n, respectively. The expanding circuit is shown in Fig. 8b, supposing it expands 2n− 2 × 2n− 2 pixels, and every pixel needs twenty-six (2n + 2)-CNOT gates (In order to facilitate the calculation, all CNOT gates are considered to have the same number of control bits). Thus, the complexity of the expanding circuit is 22n− 4 × 26 ×[12(2n + 2) − 9]. Additionally, Fig. 9 shows the complexity of single QE circuit is 4n + (12 × 2n − 9). Therefore, the complexity of the embedding circuit is 2 ×[4n + (12 × 2n − 9)] + 2 × 6. Consequently, the time complexity of the whole embedding procedure is:

$$\begin{array}{@{}rcl@{}} && \mathrm{O}\left( {\begin{array}{l} 4\times 3 + 4+ 28n + 28n + 2^{2n-4}\times 26\times [12(2n + 2)-9] \\ + 2\times \left[ {4n+(12\times 2n-9)} \right]+ 2\times 6 \end{array}} \right) \\ &=&\mathrm{O}\left( {16 + 56n + 2^{2n-4}\times 26(24n + 15)+ 56n-6} \right) \\ &=&\mathrm{O}\left( {2^{2n-4}\times (624n + 390)+ 112n + 10} \right) \approx \mathrm{O}\left( {624n\cdot 2^{2n-4}} \right) \end{array} $$
(11)

  • b. The Time Complexity of Extracting Procedure

The extracting procedure is shown in Fig. 6b. Figure 11 shows extracting circuit in which the complexity is 3 ×[4n + (12 × 2n − 9)] + 2 ×(12 × 3 − 9) + 6. The complexities of inverse Arnold scrambling |x〉, |y〉 and bit-plane scrambling are 56n, 56n and (4 × 3 + 4), respectively. Finally, the complexity of restoring circuit is 22n− 4 × 16 ×[12(2n + 2) − 9]. In summary, the time complexity of the whole extracting procedure is:

$$\begin{array}{@{}rcl@{}} && \mathrm{O}\left( {\begin{array}{l} 4\times 3 + 4+ 56n + 56n + 2^{2n-4}\times 16\times [12(2n + 2)-9] \\ + 3\times \left[ {4n+(12\times 2n-9)} \right]+ 2\times \left( {12\times 3-9} \right)+ 6 \end{array}} \right) \\ &=&\mathrm{O}\left( {16 + 112n + 2^{2n-4}\times 16(24n + 15)+ 84n + 33} \right) \\ &=&\mathrm{O}\left( {2^{2n-4}\times (384n + 240)+ 196n + 49} \right) \\ &\approx& \mathrm{O}\left( {384n\cdot 2^{2n-4}} \right) \end{array} $$
(12)

According to the above analysis, the circuit complexity of embedding and extracting procedures of the proposed scheme is higher than [24]. But this paper designed the expanding and restoring circuit realization while other researchers have not mentioned it in their gray scale image steganography scheme.

5 Conclusion

Based on LSB scheme, a novel quantum image steganography scheme is proposed, where the NEQR representation is utilized. Therein, the image sizes for cover and information are 4n × 4n and n × n, respectively. The information hiding part contains two procedures. The first procedure is the preparation of scrambling the original information image. And the other procedure is embedding work which concludes three steps: The first step is expanding the scrambled information image to the same size as the cover image; the second step is to scramble the expanded image to a meaningless image; and the third step is embedding the unordered information image into the cover image. In the information extracting part, two keys that are only possessed by the operator are used for extracting the unordered information image from the steganography image and then the inverse operation of information hiding part will be executed. The experiments show that the scheme has a good visual quality than another scheme. Furthermore, the security of the scheme is excellent according to the analysis.