A robust approach to satellite image encryption using chaotic map and circulant matrices

Abstract

In the modern era, where satellite imagery is vital for applications like ecological monitoring and national security, ensuring the safety and integrity of these data repositories is crucial. This study presents an improved satellite image encryption technique that combines the cryptographic strength of the circulant matrix in the Hill cipher with the dynamic characteristics of the hyperbolic tangent tent map, further enhanced by the Kronecker XOR product. The algorithm initiates with computing alterations by a shift amount. After preserving the leftmost pixel in each row, it executes XOR operations between alternating rows, combining the value of the current even or odd row with corresponding pixels in the adjacent rows followed by encryption using the Hill cipher. The resulting image undergoes a diffusion process utilizing a hyperbolic tangent tent map. The Kronecker XOR product operation is then applied to individual pixels to produce a secure image followed by additional diffusion with keys from the hyperbolic tangent tent map to achieve the final encrypted image. We conducted simulations using MATLAB to assess the efficiency of the proposed satellite image encryption from theoretical and statistical perspectives. The results exhibit robust encryption performance as demonstrated by metrics such as an entropy value of 7.9982, a UACI of 33.5333%, and an NPCR of 99.6038%. The experiment results demonstrate the proposed image encryption scheme's reliability, practicability, and efficiency in securing satellite images during data storage and transmission. Comprehensive testing against various attacks including correlation, histogram, chi‐square, NPCR, PSNR, UACI, SSIM, key space and key sensitivity analysis confirms the scheme's robustness, efficiency and speed. These findings verify the scheme's ability to come across the most stringent encryption and decryption standards, making it an effective solution for securing sensitive satellite image data.

INTRODUCTION

In today's data-driven world, where strict observance of digital image security and preservation is a concern, the need to provide satellite imagery with equivalent levels of protection becomes obvious. The increasing hazards arising from illegitimate use and access to satellite imagery highlight how important information security is to safeguard a nation's crucial infrastructure.¹ The 2024 Crown Strike Global Threat report states that there has been a notable surge in cyber-attacks with cloud intrusions growing by 75% over the past year.² Satellites, medical imaging systems, sensors and personal electronic gadgets are merely a few of the countless devices that constantly generate a massive quantity of images at an unprecedented rate. Networks all across the globe exchange these images for diagnostic, scientific, and analytical purposes.³ Owing to its widespread availability and affordability, the Internet is the primary means of sharing and transmitting images. Nonetheless, given the open nature of this network, it is imperative to recognize the importance of data security. To cope with this problem, encryption is the most widely used technique for securing satellite images from multiple risks during transmission, minimizing an invasion of information and security from unwanted access and illegal use. This indicates that the encryption methods support securing satellite imagery when it is being stored and transmitted.⁴

Lately, several encryption techniques have been designed to enhance the confidentiality of satellite imagery due to the extensive usage of these broadcasts. Among different methods, several algorithms have been developed especially for encoding images. One notable approach is the application of chaotic systems to enhance key space, robustness, and security. This is because traditional encryption techniques like DES (Data Encryption Standard), IDEA (International Data Encryption Algorithm), and RSA (Rivest Shamir Adleman) were initially designed to handle textual content but face challenges with images as a result of their fundamental characteristics of visual data, mainly elevated pixel correlation, and redundancies found in visual data.^5–7 Therefore, these classical algorithms are ineffective when encrypting data including large images.⁸

The new encryption techniques can cope with all the information security requirements for satellite image transmission. Chaos theory is one of the most appreciated and efficient methods for resolving these problems. Chaos-based image encryption, in contrast to conventional methods that rely on mathematical algorithms, generates a key with dual encryption and decryption capabilities by considering the inherent uncertainty and chaotic systems' randomness.⁹ Shannon proposed two key elements, confusion, and diffusion that serve as the core of cryptographic techniques widely recognized for their effectiveness and diversity.¹⁰ These elements represent the fundamental building blocks of cryptography strongly entangle the encryption key with statistical features of encrypted data, limiting opponents' attempts to decipher the key from the data to be encrypted. Substitution is an integral part of confusion techniques. Diffusion, in contrast, aims to obscure any noticeable long-term statistical traits by spreading the statistical patterns present in the original message throughout the encrypted message. Diffusion is made possible by permutation techniques that linearly rearrange input bits or bytes, generating an entirely distinct layout. Statistical analysis, known plaintext, brute force, and other sorts of attacks have all been proven completely futile against chaotic ciphers.¹¹ Chaos-based encryption schemes coupled with confusion and diffusion achieve the resilience and adaptability necessary for safeguarding multimedia and image data. Recent years have seen an extensive acceptance of chaotic systems, where they have grown to be vital in encrypting satellite images. Considering their initial values' sensitivity and pseudo randomness, these chaotic systems generated via deterministic equations have some exceptional cryptographic qualities.¹²

Pseudo-random number generations (PRNGs) plays substantial role in image encryption algorithm. They are used to create a series of arbitrary values with the goal of encoding visual information and are crucial because they can generate a large number of randomly distributed, unpredictable numbers. Encryption's principal goals of maintaining confidentiality and integrity are mainly achieved through the utilization of these number generators. The randomness and uniform distribution of the PRNG output contribute significantly for determining encryption's robustness and applicability. Therefore, choosing a reliable and secure PRNG is crucial for guaranteeing the safety and efficacy of image encoding methods.¹³

Numerous new chaos-based image encryption algorithms have been proposed. S.Behnia et al. presented a novel algorithm for image encryption based on several types of chaotic maps.¹⁴ This symmetric key cryptography methodology merges a 1D chaotic map with a typical coupled map to encrypt images with a degree of security at an acceptable speed. By merging chaotic maps, the proposed scheme leverages the advantages of a large key space and superior security ensuring a highly secure and reliable encryption solution. This method produces a cipher of the same size as the plaintext. On the other hand, Bensikaddour et al Bentoutou introduced a novel encryption method for satellite images that uses AES and discrete chaotic maps.¹⁵ These maps generate keys and create confusion to enhance security performance, particularly sensitivity to plain image. Kanwal et al. suggest an innovative encryption system-using sine and chaotic tent maps (CTMs) with Circulant matrices for image encoding and decoding.¹⁶ The three stages of this process involve a permutation phase that uses a sine map, and a diffusion phase that utilizes a CTM. The third phase includes a substitution technique, seamlessly integrating the prime Circulant matrix and Hill cipher to enhance security.

Xishun et al. suggest a novel image encryption technique that utilizes DNA computing alongside the Kronecker product.¹⁷ Their findings present their approach as a visible method for encrypting images, demonstrating that it can achieve exceptional levels of efficiency and confidentiality. Wang Xing Yuan et al. combine several encryption keys with the Kronecker product to offer an additional layer of security.¹⁸ Wu et al. present an effective image encryption approach that utilizes DNA computing and a trio of unique 1D chaotic maps, such as logistic sine map, logistic tent map, and sine tent map.¹⁹ Samsul Arifin et al. proposed a novel digital image encryption algorithm combining a logistic map, a unimodular matrix, and several Hill ciphers.²⁰ The image's pixels are shuffled using a unimodular matrix and encryption keys are computed using the logistic map. By implementing various encryption layers, the suggested algorithm may boost the safety of the image's encrypting procedure.

Research work that successfully integrates the kronecker XOR product, chaotic map, and Hill cipher for image encryption is rare. By utilizing the distinct advantages of each method, this blend of methods promises to deliver formidable security for images. Moreover, this combination is designed to fend off plaintext attacks using several secret keys. The Hill cipher is preferred for image encryption over more conventional techniques like DES, IDEA, and RSA because of its capacity to manage the high pixel correlation and redundancy that are the characteristics of an image. Because the Hill cipher applies algebraically to pixel blocks, it is more effective at upsetting the predictable patterns found in image data. Processing a group of pixels at once, this block-based method efficiently lowers redundancy and improves security, resulting in more reliable encryption for multimedia content.²¹ Symmetric key encryption – converges with a chaotic map, a form of mathematical operation that enables the creation of complex nonlinear transformations and the Kronecker product. The latter is recognized for producing randomness.^22–24

Satellite images frequently include vital details about a country's assets, infrastructures, and key landmarks. Illicit access to sensitive information could result in hazards to national security, spying, or even financial harm. Thus, the necessity to shield this private data from attackers is the driving force behind the encryption of satellite images. Since they are widely used in industries including disaster relief, crops, and defense, encryption is crucial in protecting the data because it reduces the possibility of eavesdropping during transmission.²⁵ Large-scale, multispectral data and incorporated geospatial information are characteristics of satellite images. They are sophisticated and data-rich because they cover large geographic regions at different wavelengths. For situations like catastrophe recovery and military planning where data integrity and prevention against misuse and manipulation are essential, strong encryption is necessary due to the complex and vibrant nature of geographical information.²⁶

Our proposed technique introduces novelty by integrating our new chaotic-based encryption systems into the core components of satellite image encryption. The infusion of the Hill cipher enhances the robustness while maintaining the simplicity and compactness inherent in our scheme. Leveraging chaotic maps, we strengthen resistance against advanced cryptographic attacks, particularly in the area of noise and differential cryptanalysis.

Storing or transmitting satellite images poses risks of eavesdropping, data breaches, and unauthorized access. Before uploading such data to the decentralized network, encryption is required to prevent it from unauthorized access. This study aims to integrate technologies to ensure transactions are conducted safely and securely.²⁷ Regarding the transmission of satellite images to wireless environments, there is vulnerability to significant security threats. Hence, it is highly advisable to securely transfer satellite image data to a base station using an enhanced and secure guided approach, free from modifications by unauthorized parties. This study aims to manage real-time data gathering and promptly respond to events. This approach prioritizes the enhancement of the security framework ensuring comprehensive data protection and improved utilization for the relevant authorities.²⁸

Contribution of the proposed study

We highlight the following significant contributions in this paper:

• An innovative satellite image encryption algorithm is laid out. Our methodology integrates the Hill cipher, hyperbolic tangent tent map, and Kronecker bitwise exclusive XOR product matrix techniques. Precisely, we altered the hyperbolic tangent tent map to create intricate pseudo-numbers critical to encrypted images. This modification enabled our approach to generate intricate robust pseudo-random sequences that support strong image encryption. Robust image security is achieved various strengths including the Hill Cipher, hyperbolic tangent tent map, and Kronecker xor product matrix. It is anticipated that this work will significantly advance the field of image encryption.
• The well-known Kronecker product technique is frequently utilized in digital watermarking, encoding and compression processes. In the context of the current research, we revised the Kronecker product method utilizing XOR operator rather than the product operator. This replacement makes it considerably feasible to have complicated and large keys. The byproduct of this process is that the complex keys will be merged using XOR operation with the keys produced by the hyperbolic tangent tent map. This work represents a new use of the xor operator and Kronecker product for image encryption.
• This paper also contributes by thoroughly testing and evaluating the encrypted image, which was done and compared to previous studies of a similar nature. Based on test outcomes, the suggested image encryption technique meets every requirement for successful encryption. The proposed method's effectiveness was demonstrated by contrasting it against various approaches, giving rise to secure images from any potential hazards.

The structure of this article is as follows. Section 2 gives the workflow of the proposed methodology. Section 3 contains the fundamental concepts and contextual information to comprehend this suggested image encryption. Section 4 presents the proposed framework and Section 5 is about the performance evaluation and numerical assessment. The research paper is summarized in section 6, which further provides a few suggestions for future research areas.

The encryption and decryption processes are addressed in the next section, which offers a thorough rundown of the methods used and their operating flow.

WORKFLOW OF ENCRYPTION AND DECRYPTION PROCESS

The suggested encryption method includes multiple major phases to safeguard images. Initially, a plain image is altered according to a calculated shift value. Following these alterations, paired rows undergo XOR operations with the rows' fixed left pixels playing a part in the process. Simultaneously, an intermediate cipher text is produced by generating a circulant matrix, which functions as the secret key for the Hill cipher substitution. To improve security, a hyperbolic tangent tent map also uses secret keys. After undergoing several XOR processes, the intermediate cipher text is expanded using the Kronecker XOR method. The rigorous procedure produces an image encrypted with a strong defense against unwanted access. Our suggested encryption schemes' workflow is shown in Figure 1. The decryption procedure of our suggested encryption strategy is shown in Figure 2, which outlines the process to reverse the encryption to recover the original satellite image.

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The subsequent section will provide and discuss the foundational mathematical ideas necessary to comprehend the core principles of our encryption and decryption techniques.

PRELIMINARY MATHEMATICAL CONCEPTS

The suggested image encryption framework makes use of several mathematical concepts. This section comprises an overview of them.

Chaotic tent map

The chaotic tent map is a discrete-time dynamical phenomenon²⁹ that exhibits chaotic behavior. The equation that describes the chaotic tent map is formulated as follows in Equation 1. 1 $y_{n+1}=\mathrm{ƒ}\left(y_n,\mathrm{\rho }\right)$ where $\left(y_n,\mathrm{\rho }\right)$ is defined as, $f\left(y_n,\mathrm{\rho }\right)=\left\{\begin{array}{ll}\rho \left(y_n\right)& \mathit{\text{if}}0\le y_n<0.5\\ \rho \left(1-y_n\right)& \mathit{\text{if}}0.5\le y_n<1.\end{array}\right.$

While the control parameter $\rho$ lies between [0, 2], the initial condition of the chaotic tent map is represented by $y_n$. As the system traverses via this specific range, the parameter ρ strongly impacts the system behavior, indicating how quickly it will transition from an ordered to a chaotic state. Figure 3 illustrates the chaotic tent property's bifurcation analysis.

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The following three secret keys and related control parameters can be selected for encrypting an image of 256 × 256 pixels. The first key K = ($k,k_1,k_2$) is utilized in the substitution process by randomly choosing integers from the interval [0–255] where a prime circulant matrix is generated by using k₁, k₂. By preserving this constant value inside the defined interval [0, 1], we ensure that the outcomes are robust and consistent throughout various algorithm runs. The uniformity and accuracy of the generated keys depend on this static beginning value.

Analyzing the chaotic tent map dynamic behavior, as illustrated in Figure 3A, shows that the structure covers a significant range that exhibits chaotic dynamics when it experiences the bifurcation phenomenon. Owing to its built-in randomness, sensitivity to initial values, and favorable range of chaotic parameters, the chaotic tent map serves a purpose in developing chaotic encryption algorithms. Figure 3B shows the Lyapunov exponent of CTM.

Hyperbolic tangent function

The hyperbolic tangent function, denoted as tanh(y), is a mathematical function that transforms input values within a range between −1 and 1.³⁰ It can be expressed mathematically as Equation 2. 2 $ƒ(y)=\mathit{tanh}x=\frac{e^y-e^{\text{–}y}}{e^y+e^{-y}}$ where the output is denoted as $ƒ(y)$, e represents the numerical constant and y denotes the input value. Figure 4 displays the tanh function expressed as an S –shaped graph. The curve gradually flattens as the input values get more positive or negative because the function's slope progressively approaches zero. Though this trait assists in controlling the output, it can lead to a common problem known as vanishing gradients. Tanh function has many advantages, even though the drawback of vanishing gradients is a hindrance. Because of its zero-centered output, bias issues during gradient updates are minimized, resulting in more successful convergence. By fusing hyperbolic tangent function and chaotic tent map, this study seeks to address this drawback.

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Prime circulant matrices

A circulant matrix is a unique kind of matrix in which every row experiences a cyclic shift to the right with respect to the row vector that precedes it prior, resulting in Circulant pattern within the matrix.³¹ Therefore, a Circulant matrix can be expressed as Matrix 3, 3 $\left[\begin{array}{lllll}{c}_0& c_1& c_2& \cdots & c_{n-1}\\ c_{n-1}& c_0& c_1& \cdots & c_{n-2}\\ c_{n-2}& c_{n-1}& c_0& \cdots & c_{n-3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ c_1& c_2& c_3& \cdots & c_0\end{array}\right]$

Denoted by cir $\left(c_0,c_1,c_2,\cdots c_{n-1}\right)$. Figure 5 illustrates a random matrix and a Circulant matrix.

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When the row vector's GCD equals 1, the resulting m × n circulant matrix is prime circulant. The prime circulant of a 4 × 4 circulant matrix with row vectors (a, b, c, d) is determined by gcd (a, b, c, d) = 1.

Kronecker product

The Kronecker product,¹⁷ additionally referred to as direct or tensor product, is a binary matrix operator that combines two matrices of arbitrary dimensions, an n × m matrix (referred to as $A_{n\times m}$) and a p × q matrix (referred to as $B_{p\times q}$) to generate a larger matrix with a distinctive block structure is expressed as Matrix 4. After the matrices have been altered, a new matrix having dimensions of (np) × (mq) with specific block patterns is produced. It is denoted by the symbol ⊕. $A={\left[\begin{array}{lll}{a}_{1,1}& \cdots & a_{1,m}\\ ⋮& \ddots & ⋮\\ a_{n,1}& \cdots & a_{n,m}\end{array}\right]}_{n\times m}B={\left[\begin{array}{lll}{b}_{1,1}& \cdots & b_{1,q}\\ ⋮& \ddots & ⋮\\ b_{p,1}& \cdots & b_{p,q}\end{array}\right]}_{p\times q}$

Their Kronecker product, 4 $A\oplus B={\left[\begin{array}{lll}{a}_{1,1}B& \cdots & a_{1,m}B\\ ⋮& \ddots & ⋮\\ a_{n,1}B& \cdots & a_{n,m}B\end{array}\right]}_{\mathit{np}\times \mathit{mq}}$

The kronecker product has wide applications in many disciplines particularly computer science, signal processing, quantum mechanics, and linear algebra. Originating in the 19th century, the Kronecker product concept has been thoroughly examined and utilized in various fields for a considerable time. This paper presents an innovative approach named the kronecker xor product that can be executed to enlarge an image's dimension from n × n to n² × n².

The suggested algorithm's formulation and execution will be covered in full in the next section, along with the algorithm's concept and implementation.

DEVELOPMENT AND EXECUTION OF THE PROPOSED ALGORITHM

To encode and decode messages, a cryptosystem system specifically to facilitate secure communication- uses a collection of algorithms and protocols. This study presents an enhanced encryption satellite image method that uses multiple encryption techniques. Our suggested approach commences with the confusion and diffusion techniques. The steps taken in this study are outlined in the following order.

The encryption process

• A plain RGB image with $256\times 256$ dimensions is chosen.
• Encryption Stage 1: Permutation Phase

The initial step in preserving the global spatial characteristics of the original image is to calculate the row sums. A normalized factor is used to secure uniform scaling of perturbations throughout the image. The formula outlined in Figure 6 determines the magnitude of perturbation for each with the cumulative values of all the pixels along each row given by the row sum. For making appropriate modifications, the pre-determined shift amount functions as a guide.

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• Encryption stage 2: Diffusion of Image Pixels

Every row retains its value at the leftmost pixel, the bitwise exclusive OR operations are performed for odd rows, omitting the leftmost value, between the preceding value of the current row and the corresponding pixel values in the even row. In the same way, the bitwise exclusive OR operations are performed for even rows, not including the leftmost value, between the current row's left value and corresponding pixel values in the odd row to obtain diffused image $P$. The technique of executing the exclusive bitwise OR operation upon a matrix of bits is illustrated in Figure 7.

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• Encryption stage 3: Substitution Phase using Hill Cipher and Circulant Matrices

Hill cipher²⁴ is a polygraphic substitution cipher with a foundation in linear algebra formulated by mathematician Lester S. Hill. An invertible key matrix is required to decrypt data. Archaya et al. suggested implementing a self-invertible matrix to this decryption constraint to meet this need.³² The encryption key utilized in this method is generated using circulant matrices and Algorithm 1 demonstrates the manner in which it is obtained.

1

Algorithm

Generation of Key

Input: k, k₁ and k₂ are randomly chosen integers from the interval [0, 255]; S = ($k,k_1,k_2$). Validate that gcd ($k$, 256) =1 and gcd ($k_{1,}{k}_2$) =1.

Output: Circulant Matrix $K.$

• Formulate the matrix $K_{11}$ by setting up $k_1$ and $k_2$ in a $2\times 2$ arrangement as $K_{11}=\left[\begin{array}{ll}{k}_1& k_2\\ k_2& k_1\end{array}\right]$

• Compute other matrices as follows

  $K_{12}=k\left(I{K}_{11}\right)$ mod 256

  $K_{21}=\left(k^{-1}\left(I+K_{11}\right)\right)$ mod 256

  $K_{22}=-K_{11}$ Mod 256,

  where $I$ is the $2\times 2$ identity matrix.

• Generating a self-invertible $4\times 4$ matrix as $K=\left[\begin{array}{ll}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$

The key $K$ generated in Algorithm 1 is employed throughout the substituting phase for the Hill cipher. The diffused image array, $P$ is split into $\frac{L}{4}$ sub matrices, each sized $4\times 1$ denoted by $P_i$ where $i=1,2,\dots \frac{L}{4}$ is the $i^{\mathit{th}}$ matrix. These sub matrices are multiplied by $K$ one at a time. Hill cipher is employed by means of the following formula in Equation 5. 5 $A_i=\mathrm{mod}\left(\left(P_i\times K\right),256\right)$

Combing $A_1,A_2$ and so forth up to $A\left(\frac{L}{4}\right)$ form a one- dimensional array indicated as cipher text $C$ that features all of the $A_i$ matrices.

• Encryption stage 4: Fusion of Non-linear transformation with Chaotic Dynamics for Key Generation

  The hyperbolic tangent tent map will be employed to generate secret keys $T_1$ and $T_2$ in the subsequent phase. Following that $T_1$ is exclusively xored with cipher text $C$ forming an intermediate cipher text $C_1$ which is illustrated in the following Equation 6.

6 $C_1=T_1\otimes C$

The mathematical formula that yields the secret keys $T_1$ and $T_2$ is specified in Equation 7 which contributes to rendering the secret keys more reliable and distinct. 7 $T_1\&T_2=\left\{\begin{array}{l}\left|\left.\rho \times \frac{\tanh (y)+1}{2}\right|\right.\\ \left|\left.\rho \times \left(\left.1-\frac{\tanh (y)+1}{2}\right)\right..\right|\right.\end{array}\right.\begin{array}{c}\mathit{\text{if}}0\le y<0.5\\ \mathit{\text{if}}0.5\le y<1\end{array}$

• Encryption stage 5: Kronecker XOR Matrix Expansion

  C₁ Undergoes conversion via Kronecker xor product leading to the image size expansion from $m\times m$ to $m^2\times m^2$. Bitwise exclusive OR is applied to all elements in $C_1$ preserving the other elements in $C_1$ at their original position. The image that arises as a result of this process is referred as $C_2$. A part of $C_1$ is a $2\times 2$ block matrix on which Kronecker xor product operates. This process is intended to strengthen the process's defenses against potential breaches while strengthening its confidentiality. Figure 8 depicts the expansion of a $2\times 2$ matrix into a $4\times 4$ matrix.

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• Encryption stage 6: Final Encrypted Image

  The conclusive stage results in the final encrypted image $E$ through the execution of an exclusive OR operation between the secret key $T_2$ and $C_2$ as illustrated in Equation 8 below.

8 $E=T_2\otimes C_2$

The decryption process

To retrieve the original image, we implement a reverse encryption technique. Our encryption scheme strikes an appropriate safety alongside usage, enabling a reliable decoding technique while upholding essential safety precautions. The steps involved in the decryption process are as follows

• Start with the $m$ by $n$ encrypted image $E$.
• Decryption stage 6: Utilizing the secret key $T_2$, derived from the hyperbolic tangent tent map, execute a bitwise exclusive OR operation between $T_2$ and encrypted image $E$ to obtain an intermediate image $C_2$ as followed in Equation 9.

9 $C_2=T_2\otimes E$

• Decryption stage 5: $C_1$ emerges after the intermediate image $C_2$ compresses by the kronecker xor operation.
• Decryption stage 4: The image $C_1$ and secret key $T_1$ generated by the hyperbolic tangent tent map need to be bitwise exclusively xored to produce a further new image $C$ as followed in Equation 10.

10 $C=T_1\otimes C_1$

• Decryption stage 3: Decrypt the newly derived image $C$ utilizing inverse hill cipher along with key $K$ to retrieve the undiffused image $P$ by the following formula in Equation 11.

11 $P_j=\mathit{mod}\left(\left(C_j\times K\right),256\right)$

Indicate all $P_j{\prime }_S$ in the form of 1D array $\mathit{DP}$.

• Decryption stage 2: Execute an exclusive bitwise OR operation on $\mathit{DP}$ between the leftmost pixel values of an odd row and corresponding pixel values in even rows. Likewise, execute an exclusive bitwise OR operation between the leftmost pixel value of the even row and corresponding pixel values in an odd row, preserving the leftmost value of each row in $\mathit{DP}$.
• Decryption stage 1: Compute the perturbation magnitude for each row using the cumulative pixel values along the row as a base. The pre-determined shift amount calculated as the difference between row sums and the minimum row sum divided by the normalization factor, directs the appropriate modifications to retrieve the original spatial characteristics of a plain image.

In the next section, using a variety of performance metrics and tests, we will evaluate the efficacy and efficiency of the suggested satellite image encryption technique.

PERFORMANCE ASSESSMENT OF THE PROPOSED SATELLITE IMAGE ENCRYPTION

This section focuses on thoroughly examining the suggested image cryptosystem's performance evaluation. The results of different tests will be examined to verify the security of the proposed image cryptosystem against various types of attacks. The satellite images used in this study were sourced from Digital Globe Product Samples.³³ For testing purposes, the key values are set as K = (5, 7, 11) and the value of ‘ρ’ is set to 1.985 with a precise initial value of CTM as 0.479.^34,35 Many widely used performance evaluation metrics are utilized in this work from the existing research.^13,36,37 The Table 1a displays the mathematical expressions for these metrics.

TABLE 1a Metrics for performance evaluation and their mathematical representations.

Metric	Mathematical Formulae
MSE	12 $\sum _{i=1}^m\sum _{j=1}^n\frac{{[P(i,j)-\hspace{0.17em}\hspace{0.17em}E(i,j)]}^2}{m\times n\times 3}$ Where $P(i,j)$ represents the plain image's pixel value and $E(i,j)$ represents the encrypted image's pixel value.
PSNR	13 $\log \frac{255\times 255}{\sqrt{\mathit{MSE}}}$
Entropy	14 $\sum _{i=0}^{255}p(i){\log }_2\left(\frac{1}{p(i)}\right)$ Where $p(i)$ represents the probability of whether pixel $i$ exists.
CC	15 $\gamma (x,y)=\frac{\mathit{cov}(x,y)}{\sqrt{\rho (x)}\sqrt{\rho (y)}}$ Where $\mathit{cov}(x,y)=\frac{1}{N}\sum _{I=1}^N\left(x_i-\mu (x)\right)\left(y_i-\mu (y)\right)$ $\rho (x)=\frac{1}{N}\sum _{i=1}^N{\left(x_i-\mu (x)\right)}^2$ and $\mu (x)=\frac{1}{N}\sum _{i=1}^N\left(x_i\right)$
NPCR	16 $\frac{\sum _{i=1}^m\sum _{j=1}^{n}K(i,j)}{m\times n}\times 100$ Where $K(i,j)=\left\{\begin{array}{l}0\\ 1\end{array}\right.\begin{array}{c}P(i,j)=E(i,j)\\ \mathit{\text{otherwise}}\end{array}|i\in [1,\hspace{0.17em}m]\hspace{0.17em}\hspace{0.17em}\&j\in [1,\hspace{0.17em}n]\hspace{0.17em}$
UACI	17 $\frac{1}{m\times n}\sum _{i=1}^m\sum _{j=1}^n\frac{|P(i,j)-E\hspace{0.17em}(i,j)|\hspace{0.17em}}{255}\times 100$
SSIM	18 $\frac{\left(2{\mu }_P{\mu }_E+c_1\right)\left(2{\sigma }_{\mathit{PE}}+c_2\right)}{\left({{\mu }^2}_P+{{\mu }^2}_E+c_1\right)\left({{\sigma }^2}_P+{{\sigma }^2}_E+c_2\right)}$ Where ${\mu }_P\&{\mu }_E$ is mean of plain and encrypted images, ${{\mu }^2}_P\&{{\mu }^2}_E$ is their variance, ${\sigma }_{\mathit{PE}}$ is covariance, $c_1=\left({k_1}^{2}L\right)$, $c_1=\left({k_2}^{2}L\right)$; L is dynamic range of pixel values. $k_1=0.01$ & $k_2=0.03$ by default.

Since the primary focus of our proposed system is satellite imagery, we expanded our testing to include common benchmark images like Lena and Mandrill to ensure the stability and adaptability of the suggested encryption technique. In image processing and cryptography, these images are commonly used as benchmarks. Figure 9 illustrates the performance of our algorithm on these traditional images to demonstrate that it is not only successful for satellite images but also applicable to other prevalent kinds of images used in research.

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Visual and histogram analysis

Using the human visual system (HVS) to compare an encrypted image with its plain counterpart is a fundamental technique for assessing the performance of an image cryptosystem. A variety of plain images along with their corresponding encoded forms created with the proposed image cryptosystem are shown in Figures 10–13 along with their corresponding histograms. The encrypted images' histograms exhibit nearly identical uniform distributions with the plain images' histograms each showing distinct statistical characteristics that make it challenging to identify which plain image they belong to. These findings demonstrate how well the proposed encryption algorithm protects against statistical attacks.

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The chi-square test is used to assess the uniformity of the histogram. Table 1b shows the result of ${\chi }^2$ test for different encrypted images.³⁸ It appears that the proposed algorithm accepts the null hypothesis with p-values greater than 0.05 (5% significance) for the encrypted images showing the uniformity of the histogram.³⁹ Therefore, the redundancy of the original image is completely hidden confirming the resistance against the statistical attack.⁴⁰ The ${\chi }^2$ can be computed using Equation 19. 19 ${\chi }^2=\sum _{i=1}^{256}\frac{{\left(O_i-P_i\right)}^2}{P_i}$

TABLE 1b ${\chi }^2$-test results for the assessment of histogram uniformity.

Metrics	Waterloo image	Flood in Pakistan image	Gornoural'skii image	Satellite Image
Chi-Square Value	262.9102	242.0332	270.7578	232.2969
p-value	0.5465	0.4369	0.1254	0.2465
Decision (H = 0 or 1)	0	0	0	0
Accepted	Accepted	Accepted	Accepted

In this case, Pi denotes the predicted frequency (the theoretical probability distribution for that value) and Oi is the observation frequency (the count of pixels with a certain value in the encrypted image). Note that the chi-square value for RGB images is derived from the average of the R, G, and B channels of the images.

Lossless and perceived encryption analysis

To initiate the statistical analysis, the mean squared error (MSE) and the peak signal-to-noise ratio (PSNR) are computed to access the encrypted image quality to ensure that the original image matches it precisely after an encryption algorithm has been implemented.⁴¹ MSE analyzes the variance of the original and encrypted images from one another; a higher MSE value signifies that the two images are substantially distinct from their predecessors while a relatively low MSE score means that both images are identical. MSE is inversely correlated with PSNR and forms the basis for PSNR. A low PSNR value indicates a notable distinction between the plain and the encoded images. The efficiency of the suggested image encryption scheme is demonstrated in Table 2 showcasing high MSE and low PSNR values than the methods in Refs. [42, 43]. Images with a resolution of 256 pixels by 256 pixels have been utilized for this evaluation. These results indicate that the proposed approach successfully preserves image confidentiality, emphasizing that it has the potential to advance in the field of image encryption. Beyond pixel-by-pixel variations in images, as traditional metrics like MSE and PSNR concentrate on, one such metric is the Structural Similarity Index Measure (SSIM), which assesses the degree to which two images appear in terms of structure and texture, and pixel intensity. Less image distortion is indicated by higher values obtained from the SSIM, which ranges from 0 to 1. An ideal encryption outcome should produce an SSIM value near zero, signifying that the encrypted image is completely unlike the original image. The equation provides the formula for calculating the SSIM. Its numerical values compared to other encryption schemes are demonstrated in Table 3. It should be noted that the average of the R, G, and B channels in the calculations of the PSNR, MSE, and SSIM values for the image of Lena is considered.

TABLE 2 Comparison of MSE and PSNR of Lena's image.

Image Encryption Scheme	MSE	PSNR
Ref. [42]	7747.30	9.23929
Ref. [43]	6147.51	10.3167
Proposed Scheme	8701.35	8.76267

TABLE 3 Comparison of SSIM of Lena's image.

Image Encryption Scheme	SSIM
Ref. [44]	0.000106
Ref. [45]	0.000021
Proposed Scheme	0.0075615

Information entropy analysis

Referring to the extent of disorder or randomness in the distribution of pixel concentrations within an image cipher, the concept of entropy is utilized. Higher security is indicated by greater entropy values when analyzing image encryption. An entropy value that approaches the optimum value of 8 resistant to brute force attacks. A comparison of entropy values for different encryption schemes can be observed in Table 4. The entropy calculation shows that our encryption algorithm's entropy is comparable to the standard entropy factor. This validates that the proposed cryptosystem keeps every bit of information intact. The average of the R, G, and B channels in the calculation of entropy values for the image of Lena is considered.

TABLE 4 Comparison of entropy values of Lena's images.

Image encryption scheme	Entropy values
Ref. [46]	7.8879
Ref. [47]	7.5301
Ref. [48]	7.9973
Proposed Scheme	7.9982

Correlation coefficient analysis

The correlation analysis is the method used to assess the magnitude and trajectory of two variables. With a range of $-1$ to $+1$, the correlation coefficient shows the extent to which the two variables link together and the magnitude of the relationship between them. A weaker relationship between the variables becomes apparent as the correlation coefficient gets progressively closer to 0, $+1$ indicates a positive correlation, and a perfect negative correlation is represented by a value of −1. To ensure the safe retention of the image, a strong image encryption scheme ought to produce a minimal correlation among neighboring randomly distributed pixels. We assess the pixel correlation values of the original and encrypted images in the diagonal, vertical, and horizontal directions. Figures 14–16 display the row, column, and diagonal correlation of the pixels in the original and encrypted Waterloo images.

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Table 5 presents the correlation values between adjacent pixels in the original and encrypted versions of Waterloo determined for each color channel individually. Note that the average value of the R, G, and B channels for waterloo image is used to calculate the correlation value.

TABLE 5 Correlation coefficient values for original and encrypted Waterloo image.

	Red	Green	Blue
Direction	Original	Encrypted	Original	Encrypted	Original	Encrypted
Horizontal	0.9188	0.1358	0.9119	0.1329	0.9347	0.1227
Vertical	0.9142	0.1203	0.8951	0.1461	0.9218	0.1311
Diagonal	0.8598	0.0757	0.8440	0.0708	0.8805	0.0703

Differential attack analysis

Differential attack analysis, a cryptanalytic strategy used to examine how particular modifications made to the input plaintext within the encryption algorithms impact the final resultant ciphertext.^49,50 Consequently, altering a single pixel in the original image ought to result in an entirely unique encrypted image. The number of pixel change rate (NPCR) for comparing individual pixels and the unified average change intensity (UACI) to figure out the average difference are the two tests that the literature proposes to fulfill these requirements. Encryption with a larger value of UACI is considered more beneficial. A large valued NPCR is required for a good image encryption scheme.⁵¹ These critical values have been determined as the minimal necessary condition to ensure the safety and privacy of the encrypted image through examination and testing. Consequently, it is crucial to ensure each of the UACI and NPCR scores surpass specified threshold to be able to classify an image encryption technique as safe.⁵² A contrast between the NPCR and UACI scores gathered from the present study and the ones from related studies is presented in Table 6. The average of the R, G, and B channels in the calculations of NPCR and UACI values for the image of Lena is considered.

TABLE 6 Comparability of NPCR and UACI values of Lena's images.

Image Encryption Scheme	NPCR	UACI
Ref. [53]	99.5855	30.3873
Ref. [54]	99.6200	33.6400
Proposed Scheme	99.6038	33.5333

Noise and data loss attacks

The resilience of an encryption scheme to data loss and noise is vital when transmitting cipher images over open channels. The encryption method employed in this study significantly enhances its robustness. In the case of a pixel error in the cipher image, the affected portion will fail to decrypt, and the error may be transmitted to adjacent pixels due to the diffusion property. However, the error contained within the affected portion or pixel could be limited, ensuring that the remaining pixels in the cipher image can be correctly decrypted, thereby minimizing the impact of data loss or noise, even at 90% data loss.

Within the web-based information exchange, it is important to note the risk of data loss or obscuration due to irrelevant and unrelated data which can also be known as noise. Inadequate handling of satellite imagery may diminish a satellite's ability to transmit and extract important data for further analysis or interpretation. In real-world interactions, the most important aspect of an encryption technique is its noise resistance. The suggested algorithm can withstand noise. The effectiveness of any algorithm is illustrated with both salt-and-pepper noise and Gaussian noise.⁵⁵ The NPCR and UACI methods identify the differences between the original and decrypted images. The NPCR and UACI values decrease with increasing noise resistance. The effect of introducing salt-and-pepper noise to the encrypted image generated by the proposed cryptosystem is depicted as a noise attack in Figure 17. Table 7 compares the findings of the noise attack evaluations conducted by salt and pepper. Based on NPCR and UACI values, Table 7 shows that the suggested method performs better than the method outlined in.⁵⁶ As a result, the proposed approach resists the salt and pepper noise attack better than the techniques in.⁵⁶ The findings of the Gaussian noise attack analysis with mean = 0 and variance with various degrees are shown in Figure 18. Table 8 compares the findings of the noise attack evaluations conducted by Gaussian noise based on NPCR and UACI values.

[IMAGE OMITTED. SEE PDF]

TABLE 7 Comparison of Salt-and-Pepper noise attack for the average of Lena and Mandrill Images.

	Noise 10%		Noise 25%		Noise 5%
Algorithms	NPCR	UACI	NPCR	UACI	NPCR	UACI
Ref. [55]	9.7549%	5.3951%	24.2918%	12.5001%	4.8592%	2.5034%
Proposed Scheme	9.9675%	5.0117%	24.7706%	12.4957%	54.9952%	2.4999%

[IMAGE OMITTED. SEE PDF]

TABLE 8 Comparison of Gaussian noise attack for the average of Lena and Mandrill Images.

	Variance 0.00019		Variance 0.0003		Variance 0.0005
Algorithms	NPCR	UACI	NPCR	UACI	NPCR	UACI
Ref. [57]	82.0104%	0.7685%	88.3885%	1.3320%	90.3325%	1.7234%
Proposed Scheme	99.5422%	0.0031%	99.5255%	0.0053%	99.5463%	0.0068%

Similarly, to retrieve the original image, we tested our proposed encryption scheme on an encrypted satellite image with data loss. This test was carried out to evaluate variations in data loss rates: 90%, 59%, and 20%. We assessed the decryption results of encrypted images with different degrees of data loss, as illustrated in Figure 19.

[IMAGE OMITTED. SEE PDF]

Key sensitivity analysis

An essential feature of cryptographic system evaluations, concentrating on how sensitive encrypted data is to changes in encryption. This is because even a minor alteration in the key can impact on the outcome. The encrypted image reveals no hints or clues regarding the original image.⁵⁸ With an initial value $x_0=0.479$, we assessed our proposed algorithm during the encryption phase. Later on in the decryption phase, we made a few slight adjustments, setting $x_0$ within the range of $0.479\pm {10}^{\pm 14}$ as demonstrated in Figure 20. Our findings demonstrate our encryption algorithm's increased sensitivity to key modifications, highlighting its efficacy and dependability in protecting confidential information.

[IMAGE OMITTED. SEE PDF]

Key space analysis

By using three distinct keys, the suggested encryption algorithm greatly expands the key space and increases the difficulty for unauthorized parties to decrypt the encrypted data. In addition to meeting the requirements, the suggested algorithm's use of three keys exceeds IEEE standards,⁵⁹ making it a perfect and safe choice for encryption. These keys are T₁ and T₂ produced from the hyperbolic tangent tent map, and C, which serves as a confidential key for the hill cipher. The IEEE's guidelines⁶⁰ state that strong encryption requires a key space of at least 2¹⁰⁰. A 4 × 4 circulant matrix serves as the source for the Hill cipher key. Values ranging from 0 to 255 can be assigned to each element in the key matrix. Since there are 16 elements in the 4 × 4 matrix, the number of possible keys for the Hill cipher is 256¹⁶.⁵² In the case of a hyperbolic tangent tent map (T₁ & T₂), the values of 0 or 1 can be assigned to each position in the key matrix. Consequently, there would be 2¹⁶ combinations for each key matrix. So the total key space would be 2¹²⁸ × 2¹⁶ × 2¹⁶ = 2¹⁶⁰. Table 9 demonstrates the proposed algorithm comparison with other studies.

TABLE 9 Key space comparison.

	Proposed	Ref. [32]	Ref. [61]
Key space size	2160	2149	2136

The study's major conclusions are summed up in the conclusion, along with a discussion of the consequences of the suggested encryption technique and possible directions for further research in the next section.

CONCLUSION

This work offers an innovative encryption technique that improves the protection of sensitive digital evidence including satellite images and is a noteworthy contribution to the field of digital forensics. Safeguarding information against illicit access is a major challenge in the real-world application of satellite imagery. Researchers are constantly challenged to develop novel algorithms that strengthen vital satellite data against possible cyber breaches as threats from malicious individuals keep evolving. Using chaotic systems to ensure transmission confidentiality, this paper aims to make a research contribution to the field of encrypted satellite images. The current research offers a useful instrument that can be used in forensic procedures to guarantee the confidentiality and integrity of sensitive data.

The suggested method consists of image scrambling, encryption using the Hill cipher, and the encryption of satellite images using a chaotic tent map and Kronecker xor product techniques. Specifically, the hyperbolic tangent tent map produces sophisticated, pseudo-random integers perfect for encrypting data. This contributes to the security of the suggested system and advances our knowledge of satellite image encryption. A thorough numerical analysis of multiple experimental tests has been conducted, demonstrating the long-term reliability of the recommended algorithm in opposition to different attacks, including statistical and differential attacks. The image encryption method we have suggested is a big step forward for the industry and provides a trustworthy and efficient means of protecting private data.

While the suggested plan has several benefits, there is still an opportunity for future advancements. Although classical chaotic systems have been extensively used, they possess some inherent drawbacks, including periodicity, vulnerability to phase space destruction, and low Lyapunov exponents. To overcome these limitations, researchers have been discovering ways to improve the chaotic behavior of classical systems via a process known as chaotification. The main objective of chaotification is to address these weaknesses and enhance the efficiency of chaotic encryption procedures. A possible future direction for this study could include substituting classical chaotic maps with chaotified maps, which may further improve the performance and security of the proposed encryption technique. Further, as a future study, any quantum key distribution technique⁶² can be combined for the secure key distribution with the current work.

The important findings from our performance analysis are displayed in the Table 10. The whole work including a comparison of the findings with the methods of other researchers is summarized in Table 11. Future work could enhance our encryption technique by integrating AI algorithms to further boost security. AI's impact in fields like diabetic retinopathy detection, cancer detection, and bio mathematical challenges shows its potential to create more adaptive encryption methods.^63–68 Incorporating AI-driven chaotic maps or real-time threat detection, along with quantum key distribution, could significantly improve the protection of satellite imagery and other sensitive data.

TABLE 10 Key takeaways of the proposed encryption scheme on Satellite images.

Tests	Proposed Scheme
Entropy	7.9973
MSE	8711.35
PSNR	8.66257
SSIM	0.0074515
NPCR	99.6043
UACI	33.4321

TABLE 11 Summary of the proposed work.

Tests			Algorithms
	Ref. [46]	Ref. [47]	Ref. [48]	Proposed Scheme
Entropy	7.8879	7.5301	7.9973	7.9982
	Ref. [42]	Ref. [43]	Proposed Scheme
MSE	7747.30	6147.51	8701.35
PSNR	9.23929	10.3167	8.76267
	Ref. [44]	Ref. [45]	Proposed Scheme
SSIM	0.000106	0.000021	0.0075615
	Ref. [53]	Ref. [54]	Proposed Scheme
NPCR	99.5855	99.6200	99.6038
UACI	30.3873	33.6400	33.5333
	Ref. [55]	Proposed Scheme
Salt-and-pepper noise attack
Noise 10%	NPCR	UACI	NPCR	UACI
	9.7549%	5.3951%	9.9675%	5.0117%
Noise 25%	NPCR	UACI	NPCR	UACI
	24.2918%	12.5001%	24.7706%	12.4957%
Noise 5%	NPCR	UACI	NPCR	UACI
	4.8592%	2.5034%	4.9647%	2.5049%
	Ref. [57]	Proposed Scheme
Gaussian noise attack
Variance 0.0001	NPCR	UACI	NPCR	UACI
	82.0104%	0.7685%	99.5422%	0.0031%
Variance 0.0003	NPCR	UACI	NPCR	UACI
	88.3885%	1.3320%	99.5255%	0.0053%
Variance 0.0005	NPCR	UACI	NPCR	UACI
	90.3325%	1.7234%	99.5463%	0.0068%

AUTHOR CONTRIBUTIONS

Shamsa Kanwal: Conceptualization; methodology; writing – original draft. Saba Inam: Methodology; software; data curation. Asghar Ali Shah: Methodology; software; data curation; validation. Halima Iqbal: Methodology; validation; visualization; formal analysis; data curation. Anas Bilal: Methodology; software; writing – review and editing; formal analysis; data curation. Muhammad Usman Hashmi: Validation; formal analysis; writing – review and editing; software. Raheem Sarwar: Methodology; resources; project administration; visualization; investigation.

CONFLICT OF INTEREST STATEMENT

Authors have no conflict of interest relevant to this article.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

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