In non-linear dynamical systems, which are both acyclic and non-convergent and have a highly sensitive dependency on initial values, chaotic phenomena are deterministic, stochastic-like processes. The three-dimensional Lorenz system serves as the foundation for the building of the four-dimensional Lorenz system, which uses some of its characteristics or variables to introduce the fourth dimension while maintaining the system’s ability to satisfy chaotic dynamics [36]. Its definition is shown in Eq. 1: x ̇ = α − x + y + w y ̇ = γ x − y − x z z ̇ = x y − β z w ̇ = − y z + r w . (1)

The prerequisites of possessing at least one four-dimensional phase space and at least two positive Lyapunov exponents must be met by hyperchaotic systems [35–37]. According to Wang’s method [38], the system will exhibit hyperchaotic behavior when the initial parameters are set as follows: α = 10, β = 8/3, and γ = 28; −1.52 ≤ r ≤ −0.06 [the control parameters in Eq. 1]. Additionally, the initial values of x, y, z, and w can be freely chosen. The Lorenz hyperchaotic system can be discretized using the Python software application by setting and using the Runge–Kutta method, as shown in Figure 1. r = −1 is taken as the control parameter at this point. The system’s Lyapunov index comprises λ ₁ = 0.3381, λ ₂ = 0.1586, λ ₃ = 0, and λ ₄ = −15.1752, which demonstrates that hyper-chaos has taken place.

Quantum parallelism and quantum entanglement, two fundamental concepts in quantum mechanics, can be used in quantum image processing. These concepts have benefits for image storage, storage space optimization, processing task acceleration, computing resource optimization, and information security transmission. In this paper, we adopted Jiang Nan’s generalized quantum image representation technique, also known as the generalized quantum image representation (GQIR) [38, 39], which builds on the NEQR quantum image representation [40] by storing the image through two sets of entangled quantum sequences to increase the size of the original image from 2 ⁿ × 2 ⁿ to any size H × W. From an image of size H × W and grayscale range [0, 2^(q−1)], the quantum state representation of this image can be formulated as in Eq. 2: | Ψ 〉 = 1 2 h + w ∑ Y = 0 h − 1 ∑ X = 0 w − 1 | G Y X i 〉 ⊗ | Y X 〉 , (2) where |YX⟩ represents coordinate information and G Y X i represents gray information. |YX⟩ and G Y X i are shown in Eq. 3: | Y X 〉 = | Y 〉 | X 〉 = | y 0 y 1 … y h − 1 〉 | x 0 x 1 … x w − 1 〉 , y i , x i ∈ 0,1 | G Y X i 〉 = | G Y X 0 G Y X 1 … G Y X q − 1 〉 , G Y X i ∈ 0,1 , (3) where i = 0, 1, …, q − 1. The values of h and ω can be formulated as in Eq. 4: h = log 2 H , H > 1 1 , H = 1 , w = log 2 W , W > 1 1 , W = 1 (4)

Compared with classical methods, quantum representation can significantly reduce the image storage space. For the size of 2 ⁿ × 2 ⁿ clear image, the classic image representation method needs 8 × 2 ⁿ × 2 ⁿ + n ² number of bits, compared with the number of quantum bits required for quantum image representation, 2n + q bits (q is the image color depth). Figure 2 shows a simple image presented by GQIR, in which the gray value is 2 × 2. In addition, Figure 3 represents the quantum circuit of the image. | I 〉 = 1 2 | 10011001 〉 ⊗ | 00 〉 + | 01100110 〉 ⊗ | 01 〉 + | 00110011 〉 ⊗ | 10 〉 + | 11001100 〉 ⊗ | 11 〉 .

Combined with quantum mechanical measurement theory [41], the corresponding information on image in the quantum state |Ψ⟩ can be obtained using the measurement operator L ̂ to measure the quantum state, so the measurement operator position information is given by L ̂ . L ̂ is shown in Eq. 5: L ̂ = ∑ Y = 0 H − 1 ∑ X = 0 W − 1 | Y X 〉 〈 Y X | I ⊗ q , (5) where I ^⊗q is the tensor product of the q identity matrix used for gray information on each pixel.

The gray information measurement operator C ̂ is shown in Eq. 6: C ̂ = ∑ c ′ = 0 2 q − 1 C | c ′ 〉 〈 c ′ | , (6) where c′ represents eigenvalues of C. After the operator is applied to the image, image information can be accurately observed.

The pixel coordinates are changed by the transformation matrix, and such a transformation belongs to affine transformation. The affine transformation can be expressed by Eq. 7: x ′ y ′ = a b c d x y mod N . (7)

This paper will consider the Arnold transformation as an example position transformation of image pixels [42]. For a two-dimensional Arnold transformation, suppose there is a square grid image, which has a size of N × N, represented by I (x, y), (x,y) ^T is used to represent the position coordinates of pixels. The values of x and y are integer values (x, y = 0, 1, …, N), mapping to new point addition and multiplication (mod N) via the operations in Eq. 8: x ′ y ′ = A ̂ x y mod N , A ̂ = 1 1 1 2 , (8) where x ′ , y ′ T is the coordinate of the image transformed by the Arnold transform.

In addition, Arnold transformation is a reversible transformation, which means that pixel position coordinates after transformation can be restored without error, and its inverse transformation meets the requirement of A ̂ A ̂ − 1 = A ̂ − 1 A ̂ = I . Inverse transformation can be expressed by Eq. 9: x y = A ̂ − 1 x ′ y ′ mod N , A ̂ − 1 = 2 − 1 − 1 1 . (9)

A quantum adder [43] is needed in the process of Arnold transformation of image position coordinates, as shown in Figure 4. A function that the quantum adder can achieve is |a, b⟩ → |a, a + b⟩.

It is worth noting that after passing the adder, the image is no longer rectangular and will exceed the scope. Therefore, the quantum modular N adder should also be designed based on the adder network, in which the specific implementation method is given [43].

Here, the generalized Arnold transform operator A ̂ K 1 is used to modify the encrypted image Ψ ′ . This operation enhances the security of the encryption process. The resulting image Ψ ′ ′ encapsulates the concealed grayscale information, making it highly secure and suitable for safe transmission or storage.

The decryption process, which essentially is the reverse of the encryption process, and the specific decryption steps are as follows:

This step effectively reverses the grayscale encryption process, allowing the retrieval of the original image Ψ R before encryption.

Figure 5 represents the entire flow of encryption. In the schematic representation, K ₁ is the positional encryption operator used for Arnold’s disarrangement. K ₂ and K ₃ are image pixel-value encryption operators, which are used to obtain new pixel-value encryption results by performing modulo operations on gray values and then XOR operations on the original image pixels. Finally, the encrypted image is assembled at the pixel locations to obtain a complete encrypted image. All the encryption operators are derived from four-dimensional chaotic equations.

In the experiment, we selected the following key to simulate the encryption algorithm and set the initial parameter, w t 0 α = 10, β = 8/3, γ = 28, r = −1, K 1 = x t 0 = 1.1 , K 2 = y t 0 = 2.2 , and K 3 = z t 0 = 3.3 . Figure 6 displays the encryption and decryption outcomes. The illustration demonstrates how well the encryption technique can both encode and decode the original picture.

Information entropy reflects a measure of the richness of image information. In general, the greater the image information entropy, the richer the amount of information is and the higher the quality. From the point of view of image encryption, information entropy is considered from the statistical characteristics of the whole source and represents the overall characteristics of the source in an average sense. When the information entropy of an image approaches the ideal value, it shows that the more uniform the spatial distribution of the gray image is, the more notable the encryption effect is.

For the image with a gray level of 256, the information entropy of the ciphertext image is closer to 8 bits, indicating that it has less visual information [44]. To process encrypted grayscale images, the data are first read into memory. Then, the frequency of occurrence for each grayscale level is collected by traversing each pixel. Using these frequencies, the probability of each grayscale level pixel is calculated by dividing the frequency by the total number of pixels. Finally, the information entropy is calculated using Eq. 14: H m = − ∑ i = 0 255 P m i log 2 P m i , (14) where m _i is the i-th gray level for the digital image I with 256 gray levels, and P (m _i ) is the emergence probability of m _i . Table 1 shows the comparison of the information entropy of the original image and encrypted image. The data show that the information entropy of the three images can reach above 7.999 bits after encryption, indicating that the system has better encryption and can effectively resist the statistical attack.

The image’s histogram clearly illustrates how the pixel values are distributed across the composition [47]. Figure 7 shows the comparison of the histogram distribution of an image before and after encryption. Figures 7A, B, C show the uneven distribution on the original image. After encryption, Figure 1D, E, F show that the histogram distribution of the ciphertext image is basically uniform, and by analyzing the statistical characteristics of the image or no useful statistical information can be obtained by performing any statistical analysis on it. This shows that the encryption system can withstand the histogram analysis.

The redundancy quality of the image establishes a significant link between nearby image pixels. Moreover, the correlation of adjacent pixels is often used to reflect the correlation degree of pixel values of adjacent positions of an image, including horizontal, vertical, and diagonal directions. For a good encryption algorithm, the adjacent pixel correlation of the ciphertext will approach zero. Therefore, correlation can be used as an evaluation criterion to judge the image encryption effect. The calculation method of an adjacent relation of image pixels is shown in Eq. 15: ρ xy = ∑ n , m = 1 N x n − X ̄ y m − Y ̄ ∑ n , m = 1 N x n − X ̄ 2 y m − Y ̄ 2 . (15)

In the information entropy of the original image and the encrypted image, X ̄ and Y ̄ are the average values of two adjacent pixels, N is the total number of pairs of adjacent pixels, and x _n and y _m are the values of the two adjacent pixels, respectively.

In total, 10,000 pairs of adjacent pixels are randomly selected to test the correlation in terms of distribution of adjacent pixels in horizontal, vertical, and diagonal directions. In three orientations, the correlation coefficients between plaintext and ciphertext pixels are examined. The experimental results of the correlation of adjacent pixels are shown in Table 2, and the analysis is shown in Figure 8.

By observing the data presented in Table 2, we can see that the pixel correlation of the plaintext image is very close to 1, indicating that it has a strong correlation. After the encryption algorithm, the pixel distribution of the ciphertext image is uniform, and the correlation is weak. It shows that the quantum image Arnold transformation is combined with the chaos system to control the parameters, and the encrypted image obtained after XOR operation on the related pixel method can effectively reduce the correlation between image pixels. The observed data presented in Table 2 not only underscore the enhanced security aspects of the encryption algorithm but also lend itself to a robustness analysis. The robustness of an encryption algorithm is of paramount importance to ensure that the encrypted data remain secure and intact under various conditions and potential threats. Here, we delve into a more granular assessment of the algorithm’s robustness:

It can also be seen from Table 2 that images before encryption are vulnerable to various types of attacks. However, after applying the encryption algorithm, the pixel distribution of the ciphertext image becomes more uniform, and the correlation is significantly weakened. This increased robustness against pixel-level correlation attacks is a key aspect of algorithm security.

Moreover, the combination of the quantum image Arnold transform with the chaos system-controlled parameters, followed by the XOR operation on related pixels, proves to be a robust approach to reduce the correlation between image pixels. This algorithm is designed to withstand common attacks such as differential cryptanalysis and brute-force decryption attempts. The utilization of chaos-based control parameters adds an extra layer of complexity to the encryption process, making it resistant to attacks that rely on predictable patterns.

In addition to addressing the pixel correlation, it is important to note that the algorithm also exhibits resistance to other potential vulnerabilities. For instance, it has been tested against known attacks, including differential attacks and frequency analysis, and has shown a high degree of robustness. The algorithm’s robustness is further bolstered by its ability to maintain the security of the encrypted image even when subjected to potential quantum computing-based attacks.

The plaintext sensitivity of image encryption methods is frequently evaluated using a differential attack [50]. Key sensitivity in the context of ideal multimedia encryption means that a change of one bit in the key should result in a completely different encryption result. The beginning state of the chaotic mapping and the sensitivity of the control parameters are related to the key sensitivity of chaotic cryptography in general. Sensitivity was assessed using the number pixel change rate (NPCR) and uniform average change intensity (UACI). NPCR and UACI are acronyms for the “number of pixels changed” and “average intensity of changes,” respectively, between two encrypted images.

When a pixel in a plaintext image changes, the encryption result should ideally approach the standard value in order to resist a differential attack. NPCR = 99.6094% and UACI = 33.4635% are their corresponding standard values, which can be determined using Eq. 16: N P C R : N C 1 , C 2 = ∑ i , j D i , j W × H × 100 % U A C I : U C 1 , C 2 = 1 W × H ∑ i , j C 1 i , j − C 2 i , j 255 × 100 % . (16)

The width and height of the two images are, respectively, expressed as W and H; D (i, j) is defined by Eq. 17: D i , j = 0 , C 1 i , j = C 2 i , j 1 , C 1 i , j ≠ C 2 i , j , (17) where the pixel values of the two ciphertexts at point (i, j) are represented by C ₁ (i, j) and C ₂ (i, j).

The value of K′ can be obtained by changing a bit of K, and then, two ciphertext images can be obtained using the same plaintext with the key. Table 3 represents the comparison for the NPCR and UACI values of the ciphertext before and after the change. The experimental data show that the NPCR and UACI values of our scheme are close to the ideal value.

A secure encryption method must provide a sufficiently large key space to ensure that the attacker cannot find the safe secret key in a timely manner. To survive powerful attacks, the key space should be greater than 2¹²⁸. The effective precision of chaotic system parameters in this paper can be obtained in 10⁻¹⁶. The key space for the picture algorithm can be reached in 10¹⁴⁰ ≫ 2¹²⁸. We can, therefore, conclude that the encryption system’s key space is sufficiently large to resist destructive attacks.

We also performed sensitivity analyses on related keys at the same time. Figure 9 illustrates the great sensitivity of this technique by showing that even a very minor key deviation prevents the right image from being decrypted.