An Efficient Visually Meaningful Quantum Walks-Based Encryption Scheme for Secure Data Transmission on IoT and Smart Applications

Abstract

Smart systems and technologies have become integral parts of modern society. Their ubiquity makes it paramount to prioritise securing the privacy of data transferred between smart devices. Visual encryption is a technique employed to obscure images by rendering them meaningless to evade attention during transmission. However, the astounding computing power ascribed to quantum technology implies that even the best visually encrypted systems can be effortlessly violated. Consequently, the physical realisation quantum hardware portends great danger for visually encrypted date on smart systems. To circumvent this, our study proposes the integration of quantum walks (QWs) as a cryptographic mechanism to forestall violation of the integrity of images on smart systems. Specifically, we use QW first to substitute the original image and to subsequently permutate and embed it onto the reference image. Based on this structure, our proposed quantum walks visually meaningful cryptosystem facilities confidential transmission of visual information. Simulation-based experiments validate the performance of the proposed system in terms of visual quality, efficiency, robustness, and key space sensitivity, and by that, its potential to safeguard smart systems now and as we transition to the quantum era.

1. Introduction

Smart systems are becoming integral parts of modern life, where they facilitate important internet-based services ranging from Internet of Things (IoT) to cloud storage, and many others. Sadly, this utility has propelled criminal activities to violate the integrity and confidentiality of the systems [1,2]. Cryptosystems, such as data hiding, and encryption protocols are reputed for tamper-proof security for several types of data  [3,4,5]. Whereas data encryption is primarily aimed at transforming data into unintelligible formats [6,7], visual cryptography is tailored towards safeguarding the data while retaining its meaningful state [8,9,10]. This ability to convert confidential data into secure but meaningful forms makes visual cryptography attractive in modern cryptographic applications [11].

Generally, the visual cryptographic algorithm consists of two phases: pre-encryption and embedding phases. In the pre-encryption, confidential data is permutated, substituted, and embedded onto a reference object  [12].

Recently, many visually meaningful image encryption techniques have been proposed [13,14,15,16,17,18,19], of which most are based on mathematical models where some information about the reference image is required during the decryption process. In the meantime, the inevitable realisation of quantum hardware means its astounding computing power could be exploited to violate the integrity and confidentiality of even the best cryptosystems [20]. To forestall this, quantum technology must be suffused into such systems. Quantum walks (QWs), which are the quantum equivalents of traditional (i.e., classical) random walks, exhibit remarkable stability and exhibit theoretically infinite key space allowances, properties that imbue them with robustness to resist diverse types of attacks [21].

Our study is primarily aimed at exploiting these properties to enhance the security of visual information on smart systems and devices. Furthermore, with necessary adjustments, our visual cryptosystem can be useful in safeguarding both pre- and post-quantum era smart systems. Similar efforts to employ QWs have been reported in, for example, Ref. [22], where Su and Wang presented a visual image cryptosystem that combined controlled QWs and single value decomposition (SVD). However, the performance of their protocol is impeded when the length of the controlling bit string is less than the number of nodes at the circle [21]. In this study, we ameliorate the zero probabilities associated with this by encrypting visually meaningful images with the QW prior to transmission in smart system frameworks. This way, our proposed system forestalls occurrence of zero probabilities. Furthermore, we use the QW to substitute the original image and permutate it whilst the substituted image is embedded onto the reference image via substitution of two least significant bits (LSBs) in the reference image. Moreover, using a pre-encryption phase, QWs ensure that the permutation and embedding processes are executed in one step.

To recapitulate, the main contributions of our study include:

• Introducing a new visual cryptographic mechanism that exploits quantum technologies to safeguard smart systems.
• Enhancing existing cryptosystems to overcome the weaknesses ascribed to zero probabilities exhibited when operating QWs on a circle of N vertices where the length of the controlling bit string is less than the number of nodes at the circle,
• Utilising quantum walks (QW) first to substitute the original image and then to subsequently permutate and embed it onto the reference image, and
• Integrating QWs into the encryption of visually meaningful images for secure transmission of sensitive data on smart systems.

To deliver the enumerated contributions, the rest of the paper is structured as follows: the layout of our proposed framework for confidential data transmission in smart systems is presented in Section 2, while essential precepts of QWs, which is the core of the proposed framework, are highlighted in Section 3. The foundations built in these two sections are subsequently used to construct algorithmic protocols of our visually meaningful encryption system whose details are presented and discussed in Section 4. Finally, outcomes of simulation-based experiments to assess the performance of the proposed method are reported in Section 5.

2. Proposed Framework for Secure Data Transmission in Smart Systems

With increased access and lower cost, cloud computing is becoming a basic computing platform in most domains of our daily lives. However, secure transfer of data between cloud computing platforms and Internet of Things (IoT) nodes (or devices) is still an important problem; more so, since there is still no standard medium to transfer data securely between and within smart systems [23,24]. Consequently, the need to develop more advanced techniques to safeguard data accessible to smart systems is a priority that cannot be overlook.

Meanwhile, the renewed interest and funding aimed at the realisation of physical quantum hardware portend great danger to today’s information security architectures. The anticipated perils range from exploiting critical vulnerabilities in cybersecurity to those anticipated as we make the inevitable transition to the quantum computing era. Both risks demonstrate the need to design efficient mechanisms for data protection and processing [4]. Failure to do so exposes smart systems and the data they transmit to unimaginable consequences related to privacy violation, impersonation, counterfeiting, etc. Using quantum technology to embed traditional cryptosystems, these expected criminal activities can be forestalled. Meanwhile, the quantum era will herald upgrades to subsisting data processing models, security standards, and available information technologies. In this regard, with minor necessary adjustments it is expected that our proposed system will be ready to secure such infrastructure on the new computing paradigm. Consequently, we introduce a new framework for confidential transmission of data on smart systems based on quantum walks whose envisioned architecture is presented in Figure 1.

Quantum walks is employed to safeguard data confidentiality and withstand probability attacks that could impugn data integrity and privacy in both traditional and quantum computing frameworks. Our proposed scheme offers a veritable structure to encrypt the secret data before uploading it to the cloud system, while the same QWs paired with correct secret keys facilitate recovery of confidential data during the decryption process.

As the core element of our proposed scheme, essential precepts of quantum walks are highlighted in the next section.

3. Quantum Walks (QW)

Quantum walks (QWs) are the quantum versions of classical random walks that consist of two elements: the first element is the walker space $H_s$ and the second one is the coin particle $H_p=cos\beta |0\rangle +sin\beta |1\rangle$ both $H_s$ and $H_p$ reside in a Hilbert space $H_s\otimes H_p$ [25]. For each step of running QW on a cycle of N vertices with a binary string T, a unitary transformation ${\widehat{U}}_0$ (${\widehat{U}}_1$) is performed when the ith-bit of T is 0 (1), while the unitary transformation ${\widehat{U}}_2$ is performed when the running step (i.e., the ith-step) is higher than the bit length of T [21]. The unitary transformation ${\widehat{U}}_0$ can be identified in Equation (1).

$${\widehat{U}}_0=\widehat{F}(\widehat{I}\otimes {\widehat{C}}_0)$$

(1)

where $\widehat{F}$ represents the shift operator of operating QW on a circle, which can be identified in Equation (2),

$$\widehat{F}=\sum _r^N\left(|\left(r+1\right)modN,0\rangle \langle r,0|+|\left(r-1\right)modN,1\rangle \langle r,1|\right)$$

(2)

and ${\widehat{C}}_0$ is $2\times 2$ coin operator as presented in Equation (3) [21].

$${\widehat{C}}_0=\left(\begin{array}{cc}cos{\theta }_0& sin{\theta }_0\\ sin{\theta }_0& -cos{\theta }_0\end{array}\right)$$

(3)

Similarly, ${\widehat{U}}_1$ and ${\widehat{U}}_2$ can be constructed as outlined for ${\widehat{U}}_0$. After running the QW for S steps, the probability of detecting the walker at position r can be calculated as given in Equation (4).

$$\begin{array}{cc}P(r,S)=& \sum _{b\in \left\{0,1,2\right\}}{\left|\left.〈r,0\right|{\left({\widehat{U}}_b\right)}^S|\psi {\rangle }_{\mathrm{initial}}\right|}^2\\ & +\sum _{b\in \left\{0,1,2\right\}}{\left|\left.〈r,1\right|{\left({\widehat{U}}_b\right)}^S|\psi {\rangle }_{\mathrm{initial}}\right|}^2\end{array}$$

(4)

where $|\psi {\rangle }_{\mathrm{initial}}$ is the initial state of QW.

Meanwhile, exploiting the outlined properties of QW and the potency of other quantum computing operations, algorithmic protocols to execute our cryptosystem to safeguard the smart systems as envisioned in Figure 1 are presented in the next section.

4. Proposed Encryption Approach

This section illuminates the role of QW in designing a new visually meaningful image encryption scheme. In the proposed encryption scheme, QW is first used to substitute the pristine (i.e., original) image, and it is also used to permutate and embed the substituted image onto the reference image by locating the pixel position in the reference image. Specifically, the substituted bits in the two LSBs of the located pixel are subsequently embedded onto the reference image. Figure 2 presents the outline of the encryption encryption and decryption phases of our scheme. The encryption process executed via steps enumerated in the sequel and outlined in Algorithm 1.

Algorithm 1: Encryption processes.

1.

Select initial key parameters ($T,N,S,\beta ,{\theta }_0,{\theta }_1,{\theta }_2$) that can be used to run QW for S steps on a cycle of N odd vertices ruled with a binary string T to initialise an N- dimensional probability distribution vector P. Where the primary state of the coin particle is $H_p=cos\beta |0\rangle +sin\beta |1\rangle$ and the unitary transformations ${\widehat{U}}_0$, ${\widehat{U}}_1$, and ${\widehat{U}}_2$ are constructed by ${\theta }_0$, ${\theta }_1$, and ${\theta }_2$, respectively, ($0\le \beta ,{\theta }_0,{\theta }_1,{\theta }_2\ge \pi /\phantom{\pi 2}2$).

2.

Retrieve the dimensionsl of the original image (OImg).

$$[x,y,z]\leftarrow size\left(\mathit{OImg}\right)$$

3.

Reconfigure the probability vector (P) to a matrix, then alter the size of the generated matrix to the size of the original image.

$$Pf\leftarrow P(1:fix\left(\sqrt{N}\right)\times fix\left(\sqrt{N}\right))$$

$$PK\leftarrow reshape(Pf,fix\left(\sqrt{N}\right),fix\left(\sqrt{N}\right))$$

$$RK\leftarrow resize(PK,[x,y\times z\left]\right)$$

4.

Reshape the matrix RK into an $x\times y\times z$ dimensional matrix and then convert the output into integers.

$$K\leftarrow reshape(RK,x,y,z)$$

$$Key\leftarrow fix(K\times {10}^{12}mod256)$$

5.

Substitute the original image using the generated integers.

$$SImg\leftarrow Key\oplus OImg$$

6.

Embed the substituted image (SImg) onto the reference image (RImg) using the following steps:

(a)

Resize the probability vector P to a vector of size 2x + 2y.

$$R\leftarrow resize(P,[2x+2y,1\left]\right)$$

(b)

Arrange the elements of the vector R in ascending order.

$$A\leftarrow order\left(R\right(1:2x\left)\right)$$

(c)

Retrieve the index of every element of R(1 : 2x) in A.

$$XR\leftarrow index\left(R\right(1:2x),A)$$

(d)

Repeat the last two substeps to generate YR sequence.

$$B\leftarrow order\left(R\right(2x+1:2x+2y\left)\right)$$

$$YR\leftarrow index\left(R\right(2x+1:2x+2y),B)$$

(e)

Expand the 8-bit and x × y dimension substituted image SImg of to a 2-bit and 2x × 2y dimension image.

$$XImg\leftarrow expand\left(SImg\right)$$

(f)

Replace 2LSBs of RImg(XR(i), YR(j), :) with 2bits of XImg(i, j, :) as the visual encrypted image EImg (XR(i), YR(j), :) for i = 1, 2, ..., 2x and j = 1, 2, ..., 2y.

5. Simulation Results

To evaluate the efficiency of the proposed visually encryption method, we simulated its execution on an Intel® core${}^{TM}$ i5 and 6-GB RAM laptop computer equipped with MATLAB R2016b software. Furthermore, we used images sourced from the Signal and Image Processing Institute (SIPI) [26], Laboratory for Image & Video Engineering (LIVE) [27], and Medical Image Database (MedPix) [28] as our experimental dataset samples of which are presented in Figure 3 and Figure 4. Images in Figure 3 labelled as OImg01 through OImg06 are 256 × 5256-dimension images used as original (or confidential) images. Similarly, images in Figure 4 labelled as RImg01 through RImg04 are 512 × 5512-dimension images used as reference images. Additionally, the key parameters used for running our QW are set as (N = 261, S = 521, T = “0101 1001 0110 1000 1011 0001 1101 0101 0111 0111 0011 0101”, $\beta$ = $\pi /2$, ${\theta }_0$ = $\pi /4$, ${\theta }_1$ = $\pi /6$, and ${\theta }_2$ = $\pi /3$).

The remainder of the section presents outcomes of our performance analysis covering tests for visual quality, efficiency, correlation, robustness, and key space that are employed to validate advanced cryptosystems, and later we compare some of these performances with those reported in recent similar techniques.

5.1. Visual Quality Tests

The visual quality of an encrypted image is a useful tool to assess the efficiency of any visually meaningful encryption mechanism. Outcomes of encrypted images emanating from our proposed technique are reported in Figure 5 and Figure 6, and visual inspection of those images establishes their imperceptibility and good visual quality. However, to quantitatively validate this performance, we employed the standard visual metrics of peak signal to noise ratio (PSNR) and Structural Similarity Index Metric (SSIM). These metrics can be formulated as presented in Equations (5) and (6), respectively.

$$PSNR\left(R,E\right)=20{log}_{10}\left(\frac{MA{X}_R\times \sqrt{x\times y}}{\sqrt{{\sum }_{i=0}^{x-1}{\sum }_{j=0}^{y-1}{[R(i,j)-E(i,j)]}^2}}\right)$$

(5)

$$SSIM\left(R,E\right)=\frac{\left(2{\mu }_R{\mu }_E+T_1\right)\left(2{\sigma }_{R,E}+T_2\right)}{\left({\mu }_R^2+{\mu }_E^2+T_1\right)\left({\sigma }_R^2+{\sigma }_E^2+T_2\right)}$$

(6)

where ${MAX}_R$ points to the maximum pixel value of the reference image R, $T_1$ and $T_2$ are constants, $\mu$ and $\sigma$ are the mean and variance, respectively, while E specifies the encrypted image relative to an x× y-dimensional reference image R. Generally, higher PSNR values indicate good visual quality often at levels imperceptible to the human visual system (HVS).

Table 1. Outcomes of PSNR (in dB) test for encrypted images.

Reference Image	Original Image
	OImg01	OImg02	OImg03	OImg04	OImg05	OImg06
RImg01	43.4264	43.4192	43.4203	-	-	-
RImg02	43.9165	43.9033	43.8933	-	-	-
RImg03	-	-	-	43.2824	43.2937	43.2813
RImg04	-	-	-	43.2486	43.2509	43.2394

and

Table 2. Outcomes of SSIM test for encrypted images.

Reference Image	Original Image
	OImg01	OImg02	OImg03	OImg04	OImg05	OImg06
RImg01	0.9858	0.9857	0.9856	-	-	-
RImg02	0.9824	0.9825	0.9823	-	-	-
RImg03	-	-	-	0.9981	0.9980	0.9981
RImg04	-	-	-	0.9573	0.9580	0.9577

present outcomes of the visual quality tests for our proposed scheme in terms of the PSNR (Table 1) and SSIM (Table 2) quality metrics. Defined within a range [0,1], values of SSIM closer to 1 indicate high visual quality of the encrypted image.

5.2. Efficiency Tests

To assess the efficiency of our proposed schemes, we report outcomes three analyses (histogram, correlation, and entropy) to ascertain how well the scheme blends the original and reference images for secure visually meaningful transmission and sharing.

5.2.1. Histogram Analysis

Histogram test is a significant test for evaluating the efficiency of any encryption algorithm as reflected by the frequency distribution of pixel values for the image [29]. Any practical encryption protocol should exhibit identical histograms for different cipher images. Histograms for the original and encrypted images realised using the scheme are presented in Figure 5 and Figure 6, from which we note that all images encrypted using the same reference image manifest identical histograms. Additionally, the histograms for original image (OImg01) and their encrypted versions before embedding onto the reference Baboon image are presented in Figure 7, while those for the original greyscale embedded onto the plane reference image are presented in Figure 8. We note that, prior to embedding onto the reference images, all the encrypted images exhibit identical histograms. Consequently, it can be deduced the the proposed visually meaningful image encryption approach could withstand histogram attacks.

5.2.2. Correlation Analysis

To measure the concordance between neighbouring pixels in an encrypted image, the measure of correlation coefficient between adjacent pixels is used [25]. Outcomes of correlation coefficients closer to 1 indicate that the image is meaningful. To evaluate the correlation coefficients per channel in the visually encrypted images, ${10}^4$ pairs of neighbouring pixels are selected randomly and computed using Equation (7).

$$Cf=\frac{{\sum }_{x=1}^Q\left(m_x-\overline{m}\right)\left(n_x-\overline{n}\right)}{\sqrt{{\sum }_{x=1}^Q{\left(m_x-\overline{m}\right)}^2{\sum }_{x=1}^Q{\left(n_x-\overline{n}\right)}^2}}$$

(7)

where Q denotes the full number of adjacent pixel pairs in each direction and $m_x$, $n_x$ are denote to the values of pair neighbouring pixels. Outcomes of Cf measure for images encrypted using our scheme are presented in

Table 3. Correlation coefficients for visually encrypted images.

Encrypted Image	Colour Channel	Direction
		Horizontal	Vertical	Diagonal
EImg01-1	Red	0.9678	0.9636	0.9600
	Green	0.9844	0.9821	0.9729
	Blue	0.9684	0.9643	0.9507
EImg02-1	Red	0.9676	0.9672	0.9597
	Green	0.9833	0.9815	0.9706
	Blue	0.9688	0.9667	0.9516
EImg03-1	Red	0.9640	0.9665	0.9572
	Green	0.9823	0.9836	0.9726
	Blue	0.9703	0.9650	0.9530
EImg01-2	Red	0.9548	0.9555	0.9451
	Green	0.9664	0.9711	0.9521
	Blue	0.9701	0.9725	0.9553
EImg02-2	Red	0.9553	0.9552	0.9431
	Green	0.9651	0.9731	0.9539
	Blue	0.9705	0.9708	0.9555
EImg03-2	Red	0.9528	0.9568	0.9436
	Green	0.9676	0.9711	0.9548
	Blue	0.9709	0.9708	0.9546
EImg04-3	Grey	0.7756	0.8681	0.7387
EImg05-3	Grey	0.7542	0.8603	0.7288
EImg06-3	Grey	0.7719	0.8629	0.7312
EImg04-4	Grey	0.9662	0.9680	0.9443
EImg05-4	Grey	0.9648	0.9708	0.9414
EImg06-4	Grey	0.9639	0.9683	0.9434

while the correlation distribution for encrypted image EImg01-1 is presented in Figure 9. Both outcomes corroborate claims regarding the meaningfulness of the encrypted images. Similarly, outcomes of the correlation analysis for pairings between the original images and their encrypted versions prior to embedding onto the reference image are presented in

Table 4. Correlation coefficients for pristine images and their encrypted ones before embedding onto a reference image.

Image	Colour Channel	Direction
		Horizontal	Vertical	Diagonal
OImg01	Red	0.9126	0.8716	0.8457
	Green	0.9005	0.8433	0.8152
	Blue	0.9182	0.8637	0.8222
Enc OImg01	Red	0.0013	−0.0003	0.0006
	Green	−0.0003	0.0003	−0.0008
	Blue	0.0009	−0.0003	0.0007
OImg02	Red	0.9539	0.9517	0.9165
	Green	0.9645	0.9614	0.9385
	Blue	0.9661	0.9613	0.9414
Enc OImg02	Red	0.0002	−0.0009	−0.0003
	Green	−0.0009	0.0007	−0.0002
	Blue	0.0006	−0.0001	−0.0004
OImg03	Red	0.9343	0.9666	0.9093
	Green	0.9561	0.9801	0.9423
	Blue	0.9751	0.9819	0.9619
Enc OImg03	Red	−0.0008	0.0006	−0.0009
	Green	0.0007	−0.0003	−0.0004
	Blue	−0.0001	0.0006	−0.0007
OImg04	Grey	0.9464	0.9239	0.8796
Enc OImg04	Grey	0.0007	−0.0006	0.0003
OImg05	Grey	0.9162	0.9404	0.8816
Enc OImg05	Grey	−0.0008	−0.0001	0.0001
OImg06	Grey	0.9592	0.9574	0.9143
Enc OImg06	Grey	−0.0004	0.0001	−0.0013

and the correlation distribution for encrypted image OImg04 is presented in Figure 10.

The two outcomes demonstrate that no intelligible information about the original image can be deduced, which confirms the correlation strength and efficiency of our proposed encryption scheme.

5.2.3. Entropy Analysis

Information entropy is used as a tool to assess the concentration of pixel values per bit-level in an image. For greyscale images, entropy values closer to 8 indicate the efficiency of an encryption technique [30].

Table 5. Information entropy for reference and visual encrypted images.

Image	Information Entropy
RImg01	7.6698
EImg01-1	7.7253
EImg02-1	7.7252
EImg03-1	7.7252
RImg02	7.7621
EImg01-2	7.7711
EImg02-2	7.7710
EImg03-2	7.7711
RImg03	7.3578
EImg04-3	7.3591
EImg05-3	7.3590
EImg06-3	7.3591
RImg04	6.6776
EImg04-4	6.6895
EImg05-4	6.6894
EImg06-4	6.6896

presents the entropy values for our reference and visual encrypted images. Similarly,

Table 6. Information entropy for original images and their encrypted versions before embedding them onto the reference images.

Image	Original	Encrypted
OImg01	7.5225	7.99909
OImg02	7.6658	7.99906
OImg03	7.0686	7.99915
OImg04	7.1586	7.99748
OImg05	7.6684	7.99728
OImg06	6.5233	7.99726

reports the entropy values for pairings of our original images and their encrypted versions prior to their embedding onto the reference images.

As targeted, both tables record entropy values close to an optional value of 8, which is further testimony of the efficiency of our proposed encryption scheme.

5.3. Robustness Tests: Occlusion and Data Loss Attacks

In real-world smart systems and similar applications, encrypted images are transmitted via noisy channels, which makes them susceptible to tampering, loss, and other violations. To appraise the robustness of our encryption mechanism and its ability to withstand data loss due noise addition, we considered the addition of various levels in Salt and Pepper (S&P) noise onto the encrypted images and using our decryption procedure, we evaluated for attempts to recover the original images from the violated versions.

For the occlusion attacks, we considered using cut-outs of varied sizes to occlude parts of the encrypted images. Figure 11 and Figure 12 present outcomes of the occlusion and noise attacks. The good visual quality from both outcomes manifests the robustness of our proposed encryption scheme to occlusion and noise addition attacks.

5.4. Key Sensitivity Test

Key sensitivity is an important test for the security of any cryptographic approach. It is a measure of the sensitivity of key parameters to the influence or circumvent the decryption process. To appraise the key sensitivity of our proposed scheme, we varied tiny parts of the secret key and evaluated if the protected original images could be violated. Outcomes of these tests for the indicated variations in the secret key are reported in Figure 13. Quantitatively, we employed the Number of Pixel Change Rate (NPCR) metric, which is expressed in Equation (8), to assess the key sensitivity of the proposed scheme.

$$\begin{array}{c}NPCR(OIm,DIm)=\frac{\sum _{x;y}f(i,j)}{M}\times 100\%,\hfill \\ f(i,j)=\left\{\begin{array}{cc}1,& ifOIm(i,j)\ne DIm(i,j)\\ 0,& \mathrm{otherwise}.\end{array}\right.\hfill \end{array}$$

(8)

where OIm and DIm denote the pristine and decrypted images, and M denotes the total pixels in the image. NPCR values for the original image OImg01 are presented in

Table 7. NPCR values for the original image OImg01 and the results illustrated in Figure 13.

Image Pairing	NPCR (%)
Figure 3a and Figure 13a	0
Figure 3a and Figure 13b	99.6027
Figure 3a and Figure 13c	99.6098
Figure 3a and Figure 13d	99.5890
Figure 3a and Figure 13e	99.6078
Figure 3a and Figure 13f	99.6047
Figure 3a and Figure 13g	99.6139
Figure 3a and Figure 13h	99.6241
Figure 3a and Figure 13i	99.6043

while, as noted earlier, the outcomes for different alternations to the secret decryption key are presented in Figure 13.

Both outcomes confirm the proposed schemes sensitivity to tiny modifications to its secret key as the image recovery is unsuccessful in all attempts reported (as seen in Figure 13).

5.5. Performance Analysis

To establish the efficiency of our proposed scheme, in this section we compare its performance in terms of visual quality metrics alongside recent and similar image encryption techniques. Using outcomes reported earlier for PSNR (Table 1) and SSIM (Table 2), the performance analysis in

Table 8. Comparison of the average PSNR and SSIM value for our algorithm besides other related algorithms.

Cryptosystem	PSNR(dB)	SSIM
Proposed	43.4646	0.9810
[13]	35.2058	0.9584
[14]	32.0502	0.9937
[15]	24.0488	0.6913
[16]	30.9728	0.8554
[22]	36.9028	0.9987

compares the recorded outcomes alongside those from [13,14,15,16,22].

As presented in the table, our proposed scheme performs better than most of the competing techniques in the range of 15 to 44% and up to 29.5% for the PSNR and SSIM image quality metrics.

6. Concluding Remarks

A confluence of efforts geared towards deployment of technologies on smart systems on the one hand and the invigorated interest in and accelerated march towards the quantum computing era on the other indicate not only the ubiquity, but also the vulnerability of images that we use in our everyday lives. Considering the need to safeguard the integrity and confidentiality of these diverse types of images, our study proposes an encryption scheme that exploits the potency of quantum walks to substitute the pristine images and then permutate and subsequently embed the substituted images onto a reference image. Not only does our proposed scheme provide tamperproof security for today’s computing frameworks, but it also offers safeguards to protect future smart systems that will built using quantum computing hardware. Outcomes from stimulation-based experiments reported in the study highlights the efficiency of the proposed scheme in terms of visual quality, correlation analysis, key space sensitivity and robustness. Additionally, comparisons in terms of visual quality tests indicate that the proposed scheme outperforms recent and similar schemes by as high as 44% and 29.5% for the PSNR and SSIM image quality metrics. The study and its outcomes suggest potential applications for the proposed scheme in safeguarding smart systems.