Introduction

Although it sounds futuristic, cloud computing is already part of our daily lives today. Internet networks are used as communication mediums to transfer data and add new applications [15, 18]. Amidst raised access and minimized cost, cloud computing is an attractive computing platform in most areas of everyday life. Nevertheless, secured transfer of data among internet of things devices and cloud computing platforms is still a critical problem [7, 13, 30]. As an outcome, many cloud users have their data violated or lost on cloud platforms. Hence, the necessity to design more refined techniques to protect data accessible via cloud computing is more prevalent than ever [4]. The majority of data is represented in the form of images. The security of confidential images can be achieved by utilising one of the cryptographic mechanisms including encryption and data hiding, in which image encryption concerns transforming the image from an evident pattern to an ambiguous image [5].

Chaotic maps have received a lot of attention in various scientific disciplines, particularly in developing modernistic cryptographic applications [10, 11, 20, 32, 34, 35, 38]. A chaotic map is a transformation map of a deterministic dynamic system that converts the state of the system \({S}_{0}\) into a new state \({S}_{1}\) based on the primary state \({S}_{0}\), the control parameter P and the time t [29]. Chaotic systems present the aspired characteristics of unpredictability, ergodicity, and sensitivity to their primary value(s) and control parameter(s) that meet the necessities of encryption confusion-diffusion features [24, 27]. Actually, an improper primary condition of a chaotic map can transform the dynamical system properties to non-chaotic, which indicates the conversion of the nonlinearity levels and vulnerability traps [28].

Motivation

Meanwhile, despite the significant role of chaotic maps in designing modern cryptographic mechanisms, the inevitable transition to the quantum computing era forecasts tremendous issues to the confidence and privacy of common data security mechanisms [12, 17, 23, 31]. This is credited to the staggering processing capabilities of quantum computers, which implies obstinate conventionally calculations can be achieved inside moments. This inevitably indicates the loss of privacy and confidence in a considerable many of modern information security applications including cloud computing systems. Accordingly, it is a crucial task for achieving confidentiality of data stored on cloud storage before the availability of quantum computers. Therefore, this study aims to utilise one of the quantum computational models, as a quantum-inspired system, for developing a new data confidentiality technique that can be applied in digital devices to have the capability for resisting the potential attacks from quantum and digital computers.

Quantum walks (QW) is one of the universal quantum computational models, which has fascinating nonlinear features amidst unusual sensitivity to primary conditions that can be employed to achieve chaotic dynamical behaviour [3, 41]. Accordingly, it can serve as an astounding source to provide key streams for cryptographic purposes.

Contributions

Concretely, this study investigates the integration of QW as a quantum-inspired model and Henon map as a chaotic map into designing a new image cryptosystem for cloud applications. The major contributions of this study are summarized in the following lines:

  1. 1.

    We proposed a novel image cryptosystem for cloud applications based on Gray code, QW, and Henon map,

  2. 2.

    The generated key streams from QW and Henon map are related to the plain image, therefore, our cryptosystem has a high sensitivity of any slight bit changes on the plain image,

  3. 3.

    Experimental outcomes prove that the presented image encryption approach is efficient with high security, and

  4. 4.

    The proposed cryptosystem supports the idea of integrating quantum-inspired models into predominating cryptographic mechanisms to achieve data confidentiality against unauthorised access in the transition stage to the quantum age.

Organization

The organization of this paper is coordinated as follows: the related works are discussed in the next section. The subsequent introduces the preliminary foundations on Gray code, QW, and Henon map followed by which the proposed framework for data confidentiality against unauthorised access in the transition stage to the quantum era for cloud applications is dealt. Next, the proposed image encryption approach is presented. The penultimate section provides the experimental results for the proposed cryptosystem. Ultimately, the conclusions and the future scope are outlined in the last section.

Related work

Cloud computing plays a vital task in our daily lives, in which a large amount of data is stored daily on cloud storage. The highest priority for cloud data storage is guaranteeing the security of confidential data. Javed et al. [25] proposed a novel technique for vehicle intrusion attack detection on the controller area network bus, which is based on a hybrid of the convolutional neural network model and attention-based gated recurrent unit model to reveal intrusion attacks on a controller area network bus, and Wang et al. [43] proposed a novel lightweight certificateless signature technique based on blockchain strategy. The security of confidential data can be realised through utilising one of the cryptographic mechanisms including encryption and data hiding. In [42], Vengala et al. presented a new framework for authentication, data compression, and safe data transfer in the cloud environment. However, the framework can be cracked if the eavesdropper locates the data location. In [36], Sajay et al. proposed a hybrid mechanism to enrich cloud data security using homographic and blowfish encryption. Also, in [37] Seth et al. presented a new framework for securing cloud data storage based on dual encryption and data fragmentation strategies that aim to secure the distribution of information in a multi-cloud environment, and in [40], Subramanian and Tamilselvan proposed a new cryptosystem that utilises the elliptic-curve with Diffie–Hellman mechanism to enhance the security of secret data in the cloud storage. However, the proposed algorithm is not suitable for large data.

However, the progress in quantum technology is expected to usher in the quantum age when sufficient quantum computing power is available to handle today’s best conventional cryptographic mechanisms [26]. Numerous small-scale and laboratory quantum algorithms have already been validated by simulations. There is currently a massive solicitude to the quantum field, which is supported by significant funds from governments, industry, etc. Admittedly, there is an agreement that this race for quantum power leads to the inescapable quantum age will arrive. To prevent future breaches of today’s encrypted data, it is necessary to protect it using secure quantum algorithms. This is particularly important for defence and military sectors where some data related to national security are reported to have lifespans of around 25 years [2].

To solve these important problems, it is important to introduce and outline new security methodologies for the cloud computing environment [4]. Similar efforts birthed the integration of quantum technologies into traditional cloud computing. This trend is certain to continue and grow. For example, Zhou et al. [46] presented an idea for quantum cryptography to be used for the access control dilemma in cloud computing. This idea is limited to managing cloud computing access policies using QKD protocols. Also, Singh et al. [39] added an idea of quantum computation as a service for cloud computing, consequently, everyone would be able to access the power of quantum computing without actually having it. This idea introduces quantum computing as a separate service for cloud computing and does not merge both to achieve high benefits. Similarly, Olanrewaju et al. [33] added an idea of quantum cryptography as a service for cloud computing, to guarantee a protected cloud environment between the sender and the receiver. In [18], El-Latif et al. utilised the power of QW in designing secure quantum hash function and presented its applicability to data security in fifth-generation networks environments.

However, the integration of cloud computing and quantum technologies is hard to realize today. Therefore, the solution is to utilize the capability of quantum technologies before realising the quantum era. Consequently, Abd-El-Atty et al. [4] utilised the power of QW as a quantum-inspired model for designing a novel image steganography approach for securing data transfer on cloud environments. Based on quantum-inspired QW, El-Latif et al. [17] designed a quantum-inspired blockchain model for withstanding attacks in a smart utility environment, and in [16] El-Latif et al. designed a cascaded system using QW with chaotic maps and presented its applications in substitution boxes, pseudo-random number generators, image encryption.

Preliminary knowledge

In this part, we introduce the preliminary knowledge including Gray code, quantum walks, and Henon map, which are needed for designing our proposed image cryptosystem.

Gray code

Gray code is a description of two consecutive numbers that have to vary in one bit only. This characteristic has demonstrated benefits in various applications. The description of Gray code can be represented as provided in Eq. (1).

$$\begin{aligned} G(t)=t\oplus \left( t\gg 1\right) \end{aligned}$$
(1)

here \(\gg \) signifies the binary right shift. For more explanation, a simplified example for the representation of Gray code is presented in Table 1.

Table 1 A simplified example for the representation of Gray code
Full size table

Quantum walks

QW is one of the universal quantum computational models, which has fascinating nonlinear features amidst unusual sensitivity to primary conditions that can be employed to achieve chaotic dynamical behaviour [3]. Alternating one-particle QW on a cycle has two basic components: walker space \(H_{s}\) and the coin particle \(H_{p} =\sin \alpha {\left| 1 \right\rangle } +\cos \alpha {\left| 0 \right\rangle }\), which together live in Hilbert space \(H=H_{s} \otimes H_{p}\). In each step t of acting QW on cycle of N-node, an evolution operator \({\hat{R}}_{0} \) (or \({\hat{R}}_{1}\)) is executed when the \(i^{th}\)-bit of the binary string (m) is 0 (or 1). The evolution operators \({\hat{R}}_{0}\) and \({\hat{R}}_{1}\) can be presented as given in Eq. (2).

$$\begin{aligned} \begin{array}{l} {{\hat{R}}_{0} ={\hat{E}}_{y} ({\hat{I}}\otimes {\hat{C}}_{0} ){\hat{E}}_{x} ({\hat{I}}\otimes {\hat{C}}_{0} )} \\ {{\hat{R}}_{1} ={\hat{E}}_{y} ({\hat{I}}\otimes {\hat{C}}_{1} ){\hat{E}}_{x} ({\hat{I}}\otimes {\hat{C}}_{0} )} \end{array} \end{aligned}$$
(2)

here \({\hat{E}}_{x}\) signifies the shift operator of alternating QWs on a cycle of N-node acting on x dimension, which can be defined as follows:

$$\begin{aligned} \begin{array}{*{20}{l}} {{{{{\hat{E}}}}_x}}&{}{ = \sum \limits _{x,y}^N \left( {|\left( {x + 1} \right) \bmod N,y,0\rangle \langle x,y,0|} \right) }\\ {}&{} \quad { + \sum \limits _{x,y}^N \left( {|\left( {x - 1} \right) \bmod N,y,1\rangle \langle x,y,1|} \right) }. \end{array} \end{aligned}$$
(3)

Similar \({\hat{E}}_{x} \), \({\hat{E}}_{y} \) signifies the shift operator of alternating QWs on y dimension, whereas the two operators \({\hat{C}}_{0}\) and \({\hat{C}}_{1}\) are coin operators and can be expressed as given in Eq. (4).

$$\begin{aligned} \begin{array}{l} {{\hat{C}}_{0} =\left( \begin{array}{cc} {\cos \; \beta _{0} } &{} {\sin \; \beta _{0} } \\ {\sin \; \beta _{0} } &{} {-\cos \; \beta _{0} } \end{array}\right) } \\ {{\hat{C}}_{1} =\left( \begin{array}{cc} {\cos \; \beta _{1} } &{} {-\cos \; \beta _{1} } \\ {\sin \; \beta _{1} } &{} {\sin \; \beta _{1} }. \end{array}\right) } \end{array} \end{aligned}$$
(4)

After t steps, the eventual state of \(\mathrm{|}\psi \rangle _{\textrm{initial}}\) can be stated as in Eq. (5).

$$\begin{aligned} \mathrm{|}\psi \rangle _{t} =\left( {\hat{R}}\right) ^{t} \mathrm{|}\psi \rangle _{\textrm{initial}}. \end{aligned}$$
(5)

Ultimately, the probability of locating the particle of QW at location (x, y) after t steps can be measured utilizing Eq. (6).

$$\begin{aligned} P(x,y,t)=\sum _{w\in \left\{ 0,1\right\} } \left| \langle x\textrm{,}y,w\mathrm{|}\left( {\hat{R}}\right) ^{t} \mathrm{|}\psi \rangle _{0} \right| ^{2}. \end{aligned}$$
(6)

Henon map

Henon map is one of the two-dimensional chaotic systems, which can be illustrated mathematically as presented in Eq. (7).

$$\begin{aligned} \left\{ \begin{array}{l} {x_{t+1} =y_{i} -ax_{t}^{2} +1} \\ {y_{t+1} =bx_{t} } \end{array}\right. \end{aligned}$$
(7)

here a, b signify the control parameters, and \({x}_{0}\), \({y}_{0}\) represent the primary conditions. For more details about the Henon map refer to Ref. [21].

Framework for data confidentiality on cloud-based purposes

Cloud computing provides profoundly scalable computing resources that are accessible as an online service. Amidst the fast growth of cloud computing technologies, more users are adopting cloud platforms to process and store their data. Cloud computing offers huge profits such as: savings on remote storages, processing, data sharing, saving hardware and software costs, and more. Nonetheless, there are many confidence provocations in the cloud computing context that conventional computing environments have not yet addressed [46]. In addition, security and privacy concern seriously disrupt the practical applications of cloud technologies.

In cloud systems, the intended cloud users exchanged the secret key parameters in a closed environment or via using any of the suitable asymmetric cryptographic algorithms like RSA, elliptic-curve, or Diffie–Hellman. Then, the sender encrypts the secret data using a suitable encryption algorithm and send the ciphertext to the intended receiver via cloud storage. The intended receiver receives the cipher data and starts to decrypt it using the decryption algorithm and the correct secret key parameters. In the quantum era, if any eavesdropper succeeded to hack the cloud storage and getting the ciphertext, he attempts to decrypt it. If the utilised cryptosystem is based on a mathematical model, the eavesdropper will have the ability for hacking the ciphertext to get the original plaintext. While if the utilised cryptosystem is based on a quantum model, the eavesdropper will not have the ability for hacking the ciphertext.

Therefore, we need viable quantum cryptographic techniques on digital devices for securing data stored in cloud storages to achieve data confidentiality and privacy through and after the transition stage to the quantum age. The suggested framework for data security in cloud-based applications is provided in Fig. 1. The presented framework can be utilised to transfer secret data for various applications such as Healthcare [9, 14], industrial applications [6], etc.

Fig. 1
figure 1

Framework for viable quantum cryptographic model on digital devices for securing data stored in cloud storages to achieve data confidentiality and privacy through and after the transformation stage to the quantum age

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Fig. 2
figure 2

Outline of the proposed encryption approach

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figure a

Proposed encryption approach

QW is one of the universal quantum computational models, which has fascinating nonlinear features amidst unusual sensitivity to primary conditions that can be employed to achieve chaotic dynamical behaviour. Accordingly, it can serve as an astounding source to provide key streams for cryptographic purposes. Concretely, this section investigates the integration of QW as a quantum-inspired model and Henon map as a chaotic map into designing a new image cryptosystem for cloud applications.

In the proposed cryptosystem, Gray code performed to the plain image, then some information about the outcome is utilised to operate QW for generating a probability distribution utilized in substituting the outcome of performing Gray code. After that, some information about the substituted image is utilised to iterate Henon map, then the first sequence utilised to permutate the substituted image, while the second sequence generated via Henon map with the resulting probability distribution of QW are utilised to substitute the permutated image. The outline of the suggested encryption scheme is presented in Fig. 2, and Algorithm 1 gives the encryption procedure , while the following steps give the detailed process flow for the proposed methodology.

  1. Step 1:

    Determine the initial values for key parameters (m, N, \(\alpha \), \(\beta _{0}\), \(\beta _{1}\), \(x_{0}\), \(y_{0}\), a , and b), where m, N, \(\alpha \), \(\beta _{0} \), and \(\beta _{1}\) are parameters utilised for acting alternate QW on a cycle of N-node controlled by the binary string m, \(\alpha \) is utilised for constructing the coin particle, \(\beta _{0}\) and \(\beta _{1}\) are utilised for constructing the coin operators \({\hat{C}}_{0} \) and \({\hat{C}}_{1} \), while \(x_{0} \), \(y_{0}\), a, b are used for iterating Henon map.

  2. Step 2:

    Carry out Gray code process illustrated in Eq. (1) to per pixel in original image (Og) image to get image GC.

  3. Step 3:

    Get some information about image Gc for updating the binary message m that utilized to operate QW.

    $$\begin{aligned} SPix = \sum _{i=1}^{h}\sum _{j=1}^{w}\sum _{k=1}^{d}Gc(i,j,k)\\m = append(m,\; de2bi(SPix)) \end{aligned}$$
  4. Step 4:

    Operate QW to get a probability matrix P of dimension \(N \times N\).

    $$\begin{aligned} P = QW(m,\; N,\; \alpha ,\; \beta _{0},\; \beta _{1}) \end{aligned}$$
  5. Step 5:

    Resize the probability distribution to the size of the original image \(h \times w \times d\) and reconstruct the output to a 3D matrix of into integer values as a key sequence.

    $$\begin{aligned} Ps= & {} resize(P,\;[h\;\;w\times d])\\ RP= & {} reshape(Ps,\;h,\;w,\;d)\\ {Key}_{1}= & {} fix(RP \times {10}^{12})\;\;\;\; \mod \;\; 256 \end{aligned}$$
  6. Step 6:

    Perform bitxor operation on the constructed key sequence \({Key}_{1}\) and image GC.

    $$\begin{aligned} SgQ = Gc\oplus {Key}_{1} \end{aligned}$$
  7. Step 7:

    Get some information about the substituted image SgQ and utilise this information to update initial conditions (\(x_{0}\), \(y_{0}\)).

    $$\begin{aligned} Delta= & {} \frac{\left( \sum _{i=1}^{h}\sum _{j=1}^{w}\sum _{k=1}^{d}SgQ(i,j,k)\right) \mod 256}{512}\\{} & {} \quad x_{0}= 0.5x_{0} + Delta\\{} & {} \quad y_{0}= 0.5y_{0} + Delta \end{aligned}$$
  8. Step 8:

    Iterate Henon map for \(h \times w \times d\) times using the updated initial conditions to produce two sequences X and Y.

    $$\begin{aligned}{}[X\;\;\; Y]= & {} HenonMap(x_{0},\;\;y_{0},\;\;a,\;\;b,\\{} & {} \;\;h \times w \times d) \end{aligned}$$
  9. Step 9:

    Arrange the elements of X in ascending order, then get the index of per element of the output in X.

    $$\begin{aligned} Vec= & {} sort(X)\\ PerV= & {} index(Vec,\;\;X) \end{aligned}$$
  10. Step 10:

    Reshape the elements of the substituted image SgQ to one vector, and permutate the output as given below.

    $$\begin{aligned} SgQV= & {} reshape (SgQ,\;\;h\\{} & {} \times w \times d,\;\;1) \\RgV(i)= & {} SgQV(PerV(i))\;\;For\;\; i\leftarrow 1,\; 2,\;{\dots }, \; \\ {}{} & {} \quad \times h \times w\times d \end{aligned}$$
  11. Step 11:

    Reshape the elements of the permutated image RgV to a 3D matrix.

    $$\begin{aligned} Rg = reshape(RgV,\;\;h,\;\;w,\;\;d) \end{aligned}$$
  12. Step 12:

    Reshape the elements of Y sequence to a 3D matrix, then add the elements of RP to the elements Y, and transform the output into integer values as given below.

    $$\begin{aligned} Y= & {} reshape(Y,\;\;h,\;\;w,\;\;d)\\ {Key}_{2}= & {} fix([RP+Y]\times {10}^{14})\mod \;\;256 \end{aligned}$$
  13. Step 13:

    Perform bitxor operation on the constructed key sequence \({Key}_{2}\) and the permutated image Rg to get the final cipher image.

    $$\begin{aligned} Cg = Rg\oplus {Key}_{2} \end{aligned}$$

Simulation results

To estimate the effectiveness of the suggested encryption approach, we performed the simulation on a laptop of CPU 2.5GHz, Intel \(\mathrm {{Core}^{TM}}\) i5-245M, RAM of 6GB and equipped by MATLAB R2016b. The studied dataset of images is involved four standard images with size \(512 \times 512\) (labelled as Og01, Og02, Og03, and Og04) and taken from SIPI database [1] (see Fig. 3). The original key parameters used for acting QW are initialized as: m= [1101 0101 0101 1011 0111 0000 0000 1101 0101 0101 1011], N = 19, \(\alpha \) = \(\pi \)/4, \(\beta _{0} \) = \(\pi \)/3, and \(\beta _{1} \) = \(\pi \)/6 , while the initial key parameters used to operate Henon map are initialized as: \(x_{0} \) = 0.47823, \(y_{0} \) = 0.69345, a = 1.4, and b = 0.3.

The efficacy of an image encryption approach relies crucially on its performance (the speed at which an image can be encrypted on a given computer) and it is withstanding to different attacks such as brute force, statistical and differential cryptanalysis, etc. These two important characteristics are examined in the following subsections to show the efficiency of the proposed image cryptosystem.

Fig. 3
figure 3

Investigated dataset, in which the original images are given in the top row, while the bottom row indicates the corresponding cipher images using the proposed image cryptosystem

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Statistical analyses

To demonstrate that the presented cryptosystem has the ability to resist statistical attacks, we discussed the statistical analyses such as correlation analysis, Histogram analysis, and entropy analysis in the following subsections.

Correlation analysis

There is a robust correlation per pixel with neighbouring pixels in plain images, and the correlation value is close to 1 in each direction (horizontal, vertical, and diagonal). In contrast, for encrypted images generated by a good-designed cryptographic algorithm, the correlation values must be very close to 0. To compute the correlation values of the encrypted images and their equivalent plain images, we selected 10,000 pairs of neighbouring pixels at random per direction.

$$\begin{aligned} R=\frac{\sum _{i=1}^{V}\left( O_{i} -{\bar{O}}\right) \left( C_{i} -{\bar{C}}\right) }{\sqrt{\sum _{i=1}^{V}\left( O_{i} -{\bar{O}}\right) ^{2} \sum _{i=1}^{V}\left( C_{i} -{\bar{C}}\right) ^{2} } } , \end{aligned}$$
(8)

here \(O_{i} \) and \(C_{i} \) indicate the values of neighbouring pixels, and V represents the packed number of neighbouring pixel pairs. The values of correlation for the investigated dataset are stated in Table 2, in which the correlation values of the encrypted images are very close to zero. The correlation distributions of neighbouring pixels for the Og01 image before and after ciphering are schemed in Figs. 4, 5, and 6. From the results given in Table 2 and the correlation distributions given in Figs. 4, 5, and 6, we can notice that the presented cryptosystem have the ability to withstand correlation analysis attacks.

Table 2 Outcomes of correlation coefficients for the investigated dataset
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Fig. 4
figure 4

Correlation distributions for red component of Og01 image, in where the top row represents the distributions for the red component of Og01 image, while the bottom row displays the distributions for the red component of Cg01 image

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Fig. 5
figure 5

Correlation distributions for green component of Og01 image, in where the top row represents the distributions for the green component of Og01 image, while the bottom row displays the distributions for the green component of Cg01 image

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Fig. 6
figure 6

Correlation distributions for blue component of Og01 image, in where the top row represents the distributions for the blue component of Og01 image, while the bottom row displays the distributions for the blue component of Cg01 image

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Histogram test

The frequency distribution of pixel values in an image is called histogram. A robust encryption approach must ensure the regularity of the histograms for distinct cipher images. Figure 7 provides histograms of investigated dataset , in which the histograms of plain images vary from each other, and the histograms of the analogous cipher images are identical with each other. Notwithstanding, we necessitate a numerical value to validate the histogram uniformity. Consequently, we employ the chi-square assessment (\(\chi ^{2}\)), which can be represented mathematically as follows:

$$\begin{aligned} \chi ^{2} =\sum _{y=0}^{255}\frac{\left( f_{y} -d\right) ^{2} }{d} , \end{aligned}$$
(9)

here, d signifies the image size, and \(f_{y}\) signifies the frequency of pixel value y. By assuming the significant level is \(\alpha =0.05\), then \(\chi _{\alpha }^{2}\) (255) = 293.2478. For an image, if the value of \(\chi _\textrm{test}^{2}\) is less than \(\chi _{\alpha }^{2} (255)\), the histogram for this image is uniformity, else, the image has not uniform distribution. The outcomes of \(\chi _\textrm{test}^{2}\) for the examined images are given in Table 3, in which the values of \(\chi _\textrm{test}^{2}\) for all encrypted images are less than \(\chi _{\alpha }^{2} (255)\). Thus, the proposed image cryptosystem can withstand histogram analyses attacks.

Fig. 7
figure 7

Histograms of the investigated dataset, in which the histograms of encrypted images are given in the latest three columns and are completely having an identical distribution, while the other three columns signify the histograms of original images

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Table 3 Outcomes of \(\chi ^{2} \) test for the examined dataset
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Information entropy

To assess the distribution of pixel values in an image concerning bit-level, we use the global entropy investigation and can be represented as given in Eq. (10).

$$\begin{aligned} E(Y)=\sum _{d=0}^{255}p(y_{d} )\log _{2} \frac{1}{p(y_{d} )}, \end{aligned}$$
(10)

where \(p(y_{d})\) represents the probability of \(y_{d}\). The probable values for a greyscale image are \(2^{8}\) identical value, hence the ideal entropy value is equal to 8-bit. To verify the effectiveness of the suggested cryptosystem, the entropy values of the cipher image ought to be very close to 8. However, the overall entropy does not evaluate the true randomness of the encrypted images. Therefore, the local entropy can be determined by calculating the average of the global entropy for the non-overlapping blocks (each block has 1936 pixels). Table 4 lists local and global entropy values for original images and their corresponding encrypted copies, in which all entropy values for encrypted images actually near to 8 bits. Therefore, the suggested cryptographic system is secure against entropy attacks.

Table 4 Results of local and global entropy values for the examined images
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Differential analyses

The sensitivity of plain image plays an important role to assess the effectiveness of image cryptosystems. Two tools are used to assess the original image sensitivity to tiny modifications: Uniform Average Rate of Change (UACI) and Pixel Rate of Change (NPCR). NPCR and UACI can be indicated mathematically as given below:

$$\begin{aligned} \begin{array}{l} {\textrm{NPCR}\left( \% \right) =\frac{\sum _{i;j}Df(i,j) }{V} \times 100\% ,} \\ {Df(i,j)=\left\{ \begin{array}{c} {1\, \, \textrm{when}\, \, C1(i,j)\ne C2(i,j)} \\ {0\, \, \textrm{when}\, \, C1(i,j)=C2(i,j)} \end{array}\right. } \end{array} \end{aligned}$$
(11)
$$\begin{aligned} \textrm{UACI}\left( \% \right) =\frac{1}{V} \left( \sum _{i,j}\frac{\left| C1(i,j)-C2(i,j)\right| }{255} \right) \times 100\% \end{aligned}$$
(12)

here V is the full number of pixels utilized for constructing the image, C1 and C2 are two cipher images for an original image with one-bit modification. The results of NPCR and UACI for the investigated images are given in Table 5, in which the average values of UACI and NPCR are 33.45572 and 99.61996%, respectively. This confirms that the proposed encryption scheme has an unusual sensitivity to tiny pixel changes in the plain image.

Table 5 Results of UACI and NPCR tests
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Keyspace and key sensitivity analyses

To verify that the presented cryptosystem has the capability to resist brute force attacks, in the following subsections, we discussed keyspace analysis in addition to key sensitivity analysis.

Keyspace analysis

Keyspace refers to the different keys that can be used in brute force attacks and should be sufficient to withstand such attacks. The proposed encryption approach utilises key parameters (m, N, \(\alpha \), \(\beta _{0}\), \(\beta _{1}\), \(x_{0}\), \(y_{0}\), a, and b) to operate quantum walks and iterate the Henon map throughout the encryption and decryption stages. By analysing the keyspace of only the m parameter, it seems the keyspace is infinite, but the keyspace must be finite. Assuming the computational precision of digital computers is \({10}^{-16}\), the keyspace of the proposed cryptographic system is \({10}^{144}\), which is large sufficient for any modern cryptographic system.

Key sensitivity

The sensitivity of the keys indicates that small changes in the primary keys result in large deviations in results. To assess the key sensitivity of the suggested cryptosystem mechanism, the cipher image Cg01 is decrypted with small changes to the initial keys. The key sensitivity results for the proposed approach are shown in Fig. 8.

Fig. 8
figure 8

Key sensitivity results for the proposed approach

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Noise and occlusion attacks

During data transmission through a transmission channel, noise influences the data being transferred and the data may miss some of its portions. Hence, a robust cryptographic system must be able to resist noise and occlusion attacks. To evaluate the proposed cryptographic approach alongside those attacks, we apply occlusion attacks by cropping portions of the encrypted image or adding Salt & Pepper noise to it. So let us try to restore the plain image from the imperfect encrypted image using the decryption method. Figs. 9 and 10 display the effects of occlusion and noise attacks in which the plain image is restored effectively after performing the decryption process.

Fig. 9
figure 9

Occlusion attack, in which the top row implies the defective encrypted images by cropping out some data and the bottom row shows the equivalent decrypted images

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Fig. 10
figure 10

Noise attack, in which the defective encrypted images through applying Salt & Pepper noise with varying densities are decrypted

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Time efficiency and complexity analysis

Time efficiency is an essential tool for assessing the performance of image security. Time of encryption measures the real-time taken by the proposed image security to encrypt an image. To guarantee the time efficiency of the presented cryptographic system, Table 6 states a comparison of the encryption time for the proposed scheme with the corresponding approaches for the greyscale images of various sizes. The encryption time shown in Table 6 demonstrates that the proposed image security is in favour of real-time encryption applications in the cloud.

To ensure consistency and concurrency, computational complexity analysis is performed in terms of CPU operations needed to perform the various methods. By assuming the dimensional of the original image is \(h \times w\), the estimated computational complexity for each step of the presented encryption algorithm is drawn in the following steps.

  • Step 1: Determining the initial values.

  • Step 2: (\(7 \times h \times w\)) operations required to execute Gray code operation to per pixel in the original image.

  • Step 3: (\( h \times w\)) operations required to get some information about image Gc.

  • Step 4: (\( N^2\)) operations required to operate QW.

  • Step 5: (\( h \times w\)) operations required to resize the probability distribution to the size of the original image.

  • Step 6: (\(8 \times h \times w\)) operations required to execute bit xor operation.

  • Step 7: (\( h \times w\)) operations required for getting some information about the substituted image SgQ.

  • Step 8: (\( h \times w\)) operations required to iterate Henon map.

  • Step 9: (\(h \times w \times \log (h \times w)\)) operations required for arranging the elements of X (average time complexity of quicksort algorithm) and getting the index of each element.

  • Step 10: (\( h \times w\)) operations required to permutation procedure.

  • Step 11: (\( h \times w\)) operations required to reshape the elements.

  • Step 12: (\( h \times w\)) operations required for addition and mod operations.

  • Step 13: (\(8 \times h \times w\)) operations required to execute bit xor operation.

According to the above steps, the computational complexity of the presented encryption algorithm is \(O(\textrm{max}(N^2, 8hw, hw\log (hw)))\).

Table 6 Comparison of the encryption time (in seconds) for the presented encryption cryptosystem with the corresponding approaches for greyscale images of various sizes
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Discussion

The major contribution of this study is to support the idea of integrating quantum-inspired models into predominating cryptographic mechanisms to achieve data confidentiality against unauthorised access in the transition stage to the quantum era. In this study, we proposed a novel image cryptosystem for cloud applications based on Gray code, QW, and Henon map. The efficacy of an image encryption algorithm relies crucially on its performance and its withstanding different attacks such as brute force and statistical and differential cryptanalyses.

To confirm the time efficiency of the presented cryptographic system, Table 6 presents a comparison of the encryption time for the proposed scheme with the corresponding strategies for the greyscale images of various sizes, which demonstrates that the proposed image security is in favour of real-time encryption in cloud application. And to confirm the efficiency of the proposed cryptosystem against statistical attacks, we performed correlation, Histogram, and entropy tests. The average values of correlation coefficients per direction are − 0.00007, 0.000108, and 0.00010, which confirm that the presented encryption scheme has the capability to withstand correlation analysis attacks. Also, the outcomes of \(\chi _\textrm{test}^{2}\) are given in Table 3, in which the values of \(\chi _\textrm{test}^{2}\) for all encrypted images are less than \(\chi _{\alpha }^{2} (255)\), which indicates that the presented image cryptosystem can resist histogram analyses attacks. Furthermore, Table 4 lists local and global entropy values for original images and their corresponding encrypted copies, in which all entropy values for cipher images actually near to 8 bits, which indicates that the suggested cryptographic system is secure against entropy attacks.

Furthermore, to ensure the efficiency of the proposed cryptosystem against differential attacks, we performed UACI and NPCR tests. the average values of UACI and NPCR are 33.45572 and 99.61996%, respectively, which confirm that the proposed encryption scheme has an unusual sensitivity to slight pixel modifications in the plain image. And for demonstrating the presented cryptosystem has the ability to withstand Brute force attacks, the keyspace of the proposed cryptographic system is \({10}^{144}\), which is large sufficient for any modern cryptographic technique. Also, to evaluate the proposed cryptographic approach alongside noise and occlusion attacks, we apply occlusion attacks by cropping portions of the encrypted image or adding Salt & Pepper noise to it. Figures 9 and 10 displayed the effects of these attacks in which the plain image is restored effectively after performing the decryption process.

To validate the efficiency of the suggested encryption approach alongside other related methods, Table 7 provides the average values of correlation values, global and local information entropy, UACI, NPCR, and chi-square of the suggested approach with average values of other related approaches [5, 8, 19, 44, 45]. The results stated in Table 7 demonstrate the efficiency of the suggested approach in comparison with other related approaches.

Table 7 Comparison of the suggested scheme with other related approaches
Full size table

Concluding remarks and future scope

The central contribution of this paper is to support the idea of integrating quantum-inspired models into predominating cryptographic mechanisms to achieve data confidentiality against unauthorised access in the transition stage to the quantum era. This study utilised quantum walk, as a quantum-inspired system, for developing a new data confidentiality technique that can be applied in digital devices to have the capability for resisting the potential attacks from quantum and digital computers. This article describes a new image encryption technique for cloud applications based on Gray code, QW, and Henon map. In the presented algorithm, the generated key streams from QW or Henon map is related to the plain image, thus, our cryptosystem has a high sensitivity of slight bit changes on the plain image. Experimental results proved that the proposed cryptosystem is efficient with high security in comparison with other related approaches.

But the limitations of this work are the present cryptosystem is suitable for image data confidentiality in cloud storage, not any other data types such as video, audio, document files, etc. Also, the presented cryptosystem is for fully image encryption, not partial encryption for images. In future work, we aim to modify the presented cryptosystem to be suitable for achieving audio and video confidentiality against unauthorised access in the transition stage to the quantum era.