A robust medical image steganography approach based on particle swarm optimization algorithm and quantum walks

Medical information plays an essential task in our everyday lives, in which medical data privacy and security constitute an important issue. The confidentiality of medical data can be achieved by applying one or more encryption and data hiding methods. Amidst the development of quantum computers, most medical data confidentiality techniques may be hacked because their construction is based on mathematical models. Most medical data have a long lifetime exceeding 25 years. Therefore, it is an important issue to design a new medical data hiding technique that has the capability to withstand the probable attacks from the side of quantum or digital devices. In this article, we aim to present a novel medical image steganography strategy based on quantum walks, chaotic systems, and particle swarm optimization algorithm. A 3-D chaotic system and quantum walks are utilized for operating particle swarm optimization algorithm, in which the generated velocity sequence is utilized for substituting the confidential data, and the position sequence is utilized for selecting which position in the hosting image will be employed to host the substituted confidential data. The payload capacity of the suggested mechanism is 2 bits per 1 byte, and the average value for PSNR is 44.1, which is big enough for the naked eye to not differentiate the difference between the carrier image and its stego one.


Introduction
Healthcare technologies play an essential task in our everyday lives, in which the developments in the healthcare sector aim to enhance the quality of patient care and enable patients to have greater control over their medical data [1,2]. One of the most critical issues facing healthcare developments is the confidentiality and privacy of medical data. The confidentiality of medical data can be achieved by applying one or more encryption and data hiding methods [3,4]. Encryption mechanisms aim to transform medical data from a comprehended format to an uncomprehended structure. Data hiding techniques can be categorized into two types: steganography and watermarking, in which steganography techniques aim to embed secret medical data into public data [5].
Medical images constitute the common type of medical data representation in which image encryption methods play an important role in securing medical images. However, the encrypted medical images attract attackers because the noise-like data indicates the existence of confidential data in the cipher text. To overcome this issue, image steganography is utilized to hide the data in a public cover image, inhibiting any indications of the presence of confidential communication. Therefore, various scholars aimed to design image steganographic approaches for securing medical information. For example, Loan et al. [6] presented a new data hiding technique for embedding the electronic patient records in the cover image using LSB and ISB substitution, in which secret data are encrypted utilizing the RC4 encryption mechanism. In [7], Ogundokun et al. suggested a new image steganography algorithm for embedding medical data using a modified LSB technique, which this algorithm aims to solve the acute authentication problem. Stoyanov and Stoyanov [8] proposed a medical image steganography algorithm based on a pseudorandom byte sequence generated using a nuclear spin function. In [9] Liao et al. presented a new medical image steganography strategy using the dependencies of inter-block coefficients.
There are three different factors to assess the performance of any cryptographic system, which depend on each other: visual quality, robustness, and payload capacity. Visual quality is the difference ratio between the carrier image and the stego image in which the visual quality is better when the difference is fully imperceptible to the human naked eye. Robustness is the capability of a system to preserve confidential data even if an eavesdropper perceives the existence of confidential information in the transmitted image. Payload capacity is the amount of secret data that can be embedded in the carrier image. A welldesigned image steganographic system should have good visual quality, a high payload capacity, and high security. However, when increasing the payload capacity, the stego image is distorted and the visual quality is going to be poor, which attracts eavesdroppers to the presence of confidential information in the carrier image. Therefore, payload capacity is inversely proportionate to visual quality. To solve this issue, the image steganography strategy should preserve a balance between both visual quality and payload capacity according to the application domain. Optimization strategies can be utilized to maintain the balance between payload capacity and visual quality. In [10], Ambika and Biradar presented a new medical image steganography algorithm, in which an optimization algorithm is utilized for selecting which pixels in the carrier image will host the secret medical data.
Swarm intelligence algorithms play a crucial role in designing modern image steganography approaches for providing high payload capacity and preserving the visual quality of the stego image. There are various algorithms for swarm intelligence, such as artificial bee colony [11], particle swarm optimization (PSO) [12], and ant colony optimization [13], in which the PSO algorithm has low computation time, easy-to-implement because it does not require a large number of control parameters, and has a high convergence rate. PSO concentrates essentially on a number of random particles and seeks to locate the optimal solution via recurrence. It has been utilized in various domains, including cancer classification [14], image steganography [15][16][17], image encryption [18], etc. In [15], Mohsin et al. presented an image steganography scheme using PSO algorithm, in which the role of PSO is to select the pixel position in the carrier image for hiding the confidential image, and Nipanikar et al. [16] presented an image steganography approach using sparse representation and PSO algorithm, in which the role of PSO is to select effective pixels for embedding the confidential audio signal in the carrier image. Also, Muhuri et al. [17] presented an image steganography approach based on PSO, in which the role of PSO is to locate the optimal substitution matrix for transforming the confidential object into its substituted structures.
However, one disadvantage of using PSO is that the particles become subject to early convergence, resulting in the swarm being trapped in an optimal local region and the inability to locate any new area within the local optimal solution. As a result, access to the global optimal solution becomes limited. Therefore, Jaradat et al. [19] employed logistic chaotic map with PSO to solve this problem and presented its application to image steganography, in which the role of PSO is to locate the effective pixel position in the carrier image to conceal the secret data.
Chaotic systems play an important role in designing modern security systems due to their high sensitivity to initial key parameters and their chaotic behavior. There are two categories of chaotic systems [20]: low-dimensional and high-dimensional systems. Low-dimensional chaotic systems are easy to design and have low computational complexity. However, these systems suffer from brute force attacks due to their narrow key space. High-dimensional chaotic maps are characterized by having a large key space and the ability to withstand brute force attacks. However, amidst the growth of quantum computers, most security strategies may be hacked due to their structure being based on mathematical paradigms [21].
Because of the phenomena of quantum entanglement and quantum superposition, quantum computers promised a new powerful computational paradigm unrivaled by today's most powerful digital computers. The efforts made towards the physical realization of applicable quantum devices have strengthened the belief that when quantum devices are realized, they will be able to solve various computing problems that are considered difficult to solve using available digital computers. Nonetheless, in mistaken hands, the enormous capabilities of quantum resources may be misapplied. In this manner, they form an unusual threat to modern medical information security systems. These threats differ from exploiting vulnerabilities in cyber security strategies to issues raised by the transition to the quantum computing era. Consequently, the design of modern cryptosystems needs to integrate quantum models to withstand the probable attacks from quantum devices in the near future. In this regard, risks associated with the confidential medical images would be significantly moderated or annihilated by designing quantum image steganographic algorithms. For example, El-Latif et al. [22] presented two new quantum data hiding strategies for embedding a quantum secret medical image into a quantum cover image. However, quantum steganography approaches are not effective until the effective realization of quantum computers. Notice that most medical data have a long lifetime exceeding 25 years. Therefore, it is an important issue to design a new medical data hiding technique before the entrance of the quantum era that has the ability to be performed on digital devices and withstand the probable attacks from the side of quantum or digital devices.
Among quantum models, quantum walk (QW) is a quantum computational model utilized in designing quantum algorithms [23]. QW is similar to chaos because of its high sensitivity to primary states and chaotic behavior. Also, QW has theoretically infinite initial conditions, nonperiodicity, and stability properties [24]. QW can be utilized as a quantum-inspired model for designing a new medical data hiding strategy that has the capability to resist the probable attacks from the side of quantum/digital devices. Therefore, Abd-El-Atty et al. [25] presented a new medical image steganography approach based on a quasicontrolled alternate QW, in which the quasi-QW is utilized for selecting which pixels in the carrier image will host the secret medical data. However, the presented approach in [25] embeds the secret medical data directly into the cover image without prior encryption for secret medical data.
From the issues mentioned above, we aim to present a novel medical image steganography approach using PSO, chaotic systems, and quantum walks. A 3-D chaotic system and quantum walks are utilized for operating the PSO algorithm, in which the generated velocity sequence is utilized for substituting the secret data, and the position sequence is utilized for selecting which position in the carrier image will be employed to host the substituted secret data. The simulation results of the presented algorithm prove that the proposed strategy possesses high security, high payload capacity, and good visual quality. For more illustration, we can summarize the contributions of the presented work into the following points: 1. Utilizing the benefits of quantum walks as a quantuminspired model for providing high security to medical data during and after the transition to the quantum era. 2. Utilizing the benefits of PSO algorithm to provide high payload capacity and preserve the visual quality of the stego image. 3. Simulation results of the presented algorithm prove that the proposed strategy possesses high security, high payload capacity, and good visual quality. 4. Paving the way for integrating quantum-inspired models with optimization algorithms for designing novel image steganography algorithms that have the capability to resist the probable attacks from quantum or classical platforms, provide high payload capacity, and preserve the visual quality of the stego image.
The layout of this study is organized as follows: the preliminary knowledge required for understanding the proposed strategy is given in Sect. 2. The proposed medical image steganography strategy is presented in Sect. 3, while the experimental outcomes are given in Sect. 4.

Preliminary work
In this section, we introduce the basic information for PSO algorithm, a 3-D chaotic system, and one-particle quantum walks, that are required for designing the proposed medical image steganography algorithm.

Quantum walks
QW is a quantum computational model utilized in designing quantum algorithms [23]. QW is similar to chaos because of its high sensitivity to primary states and chaotic behavior. Also, QW has theoretically infinite initial conditions, non-periodicity, and stability properties [24]. There are two essential elements of QW: walker space H p and coin H c ¼ cos a 0 j i þ sin a 1 j i, which are both lives in Hilbert space H ¼ H p H c . In each step t of operating a oneparticle QW on a circle of odd T-vertex controlled by a binary string S, the evolution operatorÊ 1 (Ê 0 ) is acted on the entire system when the t th -bit of S is 1 (0). If the t th -step overdoes the length of S, the evolution operatorÊ 2 is performed. The evolution operatorÊ 0 can be stated as in Eq. (1).
whereĤ is the shift operator of acting one-particle QW on a circle of odd T-vertex and can be stated as presented in Eq. (2).
and,Ĉ 0 is the coin operator and can be generally stated as given in Eq. (3) [26].
The evolution operatorsÊ 1 andÊ 2 can be constructed likê After t steps, the final quantum state jwi t can be defined as stated in Eq. (4).
where jwi 0 points to the primary state of the system and b 2 0; 1; 2 f g. The possibility of finding the walker at vertex d after t steps can be calculated as given in Eq. (5).

3-D chaotic system
Chaotic systems play a critical role in designing modern cryptosystems due to their simple structure, ease of implementation, and high sensitivity to their primary conditions and control parameters. Low-dimensional chaotic maps are easy to design and have low computational complexity. However, these maps suffer from brute force attacks due to their narrow key space. High-dimensional chaotic maps are characterized by having a large key space and the ability to withstand brute force attacks. Li et al. [27] proposed a novel 3-D chaotic system and introduced its application to image encryption. The proposed 3-D chaotic map is presented as given in Eq. (6).
where a is the control parameter, and p 0 , q 0 , r 0 are the initial conditions of the chaotic system. The dynamics of the chaotic system are studied in [27].

PSO
Evolutionary algorithms play a crucial task in designing modern image steganography approaches for providing high payload capacity and preserving the visual quality of the stego image. PSO is an optimization algorithm, which was designed to determine predictive controls by reducing the narrow multivariate standard [15]. Each particle in PSO embodies an individual solution. For operating PSO algorithm, each particle i has initial state, pbest (l i ), gbest (g i ), and main two elements: velocity (v i ) and position (x i ). The velocity and position for each particle can be updated using Eq. (7) [28], where j represents the dimension of the particle i, l denotes the inertia coefficient, c1 and c2 are the collective and subjective parameters, and r1 and r2 are random numbers in the interval (0,1). However, one disadvantage of using PSO is that the particles become subject to early convergence, resulting in the swarm being trapped in an ideal local region and the inability to locate any new area within the local optimal solution. As a result, access to the global optimal solution becomes limited [19]. Therefore, in this study, we customized the PSO algorithm (as given in Eq. (8)) [18] to accommodate the designing of a robust image steganographic system that has high security against potential attacks from quantum/digital devices and high payload capacity while preserving the visual quality.
where sequence W is generated from the probability distribution of acting QW, while sequences P, Q, and R are generated from iterating the 3-D chaotic system (6).

Proposed image steganography mechanism
In this section, we introduce a novel medical image steganography approach using the customized PSO algorithm, a 3-D chaotic system, and quantum walk. The role of the presented approach is to embed a confidential medical image of size h 2 Â w 2 into a cover public image of size h Â w. The 3-D chaotic system (6) and quantum walks are utilized for operating the customized PSO algorithm (8), in which the generated velocity sequence is utilized for substituting the secret data, and the position sequence is utilized for selecting which position in the carrier image will be employed to host the substituted confidential data. In what follows, the embedding and extracting procedures are described in detail.

Embedding procedure
The outline of the proposed embedding strategy is provided in Fig. 1, while the detailed steps of the embedding processes are illustrated in the next steps: Step 1: Select the primary values of key parameters (S, T, t, a, h 0 , h 1 , h 2 , p 0 , q 0 , r 0 , a, v 0 , x 0 , l, c1, c2) that required for acting QW, 3-D chaotic system (6), and the customized PSO algorithm (8). Where the key parameters for acting QW are selected as: S is a bit string of any length, T is an odd number and represents the number of the vertices in the circle, t is an integer and represents the number of steps of acting QW, and a; h 0 ; h 1 ; h 2 2 ½0; p=2 are utilized for constructing the coin particle H c ¼ cos a 0 j i þ sin a 1 j i and the evolution operatorsÊ 0 ,Ê 1 , and Ê 2 , respectively. According to Ref. [27], the key parameters for iterating 3-D chaotic system are selected as: p 0 2 ½À1:8; 1:8, q 0 2 ½À1; 0:8, r 0 2 ½À2; 2, and a 2 ½À0:0105; 0. Also according to Ref. [18], the key parameters for operating the customized PSO algorithm are selected as: l ¼ 0:1, c1 ¼ 0:5, c2 ¼ 0:5 , v 0 ¼ 0:5, and x 0 ¼ 1 (or set the optimized settings for PSO as illustrated in Ref. [17]).
Step 2: Using the selected key parameters (S, T, t, a, h 0 , h 1 , h 2 ), operate one-particle QW on a circle of odd T-vertex for t steps and controlled by the binary string S, for producing a probability vector Y.
Step 3: Resize the generated probability vector Y to the size of the cover image RIm (h Â w Â c).
Step 4: Using the selected key parameters (p 0 , q 0 , r 0 , a), iterate the chaotic system (6) for hwc times and generating three sequences P, Q, and R.
Step 6: Transform V(1: hwc 4 ) sequence into integers, and reshape the outcome to the dimensional of the confidential medical image SIm( h 2 Â w 2 Â c), then substitute the confidential image using bitwise-xor operation. ð10Þ Step 7: Expand the substituted image EIm of 8-bit and size h 2 Â w 2 to image XIm of 2-bit and size h Â w.
Step 8: Convert the expanded substituted image XIm and the host image RIm into vectors XVec and RVec, respectively. Step 9: Sort the elements of X sequence in ascending order to get the vector Z, then obtain the index per element of X in Z as a vector G.

Extracting procedure
The steps of the extracting process are the inverse of the embedding steps, in which the extracting process is outlined in Fig. 2, while the detailed steps are illustrated in the next steps: Step 1: Set the correct values of key parameters (S, T, t, a, h 0 , h 1 , h 2 , p 0 , q 0 , r 0 , a, v 0 , x 0 , l, c1, c2) that required for acting QW, 3-D chaotic system, and the customized PSO algorithm.
Step 2: Using key parameters (S, T, t, a, h 0 , h 1 , h 2 ), operate QW on a circle for producing a probability vector Y.
Step 3: Resize the generated probability vector Y to the size of the stego image Stgo (h Â w Â c).
Step 4: Using key parameters (p 0 , q 0 , r 0 , a), iterate the chaotic system (6) for hwc times and generating three sequences P, Q, and R.
Step 6: Sort the elements of X sequence in ascending order to get the vector Z, then obtain the index per element of X in Z as a vector G.

Step 7: Convert the stego image Stgo into a vector
StgoVec, and use the generated vector G for locating the pixel positions that host the encrypted secret data XVec.
Step 10: Transform V(1: hwc 4 ) sequence into integers, and reshape the outcome to the dimensional of the extracted encrypted image EIm( h 2 Â w 2 Â c), then perform bitwise-xor operation to get the confidential medical image SIm. ð16Þ

Experimental outcomes
To validate the proposed medical image steganography approach, we simulate the presented approach using software MATLAB R2016b installed on a PC with IntelÒ Core TM 2 CPU 3.00 GHz and 4-GB RAM. The dataset utilized to Fig. 2 Extracting process of the proposed image steganography strategy evaluate the presented approach is consist of two groups: the first group is utilized as cover images of dimension 512 Â 512, taken from SIPI dataset [29], and labeled as Peppers, Baboon, and Sailboat (see Fig. 3), while the second group is utilized as confidential images of dimension 256 Â 256, taken from MedPix dataset [30], and labeled as MedIm01, MedIm02, MedIm03, and MedIm04 (see Fig. 4). The key parameters utilized for acting one-particle QW on a circle are set as: S=[ 1011 1010 1011 0101 0101 1110 0101 1100 0011 1010 1000 1101 0101], T=255, t=261, a ¼ p=2, h 0 ¼ p=6, h 1 ¼ p=3, h 2 ¼ p=4, and the key parameters utilized for iterating the 3-D chaotic system (6) are set as: p 0 = 0.3579, q 0 = 0.7286, r 0 = 0.5947, a ¼ 0, while the parameters utilized for operating the customized PSO algorithm (8) are given as: v 0 = 0.5, x 0 = 1, l = 0.1, c1 = 0.5, c2 = 0.5. Several analyses are performed to validate the effectiveness of the presented steganography approach, such as image quality, security analysis, data loss attacks, and capacity analysis.

Image quality
The visual quality of the proposed approach is given in Fig. 5, in which the naked eye cannot differentiate the difference amongst the carrier image and its stego image. To measure the visual quality in quantity values, various tests are performed, such as PSNR (''Peak-Signal-to-Noise Ratio''), SSIM (''Structural Similarity Index Metric''), and UIQ (''Universal Image Quality'').

PSNR
PSNR can be represented mathematically as given in Eq. (19).
where MAX RIm refers the maximum pixel value of the cover image Rim, and Stgo refers to its equivalent stego image, where both have a size of h Â w. Notice that the naked eye cannot distinguish the difference amongst the carrier image and its stego image if the PSNR value is greater than 38. The outcomes of PSNR test are given in Table 1 for experimented images, in which all outcomes are greater than 38.

UIQ and SSIM
UIQ and SSIM are mathematical tools to detect the difference amongst cover (RIm) and stego (Stgo) images, which can be stated as follows:

Security analysis
Security acts as a vital task in any image data hiding approach. The security of the presented data hiding algorithm relies on two main factors. The first factor is based on utilizing quantum walks as a quantum technology to gain the designed algorithm the ability to resist the potential attacks from quantum devices. The other factor is based on the key space of the utilized key parameters (S, T, t, a, h 0 , h 1 , h 2 , p 0 , q 0 , r 0 , a) for acting QW and iterating the 3-D (a) (b) (c) Fig. 3 Dataset of cover images taken from SIPI dataset [29] Neural Computing and Applications (2023) 35:773-785 779 chaotic system (6), in addition to the parameters for operating the customized PSO algorithm (8). The key space only for parameter S is infinite, which may have an infinite bit string. But the key space must be finite. Consequently, if the precision calculation of digital devices is 10 À16 , the total key space for the system is 10 176 which is big enough for any secure image data hiding algorithm. To test the sensitivity of the key parameters, we attempt to extract the (a) (b) (c) (d) Fig. 4 Dataset of confidential medical images taken from MedPix dataset [30] Fig . 5 The visual quality of some of the experimented images secret image MedIm02 (Fig. 4b) from the stego image that is related to the Baboon cover image with various keys with slight modifications. The outcomes are stated visually in Fig. 6. To guarantee the high key sensitivity of the presented approach, Table 4 states the ratio of the number of pixels changed (NPCR) [31] between the secret image MedIm02 (Fig. 4b) and the stated extracted images in Fig. 6, in which the results demonstrate that the extracted images shown in Fig. 6 are totally different from the secret image MedIm02 and that the presented strategy has high key sensitivity.

Embedding capacity
The effectiveness of any image data hiding approach is based on three factors: visual quality, security, and payload capacity. The higher the payload capacity, the lower the image's visual quality, and vice versa. Therefore, it is necessary for a well-designed data hiding algorithm to possess a high payload capacity while maintaining visual quality. The payload capacity of the presented algorithm is 2-bit per 8-bit (2-bit/byte), which is satisfactory for hiding confidential large data.

Data loss analysis
It is understood that most communication channels are noisy. When data are transferred across noisy channels, these are effortlessly vandalized via occlusion attacks. Accordingly, it is crucial for a fine-designed image steganography algorithm to have the ability for withstanding occlusion attacks. To estimate the suggested image steganography approach for contra occlusion attacks, we cut data blocks of diverse rates from the stego images and then attempted to extract the secret data from defected stego images. Figure 7 shows the effects of occlusion attacks, in which the extracted secret images have acceptable visual quality and there is no loss of the visual information within the portion of the cutting block.

Discussion
The performance of the presented medical image steganographic system was evaluated from four diverse aspects: visual quality; security; occlusion attacks; and payload capacity. The visual quality of stego images is shown in Fig. 5, in which the naked eye can't differentiate the difference between the stego image and its pristine image. Also, the visual quality was measured in numerical values using PSNR, SSIM, and UIQ, in which the outcomes of PSNR were greater than 38, indicating the naked eye cannot distinguish the difference between the pristine image and its stego one. Furthermore, the histograms for differentiate stego images that have different medical confidential images embedded in the same cover image are very similar to each other (see Fig. 8). The outcomes of UIQ and SSIM for the experimented images are actually near to 1, in which the values of two extremely equivalent images are close to 1. From the aspect of embedding capacity, the payload capacity of the presented algorithm is 2-bit per 8-bit (2-bit/byte), which is satisfactory for hiding confidential large data. From the stated numerical and visual results, we can deduce that the proposed steganographic system has good payload capacity while maintaining visual quality. From the aspect of security, the security of the presented strategy relies on two main factors. The first factor is based on utilizing quantum walks as a quantum computation model to gain the designed steganographic system the ability to resist the possible attacks from quantum devices. The other factor is based on the key space of the utilized key parameters, which is 10 176 . That is big enough for any secure image data hiding algorithm. The presented medical image steganographic  system has a high sensitivity to slight modifications in primary keys, in which the results of slight modifications in the initial key when trying to extract the secret image from the stego image are stated visually in Fig. 6 and numerically in Table 4. Therefore, we can deduce that the presented system has high security and high key sensitivity. From the aspect of occlusion attacks, Fig. 7 shows the effects of occlusion attacks in which the extracted secret images have acceptable visual quality and there is no loss of the visual information within the portion of the cutting block. Therefore, we can deduce that the presented system has the ability to resist data loss attacks.
To demonstrate the efficacy of the presented data hiding approach alongside other related data hiding approaches, Table 5 provides a simple comparison among the presented steganographic algorithm and other related steganographic  [17,28] presented image steganography approaches that have good embedding capacity while preserving visual quality. However, amidst the growth of quantum computers, most security strategies may be hacked. But those algorithms [17,28] do not have the capability to withstand attacks from quantum resources. In this regard, risks associated with image steganographic systems would be significantly moderated or annihilated by designing quantum image steganographic algorithms. For example, Li et al. [32] presented a quantum steganographic algorithm based on Grey code in which its embedding capacity is 3-bit/24-bit and the quantum secret image can be extracted without any key parameter. In [33], Zhou et al. presented a new quantum steganographic scheme based on the Arnold cat map and SWAP gates, in which its payload capacity is 0.5bit per pixel, in addition extraction process requires the pristine cover image and three keys (the size of each key is the same size as the confidential image); and El-Latif et al. [22] presented a new quantum medical image steganography scheme using the logistic map, in which its payload capacity is 2-bit per pixel and the extraction process requires a key matrix whose size is equal to the size of the secret message. It is not logical to transmit a secret key whose size is greater than the secret image rather than transmit the secret message. Therefore, the authors in [25,34] presented new image steganographic systems based on QW but the secret images are embedded directly without pre-encryption. From the stated discussion above and the values provided in Table 5, we can notice the effectiveness of the presented image steganography algorithm alongside other related algorithms, and the presented encryption approach has good payload embedding capacity while preserving visual quality, high security with high key sensitivity, and the ability to resist occlusion attacks besides being applied in digital resources.

Conclusion
In this paper, we have presented a novel medical image steganography technique based on PSO, a 3-D chaotic system, and one-particle QW. The generated sequences from the 3-D chaotic system and QW are utilized for operating PSO, in which the generated velocity sequence is utilized for substituting the secret medical image, and the position sequence is utilized for selecting which position in the carrier image will be employed to host the substituted secret data. The introduced mechanism has an embedding capacity of 2 bits per 1 byte and an average PSNR of 44.1, which is large enough for the naked eye to not distinguish between the cover image and its stego image. The image steganography method presented here can be used on both greyscale and colour images. The main aim of this work is to pave the way for integrating quantum-inspired models with optimization algorithms for designing novel image steganography algorithms that have the capability to withstand the probable attacks from quantum or classical platforms, provide high payload capacity, and preserve the visual quality of the stego image. In the future, we aim to integrate PSO and a quantum model to design a new image steganography algorithm for embedding the description and treatment of patients in their medical images. Data availability The data used in the study can be accessed upon your request to the author.  3-bit/24-bit 49.0844 --The payload capacity is 3-bit/24-bit and the quantum secret image can be extracted without any key parameters [33] 0.5-bit/8-bit 50.2267 --The payload capacity is 0.5-bit per pixel, in addition extraction process requires the pristine cover image and three keys (the size of each key is the same size as the confidential image) [22] 2-bit/byte 46.3353 --The extraction process requires a key matrix whose size is equal to the size of the secret message [34] 2-bit/8-bit 43.5284 0.9695 0.9698 The secret image is embedded directly without pre-encryption [25] 2-bit/8-bit 44.2314 0.9487 0.9499 The secret image is embedded directly without pre-encryption

Declarations
Conflict of interest The author declares that he has no conflict of interest.
Ethics approval This research contains neither human nor animal studies.
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