Chaos-Based Pseudo Random Generator Function for Medical Image Encryption

Medical centers have recently increased the use of online medical image scanning. Given the importance of medical image data, encrypting this data would inevitably prevent its access and distribution [1, 2]. Also, the line between medical data and the Internet has become more open to the outside world, making it vulnerable to many hacking attempts. Hence, security development has focused on protecting medical information due to the previously unheard-of safety risks related to the protection of patients, including reports, photos, and other data [3]. Medical image encryption technology uses common encryption systems, such as RC6, AES, Blowfish, etc. block ciphers. The reliability of block ciphers is determined by the key generation mechanism and the algorithm used in the encryption process, which depends on the key length. In data encryption techniques, random chaos is the numbers that are not related to each other or any other number in the sequence to provide a potential artificial random generator [4, 5]. The features of number chaos are connected to the constraints of encryption systems. Several techniques are used in chaotic structures, including summation, sensitivity to change in the initial context, and identification of variables. The layout of random generators is essential in encryption applications [6]. Obtaining identical random numbers at the sending point and destination is a difficult problem to solve. Therefore, a program that produces a series of numbers independently is known as a pseudo-random generator (CSG) that produces an extended random pattern using a short random number as input. CSG can have a repeated process, which is a one-way process. The CSG's characteristics include a chaotic system for performing random change and responding to an initial value [7, 8].

This paper introduces the following contributions:

1. Proposed chaos depends on the pseudo-bit random generating. The RC6 encryption algorithm is employed as a function in the suggested pseudorandom bit generating, where the parameters are (k, l), and the start values are (x₀, y₀) of the Chebyshev polynomials maps are used as sub-keys for the RC6 algorithm.
2. For RC6 block ciphers, the connection between input and output blocks is completely arbitrary. This design must be one-to-one, in which case each input block is mapped to a distinct output block. Consequently, image encryption relies mainly and mostly on randomness; the randomness rate impacts the strength and robustness of any security systems and increases their complexity.
3. The proposed CSG method is further enhanced by Chebyshev polynomial maps encryption that increases privacy. A Chebyshev chaotic mapping is a technique for effective security implementation.
4. A suggested technique is used in the encryption process to encrypt the pixels with a random sequence related to the input medical picture bits.
5. Due to our keyspace being suitably wide, our design encourages resilience to brute-force searching attacks.

This study is structured as follows: Section 2 presents related works, and the preliminary plan is provided in Section 3. Section 4 outlines the general steps of the CSG process. Section 5 presents the results of the experiment and comparisons. Our efforts come to an end in Section 6.

3.1 Chebyshev polynomials and their properties

One type of nonlinear dynamic process that depends on the starting parameters is called a chaotic process. Random structures can produce highly unpredictable chaos [18]. There are many types of chaotic processes, including Henon mapping, logistic mapping, and Chebyshev polynomial mappings.

So, chaotic elements are arranged to reorder the original features and extract the permuted characteristics. The source of chaotic sequences is Chebyshev polynomial maps, which operate on the image pixel values using a nonlinear process to determine the setting parameters. Each permutation key is generated by sorting chaotic numbers in increasing or decreasing order and is defined in the following equation [19]:

$x(n+1)=T_K\left(x_n\right)=\left(k \times \arccos \left(x_n\right)\right)$          (1)

$y(m+1)=T_K\left(y_m\right)=\left(l \times \arccos \left(y_m\right)\right)$          (2)

where, $\left(x_{\mathfrak{n}}, y_{\mathfrak{m}}\right) \in[-1,1]$ and $(k, l) \in(2, \infty)$ are the control parameters.

3.2 RC6 encryption algorithm

A robust cipher should be resistant to every type of encryption analysis and system-breaking assaults for a reliable indicator of its effectiveness. RC6 stands as one of the top block cipher algorithms that were a modification of the RC5 method [20].

Three primary parameters of RC6 are w, r, and b, where w represents the word block size, r describes the encryption rounds, and b is the user-specified key length. Algorithm 1 and Algorithm 2 use four 32-bit registers that can handle 128-bit data transfer blocks and are compatible with key sizes that range between (128–2040) bits in length. RC6 has a key space large enough to prevent brute attacks [21]. Figure 1 illustrates the encryption process in the RC6 technique.

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Figure 1. RC6 encryption algorithm scheme

In this paper we used image samples from different medical image modalities, as shown in Figure 7, and the input parameters and initial values for Chebyshev maps x₀, y₀, k, l are set to 0.398, 0.456, 0.784, 0.982 successively. The primary parameters of RC6 are set as the number of round r = 20, b = 128-bits and w = 32-bits The following section examines the parameters of the performance model to evaluate the proposed algorithms.

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Figure 7. Test images

5.1 Histogram analysis

The encoding outcomes of the source and the cipher images along with the histogram are displayed in Figure 8. Histogram assessments produce the individual pixel value intensities of the histogram for the cipher and source pictures.

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Figure 8. The suggested decryption and encryption mechanism for medical pictures: Medical picture histogram, Picture encrypted, Encrypted picture histogram

So, a graph that shows the number of pixels in a given image at each varying intensity value is called a histogram. The comparison is made between the source pictures and the visual representations of the related cipher pictures. This comparison showed that the histograms of the cipher pictures are almost identical, while the histograms of the source pictures are completely different for every picture.

Moreover, a histogram is a representation that describes the total number of pixels at different intensity levels, using the axes to show the standard distribution of picture pixels. The histogram of the real image typically contains some unevenly distributed features [22].

Thus, an effective encryption technique requires stable distribution in the encrypted image. And to evaluate the success of the encryption technique, the distribution histogram shows the time numbers for the intensity values that appear in the pictures [23].

5.2 Correlation coefficients analysis

Correlation Coefficient (C.C) refers to the connection between two factors. When Image and data encryption are closely related it usually has one correlation. So that, the encrypted and source pictures have the same appearance., it means that the encryption process was not able to hide the actual image information.

Whereas the actual image and its encryption are completely different has zero correlation. The encrypted image loses some features and is very different from the actual image. When the C.C is -1, it indicates that the encrypted picture is on the opposite side of the real picture [24].

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Figure 9. Correlation coefficients for the source medical pictures

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Figure 10. Correlation coefficients for the encrypted pictures

Therefore, successful encryption leads to lower C.C. values, and are calculated using Eq. (3) by $\rho_{x y}$. Figure 9 shows the (C.C) of the source picture for the Horizontal Correlations (H.C), Vertical Correlations (V.C), and Diagonal Correlations (D.C) and Figure 10 explains the (C.C) of the encrypted picture. In Table 1, the (C.C) of the proposed approach is presented in five pictures.

$\rho_{x y}=\frac{\operatorname{con}(x, y)}{\sqrt{V(x)} \sqrt{V(y)}}$          (3)

$\begin{gathered}\operatorname{con}(x, y)=\frac{1}{N} \sum_{i=1}^N\left(\left[x_i-M(x)\right]\left[y_i-M(y)\right]\right) \\ M(x)=\frac{1}{N} \sum_{i=1}^N x_i \\ M(y)=\frac{1}{N} \sum_{i=1}^N y_i \\ (x)=\frac{1}{N} \sum_{i=1}^N\left(x_i-M(x)\right)^2 \\ V(y)=\frac{1}{N} \sum_{i=1}^N\left(y_i-M(y)\right)^2\end{gathered}$

where, $\operatorname{con}(x, y)$ represents the covariance between x and y, $(V(x)$ and $V(y))$ represent the variance of the parameters x and y, $M(x)$ and $M(y)$ describe the mean of two parameters, x and y.

Parameters x and y, $M(x)$ and $M(y)$ describe the mean of two parameters, x and y.

Table 1. C.C for the proposed technique

5.3 Analyzing the entropy

Entropy refers to the mean amount of bits in the grayscale levels of the picture, which correspond to its data value. So, a rise in entropy indicates greater ambiguity regarding the picture's contents. To analyze an arbitrary performance for the encrypted picture, entropy H(C) for the discrete two-dimensional images is determined using the entropy statistics specified as:

$H(C)=\sum_{i=0}^{N_c-1} P\left(c_i\right) \times \log _2 \frac{1}{P\left(c_i\right)}$          (4)

H(C) represents the encrypted image's entropy, whereas $P\left(c_i\right)$ describes the probability of occurrence number for every level where (i = 0–255). At the probability rate $P\left(c_i\right)$ is $2^{-8}$, when the entropy is highest, it will be equal to eight. So, the Images with more ambiguous details and minimal leakage have higher entropy. Data entropy for any image indicates that it cannot be predicted by the probability of different gray levels occurring. Therefore, the image with different gray levels represents the highest amount of unpredictability, which shows its degree of secrecy. The details of the entropy for the proposed technique are mentioned in Table 2 which shows that the entropy of the encryption picture is close to eight bits, which indicates that the proposed approach is the best [25]. In Table 2, a comparison of the entropy to both source pictures and the proposed technique is presented below.

Table 2. Entropy comparison

5.4 Definition of UACI and NPCR

Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) were initially reported according to Mao and Chen [26, 27].

The NPCR and UACI metrics are usually used to evaluate the picture security for the encryption techniques against different attackers. Greater NPCR and UACI scores indicate significant protection from distinct attacks. Let $T^1$ and $T^2$ represent the images of the ciphertext in plaintext before and after changing by one pixel. Eq. (5) defines a bipolar array A and the value of each pixel across the grid in $T^1$ and $T^2$. Eqs. (6) and (7) can mathematically define the NPCR and UACI respectively described below.

$A(i, j)=\left\{\begin{array}{l}0, \text { if } T^1(i, j)=T^2(i, j) \\ 1, \text { if } T^1(i, j) \neq T^2(i, j)\end{array}\right.$          (5)

$\operatorname{NPCR}\left(T^1, T^2\right)=\sum_{i, j} \frac{A(i, j)}{D} \times 100 \%$          (6)

$\mathrm{UACI}\left(T^1, T^2\right)=\sum_{i, j} \frac{\left|T^1(i, j)-T^2(i, j)\right|}{R \cdot D} \times 100$          (7)

where, the symbol (D) denotes the total of the pixel numbers in the ciphertext, whereas the symbol R represents the maximum pixel value. NPCR focuses on the absolute pixel number that changes during different attackers. As well as UACI focuses on the mean distinction among pair ciphertext images.

Table 3. Comparison of the NPCR and UACI metrics

NPCR has the following range from 0 to 1. When NPCR $\left(T^1, T^2\right)$ is equal to zero it indicates that every pixel has identical values for $T^1$ and $T^2$ while UACI $\left(T^1, T^2\right)$ is equal one, it indicates that every pixel of $T^1$ are updated as compared to the values of $T^2$. The critical numbers are used in both the NPCR and UACI tests. The gray value of the encrypted picture is $T^1(i, j)$, whereas the grey value in the latest encrypted picture is $T^2(i, j)$ [28]. In Table 3, a comparison of two metrics which are NPCR and UACI for the proposed technique.

5.5 Keyspace

The large keyspace prevents exhaustive searches for keys. The challenge of identifying the precise key value can be resolved by testing a variety of values until the appropriate value is determined.

Four independent variables are used in the proposed technique: two positive coefficients, l and k, and two secret initial values ($x_0$ and $x_1$). The number of different values exceeds (10¹⁴) because these variables are double-precision integers.

A keyspace of (10¹⁴)⁴ = 10⁵⁶ ≈ 2¹⁸⁶ besides, the proposed algorithm adopts an IV of 128 bits as the key which provides a total key space of (2¹⁸⁶× 2¹²⁸ = 2³¹⁴) for a single block.

Therefore, a comparison of the key space for the suggested technique and other relevant techniques is shown in Table 4. The outcomes show that the technique can successfully defend against brute force attacks by generating the input sequence from a nonlinear chaotic structure while maintaining the flexibility of this algorithm.

The main characteristics of this technique are the intrinsic nonlinearity, unpredictability, and quasirandom properties of chaotic systems to prevent linear attacks.

Table 4. Comparative analysis of the keyspace

5.6 Tests of the NIST SP 800-22

The degree of randomness of the encryption was tested using NIST SP 800-22 to analyze the chaotic nature of the random sequence produced by the proposed random generator. It relies on two initial keys (x_0,x₁) and two parameters (k, l) and generates as many random bits as necessary by the IV. Fifteen tests were created for this study to determine whether a sequence was random.

Table 5 contains a set of test results and in Table 6, the proposed approach is compared with previous research using the “Lena” picture.

For an individual test, the p-value > 0.01 means that the sequence would be considered to be random. As noted in Table 5, all p-values for the four sequences are larger than 0.01 this means they pass the NIST test successfully.

Table 5. Results of NIST SP 800-22 tests

Table 6. Comparison of the proposed approach with previous research utilizing the “Lena” picture

Although the standard image `Lena’ is non-medical image but a comparison is done in Table 6 with the related works that implement their methods on standard image `Lena’. This comparison is necessary to justify the robustness of the proposed system. We compared with standard Lena image because it is difficult to find related works implement their suggested method on the same medical images that are used by our proposed system.

Table 7 shows performance speeds of AES, RC6 and proposed method on gray scale images of different sizes of the MRI image with secret key length 128-bits. As we nots that the proposed algorithm is more complex due to the use of more rounds and operations. This adds overhead and makes proposed method slower, especially on software implementations.

Table 7. Performance speed of the proposed method with AES and RC6