1 Introduction

Over the past decade, quantum image processing has attracted the attention of many researchers due to the promising speed of quantum computing that results from astounding properties of quantum mechanics such as entanglement and quantum superposition [1]. To start quantum image processing, the first step is to encode classical images in quantum computing settings. For this purpose, several effective quantum image representation methods have been introduced to encode a classical image with logarithmically less qubits than classical pixels in the image [2,3,4,5,6,7,8,9,10]. The flexible representation of quantum images (FRQI) model [2], and the novel enhanced quantum representation of a digital grayscale image (NEQR) model [4] are considered as the fundamental quantum image representation models. The RGB multi-channel quantum image model (MCQI) [3] and the novel quantum representation of color digital images (NCQI) model [6] are extensions of the FRQI and NEQR models, respectively, to color images. These quantum image representation models encode the positional information of all pixels in a given classical image, of size \(2^n \times 2^n\), as a superposed quantum state of 2n qubits, which is entangled with a quantum state that encodes the corresponding information about the color value of all the pixels in the image. On the other hand, quantum variants of many classical image processing algorithms such as geometric transformation, image filtering, image scaling, image matching, image segmentation, image scrambling and image encryption have been developed to facilitate a comprehensive quantum analysis of classical images [11,12,13,14,15,16,17,18,19,20].

In the last two decades, the multimedia communication rate between information computing devices like smartphones, tablets, smart cameras and laptop computers etc. has increased tremendously due to the fast connectivity between them through the Internet. Therefore, designing and using effective confidential communication methods to negate unauthorized access to sensitive information has become increasingly important. Steganography is often used to communicate confidential information by hiding the secret message or image into another message or image so that it is not easily detected.

Quantum image steganography provides high processing speed due to the phenomenon of quantum parallelism and also requires much less storage resources than classical image steganography. Due to these amazing features of quantum image steganography, a number of data concealment schemes have recently been proposed in literature, which embed the given confidential information image in the quantum state representing the carrier image. The quantum image steganography algorithm proposed in Ref. [21] uses Moiré pattern during the embedding and extraction stages and conceals a binary image in the NEQR state of the given carrier image. Based on the NEQR image model, Jiang et al. [22] introduced a least significant bit (LSB) quantum steganography scheme with an embedding capacity of 1 bit for every \(2^{p+3}\) carrier bits. In contrast, the quantum steganography scheme proposed by Heidari and Farzadina [23] utilizes the Gray code and the quantum RGB image model to conceal a secret image within a carrier RGB image. The embedding capacity of this scheme is \(1\, \text {bit}/{12\, \text {bits}}\). In Ref. [24], the Arnold transform is first applied to the secret data to scramble it, and then the scrambled secret data is embedded in the NEQR state of the carrier image with an embedding capacity of \(2\, \text {bits}/{8\,\text {bits}}\). The LSB steganography scheme proposed by Zhou et al. [25] also uses the Arnold transform to scramble the secret image. Moreover, the embedding capacity of this scheme is \(1\, \text {bit}/{16\,\text {bits}}\). An efficient quantum image steganography protocol based on the NEQR model and improved exploiting modification direction algorithm was proposed by Qu et al. [26]. A modified direction-based steganography protocol for the NCQI model with payload capacity of \(1\, \text {bit}/{12\,\text {bits}}\) was proposed in [27]. Recently, Wang et al. [28] introduced a quantum image steganography scheme based on turtle shell and LSB. The embedding capacity of this scheme is \(3\, \text {bits}/{24\,\text {bits}}\). Additionally, some other high-efficiency and secure controlled quantum image steganography algorithms have been recently introduced in the literature [29,30,31].

Recently, Khan [7] analyzed the FRQI and NEQR models and concluded that the FRQI model uses only one qubit to encode the color information of the image and thus requires a large amount of projective measurements to accurately retrieve image information. On the other hand, the NEQR model uses computational basis states to encode image color information and thus does not incorporate the advantages of the quantum superposition principle in terms of storing image color information. In addition, Khan [7] has solved these issues by suggesting the improved flexible representation of quantum images model (IFRQI) that employs an effective way to encode grayscale image information in a normalized quantum state vector to enable accurate retrieval of image information through a small number of projection measurements. Similarly, the MCQI model uses only one qubit to store the color information of each red, green and blue channels of the image while NCQI model uses computational basis states to store color information of red, green and blue channels of image and does not benefit from the superposition principal in respect of storing the color information of the image.

The purpose of this article is twofold. Firstly, we extend the IFRQI model to multi-channel effective quantum representation of colored digital images model (MCEQI). Secondly, we propose a high-capacity steganography scheme based on the MCEQI model. In the proposed scheme, the red, green and blue channel information of a colored secret image is divided into four image planes \(P^{u}_{j}\; (1\le u \le 3,\,0\le j\le 3)\) with a bit depth of 2. Then, an array \(A^{u}_{j}\) of angle values encoding the color information in the plane \(P^{u}_{j}\) is prepared for all \(j\in \{0,1,2,3\}\). Finally, the encoded information is embedded in the MCEQI state of the carrier RGB image by using controlled rotations determined by the key K, generated from the fractional-order erbium-doped fiber laser chaotic system, to produce the stego-image state.

The rest of the paper is organized as follows: Sect. 2 presents a brief overview of the existing quantum image representation models. The quantum image preparation method of the proposed MCEQI model is explained in Sect. 3. Section 4 describes the fractional-order erbium-doped fiber laser chaotic system. Section 5 presents proposed quantum image steganography scheme based on the MCEQI model. The performance of the proposed steganography algorithm is evaluated in Sect. 6, while the last section presents the conclusion.

2 Related work

In this section, we explain the basic quantum image representation models which are widely used for performing the quantum analysis of the classical digital images.

2.1 FRQI

Le et al. [2] introduced the FRQI model, which benefits from the superposition principal and entanglement property of the quantum mechanics. For a grayscale image I of size \(2^n\times 2^n\), the image information is stored into a normalized quantum state as expressed in Eq. (1),

$$\begin{aligned} |I\rangle =\frac{1}{2^n}\sum _{i=0}^{2^{2n}-1}(\cos \theta _{i}|0\rangle +\sin \theta _{i}|1\rangle )\otimes |i\rangle , \end{aligned}$$
(1)

where the first sequence of 2n qubits encodes the positional information of all pixels and is entangled with one qubit that encodes the pixels grayscale value.

2.2 MCQI

Sun et al. [3] introduced an extension of the FRQI model to RGB multi-channel quantum images model (MCQI). For an RGB image I of size \(2^n\times 2^n\), the image information is stored into a normalized quantum state as expressed in Eq. (2),

$$\begin{aligned} |I\rangle =\frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}|C^{i}_{RGB}\rangle \otimes |i\rangle , \end{aligned}$$
(2)

where the state \(|C^{i}_{RGB}\rangle \) captures the color information of the red, green and blue channel pixel at the ith position and is defined as

$$\begin{aligned} |C^{i}_{RGB}\rangle= & {} \cos {\theta _R^i}|000\rangle + \cos {\theta _G^i} |001\rangle +\cos {\theta _B^i}|010\rangle + \cos {0}|011\rangle \end{aligned}$$
(3)
$$\begin{aligned}{} & {} \sin {\theta _R^i}|100\rangle + \sin {\theta _G^i} |101\rangle +\sin {\theta _B^i}|110\rangle + \sin {0}|111\rangle . \end{aligned}$$
(4)

2.3 NEQR

Zhang et al. [4] introduced the novel enhanced quantum representation of digital images (NEQR) model, which stores the color information in computational basis states. For a grayscale image I of size \(2^n\times 2^n\) and gray range in the interval \([0,2^q]\), the NEQR state is

$$\begin{aligned} |I\rangle =\frac{1}{2^n}\sum _{i=0}^{2^{2n}-1} |c^i_{q-1}\cdots c^i_{1}c^i_{0}\rangle |i\rangle . \end{aligned}$$
(5)

where \(c^i_{j} \in \{0,1\} \) for all \(0\le j\le q-1\).

2.4 NCQI

Sang et al. [6] introduced an extension of the NEQR model to novel quantum representation of color digital images (NCQI) model. For a color image I of size \(2^n\times 2^n\), the image information is stored into a normalized quantum state as expressed in Eq. (6),

$$\begin{aligned} |I\rangle =\frac{1}{2^n}\sum _{i=0}^{2^{2n}-1} |c(i)\rangle |i\rangle . \end{aligned}$$
(6)

where the state \(|c(i)\rangle \) represents color value of the pixel at the ith position encoded by the binary sequence \(r^i_{q-1}\cdots r^i_{0}g^i_{q-1}\cdots g^i_{0}b^i_{q-1}\cdots b^i_0\), i.e.,

$$\begin{aligned} \left| {c\left( i \right) } \right\rangle = \left| {\underbrace{r^i_{q - 1} \cdots r^i_0 }_{Red}\underbrace{g^i_{q - 1} \cdots g^i_0 }_{Green}\underbrace{b^i_{q - 1} \cdots b^i_0 }_{Blue}} \right\rangle \end{aligned}$$

Note that the grayscale range of the image is \([0,2^{q}-1]\).

2.5 IFRQI

Khan [7] introduced the improved flexible representation of quantum images (IFRQI) model, which encodes color information of pixels as amplitudes of qubits by using specific angle values that yield highly distinctive probability distribution in response to projective measurements. The IFRQI state of a digital image of size \(2^n\times 2^n\) with grayscale range \([0,2^{2q}-1]\) is expressed as

$$\begin{aligned} |I\rangle = \frac{1}{2^n}\sum _{i=0}^{2^{2n}-1}|c^i_{q-1} c^i_{q-2}\cdots c^i_0\rangle \otimes |i\rangle \end{aligned}$$
(7)

where quantum state \( |c^i_{q-1} c^i_{q-2}\cdots c^i_0\rangle \) captures intensity of the ith pixel, with binary representation \(b_{2q-1}b_{2q-2}\cdots b_1b_0\), which is encoded by the rule defined as

$$\begin{aligned} |c^i_j\rangle= & {} \cos ({\theta ^i_j})|0\rangle +\sin ({\theta ^i_j})|1\rangle ,\; \theta ^i_j \nonumber \\= & {} \left\{ \begin{array}{ll} 0, &{} \textit{if}\; b^i_{2j+1}b^i_{2j}=00 \\ \frac{\pi }{5}, &{} {if}\; b^i_{2j+1}b^i_{2j}=01 \\ \frac{\pi }{2}-\frac{\pi }{5}, &{} {if}\; b^i_{2j+1}b^i_{2j}=10 \\ \frac{\pi }{2}, &{} {if}\; b^i_{2j+1}b^i_{2j}=11 \end{array} \right. ,\quad 0\le j \le q-1. \end{aligned}$$
(8)

3 Multi-channel effective quantum representation of a color digital image (MCEQI)

Khan [7] performed a rigorous analyses of the FRQI model and NEQR model and concluded that the FRQI model and NEQR model have their respective limitations when it comes to representing color information in quantum images. On the one hand, the FRQI model uses a single qubit to represent the probabilities of different colors, which can lead to a loss of information and a need for a large number of measurements. On the other hand, the NEQR model does not fully exploit the advantages of quantum superposition in representing color information, which can result in space complexity issues. Likewise, the MCQI and NCQI models take different approaches to representing color information in quantum images. The MCQI model uses only one qubit per color channel, which could potentially lead to information loss or a need for additional measurements. Meanwhile, the NCQI model uses computational basis states to represent the colors, which may offer a more accurate representation but could also require more qubits and increase the overall complexity of the quantum image representation.

In this section, we introduce a natural extension of the IFRQI model to store color digital images on quantum computers. For a digital color image I of size \(2^n\times 2^n\) with a bit depth given by 2p for the red, green and blue channels, the MCEQI state is defined as

$$\begin{aligned} |I\rangle = \frac{1}{2^n}\sum _{i=0}^{2^{2n}-1}|c(i)\rangle \otimes |i\rangle \end{aligned}$$
(9)

where quantum state \( |c(i)\rangle =|c^{i,1}_{p-1} c^{i,1}_{p-2}\cdots c^{i,1}_{0} c^{i,2}_{p-1} c^{i,2}_{p-2}\cdots c^{i,2}_{0} c^{i,3}_{p-1} c^{i,3}_{p-2}\cdots c^{i,3}_0\rangle \) encodes the color information of the ith pixel. The binary representation of the ith pixel is \(r^i_{2p-1} r^i_{2p-2}\cdots r^i_0 g^i_{2p-1} g^i_{2p-2}\cdots g^i_0 b^i_{2p-1} b^i_{2p-2}\cdots b^i_0\). The encoding rule is defined as follows

$$\begin{aligned}&\nonumber |c^{i,u}_j\rangle = \cos ({\theta ^{i,u}_j})|0\rangle +\sin ({\theta ^{i,u}_j})|1\rangle ,\quad \text {and} \\&\theta ^{i,u}_j = \left\{ \begin{array}{ll} 0, &{} {if}\; e^{i,u}_{2j+1}e^{i,u}_{2j}=00 \\ \frac{\pi }{5}, &{} {if}\; e^{i,u}_{2j+1}e^{i,u}_{2j}=01 \\ \frac{\pi }{2}-\frac{\pi }{5}, &{} {if}\; e^{i,u}_{2j+1}e^{i,u}_{2j}=10 \\ \frac{\pi }{2}, &{} {if}\; e^{i,u}_{2j+1}e^{i,u}_{2j}=11 \end{array} \right. ,\quad 1\le u \le 3,\quad 0\le j \le p-1, \end{aligned}$$
(10)

where \(e^{i,1}_k=r^{i}_k\), \(e^{i,2}_k=g^{i}_k\), and \(e^{i,3}_k=b^{i}_k\) for all \(0\le k \le 2p-1\). Figure 1 shows a \(2\times 2\) color image with a bit depth of 24, the angle values encoding the color information of the red, green and blue channels and the corresponding MCEQI state.

Fig. 1
figure 1

An example image with rotation angles and the corresponding MCEQI state

Full size image

3.1 Quantum image preparation

The quantum image preparation method begins with the initialization of quantum states, which are then transformed into the desired quantum states through unitary operations. To explain the proposed quantum image preparation method, we consider an RGB image I of size \(s=2^n \times 2^n\), where the red, green, and blue channels have a bit depth of 2p. The MCEQI state that encodes the image I is represented as \(I_q\) and requires \(3p+2n\) qubits.

The initialized quantum state of the proposed quantum image preparation method is as follows.

$$\begin{aligned} |\Phi _0\rangle =|0\rangle ^{\otimes (3p+2n)}. \end{aligned}$$
(11)

The following two steps transform the initialized state \(|\Phi _0\rangle \) into the MCEQI state \( I_q \).

3.2 Step 1

In this step, we apply the unitary transformation \(T_1\), given in (12), on the state \(|\Phi _0\rangle \) and obtain the quantum state \(|\Phi _1\rangle \) as given in (13),

$$\begin{aligned} \mathcal {T}_1= & {} I^{\otimes 3p} \otimes H^{\otimes 2n} \end{aligned}$$
(12)
$$\begin{aligned} \nonumber \mathcal {T}_1|\Phi \rangle _{0}= & {} I^{\otimes 3p} \otimes H^{\otimes 2n}|0\rangle ^{\otimes (3p+2n)} \\ \nonumber= & {} \frac{1}{2^n} I|0\rangle ^{\otimes {3p}} \otimes H|0\rangle ^{\otimes {2n}} \\ |\Phi _1\rangle= & {} \frac{1}{2^n} |0\rangle ^{\otimes 3p}\otimes \sum _{i=0}^{s-1}|i\rangle \end{aligned}$$
(13)

where H denotes the Hadamard transformation that has matrix representation as \(H=\frac{1}{\sqrt{2}}\left( \begin{array}{cc} 1 &{} 1 \\ 1 &{} -1 \\ \end{array} \right) \), and \(I=\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \\ \end{array} \right) \) is the identity matrix of order 2.

3.3 Step 2

In this step we encode the color information of all pixels of the image I into quantum state of 3p qubits by applying \(2^{2n}\) generalized controlled rotations on the intermediate state \(|\Phi _1\rangle \). In particular, the application of the generalized controlled rotation \( \mathcal {T}^{i} \), given in (14), on the state \(|\Phi \rangle _1\) embeds the color value of the pixel at ith position of the image I.

$$\begin{aligned} \mathcal {T}^{i}= & {} \bigg (I^{\otimes 3p}\otimes \sum _{v=0, v \ne i}^{s-1} |v\rangle \langle v|\bigg )+ T^{i} \otimes |i\rangle \langle i| \end{aligned}$$
(14)

where quantum operation \( T^{i} \) is defined as

$$\begin{aligned} T^{i}= & {} \otimes _{j=0}^{p-1}T^{i,1}_{j}\otimes _{j=0}^{p-1}T^{i,2}_{j}\otimes _{j=0}^{p-1}T^{i,3}_{j} . \end{aligned}$$
(15)

The color encoding operation \(T^i\) is a composition of 3p rotation operations \(R_{y}(2\theta ^{i,u}_j)=\left( \begin{array}{cc} \cos {\theta ^{i,u}_j} &{} -\sin {\theta ^{i,u}_j} \\ \sin {\theta ^{i,u}_j} &{} \cos {\theta ^{i,u}_j} \\ \end{array} \right) \), which can expressed as

$$\begin{aligned} T^{i,u}_j(|0\rangle )= & {} R_{y}(2\theta ^{i,u}_j)|0\rangle \end{aligned}$$
(16)

where \(1\le u\le 3\). Note that, if \(\theta ^{i,u}_j \ne 0 \), then \( T^{i,u}_j \) is a multi-controlled rotation \( C^{2n}(R_{y}(2\theta ^{i,u}_j)) \), which is component of \(\mathcal {T}^{i}\) (14). Otherwise, \(T^{i,u}_j\) is the identity operation and has no effect on the quantum state \(|\Phi \rangle _1\). Thus, the operation \( \mathcal {T}^{i} \) stores the color value of the ith pixel in the image I. The output of the state \(|\Phi \rangle _1\) after application of the color setting operation \(\mathcal {T}^{i}\) on it is expressed as follows.

$$\begin{aligned} \nonumber \mathcal {T}^{i}|\Phi _{1}\rangle= & {} \mathcal {T}^{i}\bigg (\frac{1}{2^n}\sum _{i=0}^{s-1} |0\rangle ^{\otimes 3p} |i\rangle \bigg ) \\ \nonumber= & {} \bigg (I^{\otimes 3p}\otimes \sum _{v=0, v \ne i}^{s-1} |v\rangle \langle v|+ T^{i} \otimes |i\rangle \langle i| \bigg ) \bigg (\frac{1}{2^n}\sum _{v=0}^{s-1}|0\rangle ^{\otimes 3p} |v\rangle \bigg ) \\ \nonumber= & {} \frac{1}{2^n} \bigg [ \bigg (I^{\otimes 3p} \otimes \sum _{v=0,v \ne i}^{s-1}|v\rangle \langle v|\bigg )\bigg (\sum _{v=0}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle \bigg ) \\ \nonumber{} & {} +\, \bigg (T^{i} \otimes |i\rangle \langle i|\bigg )\bigg (\sum _{v=0}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle \bigg )\bigg ] \\ \nonumber= & {} \frac{1}{2^n} \bigg [\sum _{v=0, v\ne i}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle + T^{i}|0\rangle ^{\otimes 3p} \otimes |i\rangle \bigg ] \\ \nonumber= & {} \frac{1}{2^n} \bigg [ \sum _{v=0, v\ne i}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle + (\otimes _{j=0}^{p-1}T^{i,1}_{j}\otimes _{j=0}^{p-1}T^{i,2}_{j}\otimes _{j=0}^{p-1}T^{i,3}_{j})(|0\rangle ^{\otimes 3p})\otimes |i\rangle \bigg ] \\= & {} \frac{1}{2^n} \bigg [ \sum _{v=0, v\ne i}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle + |c^{i,1}_{p-1}\cdots c^{i,1}_{0} c^{i,2}_{p-1}\cdots c^{i,2}_{0} c^{i,3}_{p-1}\cdots c^{i,3}_0\rangle \otimes |i\rangle \bigg ] \end{aligned}$$
(17)
$$\begin{aligned}= & {} \frac{1}{2^n} \bigg [ \sum _{v=0, v\ne i}^{s-1} |0\rangle ^{\otimes 3p} |v\rangle + |c(i)\rangle \otimes |i\rangle \bigg ] \end{aligned}$$
(18)

The quantum operation of step 2 is given as in (19).

$$\begin{aligned} \mathcal {T}_{2}= & {} \prod _{i=0}^{s-1}\mathcal {T}^{i} \end{aligned}$$
(19)

After applying \( \mathcal {T}_2 \) on the state \( |\Phi \rangle _{1} \), we obtain the MCEQI state \(I_q\) as follows

$$\begin{aligned} |I_q\rangle= & {} \frac{1}{2^n}\sum _{i=0}^{s-1}|c(i)\rangle \otimes |i\rangle \end{aligned}$$
(20)

where the quantum state \( |c(i)\rangle =|c^{i,1}_{p-1} c^{i,1}_{p-2}\cdots c^{i,1}_{0} c^{i,2}_{p-1} c^{i,2}_{p-2}\cdots c^{i,2}_{0} c^{i,3}_{p-1} c^{i,3}_{p-2}\cdots c^{i,3}_0\rangle \) encodes color information of the ith pixel, which has a classical binary representation as \(r^i_{2p-1} r^i_{2p-2}\cdots r^i_0 g^i_{2p-1} g^i_{2p-2}\cdots g^i_0 b^i_{2p-1} b^i_{2p-2}\cdots b^i_0\). This encoding is done using the rule defined in Eq. (10). The quantum circuit shown in Fig. 2 transforms the initialized state to the MCEQI state, which encodes both the positional and color information of a \(2\times 2\) color image. Each channel (red, green, and blue) of the image has a bit depth of 8. In the circuit, the states PI and CI, as illustrated in Fig. 2, represent the positional and color information of all pixels in the image, respectively. For brevity, the labels \(R_y\) have been omitted, and only half of the angle values for the rotation operations used in the circuit are provided in Fig. 2.

Fig. 2
figure 2

Quantum circuit that transforms the initialized state to MCEQI state for a colored digital image of size \(2\times 2\)

Full size image

3.4 Time complexity of the MCEQI model

The time complexity of the proposed image preparation method is defined as the number of basic unitary operations required to transform the initialized quantum state into MCEQI state. The time complexity of the first step of the proposed method is \(3p+2n\) while the unitary transform of the second step is the composition of \( 2^{2n} \) operations \( \mathcal {T}^{i} \) (15), where \(i=0,1,...,2^{2n}-1\). And, the time complexity of \(\mathcal {T}^{i}\) is O(3pn) [7]. Accordingly, the time complexity of the second step of the proposed quantum image representation method is \(O(3pn2^{2n})\). For a classical RGB image of size \(2^n\times 2^n\) and with bit a depth of the red, green and blue planes given by 2p, a comparison of the proposed model with the existing quantum colored image models is presented in Table 1. It is evident from Table 1 that the MCEQI model is comparable with the existing quantum color image representation models in respect of time and space complexity.

Table 1 Comparison of the time and space complexity of the proposed model with existing quantum colored image representation models
Full size table

4 Fractional-order double-ring erbium-doped fiber laser chaotic system

Luo et al. demonstrated that erbium-doped fiber-ring lasers are capable of exhibiting chaotic behavior [32]. This significant finding has paved the way for a new field of investigation, which seeks to explore the potential use of such chaotic systems for cryptographic purposes. The double-ring erbium-doped (DRED) fiber laser system, composed of two erbium-doped fiber rings (Ring u and v) coupled by a coupler \(C_0\), is shown in Fig. 3. The two ring resonators are also coupled to a separate standard fiber through wavelength-division-multiplexing (WDM) couplers \(C_u\) and \(C_v\). With the help of the coupler \(C_0\), the laser field in both rings is frequency-locked, resulting in a phase change of \(\pi /2\), which leads to the coupling of the laser in one ring into the other. The mathematical description of the dynamic equation governing the behavior of the DRED fiber laser system is as follows.

$$\begin{aligned} \left\{ \begin{array}{l} \dot{E}_u= -k_u(E_u +C_0 E_v) + g_u E_u D_u \\ \dot{E}_v= -k_v(E_v -C_0 E_u) + g_v E_v D_v \\ \dot{D}_u= -(1+I_{Pu}+E^{2}_u)D_u+I_{Pu}-1 \\ \dot{D}_v= -(1+I_{Pv}+E^{2}_v)D_v+ I_{Pv}-1 \end{array},\right. \end{aligned}$$
(21)

In the above description, \(E_u\) and \(E_v\) denote the normalized output laser intensity of Ring u and Ring v. The pump field strengths of Ring u and v are represented by \(I_{Pu}\) and \(I_{Pv}\), respectively, while \(D_u\) and \(D_v\) denote the number of normalized reversed particles. \(C_0\) represents the coupling coefficient of the directional coupler to \(1.55 \upmu \textrm{m}\) wavelength light waves, and \(k_u\) and \(k_v\) denote the loss coefficients of the two lasers respectively. Furthermore, the gain coefficients of Ring u and v are expressed using \(g_u\) and \(g_v\).

Fig. 3
figure 3

The erbium-doped fiber double-ring laser system

Full size image

In general, the variables in the dimensionless state equation exhibit a dynamic range that exceeds the linear range of the op-amp. As the linear range of the op-amp is limited to \(\pm 13.5\)v, the actual amplifier can not be realized. In order to achieve the desired amplification factor of the operational amplifier, it is necessary to regulate the output voltage of each stage within \( \pm 10\)v. To achieve this, the variables and control parameters are adjusted as follows: \(E_u\) is replaced with 10x, \(E_v\) with 10y, \(D_u\) with z/10, \(D_v\) with w/10, t with t/100, \(k_u\) with 100a, \(C_0\) with b, \(g_u\) with 1000c, \(k_v\) with 100d, \(g_v\) with 1000f, \(I_{Pu}\) with g, and \(I_{Pv}\) with k. By substituting these adjusted variables into the dynamic equation of the DRED fiber laser system (21), the state Eq. (22) is obtained.

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}= -a(x+by)+cxz \\ \dot{y}= -d(y-bx)+fyw \\ \dot{z}= -\frac{(1+g+100x^2)z}{100} +\frac{g-1}{10} \\ \dot{w}= -\frac{(1+k+100y^2)w}{100} +\frac{k-1}{10} \end{array},\right. \end{aligned}$$
(22)

Chaotic behavior can be observed in the double-ring erbium-doped fiber laser system (22) when the parameters a, b, c, d, f, g, and k satisfy the conditions \(a\in [8, 11.5]\), \(b\in [0.16, 0.5]\), \(c \in [5.5, 11]\), \(d \in [8, 10.3]\), \(f \in [4.5, 7.3]\), \(g\in [3, 5.3]\), and \(k \in [4.5, 7.2]\). A detailed discussion on the parameter values that give rise to chaotic behavior in system (22) is available in [32, 33].

The mathematical system of fractional differential equations that models the double-ring erbium-doped fiber laser chaotic dynamical system is recently studied by Li et al. [33]. The governing equations of this chaotic system are given as in Eq. (23),

$$\begin{aligned} \left\{ \begin{array}{l} \frac{d^{\lambda _1}x}{dt^{\lambda _1}}= -a(x+by)+cxz \\ \frac{d^{\lambda _2}y}{dt^{\lambda _2}}= -d(y-bx)+fyw \\ \frac{d^{\lambda _3}z}{dt^{\lambda _3}}= -\frac{(1+g+100x^2)z}{100} +\frac{g-1}{10} \\ \frac{d^{\lambda _4}w}{dt^{\lambda _4}}= -\frac{(1+k+100y^2)w}{100} +\frac{k-1}{10} \end{array},\right. \end{aligned}$$
(23)

where \(x,\,y,\,z,\,w\) are state variables, and \(a,\, b,\,c,\,d,\,f,\,g,\,k \) are parameters, and \(\lambda _i\, (i=1,2,3, 4)\) denotes the order of Caputo fractional derivative defined as in Eq. (24).

$$\begin{aligned} \frac{d^{\lambda }h(t)}{dt^{\lambda }}= & {} \frac{1}{\Gamma (n-\lambda )} \int _{0}^{t} {\frac{h^{n}(\tau )}{(t-\tau )^{\lambda -n+1}} d\tau }, \; \text {for}\; n-1<\lambda <n. \end{aligned}$$
(24)

Equation (24) defines the fractional derivative of order \(\lambda \) of the function h(t) in which \(\Gamma (s)=\int _{0}^{\infty } e^{-t} t^{s-1} dt\) is the Euler’s Gamma function. By employing the finite forward difference scheme, the fractional derivative of order \(\lambda \), for \(\lambda \in (0,1)\), of h(t) can be approximated as:

$$\begin{aligned} \frac{d^{\lambda }h(t_{u+1})}{dt^{\lambda }}= & {} \frac{1}{\Delta t^{\lambda } \Gamma (2-\lambda )} \left[ h(t_{u+1})+\sum _{j=0}^{u-1}(b_{j+1}-b_j)h(t_{u-j})-b_u h(t_0)\right] \qquad \end{aligned}$$
(25)

where \(\Delta t\) is the step size and \(t_u=t_0+u\Delta t\) and \(b_j=(j+1)^{1-\lambda }-j^{1-\lambda }\). Using (25) in (23) and simplifying the expressions we obtain

$$\begin{aligned} \nonumber x(t_{u+1})= & {} \frac{1}{1+aA_1}\Bigg [\sum _{j=0}^{u-1}(b_j^1-b_{j+1}^1)x(t_{u-j}) + b_u^1 x(t_0)-abA_1y(t_u)\\ \nonumber{} & {} +\,cA_1x(t_u)z(t_u)\Bigg ]\\ \nonumber y(t_{u+1})= & {} \frac{1}{1+dA_2}\Bigg [\sum _{j=0}^{u-1}(b_j^2-b_{j+1}^2)y(t_{u-j}) + b_u^2 y(t_0)-dbA_2x(t_u)\\ \nonumber{} & {} +\,fA_2y(t_u)w(t_u)\Bigg ]\\ \nonumber z(t_{u+1})= & {} \frac{1}{1+0.01(1+g)A_3}\Bigg [\sum _{j=0}^{u-1}(b_j^3-b_{j+1}^3)z(t_{u-j}) + b_u^3 z(t_0)\\ \nonumber{} & {} -\,A_3(x(t_u))^2z(t_u)+0.1(g-1)A_3\Bigg ]\\ \nonumber w(t_{u+1})= & {} \frac{1}{1+0.01(1+k)A_4}\Bigg [\sum _{j=0}^{u-1}(b_j^4-b_{j+1}^4)w(t_{u-j}) + b_u^4 w(t_0)\\{} & {} -\,A_4(y(t_u))^2w(t_u)+0.1(k-1)A_4\Bigg ] \end{aligned}$$
(26)

where \(A_i=\Delta t^{\lambda _i} \Gamma (2-\lambda _i),\, b_j^i=(j+1)^{1-\lambda _i}-j^{1-\lambda _i}, \text {for} \,(i=1,2,3,4)\) and parameter values involved in (23) are chosen to be \(a=8.2,\; b=0.2,\; c=10,\; d=10,\; f=5,\; g=5,\; k=5\), state trajectories corresponding to solution (x, y, z, w) of (23) exhibit chaotic behaviour. For \(\lambda _i=0.90 \;(i=1,2,3,4)\) and initial values \(x=0.01,\; y=0.01,\; z=0.01,\; w=0.01\), plot of the state trajectories in \(xy-\) plane, \(xz-\) plane, \(xw-\) plane, \(yz-\) plane, \(yw-\) plane, and \(zw-\) plane are shown in Fig. 4.

Fig. 4
figure 4

For \(x=0.01, y=0.01, z=0.01, w=0.01\), plot of state trajectories of fractional-order double-ring erbium-doped fiber chaotic system (23)

Full size image

5 Proposed steganography scheme based on the MCEQI model

In the proposed steganography scheme, we embed an RGB secret image of size \(2^n\times 2^n\) in a colored carrier image of size \(2^{n+1}\times 2^{n+1}\). The color range of each red, green and blue channels is considered to be [0, 255]. Figure 5 shows the flowchart of the proposed embedding and extracting algorithms.

Fig. 5
figure 5

Flowchart of proposed steganography embedding and extracting procedures

Full size image

5.1 Preparation procedure

The proposed steganography scheme starts with the iteration of the fractional-order double-ring erbium-doped fiber laser chaotic system (23) against the chosen initial conditions, which serve as the keys of the system, to generate an array of length \(4\times 2^{2n}\), called key \(K=[K_0 \; K_1 \; K_2\; K_3]^T\), where \(K_j\) (\(0\le j\le 3)\) is a sequence of length \(2^{2n}\) containing all distinct integer values in the range \([0,2^{2n}-1]\). Moreover, prior to the embedding stage, the user is also required to decompose the information of each color channel (red, green, and blue), denoted by \(P^1, P^2\), and \(P^3\), respectively, of the secret RGB image into four planes \(P_{j}^{u} (0\le j \le 3, 1\le u\le 3) \), each with a bit depth of 2. The corresponding arrays \(A_{j}^{u} (0\le j \le 3, 1\le u\le 3)\) of angle values that encode the 2-bit color information are also determined. If the ith pixel, according to the ordering of pixels introduced by Khan [7], of the secret information image has the binary representation as \(r^i_{7} r^i_{6}\cdots r^i_0 g^i_{7} g^i_{6}\cdots g^i_0 b^i_{7} b^i_{6}\cdots b^i_0\), then the color value at the ith position of the plane \(P_j^u\) is \(e^{i,u}_{2j+1}e^{i,u}_{2j}\) for all \(0\le j \le 3\), where \(e^{i,1}_{2j+1}e^{i,1}_{2j}=r^{i}_{2j+1}r^{i}_{2j}\), \(e^{i,2}_{2j+1}e^{i,2}_{2j}=g^{i}_{2j+1}g^{i}_{2j}\), \(e^{i,3}_{2j+1}e^{i,3}_{2j}=b^{i}_{2j+1}b^{i}_{2j}\). And, the corresponding angle value at the ith position of \(A^{u}_{j}\) is determined by the rule defined in Eq. (10). Figure 6 shows the four secret information planes \(P^{u}_{j}\) (\(0\le j \le 3\), \(u\in \{1,2,3\} )\) of the red, green and blue channels. It also presents the corresponding arrays \(A^{u}_{j}\) (\(0\le j \le 3\)) of rotation angles for a generic secret information image of size \(2\times 2\). In this image, each pixel (indexed as i) is represented by a binary sequence \(r^i_{7} r^i_{6}\cdots r^i_0 g^i_{7} g^i_{6}\cdots g^i_0 b^i_{7} b^i_{6}\cdots b^i_0\), where \(r^i_{7} r^i_{6}\cdots r^i_0\) denotes the binary representation of the red channel, \(g^i_{7} g^i_{6}\cdots g^i_0\) represents the green channel, and \(b^i_{7} b^i_{6}\cdots b^i_0\) represents the blue channel. These representations hold true for all i values in the set \(\{0, 1,2,3\}\). Note that the planes \(P^1_0,\; P^2_0\) and \(P^3_0\) contain each pair comprising of two least significant bits of the red, green and blue channels, respectively, of every pixel in the secret image. Likewise, the planes \(P^1_3,\; P^2_3\) and \(P^3_3\) contain each pair comprising of two most significant bits of the red, green and blue channels, respectively, of every pixel in the secret image. Furthermore, the user is also required to prepare the MCEQI state of the cover information image prior to the embedding stage.

Fig. 6
figure 6

Image planes of bit depth 2 and arrays of rotation angles for a generic \(2\times 2\) secret color image with the red, green, and blue channel ranges of [0, 255]

Full size image

5.2 Embedding scheme

In this section, we explain the proposed steganography scheme which can be divided into four steps that are given below.

Inputs: a secret RGB image S of size \(2^n\times 2^n\), a cover RGB image C of size \(2^{n+1}\times 2^{n+1}\), and the initial values (x(0), y(0), z(0), w(0)) together with \(\lambda _i\in [0.9,1]\), \(i=1,2,3,4\), representing the orders of fractional derivatives for the fractional-order double-ring erbium-doped chaotic system (23).

Output: the stego-image state \(|S_C\rangle \).

Step 1 Prepare the MCEQI state \(|C\rangle \), given as in Eq. (27), for the cover RGB image C by the method described in Sect. 3.

$$\begin{aligned} | C \rangle= & {} \frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n+2}-1}|c(i)\rangle \otimes |i\rangle \nonumber \\= & {} \frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}|c(2^{2n}\cdot 0+i)\rangle \otimes |{2^{2n}\cdot 0}+i\rangle \nonumber \\{} & {} +\,\frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}|c(2^{2n}\cdot 1+i)\rangle \otimes |{2^{2n}\cdot 1}+i\rangle \nonumber \\{} & {} +\,\frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}|c(2^{2n}\cdot 2+i)\rangle \otimes |{2^{2n}\cdot 2}+i\rangle \nonumber \\{} & {} +\,\frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}|c(2^{2n}\cdot 3+i)\rangle \otimes |{2^{2n}\cdot 3}+i\rangle \nonumber \\= & {} \frac{1}{2^{n+1}}\sum _{i=0}^{2^{2n}-1}\sum _{k=0}^{3}|c(2^{2n}\cdot k+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \end{aligned}$$
(27)

where \(|c(i)\rangle =|c^{i,1}_{3}c^{i,1}_{2}c^{i,1}_{1}c^{i,1}_{0}c^{i,2}_{3}c^{i,2}_{2}c^{i,2}_{1}c^{i,2}_{0}c^{i,3}_{3}c^{i,3}_{2}c^{i,3}_{1}c^{i,3}_{0}\rangle \), and \( |c^{i,u}_{j}\rangle = \cos (\xi ^{i,u}_{j}) |0\rangle + \sin (\xi ^{i,u}_{j}) |1\rangle \), defined as in Eq. (10), for all \(0\le i\le 2^{2n+2}-1\), \(0\le j\le 3\) and \(1\le u \le 3\).

Step 2 Iterate the fractional-order double-ring erbium-doped laser chaotic system (23) with the chosen initial values \(\left( x(0), y(0),z(0),w(0)\right) \), and the orders of fractional derivatives \((\lambda _1,\lambda _2,\lambda _3,\lambda _4)\). Obtain a vector \((\bar{x}(i), \bar{y}(i), \bar{z}(i), \bar{w}(i))\) of integers from the state vector (x(i), y(i), z(i), w(i)) as given in Eqs. (28)–(31), respectively and append each of its distinct component with the corresponding subkey of the key \(K=[\begin{array}{cccc} K_0&K_1&K_1&K_3 \end{array}]^T\) until length of K is \(4\times 2^{2n}\). For instance, if \(\bar{x}(i)\) is not listed in the subkey \(K_1\), then update \(K_1\) by appending \(\bar{x}(i)\) with \(K_1\). Otherwise, ignore \(\bar{x}(i)\), and repeat this process for \(\bar{x}(i+1)\), which corresponds to the next state vector \((x(i+1), y(i+1), z(i+1), w(i+1))\), until length of \(K_1\) is \(2^{2n}\).

$$\begin{aligned} \bar{x}(i)=\Big \lfloor \bmod \big (x(i)\times 10^{14},2^{2n} \big ) \Big \rfloor \end{aligned}$$
(28)
$$\begin{aligned} \bar{y}(i)=\Big \lfloor \bmod \big (y(i)\times 10^{14},2^{2n} \big ) \Big \rfloor \end{aligned}$$
(29)
$$\begin{aligned} \bar{z}(i)=\Big \lfloor \bmod \big (z(i)\times 10^{14},2^{2n} \big ) \Big \rfloor \end{aligned}$$
(30)
$$\begin{aligned} \bar{w}(i)=\Big \lfloor \bmod \big (w(i)\times 10^{14},2^{2n} \big ) \Big \rfloor \end{aligned}$$
(31)

In the above equations, \(\lfloor \; \rfloor \) denotes the floor function.

Step 3 Decompose the secret information of each red, green and blue channels of the image S into four planes \(P^u_j\) (\(0\le j \le 3\), \(1\le u \le 3\)) of bit depth given by 2. Then prepare the associated arrays \(A^u_j\) (\(0\le j \le 3\), \(1\le u \le 3\)) of rotation angles, which encode the 2-bit color information in \(P^u_j\) (\(1\le j \le 3\), \(1\le u \le 3\)).

Step 4 Define the sub-operation

$$\begin{aligned} \overline{R}^{v}_{j}= & {} \bigg (I^{\otimes 12}\otimes \sum _{i=0, i \ne j \cdot 2^{2n} + v}^{2^{2n+2}-1} |i\rangle \langle i|\bigg )+ L^{v}_j \otimes |j \cdot 2^{2n} + v\rangle \langle j \cdot 2^{2n} + v| \end{aligned}$$
(32)

where \( L^{v}_{j} =I^{\otimes 3}\otimes R_{y}(2\theta ^{K_j(v),1}_j) \otimes I^{\otimes 3}\otimes R_{y}(2\theta ^{K_j(v),2}_j) \otimes I^{\otimes 3}\otimes R_{y}(2\theta ^{K_j(v),3}_j) \), for \(0\le j \le 3\) and \(0\le v\le 2^{2n}-1\), and \( R_{y}(2\theta ^{K_j(v), u}_j) \) represents the rotation operation that embeds the secret information using the angle \(\theta ^{K_j(v),u}_j \) in the qubit encoding the color information of the red, green, and blue channels as \(|c^{j\cdot 2^{2n}+v, u}_0\rangle \) for \(u=1,2\), and 3 respectively. The application of \(L^{v}_{j}\) on \(|c(j \cdot 2^{2n} + v)\rangle =|c(t)\rangle \), \(t=j \cdot 2^{2n} + v\) outputs the state as given in Eq. (33).

$$\begin{aligned} \nonumber L^{v}_{j}|c(t)\rangle= & {} \bigg (I^{\otimes 3}\otimes R_{y}(2\theta ^{K_j(v),1}_j) \otimes I^{\otimes 3} \otimes R_{y}(2\theta ^{K_j(v),2}_j) \\ \nonumber{} & {} \otimes I^{\otimes 3}\otimes R_{y}(2\theta ^{K_j(v),3}_j)\bigg )|c^{t,1}_{3}c^{t,1}_{2}c^{t,1}_{1} c^{t,1}_{0}c^{t,2}_{3}c^{t,2}_{2}c^{t,2}_{1} c^{t,2}_{0}c^{t,3}_{3}c^{t,3}_{2}c^{t,3}_{1} c^{t,3}_{0}\rangle \\ \nonumber= & {} |c^{t,1}_{3}c^{t,1}_{2}c^{t,1}_{1} {\overline{c}}^{t,1}_{0}c^{t,2}_{3}c^{t,2}_{2}c^{t,2}_{1} {\overline{c}}^{t,2}_{0}c^{t,3}_{3}c^{t,3}_{2}c^{t,3}_{1} {\overline{c}}^{t,3}_{0}\rangle \\= & {} |\overline{c}(t)\rangle \end{aligned}$$
(33)

where \(|{\overline{c}}^{t,u}_{0}\rangle = \cos ({\xi ^{t,u}_{0}} + {\theta ^{K_j(v),u}_j}) |0\rangle + \sin ({\xi ^{t,u}_{0}} + {\theta ^{K_j(v),u}_j}) |1\rangle \), for all \(1\le u \le 3\), and \(|c^{t}_{j}\rangle =\cos ({\xi ^{t,u}_{j}}) |0\rangle + \sin ({\xi ^{t,u}_{j}}) |0\rangle \), for all \(t=j \cdot 2^{2n} + v,\, 1\le u, j \le 3\), \(0\le v\le 2^{2n}-1\). The action of the controlled rotation sub-operation \({\overline{R}}^{v}_{j}\) on the cover image state \(|C\rangle \) results in the embedding of secret information encoded as \(\theta ^{K_j(v),u}_j;(1\le u\le 3)\) in the least significant qubit of the quantum states that capture color information of the red, green, and blue channels, denoted as \(|c^{j\cdot 2^{2n}+v, u}_0\rangle \) for \(u=1,2\), and 3 respectively. The process can be described as follows.

$$\begin{aligned} \nonumber \overline{R}^{v}_j |C\rangle= & {} \overline{R}^{v}_j \bigg (\frac{1}{2^{n+1}} \sum _{i=0}^{2^{2n}-1}\sum _{k=0}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \bigg )\\ \nonumber= & {} \frac{1}{2^{n+1}}\overline{R}^{v}_j \bigg (\sum _{i=0,i\ne v}^{2^{2n} -1} \sum _{k=0,k\ne j}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber{} & {} +\,|c({2^{2n}\cdot j}+v)\rangle \otimes |{2^{2n}\cdot j}+v\rangle \bigg )\\ \nonumber= & {} \frac{1}{2^{n+1}} \bigg (\sum _{i=0,i\ne v}^{2^{2n} -1} \sum _{k=0,k\ne j}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber{} & {} +\,L^{v}_{j}|c({2^{2n}\cdot j}+v)\rangle \otimes |{2^{2n}\cdot j}+v\rangle \bigg )\\ \nonumber= & {} \frac{1}{2^{n+1}} \bigg (\sum _{i=0,i\ne v}^{2^{2n} -1} \sum _{k=0,k\ne j}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber{} & {} +\, |\overline{c}({2^{2n}\cdot j}+v)\rangle \otimes |{2^{2n}\cdot j}+v\rangle \bigg ) \end{aligned}$$

The action of \(\overline{R}^{w}_l\) on state (34) gives the state (34).

$$\begin{aligned} \nonumber \overline{R}^{w}_l \overline{R}^{v}_j |C\rangle= & {} \overline{R}^{w}_l \bigg (\overline{R}^{v}_j \bigg (\frac{1}{2^{n+1}} \sum _{i=0}^{2^{2n}-1}\sum _{k=0}^{3}|c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \bigg ) \bigg )\\ \nonumber= & {} \overline{R}^{w}_l \frac{1}{2^{n+1}} \bigg (\sum _{i=0,i\ne v}^{2^{2n} -1} \sum _{k=0,k\ne j}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber{} & {} +\, |\overline{c}({2^{2n}\cdot j}+v)\rangle \otimes |{2^{2n}\cdot j}+v\rangle \bigg )\\ \nonumber= & {} \frac{1}{2^{n+1}} \bigg (\sum _{i=0,i\ne v,w}^{2^{2n} -1} \sum _{k=0,k\ne j, l}^{3} |c({2^{2n}\cdot k}+i)\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber{} & {} +\, |\overline{c}({2^{2n}\cdot j}+v)\rangle \otimes |{2^{2n}\cdot j}+v\rangle \\{} & {} +\, |\overline{c}({2^{2n}\cdot l}+w)\rangle \otimes |{2^{2n}\cdot l}+w\rangle \bigg ) \end{aligned}$$
(34)

Apply the composite operation \(\overline{R}\) (35) on the cover image state \(|C\rangle \), to obtain the stego-image state \(|S_C\rangle \) (36).

$$\begin{aligned} \overline{R}= & {} \prod _{j=0}^{3}\prod _{v=0}^{2^{2n}-1} {\overline{R}}^{v}_{j} \end{aligned}$$
(35)
$$\begin{aligned} \nonumber \overline{R} (|C\rangle )= & {} \prod _{j=0}^{3}\prod _{v=0}^{2^{2n}-1} {\overline{R}}^{v}_{j} (|C\rangle ) \\ \nonumber= & {} \frac{1}{2^{n+1}} \sum _{i=0}^{2^{2n} -1}\sum _{k=0}^{3} L^i_k\big (|c({{2^{2n}\cdot k}+i})\rangle \big )\otimes |{2^{2n}\cdot k}+i\rangle \\ \nonumber= & {} \frac{1}{2^{n+1}} \sum _{i=0}^{2^{2n} -1}\sum _{k=0}^{3} |\overline{c}({{2^{2n}\cdot k}+i})\rangle \otimes |{2^{2n}\cdot k}+i\rangle \\= & {} |S_C\rangle \end{aligned}$$
(36)

Fig. 7 shows embedding circuit for a generic secret image, given in Fig. 6, against key \(K=[K_0 \; K_1 \; K_2 \; K_3]\), where \(K_0=[3\; 2\; 0\; 1]\), \(K_1=[3\; 2\; 1\; 0]\), \(K_2=[2\; 1\; 3 \; 0]\), \(K_3=[1\; 3\; 0\; 2]\).

Fig. 7
figure 7

Embedding circuit for a generic secret color image of size \(2\times 2\), given in Fig. 6, against key K

Full size image

5.3 Extracting procedure

The extracting procedure of the proposed steganography scheme involves the following four steps.

Inputs: The stego-image state \(|S_C\rangle \), the cover RGB image C, and the keys, x(0), y(0), z(0), w(0), \(\lambda _1\), \(\lambda _2\), \(\lambda _3\), \(\lambda _4\).

Output: The secret RGB image S.

Step 1 Iterate the system (23) with the preliminary conditions and generate the key \(K=[K_0 \; K_1 \; K_2 \; K_3]^T\) as described in the embedding procedure.

Step 2 Perform the projective measurements on identical copies of the stego-image state \(|S_C\rangle \) and obtain a probability distribution of all possible computational basis \(|p_1\rangle \otimes |p_2\rangle \otimes |p_3\rangle \otimes |i\rangle \) (\(0\le i \le 2^{2n+2}-1, 0\le p_1,\,p_2,\,p_3 \le 15\)). Then extract the angle values \(\xi ^{j\cdot 2^{2n}+v,u}_0 + \theta ^{K_j(v),u}_j \) (\(0\le v \le 2^{2n}-1,\, 0\le j \le 3,\, 1\le u\le 3\)) stored in the stego MCEQI state as \(|{\overline{c}}({j\cdot 2^{2n}+v})\rangle \).

Step 3 Obtain the four arrays \(A^u_j\) of rotation angles \(\theta ^{K_j(v),u}_j\) of each red, green and blue channels of the secret RGB image after subtracting \(\xi ^{j\cdot 2^{2n}+v,u}_0\), the rotation angles of the cover RGB image, from \(\xi ^{j\cdot 2^{2n}+v,u}_0 + \theta ^{K_j(v),u}_j \), for all \(0\le v \le 2^{2n}-1,\, 0\le j \le 3,\, 1\le u \le 3\).

Step 4 Construct the inverse key \(K'=[K^{\prime }_{0} \; K^{\prime }_{1} \; K^{\prime }_{2} \; K^{\prime }_{3}]^T\) with the property \(K_j\circ (K^{\prime }_{j})=K^{\prime }_{j} \circ (K_j)=Id\), where Id is the identity mapping on \(\{0,1,2,\ldots , 2^{2n}-1\}\), and descramble the position of the rotation angles in array \(A^{u}_j\) so that the angle at the vth position is \(\theta ^{K^{\prime }_{j}(K_j(v)),u}_j=\theta ^{Id(v),u}_j=\theta ^{v,u}_j\). Then decode the angle value \(\theta ^{v,u}_j\) (\(0\le v\le 2^{2n}-1,\, 0\le j \le 3,\, 1\le u \le 3\)) by the rule defined in Eq. (10) and obtain the secret information planes \(P^u_j \) (\(0\le j \le 3\)). Lastly, compose the secret information planes \(P^u_j\) (\(0\le j \le 3,\, 0\le j \le 3\)) and obtain the secret RGB image C.

6 Simulation and performance comparison

We have performed simulation experiments on a classical computer using the software MATLAB. The three sampled secret RGB images, each of size \(2^7 \times 2^7\), used in the experiments are shown in Fig. 8.

Fig. 8
figure 8

Secret information images

Full size image

6.1 Visual Quality

A robust steganography algorithm should generate a steganography image, which is indistinguishable, at least visually, from the corresponding cover image. The four cover images and their histograms are shown in Fig. 9m–n. In addition, Fig. 9o–p show the stego images obtained by embedding the secret Female image in the cover images, and their corresponding histograms. It is seen from Fig. 9 that the cover and corresponding stego images are indistinguishable.

Fig. 9
figure 9

m The four cover images with size \(2^8 \times 2^8\); n corresponding histograms of cover images; o the corresponding stego images that hide the secret Female image; p corresponding histograms of stego images

Full size image

Also, the peak-signal-to-noise ratio (PSNR) metric, formulated as in Eq. (37), is commonly used to quantify the visual effects of a steganography scheme.

$$\begin{aligned} PSNR= & {} 20 \log _{10}\Bigg (\frac{MAX_l}{\sqrt{\frac{1}{ m \times n\times 3}\sum _{i=1}^m\sum _{j=1}^n \sum _{k=l}^3[S(i,j,k)-C(i,j,k)]^2}} \Bigg ) \nonumber \\ \end{aligned}$$
(37)

In above equation S and C denote the stego-image and the corresponding cover image, respectively, each with size \(m\times n\), and \(MAX_l\) is the maximum possible intensity value of the pixels of the image. The value of PSNR increases when the similarity between the steganography image and its cover image increases. That is, the more the stego-image is similar to its cover image, the higher is the value of PSNR. And, as the two least significant bits of each red, green and blue values of every pixel of the cover image encode two bits of the secret information, therefore, the minimum theoretical value of PSNR for the proposed steganography algorithm is 38.58 dB, which is comparable with the PSNR value of 30 dB [21]. However, the values of PSNR for the sample images are listed in Table 2. It is evident from Table 2 that for the proposed steganography algorithm the average experiential value of PSNR is 44 dB, which is comparable with the PSNR values of 30 dB [21] and 35.51 dB [34].

6.1.1 Universal image quality

The Universal Image Quality (UIQ) metric is used to measure the degree of distortion in a stego-image S in comparison to its corresponding cover image C. The UIQ metric is defined as

$$\begin{aligned} UIQ(C,S)= \frac{4\mu _{C}\mu _{S}\sigma _{C,S}}{(\mu ^2_{C}+\mu ^2_{S})+(\sigma ^2_{C}+\sigma ^2_{S})} \end{aligned}$$
(38)

where \(\mu \) and \(\sigma \) are the mean and the variance, respectively. The scale of UIQ values ranges between -1 and 1. When the UIQ value approaches 1, the stego-image and cover image are more alike. Various test cover images and their stego-image counterparts have been evaluated using the UIQ test, and the UIQ values obtained from the analysis are presented in Table 3. The results from Table 3 indicate that the UIQ values are close to 1.

Table 2 PSNR values (in dB)
Full size table

6.1.2 Structural similarity index matrix

The degree of similarity between the cover image and stego-image is commonly assessed using the Structural Similarity Index Matrix (SSIM) metric. The SSIM metric is defined as

$$\begin{aligned} SSIM(C,S)= \frac{(2\mu _{C}\mu _{S}+c_1)(2\sigma _{C,S}+c_2)}{(\mu ^2_{C}+\mu ^2_{S}+c_1)+(\sigma ^2_{C}+\sigma ^2_{S}+c_2)} \end{aligned}$$
(39)

where \(c_1\) and \(c_2\) are constants, \(\mu \) and \(\sigma \) are mean and variance, respectively. To assess the effectiveness of the proposed steganography algorithm, the SSIM metric was applied to multiple cover images and their corresponding stego images, generating a set of SSIM values that are presented in Table 3. The obtained SSIM values closely approach the optimal value of 1, indicating a high level of similarity between the cover image and stego-image. These results suggest the robustness of the proposed steganography algorithm.

6.1.3 Normalized cross correlation

Normalized Cross Correlation (NCC) metric is widely used to assess the sameness of the RGB stego-image S and the RGB cover image C. The NCC is defined as

$$\begin{aligned} NCC(C,S) = \frac{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k)-\mu _C)(S(i,j,k)-\mu _S)}{\sqrt{A} \sqrt{B}} \end{aligned}$$
(40)

where \(A=\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k)-\mu _C)^2\), \(B=\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(S(i,j,k)-\mu _S)^2\), \(m\times n\) is the size of C and S, \(\mu _C\) and \(\mu _S\) are mean values of C and S, respectively. C(i, j, k) and S(i, j, k) are the color values of the k-th channel pixels at the position (i, j) in the cover image C and the stego-image S, respectively. The NCC values range from \(-1\) to 1, and values closer to 1 indicate better matching of the stego-image with the cover image. The NCC test has been performed on different cover images and their stego images, and the obtained NCC values are listed in Table 3. The obtained NCC values are close to 1, which indicates that the proposed steganography algorithm resists NCC-based attacks.

6.1.4 Normalized absolute error

The degree of similarity between an RGB cover image C and its stego-image S, with size \(m\times n\), can be quantified using the normalized absolute error (NAE) metric. If C and S are of size \(m\times n\), then the NAE between C and S is

$$\begin{aligned} NAE(C,S) = \frac{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k)-S(i,j,k))}{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}C(i,j,k)}. \end{aligned}$$
(41)

When the stego-image S resembles the cover image C, the NAE value between them tends to be close to zero. The NAE values between different test cover images and their stego images are given in Table 3.

6.1.5 Average Difference

Average difference (AD) between an RGB cover image C and its stego-image S, with size \(m\times n\), is defined as

$$\begin{aligned} AD(C,S) = \frac{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k)-S(i,j,k))}{m\times n\times 3}. \end{aligned}$$
(42)

Generally, AD values that are closer to zero suggest that the differences between the stego-image and the cover image are negligible. Table 3 displays the AD values computed for various cover images and their corresponding stego images, which are all close to zero.

6.1.6 Image fidelity

Image fidelity (IF) metric is commonly used to distinguish the stego-image S and the cover image C. If C and S are of size \(m\times n\), then the IF between C and S is defined as

$$\begin{aligned} IF(C,S) = 1-\frac{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k)-S(i,j,k))^2}{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k))^2}. \end{aligned}$$
(43)

Generally, an IF value that is closer to 1 indicates a high degree of similarity between the cover image C and its stego-image S. Table 3 reports the IF values computed for various cover images and their corresponding stego images, which are all greater than 0.99, suggesting a high level of similarity between them.

6.1.7 Maximum difference

Maximum difference (MD) metric quantifies the distortion of the stego-image S relative to the cover image C. The MD between C and S of size \(m\times n\) is

$$\begin{aligned} MD(C,S) = max \{|C(i,j,k)-S(i,j,k)|: 1\le i\le m,\, 1 \le j \le n, \, 1 \le k \le 3 \}. \nonumber \\ \end{aligned}$$
(44)

where max is the maximum value function and abs is the absolute value function. The MD metric measures the difference between the stego-image and cover image grayscale values, which range from 0 to 255. A small MD value indicates a good steganography algorithm. For the proposed steganography algorithm, the maximum value of MD between the stego-image and the cover image is 3. Table 3 lists the MD values computed for the test images used in our experiment.

6.1.8 Structural content

Structural content (SC) between the stego-image S and the cover image C is

$$\begin{aligned} SC(C,S) = \frac{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(S(i,j,k))^2}{\sum ^m_{i=1}\sum ^n_{j=1}\sum ^3_{k=1}(C(i,j,k))^2}. \end{aligned}$$
(45)

Generally, an SC value closer to 1 indicates that the difference between the stego-image and the cover image is negligible. Table 3 reports the SC values computed for the test images and their corresponding stego images, which are all found to be in close proximity to 1, indicating that the distortion of the stego-image relative to the cover image is insignificant.

Table 3 UIQ, SSIM, NCC, NAE, AD, IF, MD and SC values
Full size table

6.2 Embedding capacity

The embedding capacity of a steganography scheme is usually defined as the ratio between the number of secret information bits and the number pixels of a cover image. This definition does not provide the exact amount of resource consumption as bit depth of pixel values of a cover image differs for different image representation models. In this work, we define the embedding capacity of a steganography scheme as the ratio between the number of secret information bits and the number of bits in the cover image. The embedding capacity of the proposed scheme is given as follows:

$$\begin{aligned} \nonumber C= & {} \frac{\text {number of secret information bits}}{\text {number of cover image pixels}}\\ \nonumber= & {} \frac{3\times 8\times 2^{n}\times 2^{n}}{2^{n+1}\times 2^{n+1}}\\= & {} 6 \end{aligned}$$
(46)

or

$$\begin{aligned} \nonumber C= & {} \frac{\text {number of secret information bits}}{\text {number of cover information bits}}\\ \nonumber= & {} \frac{3\times 8\times 2^{n}\times 2^{n}}{3\times 8\times 2^{n+1}\times 2^{n+1}}\\= & {} \frac{1}{4}. \end{aligned}$$
(47)

The comparison in respect of embedding capacity, resource consumption and PSNR results of the proposed steganography scheme with some existing steganography schemes is provided in Table 4. It is evident from this table that the proposed steganography scheme has high embedding capacity and uses only 12 qubits to store the color information of all pixels of the stego-image.

Table 4 Comparison of embedding capacity, quantum resourse consumption, and PSNR results among various existing steganography schemes; b/p means bit per pixel and sb/cb means secret information bit per cover information bit
Full size table

6.3 Complexity

The first three steps of the proposed steganography scheme described in the Sect. 5 are performed on a classical computer prior to the last embedding step. In the proposed embedding stage, the secret information of a RGB image of size \(2^n\times 2^n\) is embedded in the MCEQI state of a cover RGB image of size \(2^{n+1}\times 2^{n+1}\) using the multi-controlled operation \(\overline{R}^{v}_j\) (32) which is a composition of three generalized controlled rotations \(C^{2n+2}(R_{y}(2\theta ^{K_j(v),u}_j)\), (\(1\le u\le 3\)), determined by the key K generated from the chaotic system. Since, the complexity of generalized controlled rotation \(C^{2n+2}(R_{y}(2\theta ^{K_j(v),u}_j)\) is 8n [7], therefore, the complexity of \(\overline{R}^{v}_j\) is 24n, for all \(0\le v\le 2^{2n}-1\) and \(0\le j \le 3\). Moreover, the whole secret information is embedded in the MCEQI state of the cover image by applying \(2^{2n+2}\) operations \(\overline{R}_{v}^j\) (\(0\le v\le 2^{2n}-1, 0\le j\le 3\), thus the time complexity of the whole embedding step is \(24n 2^{2n+2}\).

6.4 Security analysis

6.4.1 Key sensitivity analysis

To safeguard the security of a cryptographic algorithm, it is crucial that the algorithm exhibits a high degree of sensitivity to secret keys. This means that even a slight variation between two keys should result in entirely different restored images compared to their corresponding stego images. To test the key sensitivity of a proposed image steganography algorithm, we have performed an experiment on the stego-image of “Airplane”, which had concealed the “Female” image, and used slightly altered parameter values and initial conditions compared to the values, used in Sect. 4, during the retrieval process. The recovered secret images are shown in Fig. 10, providing clear evidence that the proposed steganography algorithm exhibits a significant degree of key sensitivity.

Fig. 10
figure 10

The recovered secret images using slightly modified keys

Full size image

6.5 Robustness to noise attack

As the availability of a quantum environment for practical use is still limited, the efficacy of the suggested steganography algorithm’s anti-noise attack capabilities has been evaluated in a classical environment. The results of the evaluation, which include the stego images and and their corresponding extracted secret images under 0.05 salt and pepper noise, are shown in Fig. 11. Based on Fig. 11, it can be inferred that the proposed algorithm can resist noise attacks as it allows retrieval of the vital information of the secret image even if the stego-image is disturbed by noise. The results indicate that the algorithm holds potential for maintaining the confidentiality of the secret image despite the presence of noise.

Fig. 11
figure 11

a–l are the stego images obtained after adding pepper noise with intensity of 0.05 to the stego images, “Goldhill”, “Airplane”, “Peppers”, and “Baboon”, that hide secret images “Female”, “House” and “Sailboat”; m–x are the extracted secret images

Full size image

7 Conclusion

We have proposed a novel multi-channel effective representation of quantum images (MCEQI) and developed a high-capacity steganography scheme based on this model. Our proposed scheme divided the secret image into four planes of bit depth given by 2, and then encoded the 2-bit information of each plane into an array of angle values using the MCEQI model. The key-controlled rotations with these angle values were applied to the MCEQI state of the cover image to obtain the state of the steganography image. The inverse process was used to extract the secret image.

In future work, there are several potential avenues for research. Firstly, we recommend exploring ways to increase the embedding capacity of the scheme while maintaining the same level of performance in terms of time complexity and visual effects. Secondly, the proposed scheme uses a fixed bit depth of 2 for the planes of each color channel. It would be interesting to investigate the impact of varying the bit depth on the embedding capacity and visual quality of the steganography image. Thirdly, the MCEQI model has potential applications beyond steganography, such as in quantum image processing, quantum cryptography, and quantum machine learning. Future work could explore these applications and investigate the performance of the MCEQI model in these areas.