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There has been a significant increase in the demand for secure image storage in healthcare organizations in recent years. Encryption is used to address the challenge of encrypting sizeable digital image files, as full encryption can be computationally expensive and take a long time to process. In this paper, partial and selective encryption is proposed for medical images. First, deep learning based on U-Net is used to localize a tumor region called (ROI) a region of interest. A diffusion phase of the proposed system handles pixel values and positions based on linear and hyperchaotic systems. It includes converting an image's pixel values and repositioning pixels in a predetermined order. In the confusion phase, one level of Integer Discrete Wavelet Transform (IWT) is applied to divide the scrambled region into four sub-bands. Then, a Feistel network based on polynomial-based secret image sharing (SIS) encrypts the lowest frequency band only while the three bands LH, HL, and HH are diffused using a mapping technique based on the Morton scan to swap coefficients positions and then confused based on the hyperchaotic system. The culmination of these techniques results in generating a test image cipher characterized by robust confusion and diffusion properties. Importantly, this methodology has yielded remarkable results, reducing the encryption time by up to 96%. This efficiency is achieved without compromising the security or quality of the encrypted medical images. as high entropy is attained post-encryption. Furthermore, by employing the Integer Discrete Wavelet Transform (IWT), the integrity and fidelity of the encrypted images remain uncompromised. Additionally, to bolster the level of confusion in the encryption process, a substantial key space of 2^{1628} has been employed, further enhancing the resilience of the encryption method.
medical image, image encryption, IWT, partial encryption, secret image sharing, hyperchaotic system
The proper handling of medical images is of utmost importance due to the sensitive patient and diagnostic information they contain. These images are shared among hospitals, doctors, and patients over public networks, making it essential to have strong security measures for their storage and transmission to protect patients' privacy. However, traditional cryptographic techniques are insufficient for encrypting images due to the unique characteristics of digital image data, such as high redundancy, pixel correlation, and large size. This has increased demand for advanced and specialized image encryption algorithms, as conventional methods are no longer reliable. As a result, many low-complexity encryption techniques explicitly tailored for medical images have been developed to meet this challenge [1].
In contrast to conventional full-image encryption approaches, the proposed method focuses on encrypting only specific portions of the secret image. This selective encryption is grounded because only certain parts of the secret image are sensitive or require confidentiality. Determining the target portion can be manual or automatic [2]. Recent years have witnessed significant interest in chaotic systems, maps, attractors, and sequences within the research community [3, 4]. Applications of chaotic encryption techniques have been widely used for security purposes in various domains, including smart grids and communication systems [5, 6].
Additionally, these techniques are extensively utilized to safeguard different types of content, such as images [7, 8]. Chaotic image encryption algorithms offer the advantage of being easily implementable in software and hardware, contributing to their popularity among scholars. These algorithms typically rely on scrambling and diffusion techniques to obscure the statistical properties of plain images, thereby enhancing the statistical quality of the resulting ciphertext [9]. Transform domain encryption, such as the use of (DCT) discrete cosine transform [10], (FRFT) fractional Fourier transform [11], or (DWT) discrete wavelet transform [12], offers increased security compared to spatial encryption. However, the complexity of such algorithms presents a challenge for potential attackers aiming to decode the data illegally. The presented paper introduces a novel symmetric encryption algorithm that leverages the benefits of transform domain encryption. The proposed work has made significant contributions, which are outlined below:
·A practical image encryption model designed for medical images.
·An efficient diffusion model is adopted based on a linear system and chaotic map.
·Implementation of polynomial-based secret image sharing for encryption of the lowest frequency band.
·Utilization of partial Encryption to address the computational overhead and processing time challenges posed by sizeable digital image sizes.
·Repositioning of pixels within the Region of Interest (ROI) through linear and hyperchaotic systems.
·Creation of a partial encryption system that amalgamates the strengths of the hyperchaotic map, secret image sharing, integer discrete wavelet transforms, and hash function. This system is highly dependent on the initial values and can be easily implemented.
The proposed encryption can cipher all medical images, such as X-rays, CT scans, and MRIs. It can safeguard patients' private and sensitive information within medical images from unauthorized access and maintain confidentiality. A partial or selective encryption scheme is utilized to enhance the speed of encryption medical image algorithms.
An increasing number of medical equipment, such as insulin pumps and pacemakers, is connected to the Internet and other networks. Encrypting the data transmitted between these devices and other systems is essential to protect patient information from unauthorized access. This security measure can help safeguard sensitive medical information and prevent data breaches.
The paper's structure can be outlined as follows: Section 2 will discuss related works, while Section 3 will explain Preliminaries. Section 4 will elaborate on the proposed method, and Section 5 will present the experimental results and analysis. Lastly, Section 6 will outline the conclusions.
Various industries, including healthcare, have significantly transformed the modern digitalization landscape. As a vital aspect of this evolution, telemedicine has revolutionized the exchange of medical information, mainly by sharing medical images. However, the convenience of such information exchange comes with the challenge of ensuring robust confidentiality and security. This is especially critical in the healthcare sector, where electronic health records contain sensitive personal information. Unauthorized access to medical images can lead to fraudulent activities, identity theft, and other privacy breaches. Consequently, there is a pressing need for advanced encryption methods to safeguard these valuable assets.
Overcoming Security Challenges In response to these security concerns, researchers have delved into medical image encryption, seeking innovative solutions to ensure the integrity and confidentiality of patient data. Several works in this domain have explored diverse techniques, each aiming to address specific security and efficiency challenges.
The proliferation of medical information exchange, particularly through medical images, necessitates robust confidentiality measures in telemedicine [13]. This study, conducted in 2021, introduces a novel cryptosystem tailored for telemedicine applications. It employs a hybrid approach to encrypt diverse medical images, enhancing security and key space. By individually scrambling two plain images using chaotic maps, they are then fused using Cramer's rule-based mathematical expression, bolstering the cryptosystem's resilience. Image encryption involves permutation and diffusion via Logistic-May and Henon maps and Logistic-Sine scrambling. Security analysis and experimental simulations confirm the algorithm's efficacy in maintaining robust encryption.
In 2021, a recent study [14] introduced a new medical image encryption algorithm that uses DNA cryptography and is highly robust. The approach involves image masking before encryption, enhancing the original image's randomness without requiring a key. The process includes confusion and diffusion on the masked image, rendering the resulting cipher-image devoid of perceptual or statistical information. The proposed algorithm employs the confusion step in the generalized Arnold's Cat Map and introduces a unique diffusion operating on pixel and DNA-plane levels. The DNA encoding, decoding, and XOR rules are selected based on the values generated by the Chaotic 2D-Logistic Sine Coupling Map. This approach enhances resistance against brute force and statistical attacks. The cipher image remains impervious to intrusion without the correct key while maintaining efficient decryption with the valid key. Extensive testing showcases its effectiveness on medical and natural images, exhibiting low inter-pixel correlation and high entropy. The algorithm also demonstrates robustness against various attacks, ensuring a secure image cryptosystem.
A lightweight cryptosystem has been introduced in the study [15], which integrates the Henon chaotic map, Chen's chaotic system and Brownian motion to provide robust encryption for medical images. The proposed approach has been thoroughly analyzed, covering a range of factors, including histogram analysis, pixel correlation, contrast, mean square error, energy, homogeneity, NIST compliance, entropy, pixel change rate, average intensity change, signal-to-noise ratio, and time complexity. Based on experimental results, this method is highly effective in securing confidential patient image data, making it a reliable solution for medical image encryption.
In 2020, this work [16] addressed the secure transmission of medical images, preserving sensitive regions through an innovative algorithm. Key areas are identified, and data hiding with texture analysis is employed, safeguarded by an Arnold transformation. A QR code replaces the original key regions, ensuring authorized access. The method ensures secure storage and recovery of patient data while controlling image quality. Experimental validation confirms its effectiveness.
In 2020, Yan et al. [17] introduced an Adaptive Partial Image Secret Sharing (APISS) scheme using salience detection, image inpainting, and linear congruence. Unlike classic Image Secret Sharing (ISS), APISS encrypts only the sensitive part of a secret image. The scheme automatically detects the salient part and encrypts it into meaningful shares using linear congruence and image inpainting. Decryption involves progressively adding shares until full lossless decryption is achieved. The method offers efficiency benefits and is validated through experiments.
In 2020, George et al. [18] proposed a selective image encryption approach to enhance efficiency. Unlike traditional methods, which encrypt the entire image, this technique encrypts specific portions to reduce computational complexity and encryption/decryption time. It employs a Discrete Cosine Transform (DCT)--b-based compression algorithm in YCbCr and RGB colour spaces. Results indicate improved image quality and compression ratios using YCbCr over RGB. 1-D DCT reduces transform time significantly compared to 2-D DCT. Adaptive scalar quantization preserves visual quality. DC coefficients are differentially encoded, and AC coefficients are compressed using Run Length Encoding (RLE). Encryption is applied selectively to these bulks, substantially reducing decryption time. The proposed approach efficiently balances compression and encryption, enhancing overall performance.
In 2022, Noori Ghanim and Raheem Khoja [19] applied a one-level Discrete Wavelet Transform to coloured images, dividing them into four subbands. A Gabor filter and K-means clustering enables texture segmentation on the lowest frequency band, serving as a novel scrambling method. Scrambled segments are then encrypted using alternating AES and RC4 Algorithms, involving XOR operations with the key of the other algorithm. Rigorous security and performance analyses reveal the method's resistance to various attacks and its potential for real-time image transmission and encryption.
In 2022, Yousif et al. [20] introduced enhanced criteria for medical image encryption. It eliminates pixel correlation by scrambling and diffusing via polynomial-based secret sharing. Integer wavelet transform encrypts image frequencies with AES, ensuring robust security. The LL part becomes the diffusion image, secured using inverse Haar wavelet transform. Reversing and shuffling frequencies restore the original image. The extensive statistical evaluation confirms the algorithm's resistance against attacks, surpassing other image cryptography methods in security.
In 2022, Salman et al. [21] introduced a secure and efficient algorithm that encrypts predetermined regions to minimize encryption, decryption complexity, and time. Image processing divides images into (ROI) and (RONI); polynomial-based secret image sharing and chaotic mapping techniques are used to encrypt the Region of Interest (ROI) component. This approach ensures effective diffusion and confusion, enhancing concealment. A firm reliance on key improvement is seen in secure decryption. The encryption solution's breadth thwarts brute-force attacks, addressing protection issues during network transmission of medical images.
In 2022, Kiran and Parameshachari [22] presented a low-complexity Block Cipher-based Region of Interest (ROI) medical image encryption with multiple maps. ROI extraction using Laplacian edge detection identifies essential regions. Arnold's cat map circularly permutes ROI, and then the duffling system encrypts it while preserving unimportant parts. The proposed algorithm exhibits robust security against attacks, outperforming recent chaotic encryption methods. Experiments validate its efficacy in safeguarding medical image confidentiality.
In 2023, Natsheh et al. [23] addressed the need for efficient yet robust DICOM image security. A novel selective encryption approach is introduced, automatically identifying regions using pixel thresholding segmentation and then encrypting them based on importance. Single and multi-frame DICOM images benefit from adaptive two-region Encryption, with the Region of Background (ROB) using light encryption and the Region of Interest (ROI) using advanced encryption. Approach I for multi-frame DICOM images achieves substantial time savings and compression, while Approach II for single-frame images enhances efficiency through multi-region selective encryption. Comprehensive cryptanalysis metrics confirm the methods' robustness against diverse attacks. The approach I use shows superior performance based on estimated processing time.
In 2023, Al-Barzinji and Abd Abrahim Mosslah [24] introduced a novel partial image encryption approach focusing on DCT coefficient frequencies. Encrypting all data in specific applications can be impractical due to decryption processing demands. Partial encoding offers a solution, meeting application needs without overwhelming processing. The proposed method preserves public bit value and aligns with the JPEG format. It achieves enforceability, spatial selectivity, self-sufficiency, and coordination compliance, addressing real-time application requirements. The approach's effectiveness is compared to other technologies like quadtree AES.
In 2021, Harshitha et al. [25] proposed a method for encrypting images to protect medical image data. This is achieved using a chaotic logistic map and a linear feedback shift register (LFSR) to generate pseudo-random sequences. The generated sequences are combined using an XOR operation to create an encryption key. This key, along with security transformations, encrypts medical images, benefiting from the chaotic maps' random behaviour and sensitivity to initial conditions for added security. The proposed approach effectively secures diverse medical image formats and resists brute force and man-in-the-middle attacks.
In 2021, El-Shafai et al. [26] addressed the need for medical image encryption in telemedicine and healthcare by presenting an efficient cryptosystem that leverages DNA rules and chaos maps. The proposed approach utilizes a logistic chaos map, a piecewise linear chaotic map (PWLCM), and DNA encoding. The process involves generating a secret key image using PWLCM, encoding images using DNA rules and chaos maps, and iterating through columns for optimal ciphering. Experimental results demonstrate the cryptosystem's robust security, acceptable processing time, and resilience against various attack types.
These endeavours collectively demonstrate the ongoing commitment of researchers to enhancing the security landscape of medical image encryption, catering to the unique demands of telemedicine and healthcare systems.
In conclusion, the increasing reliance on digital technologies in healthcare and telemedicine mandates robust security measures to safeguard sensitive medical images and patient data. The diverse approaches presented in the related works exemplify the innovative strategies employed to address these challenges. From hybrid encryption schemes to DNA-based cryptography and selective encryption methods, each approach contributes to the evolving landscape of medical image security. As telemedicine continues to grow, these advancements promise to ensure patient privacy, data integrity, and secure image communication within the healthcare ecosystem.
As shown in Table 1, the previous work had some gaps in time consumption, lack of efficient similarity measurement, absence of partial methods, and the need for more efficient encryption techniques to provide better entropy levels. In this proposed work, a new and novel approach to providing security for medical images is presented by exploiting and employing the idea of secret image sharing based on polynomials, hyperchaotic maps, and DWT transform for selective and partial image encryption. The proposed algorithm aims to reduce space and bandwidth usage by localizing the vital area (ROI) and focusing on the essential ROI information concentrated in the low parts of frequent DWT. This valuable information is used as coefficients of the secret image-sharing polynomials. The frequency domain is advantageous for encryption as it simplifies finding the crucial areas that need to be encrypted. Connecting between spatial domain pixel values and frequency domain values is challenging, which makes the image safe from numerous assaults, including known plaintext attacks and Ciphertext-only attacks. The DWT reconstruction processes make it impossible to discern between sections that are encrypted and those that are not. To overcome the security issue for high frequencies of DWT (I.H, HL, and HH), the diffusion step is used to hide the correct locations of the transformed image coefficients of these bands.
In the proposed method, the SHA-512 hash values were employed as the system's initial parameters generator for plaintext sensitivity, and a hyperchaotic was used as a subkeys generator for the polynomials as coefficients. High-dimensional hyperchaotic systems are better suited for image encryption due to their intricate structure and reliability in generating suitable. In addition, the algorithm uses the Morton scan traversal technique to strengthen the system.
The suggested encryption was more secure as a result of all these actions than many others that have been documented in the literature. Furthermore, as far as we know, none of the image ciphers found in literature have been created utilizing these acts. Many excellent features of the suggested scheme are evident, such as its high randomness, low processing cost, huge key space, flexible parameter space, significant sensitivity to plaintext and keys, and quick execution speed. Consequently, the suggested plan protects private digital photos against cryptographic assaults.
Table 1. Summary table of related works
Ref. |
Methods |
Limitations |
Achievements |
All/Partial Encryption |
[15] |
Using Henon Chaotic Map (HCM) Brownian Motion (BM) Chen Chaotic system (CCS) |
time-consuming using two different chaotic maps do not use key |
the good Correlation coefficient of the encrypted image is closer to the ideal entropy cipher image |
All |
[19] |
Discrete Wavelet Transform K-means AES and RC4 |
The entropy of the encrypted image is not closer to the ideal time-consuming |
against ciphertext-only attacks. Using real-time application |
All |
[20] |
Discrete Wavelet Transform AES |
using for 2D images |
against ciphertext-only attacks. known plaintext attacks lower security and the file size |
All |
[21] |
Quadratic chaotic map Polynomial-based SIS |
time-consuming for larger images using for 2D images |
reduce the encryption/decryption time, and enhance image processing methods. |
Partial |
[23] |
Selection mechanisms compression AES Lightweight |
time-consuming not using similarity measurement do not use segmentation methods |
combination of lossless compression and segmentation |
All |
[24] |
DCT multiple encryptions Pluralism |
need more than 2 points' to encrypt the image |
trade-off time and power usage reduce the encrypted and decrypted time |
All |
[27] |
Selective Encryption Sharing & Embedding |
time-consuming |
Distinguish sensitive and non-sensitive information of plain images before making encryption |
All / Partial |
3.1 Secret image sharing
Thien and Lin's contribution in 2002 [28] brought forth a novel concept by integrating the secret-sharing scheme proposed by Shamir [29]. In their approach, they harnessed the potential of the secret image to generate a set of n shared images, where restoring the original secret image necessitates k or more shared images. Impressively, k-1 shares remain insufficient to unveil any details regarding the secret image. A key innovation of their method lies in utilizing subsequent pixel values as the coefficients in the polynomial equation, deviating from conventional random numbers. This innovative approach led to a reduction in the size of shared images by a factor of 1/k. e importance of robust security considerations in the design of image-sharing methods.
Thien and Lin's image secret-sharing scheme involves sharing a secret image (S) among participants. The group comprises n participants, each assigned a unique identifier (id_{1}, id_{2}, … , in) in a finite field. The allocator selects n distinct non-zero elements as identifiers and makes them public. The secret image has w pixels, and the allocator takes the grayscale value of the first k pixels in S as the value of a_{0}, a_{1}, … , a_{k−1}. These values are used to form a polynomial. The threshold for the scheme is $\varnothing$, and the correspondence between each participant and their identifier is also public and can be expressed using the following equation:
$\begin{gathered}f(x)=\left(a_0+a_1 x+a_2 x^2+\cdots+\right. \left.a_{(\varnothing-1)} x^{(\emptyset-1)}\right) mod p\end{gathered}$ (1)
The allocator generates n secrets to distribute among n participants, with each secret corresponding to the identifier of the first pixel (S^{i}_{1}) in each shadow image.
$S^i=\mathrm{f}\left(\mathrm{id}_1\right), i=1, \ldots, n$ (2)
The allocator performs the previously described operation on the kth+1 to 2kth pixel in S using the values of a_{0}, a_{1}, …, a_{k}_{−1} to generate S^{i}_{2}, i=1, … , n. The secret sharing process is repeated for all pixels in S, resulting in S^{i}_{w}_{/k}, i=1, … , n.
Thien and Lin's scheme for recovering secret images involves using k shadow images (S_{1}, S_{2}, …, S_{n}) and their corresponding $I D s$ (id_{1}, id_{2}, …, id_{n}) to recover the secret image. This is done by extracting the 1^{st} pixel of each shadow image (S_{1}, S_{2}, …, S_{n}) and applying the formula of Lagrange's interpolation.
$f(x)=\sum_{i=1}^{\emptyset} f\left(x_i\right) \prod_{h \neq i}^{1<h<\emptyset} \frac{\left(x-x_h\right)}{\left(x_i-x_h\right)}$ (3)
have
$f(x)=\sum_{i=1}^{\emptyset} S^i 1 \prod_{h \neq i}^{1<h<\emptyset} \frac{\left(x-i d_h\right)}{\left(i d_i-i d_h\right)}$ (4)
To obtain the grayscale value first to the K^{th} pixel of a secret image as a secret message (a_{0}, a_{1}, …, a_{k}_{−1}), one can use f(x) coefficients. For Lagrange's interpolation, the 2^{nd} pixel of each shadow image (S^{1}_{2}, S^{2}_{2}, …, S^{k}_{2}) can be obtained.
$f(x)=\sum_{i=1}^{\emptyset} S^i 2 \prod_{h \neq i}^{1<h<\emptyset} \frac{\left(x-i d_h\right)}{\left(i d_i-i d_h\right)}$ (5)
The secret image, S, can use the coefficients of the polynomial function f(x). need to extract a_{0}, a_{1}, … , a_{k}_{−1} as the kth+1st to 2kth pixels from f(x). This process is repeated until we have recovered all the pixels in $S$, resulting in the complete secret image. The reference [30] explains this process in detail.
For a better understanding, refer to Figure 1, which illustrates the image secret-sharing construction process for (2, 4) with k=2 and n=4. This method enables the creation of a first-order polynomial function.
$S x(i, j)=(110+112 x)(mod \ 251, 3)$ (6)
The first two-pixel values in the Lena image are 110 and 112 [31].
Figure 1. Thien and Lin’s secret sharing scheme [31]
In the proposed system, use (n, n) secret image sharing with n=8 to generate eight shares as cipher output for the block of eight bytes.
3.2 The hyperchaotic system
The Hyperchaotic System employed in this study is derived from the dynamic system proposed by Natiq et al. [32], which is an enhancement of the system introduced by Liu et al. [33]. The key modification in this hyperchaotic system involves substituting the original cubic nonlinearity nonlinearity with a hyperbolic sine nonlinearity nonlinearity. This change is motivated by the desire to streamline the practical design of electronic circuits, as the hyperbolic sine nonlinearity can be easily implemented using two diodes connected antiparallel. In contrast, cubic nonlinearity nonlinearity requires analogue multipliers, which can introduce complexities and challenges in circuit realization.
The mathematical representation of this hyperchaotic system is as follows:
$\left\{\begin{array}{c}x_1=-\beta_1\left(x_2+0.2\left(x_1-\varepsilon \sinh \left(x_1\right)\right)\right) \\ x_2=\beta_2 * x_1-x_2+x_3+x_4 \\ x_3=\beta_3 * x_2-x_4 \\ x_4=-\beta_4 * x_1\end{array}\right\}$ (7)
In the equations above, $x_1, x_2, x_3$, and $x_4$ represent the system's state variables, while $\beta_1, \beta_2, \beta_3$, and $\beta_4$ are positive parameters. Notably, the complex behaviour exhibited by this hyperchaotic system is primarily associated with the state variable $x_1$, which is linked to the hyperbolic sine nonlinearity.
This modified hyperchaotic system offers a more feasible approach to implementing chaotic behaviour in practical electronic circuits by introducing the hyperbolic sine nonlinearity and its relatively simple realization through diodes. This trade-off between theoretical complexity and practicality is a common consideration in chaos-based research, ensuring that chaotic systems can be effectively harnessed for real-world applications such as secure communication and encryption.
3.3 Morton scan
Morton scan, also known as the Z-order curve, is a method used to traverse and index multi-dimensional data in a linear, one-dimensional sequence. It is beneficial for dividing a 2D space into smaller sub-blocks and then scanning through those sub-blocks in a manner that preserves spatial locality. This is achieved by interleaving the bits of the 2D coordinates of each point to create a single index that reflects the spatial relationship. Coordinates to Index Conversion: Consider a 2D grid where each cell has an X and Y coordinate. To convert these coordinates into a Morton index, you interleave the bits of the X and Y coordinates. This means you take the least significant bit of the X coordinate, then the least significant bit of the Y coordinate, then the second least significant bit of X, the second least significant bit of Y, and so on, until you have used all the bits of both coordinates. This interleaved bit sequence forms the Morton index. Spatial Locality: The essential advantage of the Morton scan is that nearby points in 2D space are likely to have similar Morton indices, which preserves their spatial proximity even in the one-dimensional index space. Sub-Block Division: When applying the Morton scan to sub-blocks, you divide the 2D space into smaller rectangular regions (sub-blocks). Then, you convert each sub-blocks top-left corner coordinates into a Morton index. By scanning through these Morton indices, you traverse through the sub-blocks in a way that maintains their spatial relationship. In your context of applying the Morton scan for each sub-block, you are to use this technique for indexing or accessing data stored in a grid or a 2D structure. Using the Morton scan, you can improve cache utilization and data locality when accessing neighbouring sub-blocks, which can be especially useful in image processing, spatial databases, and similar applications, as illustrated in Figure 2.
Figure 2. Z-order curve "Morton Scan"
3.4 Feistel network
The Feistel cipher takes Horst Feistel's name and utilizes a symmetric key architecture. It reverses the critical application using nearly the same encryption and decryption procedure. Figure 3 shows a Feistel round function. Figure 3 splits an n-bit block into two parts, L (left) and R (right). The right half is next subjected to a function F(R, K), and the outcome is XORed with the left half (this is the first involution):
$\begin{gathered}L_{i+1}=R_i \\ R_{i+1}=L_i \oplus \mathrm{F}\left(K_i, R_i\right)\end{gathered}$ (8)
The round subkey, or K, generated by the key scheduling algorithm may change from one round to the next. After that, the procedure is repeated with the halves switched (the second involution). The function (F) applied need not be reversible [34].
Figure 3. Feistel cipher structure
The Feistel network is used in the proposed system to cypher 16 bytes, i.e., (4×4) bytes as input, which is split into two parts of 8 bytes, L and R. The F function is performed (8,8) secret image sharing based on polynomials. The Key K_{i} is supplied from key generation steps based on hyperchaotic maps.
3.5 Linear system
The system proposed in reference [35] uses a set of linear equations to generate shares. Each share comprises a secret set of k integer numbers, denoted as a_{ij}, where i ranges from 1 to n and j ranges from 1 to k. Here, n represents the total number of shares, and k denotes the minimum number that can be used to recover the image data. To put it simply, given a block of image data {Vj| j=1...k}, the i^{th} share is calculated using the following linear equations:
$S_i=\sum_{j=1}^k a_{i j} V_J \ mod \ 256$ (9)
This passage refers to retrieving data from a block using shares generated from the block. In this process, S_{i} represents the i^{th} generated share for the block V(). The coefficient represents the i^{th} coefficient belonging to the linear equation representing the share. By collecting k shares from n the total shares (i.e., {S_{i}| i=1...k}), the inverse matrix of A={a_{ij}| i, j=1...k } the method mentioned above can be employed to retrieve the values of V precisely ().ensuring accurate reconstruction of the original data.
$V=\mathrm{A}^{-1} S$ (10)
The range of shared values, S_{i}, is confined within [0..255]. Consequently, the equation mentioned above can be expressed as follows:
$\sum_{i=1}^k a_{i j} V_j=S_i+256 p_i$ (11)
During retrieval, share values are adjusted based on specific integer division rules. Here, p_{i} represents an integer value that does not contribute to the shared values.
Algorithm (1): (k, n)-Threshold Reveal Phase |
Input: k: denoted the number of allied shares. k shares: An array consisting of k bytes. Output: V: An array containing secret bytes retrieved. |
Step 1: Select the shares that correspond to indexes {n1, n2, ..., nk}, ensuring only one secret byte value is get from each chosen share (i.e., {Sm| m=1, 2, …}(. Step 2: Create the coefficients matrix, A'(), for the linear equations, where $a_{m l}^{\prime}=a_{n_m l}$ where, $a^{\prime}{ }_{m l} \in \mathrm{A}^{\prime}$ and $a_{n_{m l}} \in \mathrm{A}$ m=1, 2, ..., k and l=1, 2, ..., k. Step 3: To compute the determinant value of matrix A, we can use the formula D = det(A). Once we have obtained the determinant value, we can derive the complementary matrix C. The condition must be met for all values of j that fall within the range of 1 to k. $\sum_{i=1}^k a_{i j} C_{i j}=\sum_{j=1}^k a_{i j} C_{i j}=D$ (12) where, C_{ij} equals the determinant of the reduced matrix C (with the ith row and column removed), multiplied by (-1)^{i+j}. Step 4: Determine the values of the retrieved secret bytes {V'_{j}| j=1, ..., k]} using: $V_j^{\prime}=\frac{1}{D}\left\{\left(\sum_{i=1}^k C_{i j} S_{n i}\right)+w_j\right\}$ (13) w_{j} is an integer value is multiples of 256 as: $w_j=256 \sum_{i=1}^k C_{i j} p_i$ (14) |
The proposed system consists of tumor segmentation (i.e., ROI) and ROI Encryption. The details of the two stages are illustrated in the following sub-sections.
4.1 Segmentation of region of interest (ROI)
This study uses U-Net architecture to present an automatic brain tumor segmentation on magnetic resonance imaging. The tumor segmentation block diagram based on U-Net is shown in Figure 4.
Figure 5 shows the original images, their masks, and predication images.
Figure 4. Tumor segmentation based on U-Net
Figure 5. Examples of brain tumor segmentation
4.2 Proposed partial image encryption
The proposed algorithm for partial image encryption is a selective and partial method for encrypting medical images. The algorithm uses the Integer Discrete Wavelet Transform (IDWT), a chaotic system, and secret image sharing. This method helps to address security concerns arising from high inter-pixel correlation and large data capacity when encrypting an image. A suitable encryption technique should diminish this correlation, obscure positions of pixels, and render the original image cryptic and disorderly, thereby haphazardly randomizing and disarranging pixel locations. In the presented diffusion approach, the image is perturbed through a linear system by generating random variables to displace the positions of all pixels in the plaintext image. The IDWT is applied to the perturbed image, wherein the low-frequency coefficients can be utilized to construct a blurred approximation of the original image.
In contrast, the high-frequency coefficients solely contribute to the finer image details. Therefore, encrypting only the low-frequency components might be of interest. The proposed image encryption process encompasses key generation, image permutation based on a chaotic map, and image diffusion using secret image sharing.
4.3 Key generation
The process of generating keys uses both hash coding and a hyperchaotic map. Specifically, the hash function takes features from the area of interest using SHA-512, resulting in a 512-bit hash value [36]. As SHA-512 is permanent, it resists various attacks, including plaintext attacks. The algorithm for generating keys in the encryption scheme using SHA-512 is outlined in Algorithm 2. High-dimensional hyperchaotic systems have a more complex structure than low-dimensional chaotic systems, resulting in more robust randomness of the generated sequence, making them ideal for image encryption.
In this encryption method, the secret keys comprise the initial values and parameters of the hyperchaotic system. Using pixels within the region of interest (ROI) as the primary private key source enhances security.
Algorithm (2): Proposed Key Generation |
Input: ROI, S_{key } // Secret encryption key Output: k_{i} i=1,..,n // n is the number of sub-blocks |
Step 1: Calculate SHA-512 of ROI to generate a vector with a 512-bit hash value. $\begin{gathered}\text { hash=SHA-512 (ROI) } \\ \text { hash }=\left\{h_l, h_2, \ldots, h_{32}\right\}\end{gathered}$ (15) It should be noted that when extracting features, even minor changes can significantly impact the outcome. For instance, a one-bit change in the Region of Interest (ROI) can cause hash values to change drastically. To mitigate this issue, four values are produced using the XOR operation (⊕) on the hash values, as shown in the equations below: $\begin{aligned} & S_1=\mathrm{h}_1 \oplus h_2 \bigoplus \ldots \bigoplus h_8 \\ & S_2=h_9 \oplus h_{10} \bigoplus \ldots \bigoplus h_{16} \\ & S_3=h_{17} \bigoplus h_{18} \bigoplus \ldots \bigoplus h_{24} \\ & S_4=h_{25} \oplus h_{10} \bigoplus \ldots \bigoplus h_{32}\end{aligned}$ (16) Step 2: Suppose the initial keys $x_1, x_2, x_3$, and $x_4$ and positive parameters are $\beta_1, \beta_2, \beta_3$, and $\beta_4$. are randomly chosen for the hyperchaotic system in Eq. (1). After that, the first keys are updated affording to the plain image pixel value as follows: $\begin{aligned} X^{\prime} & =\left(x_1 \bmod S_1\right) / 256 \\ Y^{\prime} & =\left(x_2 \bmod S_2\right) / 256 \\ Z^{\prime} & =\left(x_3 \bmod S_3\right) / 256 \\ W^{\prime} & =\left(x_4 \bmod S_4\right) / 256\end{aligned}$ (17) Step 3: Normalize the four variables $X^{\prime}, Y^{\prime}, Z^{\prime}$, and $W^{\prime}$ to the range [0, 1]. Step 4: The hyperchaotic system is firstly iterated (100 times) to generate four chaotic sequences, denoted as $X^{\prime}, Y^{\prime}, Z^{\prime}$, and $W^{\prime}$, respectively; the first 100 elements of each sequence are discarded. This practice enhances the sensitivity of the initial values and parameters of the map, effectively mitigating transient effects. Step 5: Afterward, the chaotic sequences $X^{\prime}$ and $Y^{\prime}$ are generated. Then, a key extension method is performed by multiplying $X^{\prime}$ (8, 1) and $Y^{\prime}$ (1, 8), and a chaotic matrix Chaotic1 (8, 8) is obtained. Step 6: Repeat step 5 with Z′ and W' to obtain Chaotic2 (8, 8). Step 7: The chaotic matrix Chaotic1 and Chaotic2 are handled together with UserKey to construct a chaotic matrix k_{1}and k_{2}of size 8×8 as expressed in Eqs (18) and (19). $\mathrm{k}_1=((Chaotic 1 \times 1000) \times UserKey ) \bmod 256$ (18) $\mathrm{k}_2=((Chaotic 2 \times 1000) \times UserKey ) \bmod 256$ (19) where, UserKey is a user secret key with size 8×8. Step 8: Repeat steps from step 2 to step 6 to generate k_{i} i=1, ..., n. For each value of UserKey, a unique chaotic sequence of Chaotic is generated. |
4.4 Image diffusion phase
The proposed diffusion phase employs a set of linear equations with n=2 and k=2. The linear transformation presented in Eq. (20) can detach a high pixel correlation.
$\left[\begin{array}{l}x^{\prime} \\ y^{\prime}\end{array}\right]=A\left[\begin{array}{l}x \\ y\end{array}\right] mod \ N=\left[\begin{array}{ll}x_1 & x_2 \\ x_3 & x_4\end{array}\right] \times\left[\begin{array}{l}x \\ y\end{array}\right] mod \ N$ (20)
where, (x, y) represents the original pixel position, (x', y') denotes the transformed pixel position, and both (x', y') and (x, y) belong to the range [1, N]×[1, N]. The image is divided into N sub-blocks, and the $x_1, x_2, x_3$, and $x_4$ are hyperchaotic system variables generated by Eq. (7) as secret keys. The ROI is first rearranged in a symmetric matrix as input to the diffusion phase.
Besides using linear equations, modular algebra controls the increasing index size. According to Eq. (15), the range of indices values $\left[\begin{array}{l}x^{\prime} \\ y^{\prime}\end{array}\right]$ is [0... N].
$\left[\begin{array}{ll}x_1 & x_2 \\ x_3 & x_4\end{array}\right] \times\left[\begin{array}{l}x \\ y\end{array}\right] mod \ N=\left[\begin{array}{l}x^{\prime}+p_i \\ y^{\prime}+p_i\end{array}\right]$ (21)
P_{i} is an integer value that will not be included in the index values. During retrieval, its values will be adjusted according to specific integer division rules. Figure 6 shows an example of the proposed image diffusion.
Example 1:
Let A an is 8×8 image; the number of 2×2 subblocks is 16 and N=7. Let x_{1}=13, x_{2}=29, x_{3}=53, and x_{4}=31. Let x=5 and y=1. According to the Eq. (8):
$\begin{gathered}{\left[\begin{array}{l}x^{\prime} \\ y^{\prime}\end{array}\right]=A\left[\begin{array}{l}x \\ y\end{array}\right] mod \ 7=\left[\begin{array}{ll}13 & 29 \\ 53 & 31\end{array}\right] \times\left[\begin{array}{l}5 \\ 1\end{array}\right] mod \ 7} \\ 13 \times 5+29 \times 1=mod \ 7=3 \\ 53 \times 5+31 \times 1=mod \ 7=3 \\ \mathrm{X}^{\prime}=3, \mathrm{y}^{\prime}=3\end{gathered}$
Let x=7 and y=5. According to the Eq. (1):
$\begin{gathered}{\left[\begin{array}{l}x^{\prime} \\ y^{\prime}\end{array}\right]=A\left[\begin{array}{l}x \\ y\end{array}\right] mod \ 7=\left[\begin{array}{ll}13 & 29 \\ 53 & 31\end{array}\right] \times\left[\begin{array}{l}7 \\ 5\end{array}\right] \ mod \ 7} \\ 13 \times 7+29 \times 5=mod \ 7=5 \\ 53 \times 7+31 \times 5=mod \ 7=1 \\ \mathrm{x}^{\prime}=5, \mathrm{Y}^{\prime}=1\end{gathered}$
As a result, the pixel at location (5,1) is placed at location (3,3), and the pixel at location (7,5) is placed at location (5,1).
Figure 6. Example of the proposed diffusion phase
4.5 Image confusion phase
In this phase, selective and partial image encryption is used. The ROI is divided into 8×8 sub-blocks, and IDWT is applied for each block. The low-frequency coefficients LL of size 4×4 are only encrypted using (n,n) polynomial-based secret image sharing. The LL band for each block is passed to the Feistel Network FN, and then the inverse of IDWT is applied to the encrypted LL′ bands to generate C_{i}, i=1, ..., n, where n is the number of subblocks. The high-frequency LH, HL, and HH are diffused using the Morton scan to swap coefficient positions and then confused based on the chaotic map. The block diagram of the proposed image confusion phase is shown in Figure 7.
The Feistel network is used in the proposed system to cipher 16 bytes, i.e., (4×4) bytes as input, split into two parts of 8, L, and R. The F function performs (8,8) secret image sharing based on polynomials. The Key K_{i} is supplied from key generation steps based on hyperchaotic maps. The Feistel network performs two secret image-sharing rounds (8,8), as illustrated in Figure 8.
Algorithm (3): Proposed Confusion Phase |
Input ROI, k_{i} i=1, ..., n // n is the number of sub-blocks, Width, Height, blocks divided by 8 Output EROI |
Step 1: Divide the ROI into subblocks of size 64 bits (8×8). Step 2: For i=1 to Width For j=1 to Height Get sub-block Blk Apply IDWT on the Blk. Pass LL band to FN to construct polynomials to calculate the encrypted coefficients such as the following: $\begin{gathered}c_1=l l_1 k_{11}+l l_2 k_{12}+l l_3 k_{14}+l l_4 k_{14}+l l_5 k_{15}+l l_6 k_{16} \\ +l l_7 k_{17}+l l_8 k_{18} mod \ 2 51 \\ c_2=l l_1 k_{21}+l l_2 k_{22}+l l_3 k_{24}+l l_4 k_{24}+l l_5 k_{25}+l l_6 k_{26} \\ +l l_7 k_{27}+l l_8 k_{28} mod \ 251 \\ \cdot \\ \cdot \\ \cdot \\ c_8=l l_1 k_{81}+l l_2 k_{82}+l l_3 k_{84}+l l_4 k_{84}+l l_5 k_{85}+l l_6 k_{86}+ \\ l l_7 k_{87}+l l_8 k_{88} mod \ 251\end{gathered}$ Rearranged the encrypted coefficients in LL′ To generate vector S, apply a Morton scan to each sub-block with the remaining coefficients of the LH, HL, and HH bands. Apply confusion on the S vector based on the chaotic sequence X′ to generate the S′ vector. Rearranged the S′ vector into LH′, HL′ and HH′_{.} bands Apply the inverse of IDWT to generate Cll. Put the subblock Cll in the EROI. EndFor j EndFor i Step 3: The output is the EROI. |
Figure 7. Block diagram of the proposed confusion phase
Figure 8. Block diagram of the proposed Feistel network
Example to show the (8,8) Polynomial-based SIS.
Let b be a block from diffused image DiffROI of size 8 bytes, as shown below:
Band LL
ll_{1} |
ll_{2} |
Ll_{3} |
ll_{4} |
ll_{5} |
ll_{6} |
ll_{7} |
ll_{8} |
Let k be a secret key block of size 8×8 (i.e., 512 bits).
Key k_{1}
k_{11} |
k_{12} |
k_{13} |
k_{14} |
k_{15} |
k_{16} |
k_{17} |
k_{18} |
k_{21} |
k_{22} |
k_{23} |
k_{24} |
k_{25} |
k_{26} |
k_{27} |
k_{28} |
k_{41} |
k_{42} |
k_{43} |
k_{44} |
k_{45} |
k_{46} |
k_{47} |
k_{48} |
k_{41} |
k_{42} |
k_{43} |
k_{44} |
k_{45} |
k_{46} |
k_{47} |
k_{48} |
k_{51} |
k_{52} |
k_{53} |
k_{54} |
k_{55} |
k_{56} |
k_{57} |
k_{58} |
k_{61} |
k_{62} |
k_{63} |
k_{64} |
k_{65} |
k_{66} |
k_{67} |
k_{68} |
k_{71} |
k_{72} |
k_{73} |
k_{74} |
k_{75} |
k_{76} |
k_{77} |
k_{78} |
k_{81} |
k_{82} |
k_{83} |
k_{84} |
k_{85} |
k_{86} |
k_{87} |
k_{88} |
Construct (8,8) Polynomial-based SIS for each subblock {ll_{j} |j=1 ..., 8} to generate 8 bytes in encrypted block {cll_{j}| j=1 ..., 8}; utilize the following polynomials to compute the encrypted first row:
$\begin{gathered}c_1=l l_1 k_{11}+l l_2 k_{12}+l l_3 k_{14}+l l_4 k_{14}+l l_5 k_{15}+l l_6 k_{16} \\ +l l_7 k_{17}+l l_8 k_{18} \\ c_2=l l_1 k_{21}+l l_2 k_{22}+l l_3 k_{24}+l l_4 k_{24}+l l_5 k_{25}+l l_6 k_{26} \\ +l l_7 k_{27}+l l_8 k_{28} \\ \cdot \\ \cdot \\ \cdot\\ c_8=l l_1 k_{81}+l l_2 k_{82}+l l_3 k_{84}+l l_4 k_{84}+l l_5 k_{85}+l l_6 k_{86} \\ +l l_7 k_{87}+l l_8 k_{88}\end{gathered}$
Cipher band LL′
cll_{11} |
cll_{12} |
cll_{13} |
cll_{14} |
cll_{15} |
cll_{16} |
cll_{17} |
cll_{18} |
This section uses various validation metrics to conduct performance and security analyses.
5.1 Security analysis
The main focus of the design philosophy behind this work was to provide a high level of security. Therefore, we designed the proposed cipher with substantial security aspects. It has been created based on the theory of provable security, which protects against brute-force attacks, matching plaintext-ciphertext, differential cryptanalysis and linear cryptanalysis.
The only way to make a block cipher withstand a brute-force attack is to choose input key k so large that a brute-force attack becomes practically infeasible. The key length k of the proposed algorithm is 2^{1628}, as computed in the next section. On the other hand, the length block size of 512 bits is long enough to resist the matching plaintext-ciphertext
The frequency domain has numerous benefits for encryption. Finding the crucial areas that need to be encrypted is simpler. Connecting between spatial domain pixel values and frequency domain values is challenging. As a result, the image is safe from numerous assaults, including known plaintext attacks and ciphertext-only attacks.
The security analysis has shown that the Linear system possesses a security property. According to this property, "Any k-1 or fewer shadows cannot obtain enough information to reveal the secret image." In our scheme, for a secret colour image of size 512×512, there are a certain number of possibilities to correctly guess the combined coefficients a_{ij} (where i∈[1, n] and j∈[1, k], with n representing the total number of shares and k being the minimum number required to retrieve the image data), along with a block of image data {V_{j}|j=1, ..., k}.
$BrutForce =(8 \times k)^Q \log _2\left(R\left(a_{i j}\right)\right)^k$ (22)
where,
$Q=\frac{width \times hight}{k}$ (23)
R: Range of a's
Example: Let the size of the secret colour image is 512×512, then the probability of guessing the right block of image data {V_{j}|j=1...k} when k=2 is:
(8×2)^{( 512}^{×}^{512)/2}=2^{524288}
Moreover, the probability of guessing the right combined coefficients a_{ij} when R(a_{ij})=256 is:
$\log _2\left((256)^2\right)=\log _2\left(2^{16}\right)=16$
It is essential to understand that the likelihood of guessing the combined coefficients a_{ij} is much lower compared to the probability of correctly guessing a block of image data. This probability is affected by the size of the image and the value of k, while the coefficients a_{ij} are distinct for each share throughout the entire image.
In addition to using the image diffusion phase to decorrelate pixels prior to applying the (n, n) secret image sharing scheme, Discrete Wavelet Transformation (DWT) is utilized to process a secret image. After applying a transform to the data, a combination procedure decorrelates the transform coefficients and improves security.
Regarding polynomial-based image secret sharing, recovering a secret image involves using Lagrange's interpolation to reconstruct the polynomial and obtain the grayscale values of the original secret image. However, retrieving every pixel of the secret image requires solving multiple linear systems, which can be computationally intensive and time-consuming due to the large number of secret image pixels and the significant amount of data involved.
Key-dependent permutation operations enhance the proposed algorithm's security. This involves exchanging input bits under the control of subkeys to destroy the additive difference and protect it against linear and differential cryptanalysis. Each algorithm phase depends on its key, which prevents fixed output and increases its nonlinearity.
5.2 Key space analysis
The key space is fundamental to any encryption scheme, providing vital resistance against brute force attacks. The proposed system utilizes eight independent variables, consisting of four secret initial values $\left(x_1, x_2, x_3\right.$, and $x_4$) along with four positive parameters $\left(\beta_1, \beta_2, \beta_3\right.$, and $\left.\beta_4\right)$. Consequently, these symbols collectively constitute the key space [37]. However, due to the utilization of double-precision numbers for these variables, the count of distinct values surpasses 10^{14}. The proposed approach employs the hyperchaotic system thrice, each incorporating diverse initial and parameter values, thereby establishing a key space of ((10^{14})^{8})^{3 }=10^{336} ≈ 2^{1116 }for a single block. Furthermore, the proposed algorithm adopts a user key spanning 512 bits, resulting in a total key space of 2^{1116} × 2^{512}= 2^{1628}. Table 2 offers a comparative analysis of the key space between the proposed system and other relevant methodologies.
Notably, the employed algorithm exhibits a larger key space than most algorithms under comparison. This outcome ^{signifies} that the scheme has a robust capability to thwart brute-force attacks effectively. Moreover, incorporating a nonlinear chaotic system to generate the input sequence within this algorithm enhances its resilience. The inherent nonlinearity, unpredictability, and pseudo-random characteristics of chaotic systems contribute significantly to countering linear attacks.
Table 2. Key space comparison between the proposed scheme and other schemes
Algorithm |
Key Space |
Proposed system |
2^{1628} |
[21] |
2^{465} |
[38] |
2^{348} |
[39] |
2^{197} |
[40] |
2^{425} |
[41] |
2^{298} |
[42] |
2^{197} |
[43] |
2^{199} |
5.3 Statistical analysis
The application of statistical analysis emerges as a critical metric within the domain of image encryption technology. Researchers have carried out two distinct categories of tests: histogram analysis and correlation coefficient analysis.
5.3.1 Histogram analysis
The histogram of a given image illustrates the distribution of pixel intensity values. In the case of an unencrypted image, the histogram typically exhibits slanting bars, a characteristic that could potentially be exploited by malicious parties to glean information about the image. To counter such statistical attacks, an encryption scheme must possess the ability to transform these slanting bars into well-organized bars with a distribution that closely resembles the original. By doing so, the scheme prevents hackers from extracting meaningful information. The histograms of both the unencrypted and encrypted images of Lena are depicted in Figure 9. In particular, Figure 9(e) displays the curved slanting bars in the histogram of the unencrypted and encrypted Lena image. The histogram transformed into a well-organized, evenly distributed plain bar in the encrypted counterpart. These organized plain bars offer robust resilience against histogram attacks, underscoring the efficiency of the proposed encryption scheme.
Figure 9. (a) Original & encrypted images, (b) horizontal, (c) vertical, (d) diagonal correlations of original & encrypted images, (e) Histogram of original & Encrypted images
Furthermore, the proposed algorithm was applied to several images of brain scans, specifically after extracting the region of interest (ROI) from each image. This process is illustrated in Figure 10.
Figure 10. (a) Original image, (b) ROI, (c) Histogram of ROI, (d) Encrypted image, (e) Histogram of the encrypted image
5.3.2 Analysis of the correlation coefficient
For natural and unencrypted images, pixel arrangement follows systematic patterns, with neighbouring pixels exhibiting solid correlations. The correlation coefficient (CC) serves as a crucial security parameter, quantifying the inter-pixel correlation within an image. These adjacent pixels can be aligned vertically, diagonally or horizontally. Image encryption methods are designed to disrupt these patterns of adjacent pixels. To evaluate the correlation coefficient (CC) within the proposed encryption scheme, 10,000 pairs of consecutive pixels were randomly selected from the encrypted and original images. CC was computed using the following equation:
$C C=\frac{A \sum_{l=1}^A\left(x_1 \times y_1\right)-\sum_{l=1}^A x_1 \times \sum_{l=1}^A y_1}{\sqrt{\left(A \sum_{l=1}^A x_1^2-\left(\sum_{l=1}^A x_1^2\right)\right)\left(A \sum_{l=1}^A y_1^2-\left(\sum_{l=1}^A y_1^2\right)\right)}}$
where, A denotes the number of pixels; the neighbouring pixels are called x and y.
The fully encrypted image showed correlation coefficients approaching zero in all directions. Furthermore, the partially encrypted image's correlation coefficients fell from 0 to 1. This observation underscores the algorithm's resilience against geometric attacks.
Figure 11. Correlation coefficients were calculated for the original medical images using the proposed algorithm, including (a) Original image, (b) Region of interest (ROI), (c) Horizontal correlations, (d) Vertical correlations, and (e) Diagonal correlations
Figure 12. Correlation coefficients for the encrypted medical images utilizing the proposed algorithm (a) ROI, (b) Encrypted image, (c) horizontal, (d) vertical, (e) diagonal correlations
However, the correlation coefficients for the encrypted and original images were analyzed in three directions to provide a comprehensive understanding. The graphical representations of these correlation coefficients can be found in Figure 11 and Figure 12.
5.3.3 Entropy
Entropy serves as a means to assess the degree of randomness or unpredictability exhibited by pixel values within an image. It aids in quantifying the level of disorder present in a cryptogram, thereby providing insight into the effectiveness of diffusion and confusion processes within encryption. Mathematically, entropy could be calculated using the equation below [44]:
$E=\sum_{i=0}^n p_i \log _2\left(p_i\right)$
where, p_{i} represents the probability of the occurrence of pixel value i in the image.
A cryptogram approaching a value of 8 or equaling it signifies a more robust implementation of diffusion and confusion, rendering it more resilient to entropy attacks. This heightened level of randomness within the cryptograms enhances their security by making them inherently more challenging to predict or decipher. Furthermore, as illustrated in Table 3, the results of entropy analysis provide insights into the security and strength of encryption methods.
Table 3. The entropy of the original and encrypted images has been calculated using the proposed algorithm
Image |
Original Image |
Encryption Image |
1^{st} Image |
5.2350 |
7.9970 |
2^{nd} Image |
5.7419 |
7.9967 |
3^{rd} Image |
5.5482 |
7.9973 |
4^{th} Image |
5.3695 |
7.9969 |
5^{th} Image |
6.5444 |
7.9966 |
6^{th} Image |
6.8078 |
7.9975 |
7^{th} Image |
6.5210 |
7.9965 |
8^{th} Image |
6.3950 |
7.9965 |
Table 3 presents the entropy values for eight images before and after encryption. The entropy values were computed to evaluate the images' degree of randomness and disorder. This analysis provides a comprehensive understanding of how encryption processes impact the entropy of the images, shedding light on the effectiveness of the encryption scheme in preserving security and minimizing predictability.
The table demonstrates the impact of encryption on the entropy values, with post-encryption values consistently approaching or reaching the desirable value of 8. This result highlights the efficacy of the encryption method in improving the randomness and security of the encrypted images.
5.3.4 NPCR (Number of Pixel Change Rate)
The Normalized Pixel Change Rate (NPCR) is a metric that measures how effective an encryption algorithm is in preventing differential attacks. It calculates the percentage of changes in pixel locations between two images. A high NPCR value, close to 100%, indicates a significant difference between the two images, implying that they have undergone substantial transformation due to encryption. This increased level of variation enhances the security against differential attacks, as it becomes increasingly difficult for attackers to extract significant information from encrypted data. A near-100% NPCR value indicates the encryption's effectiveness in introducing significant alterations to image pixel positions, thereby enhancing security. Table 4 presents the analysis of NPCR results for eight tested images below. It was found that the NPCR rate for all images was 99%, indicating that the encryption process significantly altered the locations of the pixels. This result confirms that the encryption scheme is robust against differential attacks, which enhances the system's overall security.
5.3.5 UACI (Unified Average Changed Intensity)
The average is calculated from the differences in pixel intensity between the two images. The expected outcome is that this calculation yields a result greater than 33%. Normalized Unified Average Change Intensity (UACI) and Pixel Change Rate (NPCR) were computed using two cryptograms with nearly identical keys, except for a slight alteration in the least significant decimal digit. The NPCR values obtained exceeded 99%, while the UACI values surpassed 33%. This implies that the proposed system generates a significantly distinct cryptogram when provided with a plain image and a minor alteration. The findings are summarized in Table 4.
The proposed method's effectiveness was evaluated on a standard image (Lena), using different parameters such as entropy, correlation analysis, unified average change intensity (UACI), and normalized pixel change rate (NPCR) in Table 4. The results of this evaluation are presented, where the histogram and correlation coefficient distributions of the proposed method demonstrate positive outcomes. The UACI and NPCR values obtained through this method are comparable to those obtained through polynomial-based secret image sharing and chaotic maps encryption approaches. One significant achievement of this method is that the information entropy value obtained is also very close to the ideal value. Moreover, all tests were completed successfully, which supports the claim that the suggested approach is reliable and secure.
Table 4. The entropy of the original and encrypted images using the proposed algorithm
Image |
NPCR |
UACI |
1^{st} Image |
99.6292 |
37.3938 |
2^{nd} Image |
99.6002 |
35.7715 |
3^{rd} Image |
99.5972 |
35.4818 |
4^{th} Image |
99.6460 |
34.0463 |
5^{th} Image |
99.6384 |
32.5668 |
6^{th} Image |
99.6292 |
31.7352 |
7^{th} Image |
99.6414 |
29.0743 |
8^{th} Image |
99.6490 |
28.7795 |
5.3.6 Time complexity
Selective image encryption is a crucial aspect of medical imaging as it helps to reduce both time and cost. Many current encryption methods are complex and rely on traditional techniques. The proposed algorithm ensured the privacy of the ROI; there is a trade-off between time complexity and security requirements compared with full image encryption. Selective image encryption presents several advantages compared to the network transform method. However, it still faces security challenges since only specific parts of an image receive sufficient protection. The level of security provided by encryption depends on the intended application and meeting the necessary criteria. Delve into the critical aspect of time complexity concerning implementing Partial Image Encryption. Time complexity plays a pivotal role in evaluating the efficiency of cryptographic algorithms. In this context, it explicitly addresses how execution time was optimized by applying Partial Image Encryption techniques. The empirical results of these optimizations are meticulously presented in Table 5 for comprehensive analysis.
Table 5. Comparison of the suggested algorithm with alternative techniques for the "Lena" image
Correlation |
||
Horizontal |
Vertical |
Diagonal |
-0.0028 |
0.0171 |
0.0022 |
-0.0359 |
0.0020 |
0.0019 |
/ |
/ |
/ |
0.0020 |
0.0105 |
0.0019 |
0.0023 |
0.0019 |
0.0011 |
0.0030 |
0.0024 |
0.0034 |
-0.0016 |
0.0028 |
-0.0006 |
-0.0008 |
0.0064 |
-0.0223 |
Table 6 provides a detailed comparison of the execution times for the ROI and the encryption of the original image. The results indicate a significant decrease in execution time when using the Partial Image Encryption technique compared to encrypting the original image. This achievement is highly practical and efficient in real-world scenarios. The reduction in execution time is due to the selective encryption of specific regions of interest (ROI) within the image, resulting in a streamlined process that maintains security and efficiency. This optimization enhances the overall user experience and holds promise for applications that require fast encryption and decryption, such as real-time image transmission and secure data storage.
Table 6. The time required for both the ROI and complete image encryption
Image |
Size (Original Image) (KB) |
Size (ROI) (KB) |
Time of ROI (sec) |
Time of Localization |
Time of Encrypt Original Image (sec) |
1^{st} Image |
71.9 |
8.09 |
0.10384 |
0.026 |
2.3737 |
2^{nd} Image |
59.6 |
20.4 |
0.35629 |
0.053 |
2.2707 |
3^{rd} Image |
64.4 |
11.3 |
0.23348 |
0.029 |
2.3045 |
4^{th} Image |
73.1 |
17.8 |
0.33749 |
0.033 |
2.4280 |
5^{th} Image |
56.5 |
10.6 |
0.15459 |
0.027 |
2.2364 |
This study has addressed the formidable challenge of securing medical images, which often encompass sensitive patient and diagnostic data that traverse public networks. Conventional cryptographic methods have proven inadequate due to the unique attributes of digital images, including their substantial size, inherent redundancy, and pixel correlation. To confront this challenge, an innovative approach to image encryption was introduced, amalgamating multiple techniques to bolster security. The proposed method optimizes security and efficiency by selectively encrypting specific segments of the confidential image. An encryption mechanism was crafted using the principles of chaotic systems, maps, and attractors, imparting high levels of randomness and unpredictability. The integration of polynomial-based secret image sharing, coupled with partial encryption, further enhances the resilience of the proposed approach against potential attacks.
Furthermore, the power of transform domain encryption was harnessed, capitalizing on techniques like discrete wavelet transform to enhance security without introducing undue algorithmic complexity. A symmetric encryption algorithm strengthens the overall security architecture while ensuring practical implementation. This paper has introduced a pragmatic image encryption model customized for medical images, introducing an efficient diffusion model based on a linear system and chaotic map. This study effectively mitigated the computational overhead of sizeable digital image sizes by incorporating partial encryption and pixel rearrangement within the Region of Interest (ROI).
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