Integrated Framework Design of Hybrid Security Approach to Safeguard High-Dimensional Data Transmission

Table 1. Comparison of Advanced Encryption Standard (AES), chaos-based and hybrid image encryption methodology in terms of computational speed, cost of implementation and statistical security measures (entropy and correlation coefficient)

The proposed work introduces a novel computational model that combines bio-inspired algorithmic strategy with chaotic systems to protect HDD transmission within an encrypted domain. The proposed work also aims to offer simple, optimized operation that significantly improves the performance of encryption modelling in HDD and protects the confidentiality and integrity of the HDD from a security point of view during transmission in bandwidth-intensive applications. 

4.1 High-dimensional image data pre-processing for encrypted domain

The proposed work considers a high-dimensional image data $I_{H D D}$ for transmission and applies pre-processing operation over the $I_{H D D} \in \mathbb{R}^{m \times n \times 3} / \mathbb{R}^{m \times n}$. The preprocessing operation in the encrypted module locates the image data $I_{H D D}$ and further performs a mapping operation on $I_{H D D}$ in the form of $\mathrm{f}_{\text {map}}$: $\mathcal{F} \rightarrow \mathbb{R}^{m \times n \times 3} / \mathbb{R}^{m \times n}$ where $\mathcal{F}$ belongs to the set of valid high-dimensional image file locations. The mapping function generates $\mathbb{R}^{m \times n \times 3} / \mathbb{R}^{m \times n}$ in the form of numerical matrix of pixel attributes. The preprocessing operation in the proposed security framework also checks the dimensionality of the $I_{H D D} \in \mathbb{R}^{m \times n \times 3}$ and convert it into $I_{H D D} \in \mathbb{R}^{m \times n}$. This computing step basically reduces dimensionality while preserving the luminance that is critical for computational efficiency. The proposed hybrid security approach also exploits correlation estimation across spatial domains of the image data $I_{H D D} \in \mathbb{R}^{m \times n}$. For any pixel $p(x, y) \in I_{H D D} \in \mathbb{R}^{m \times n}$, three correlation types are computed to estimate the local spatial dependency.

4.2 Correlation analysis before high-dimensional data encryption 

Natural high-dimensional image data often exhibit high spatial correlation among complex neighbouring pixels (horizontal, vertical, diagonal). This reflect a highly structured and predictable pixel layout which is vulnerable to various forms of statistical attacks. The prime objective of the proposed security strategy is to eliminate such predictability. It has to be noted that high residual correlation in the encrypted high-dimensional image can be exploited by statistical attacks which further reveals the structure or patterns of the original image. Therefore, the need arises to validate encryption system’s resilience to statistical analysis. In the first computing step the proposed system applies a functionalmodule of $\mathrm{f}_{H C}(x)$ that enables estimation of correlation among horizontal pixel entities in $p_h(x, y) \in I_{H D D} \in \mathbb{R}^{m \times n}$. Let $I_{H D D} \in \mathbb{R}^{m \times n}$ is an image matrix from HDD, where $m$ refers to the number of rows and $n$ refers to the number of columns. The horizontal pixel pairs in $I_{H D D} \in \mathbb{R}^{m \times n}$ defined as $\left\{\left(x_h, y_h\right)\right\}$ where $x_h=I_{H D D}(\mathrm{i}, \mathrm{j})$ and $y_h=I_{H D D}(\mathrm{i}, \mathrm{j}+1)$. Here $i$: $\forall i=1 \rightarrow m-1$ and $j$: $\forall j=1 \rightarrow n-1$. The proposed method reduces the computational load in the execution by introducing an empirical constant value of $h$. The analytical modeling for the estimation of horizontal correlation initially computes the mean values of $x_h$ and $y_h$ in the form of $E_x$ and $E_y$ and further estimate the variance considering the following mathematical modeling. 

$\begin{gathered}D_x=1 / n \sum_{h=1}^n\left(x_h-E_x\right)^2, \\ D_y=1 / n \sum_{h=1}^n\left(y_h-E_y\right)^2\end{gathered}$               (1)

Further, the computing operation also considers estimating the covariance $cov_{xy}$ which is also computed considering the following mathematical operation. 

$\operatorname{cov}_{x y}=1 / n \sum_{h=1}^n\left(x_h-E_x\right)\left(y_h-E_y\right)$         (2)

Finally, the horizontal correlation coefficient is estimated as follows: 

$r_{x y}^H=\left|\operatorname{cov}_{x y}\right| / \sqrt{D_x, D_y}$          (3)

Here in the proposed security strategy Pearson correlation coefficient is applied over horizontal adjacent pixels that measures how similar neighbouring pixel values are across the rows of the HDD. The goal of encryption is to retain low redundancy so that $r_{x y}^{(\text {encrypted})} \approx 0$. 

Similarly, the security approach also considers estimating the correlation between vertical pixel pairs $\left\{\left(x_v, y_v\right)\right\}$ in $I_{H D D} \in \mathbb{R}^{m \times n}$. Here $x_v=I_{H D D}(i, j)$ and $y_v=I_{H D D}(\mathrm{i}+1, \mathrm{j})$. Further considering the Pearson correlation coefficient estimation the statistical similarity between vertically adjacent pixels is measured in the form of $r_{x y}^V$. The process also further computes the correlation between diagonally adjacent pixels where $x_d=I_{H D D}(i, j)$ and $y_d=I_{H D D}(\mathrm{i}+1, \mathrm{j}+1)$. The diagonal correlation $r_{x y}^D$ verifies how pixel intensity is related along diagonal paths. Similar to vertical and horizontal correlation, high values indicate structural regularity, while low values signify high confusion after encryption, which corresponds to optimal security. Like, vertical/horizontal correlation here, the high values imply structural regularity, whereas low values indicate post-encryption high confusion and optimal security. The proposed security module further computes the global correlation measures, which consider the average of three directions as follows: 

$r_x^{\prime}=1 / 3\left(r_{x y}^H+r_{x y}^V+r_{x y}^D\right)$           (4)

This provides an average estimation of pixel dependency that also implies how redundant or predictable the data is before applying the proposed encryption strategy. Here the quantitative measure of pixel dependencies offers a baseline prior to applying the chaos-based encryption modeling. 

4.3 Partitioning operation in the context of high-dimensional data security 

This partitioning operation is also a pre-processing operation prior to encryption. It helps in enabling parallel encryption, and also localizes correlation analysis which supports modular encryption architecture in hybrid secure schemes. The proposed security model considers $I_{H D D} \in \mathbb{R}^{m \times n}$ and further split the data into four different quadrants, considering the midpoints of $m_2=\lfloor m / 2\rfloor$ and $n_2=\lfloor n / 2\rfloor$. It computes the top-left partition in the form of $I_1=I_{H D D} \in \mathbb{R}^{m \times n}\left(1 \rightarrow m_2, 1 \rightarrow n_2\right)$, top-right partition in the form of $I_2=I_{H D D} \in \mathbb{R}^{m \times n}\left(1 \rightarrow m_2, n_2+1 \rightarrow n\right)$. Further, it also computes the bottom-left partition in the form of $I_3=I_{H D D} \in \mathbb{R}^{m \times n}\left(m_2+1 \rightarrow m, 1 \rightarrow n_2\right)$ and the bottom-right partition of $\quad I_4=I_{H D D} \in \mathbb{R}^{m \times n}\left(m_2+1 \rightarrow m, n_2+1 \rightarrow n\right)$. The operation results are further visualized as $\mathrm{U}_{i=1}^4 I \in \mathbb{R}^{m_2 \times n_2}$.

4.4 Chaos seed initialization for high-dimensional data 

The security modeling further enables a chaotic system parameter initialization mechanism, which is customized considering the pixel attributes from the partitioned image data quadrants. Let each partition in $\bigcup_{i=1}^k I$ where $k=1 \rightarrow 4$ is of $m_k \times n_k$ size. Here, the approach considers the first 5 grayscale pixel intensity values $p_i$ from partition $I_k$ to derive the chaotic seed. The computation modeling in this phase consists of three core stages, which are binary representation computation, decimal conversion and normalization operations. Here each pixel $p_i \in[0,255]$ is further represented in an 8 -bit binary representation as $B_i \in\{0,1\}^8$. Further, the process stacks them into a form of binary stream of length 40 as $B_k \in\{0,1\}^{40}$. Further, the process computes a 40-bit decimal number in the form of $U_k$ which is computed considering the following Eq. (5):

$U_k=\sum_{i=1}^{40} B_k^{(i)} \cdot 2^{40-i}$             (5)

Here $B_k^{(i)} \in\{0,1\}$ refers to the $i^{\text {th}}$ bit in the 40-bit stream for the partition $k$. Further, the normalization operation takes place over $U_k$ that generates $U 0_k$. This computing step in the HDD transmission for several key reasons, including: 

• Deriving initial chaotic seed values directly from image pixel intensities, the encryption operation becomes data-dependent. 
• Even a small variation in the high-dimensional image alters the chaotic seed computation that generates completely different encrypted outputs. 

This strategy also enhances the sensitivity and unpredictability in encrypted mechanism. That means this step transform static image features into highly sensitive and unpredictable control parameters for chaos-based encryption modeling. It also helps coupling the data and encryption key that ensures high security, strong diffusion and robustness against various forms of statistical attacks.

4.5 Customized chaotic functional design in hybrid security strategy 

In this step, the proposed security framework also computes the chaotic sequence, which makes use of a confidential key attribute $(K \in \mathbb{R})$. The chaotic charting and encryption module in this proposed security strategy utilizes the strength factors of logistic maps and bitwise XOR operations to secure the high-dimensional image data. The process considers a partitioned data sequence in the form of $I_1 . I_2, I_3, I_4 \in \mathbb{R}^{m \times n}$. Further, it exploits the logistic map to generate the Chaotic sequence. The secret key $K \in \mathbb{R}$ seed the pseudorandom number generator, and further it generates permutation vector $P_{v e c}^K$ that shuffle the pixel indices.

4.6 Chaotic sequence generation in hybrid security approach

The popularity of chaotic systems arises due to their potential deterministic, complex and pseudo-random behaviour. Here deterministic implies that the system follows precise mathematical rules without any randomness. Also, here pseudo-random behaviour exhibits that the output appears random and unpredictable especially over time, even though it’s fully determined by initial conditions. The characteristic- deterministic yet unpredictable makes chaotic systems suitable for HDD encryption especially images with complex pixel distributions. Chaos functions like logistic maps are also known for their sensitivity to initial conditions and pseudo-randomness which also helps in high-dimensional image data encryption. The Chaotic function in the design and modeling of the proposed security framework follows logistic map and is defined as follows: 

$x_{n+1}=r x_n\left(1-x_n\right)$            (6)

Here $x_n \in(0,1)$ is considered a state at the iteration of $n$. $r \in \sigma, k$ refers to the factor that controls the behaviour of the function. The function is considered chaotic if $r>\sigma$. Here $x_0$ is the initial condition in the Chaotic functional system. In the proposed security framework for each quadrant, the logistic map enables the generation of a chaotic sequence which can be computed considering the following Eq. (7).

$x_n=r x_{n-1} \cdot\left(1-x_{n-1}\right)$ where $0<x_0<1, r=4$          (7)

Here $x_0$ refers to the initial value, $r$ is a control parameter and the sequence is generated up to $N=n_r * n_c$ values. Here $n_r$ refers to $m$ and $n_c$ refers to $n$ in reference to the above computing stages. The above function is computed for 1 to $\left(n_r * n_c-1\right)$ values.

Further, the computation generates a chaotic sequence in the form of $\left\{x_n\right\}_{n=1}^N$. Further outcome of the key-based permutation vector $P_{v e c}^K$ is used to perform the index-based permutation and extraction as shown in the following Eq. (8).

$E^{(k)}=x_n^{(k)}\left(P_{v e c}^K\right)_{1 \rightarrow n_r * n_c}$             (8)

Here the chaotic sequence is re-ordered with respect to the permutation vector representation. Further the proposed work performs dimension normalization for data matrix $T \in \mathbb{R}^{4 \times N}$ where all encrypted vectors $E^{(k)}$ are further appended. Here the approach applies matrix formation via zero-padding shorter rows during the appending operations. The computation of $T \in \mathbb{R}^{4 \times N}$ is performed considering the following mathematical expression. Let $E^{(k)} \in \mathbb{R}^m$ then

$T_{(i,:)}=\left\{\begin{array}{c}E^{(k)}, \quad \operatorname{len}(T)=M \\ {\left[E^{(k)}, 0 \ldots .0\right], M<N} \\ \text { truncate or adjust}, M>N\end{array}\right.$            (9)

Here $M$ is the length of chaotic vector $E^{(k)}$ for each quadrant and $n$ refers to the target row width considering the max of $E^{(k)}$ lengths. The value of length of $M=n_r * n_c / 4$. The computation also helps in generating the chaotic encrypted keys in this security approach. The process also performs chaotic sequence re-ordering for the ease of computation. The following Figure 1 shows the computational steps involved in generating the chaotic sequence of HDD.

1.png

Figure 1. Chaotic sequence generation in hybrid security approach

4.7 Chaos-based encryption mechanism 

It is further used to form the encryption key stream. Here, each partition uses a different seed, such as $x_n^{(1)}$ uses $U_{01}$ for $I_1, x_n^{(2)}$ uses $U_{02}$ for $I_2, x_n^{(3)}$ uses $U_{03}$ for $I_3$ and $x_n^{(4)}$ uses $U_{04}$ for $I_4$. Further, the proposed work considers the pixel shuffling operation via the permutation index. It computes a permutation vector $P_{v e c} \in \mathbb{Z}^{n_r, n_c}$. This vector further helps permute the indices for each partition's chaotic sequence and generates $\mathrm{X}^{\text {perm}}$. The computation process also enables sequence normalization, which scales the chaotic sequence of HDD to the grayscale range. Finally, an encryption mask is generated in the form of $E_{\text {mask}} \in \mathbb{Z}^{n_r, n_c}$. Further, each individual partition of $I^{(i)} \in \mathbb{Z}^{n_r, n_c}$ for $\bigcup_{i=1}^4 I \in \mathbb{R}^{m_2 \times n_2}$ is encrypted considering the chaotic XOR operations, block diffusion $\Phi($. using forward chaining for each $I_E^{(i)}$ and global diffusion $\Psi($.$) ,$ as shown in the following Eq. (10) and (11) respectively.

$I_E^{(i)}=\Phi\left(I^{(i)} \oplus \bmod \left(\left[X(I X) \times 10^{14}\right], 256\right)\right)$            (10)

$I_E=\Psi\left(\bigcup_{i=1}^4 I_E^{(i)}\right)$           (11)

As in the proposed security approach pseudo-random generator $\left(\varepsilon_G(K)\right)$ outcome is seeded with $K \in \mathbb{R}$. Therefore, any adversaries' not having $K \in \mathbb{R}$ unable to reproduce the same permutation or encryption sequence. Therefore, while combined with chaotic maps, $\varepsilon_G(K)$ provides unpredictability and also strengthen the encryption process. Here in $I_E^{(i)}$ the first five pixels are typically preserved as header for the purpose of synchronization. Finally, the proposed computational approach reconstructs the full encrypted HDD $I_E^{(i)}$ considering four different encrypted quadrants which is shown as follows:

$I_E^{(i)}=\left[\begin{array}{ll}I_E^{(1)} & I_E^{(2)} \\ I_E^{(3)} & I_E^{(4)}\end{array}\right]$           (12)

In the proposed system, zero padding plays a key role in matrix formation and positively influences the encryption modeling. Existing chaos based encryption schemes typically represent data as one dimensional vectors. In contrast, the proposed security modeling applies encryption operation like bitxor where the need for matching the shape of the original high-dimensional image blocks arise. To meet this requirement, a matrix reshaping operation is employed. When the chaotic sequence, after permutation and quantization, does not match the size of the image blocks, zero padding resolves the dimensional inconsistency. Furthermore, zero padding ensures that:

•     All encrypted blocks have a uniform size;
•     They can be correctly placed back into the global image matrix $I_E^{(i)}$.

This approach also helps eliminating the indexing errors and support smoother reconstruction of encrypted high-dimensional image in the proposed system. The analytical design of the chaotic-based encryption algorithm is shown as below: 

The proposed hybrid security strategy also synergizes a bio-inspired algorithmic modeling with the above customized encryption module which is basically a chaos-based encryption system. It also helps forming a hybrid approach that addresses the growing security needs of bandwidth-intensive applications. The security framework- applies a custom chaos-based encryption strategy as high-dimensional images benefit from light-weight chaotic-encryption methods owing to their larger size and need for fast processing. The proposed work also further integrates a novel bio-inspired strategy for effectively tuning the encryption parameters that also reduces computational burden to a longer run. The encrypted data chunks are considered an initial population and evolve them towards optimized outcome to enable adaptive security enhancement.

4.8 Bio-inspired optimization integration 

The proposed security framework further considers $I_E^{(i)} \in \mathbb{R}^{n_r, n_c}$ and further also applies image partitioning operation and split the encrypted image into equal-sized quadrants. It is represented as follows:

$I_E^{(i)}=\left[\begin{array}{ll}q_1 & q_2 \\ q_3 & q_4\end{array}\right], q_i \in \mathbb{R}^{n_r / 2 \times {n_c} / 2}$          (13)

The approach further applies a partition recombination modeling, which is also referred to as a bio-inspired shuffling strategy that generates four different recombinations of quadrants in the form of $\bigcup_{i=1}^4 S_i \in \mathbb{R}^{n_r / 2 \times n_c / 2}$. Here for each configuration of $S_i$, the system further computes correlation estimation in the form of $r_{x y}^H, r_{x y}^V, r_{x y}^D$ and also further estimates the average correlation $R_x^{\prime}$ for each shuffled image matrix. The system also formulates optimization criteria to obtain the $G_{\text {opt}}$ that is represented as follows:

$G_{\text {opt}}=\arg \min _i R_x^{\prime}(i), \forall i \in 1 \rightarrow 4$        (14)

The security approach further computes the information entropy, considering a customized function of $\mathrm{f}(H)$ for the considered high-dimensional image data. The computation is carried out considering the following mathematical operation. 

$\mathrm{f}(H)_{I_E^{(i)}}=-\sum_{i=0}^{255} p_i \log _2 p_i$         (15)

where, $p_i$ refers to the probability of gray-level $i$ in the image of $I_E^{(i)}$. Further the shuffled image obtained from $G_{\text {opt}}$ is used as cipher image. The proposed strategy aims to optimize the process while striving to achieve lower correlation values. Here, low correlation via $G_{\text {opt}}$ implies better encryption. Also, the entropy estimation here measures the unpredictability, where the study maximizes the function $f(H){ }_{I_E^{(i)}}$ to generate a stronger cipher prior to network transmission. Here the index of the optimal pattern $i$ is stored in the last pixel of $I_E\left(n_r, n_c\right)=i$. Further, the system communicates the optimized encrypted HDD to the receiver side via bandwidthfriendly transmission. In the proposed hybrid security framework, the optimization policy is integrated with the chaos-based encryption strategy to optimize the encrypted images and further it helps in determining the one with the highest entropy and lowest adjacent pixel correlation as the final cipher image. Here the proposed work employs a genetic algorithm (GA) that refines the encrypted images while considering the following key aspects: 

•     Fitness Evaluation: This criterion is assessed considering entropy and pixel correlation metrics. 
•     Selection and Crossover: The approach combines high fitness images to produce new generations. 
•     Termination: The process continues until the improvements plateau and till it ensures optimal encryption quality. 

4.9 Hybrid security decryption approach 

In the decryption pipeline of the proposed work, the study considers the encrypted high-dimensional image data of $I_E^{(i)}$ and further applies the bio-inspired optimization to re-order $q_i \in \mathbb{R}^{n_r / 2 \times n_c / 2}$. It also further applies the complementary $(2 \times 2)$ portioning operation and further computes the chaotic initialized seed value. Further, considering the secret key entity of $K \in \mathbb{R}$ seeds from the authorized user side, the system applies a chaotic charting strategy to re-generate the $T \in \mathbb{R}^{4 \times N}$ which further undergoes the decryption operation to reconstruct the original image. The evaluation metrics used in the proposed security approach consider entropy analysis, correlation mapping and confidential key-sensitivity, which are further assessed in the next segment of the study.