Genetic Crossover at the RNA Level for Secure Medical Image Encryption

Recently, an algorithm using DNA displacements to generate summary information as initial values for a four-dimensional metastable hyper chaotic system was introduced by Liang et al. [22]. Similarly, Zhu et al. [23] proposed an algorithm that utilizes DNA encoding and scrambling operations between blocks to enhance the scrambling effect. Furthermore, Zhu et al. [24] proposed an algorithm based on parallel DNA encoding to address the limitations of popular DNA encoding-based image encryption algorithms. In the study of Yao et al. [25], the authors proposed a centered around DNA storage image encryption algorithm that utilizes the information processing mechanism of molecular biology for pixel replacement through genetic hybridization. This method achieves double diffusion using pixel diffusion and genetic mutations. Furthermore, Qobbi et al. [26] describes a new image encryption algorithm that includes obfuscation and genetic operations. In the study of Mansoor and Parah [27], a new algorithm is proposed that integrates two one-dimensional chaotic graphs (logic graph and tent graph) to generate pseudo-random sequences for DNA encoding. Furthermore, Alawida et al. [28] introduced an algorithm that combines DNA computing and finite state machines (FSM) to design key plans with high flexibility and statistical randomness while achieving confusion and diffusion properties. Zhang et al. [29] proposed a bit-level and pixel-level dual-arrangement image encryption scheme based on DNA encoding to solve the problem of complete disorder of adjacent pixels. Gera and Agrawal [30] proposes an algorithm that combines discrete 4D hyper chaotic graphs and DNA encoding to improve encryption performance and distribution. In the study of Zhang et al. [31], the DNA notation is conventional and used in a static manner under the control of a single table, employing only the XOR operator. It is essentially a computation within Z⁄4Z. Furthermore, this paper utilizes a single chaotic map (logistic), making it vulnerable to brute force attacks due to the calculation size being 2⁶⁴. Additionally, the authors do not specify the hash functions used in the RNA code. A new image encryption algorithm based on JRPS (Joint RNA-level permutation and substitution) is proposed that can resist various attacks. Furthermore, in the equation for constructing the new two-dimensional map, the author uses the bit to write, introduced an algorithm that uses DNA and subsequent RNA encoding to improve the security of biometric information [32]. Sun et al. [33] uses the hash function for the LS2MAP map value, and these steps are less clear in the paper.

$\left\{\begin{array}{c}x_{n+1}=\sin \left(\frac{a}{x_i}\right) \cos \left(\frac{b}{x_i}\right) \\ y_{n+1}=\frac{x_{n+1} \cos \left(\frac{b}{y_i}\right)}{\cos \left(\frac{b}{x_i}\right)}\end{array}\right.$

The author must Demonstrate that this new map is chaotic and exhibits the properties of chaos or provide a reference. The author describes RNA using 4 nucleotides; however, RNA is defined by only 3 nucleotides. Proposed a chaotic image encryption scheme based on RNA operations and cardioid method. Tahbaz et al. [34] employs a hash function for the LS2MAP map value, and these steps are less clear in the proposed article. In the equation for constructing the new two-dimensional map, the author uses the bit to write. the authors proposed a new hybrid approach involving Magical Chaos (MSC) and RNA codons for image encryption. Finally, Jarjar et al. [35] proposed a new satellite image encryption algorithm based on a seven-dimensional complex chaotic system and RNA calculation.

Based on chaos theory [22], our method relies on two most commonly used chaotic maps in the field of cryptography. Upon completion of this study, a thorough analysis of the performance inherent to our approach will be conducted and compared to other systems within the same framework. Our approach revolves around the following axes:

4.1 Chaotic sequences design

Our algorithm utilizes the two most widely used chaotic maps in the field of cryptography for their ease of configuration and their extreme sensitivity to initial conditions, namely the logistic map and the PWLCM map.

4.1.1 Logistic map

The logistic map is a recurrent sequence determined by a second-degree elementary polynomial, given by Eq. (1).

$\left\{\begin{array}{c}\left.\left.\vartheta_0 \in\right] 0,5 ; 1[, \eta \in] 3,75 ; 4\right] \\ \vartheta_{n+1}=\eta \vartheta_n\left(1-\vartheta_n\right)\end{array}\right.$           (1)

where, $\vartheta_0$ is the initial condition, $\eta$ is the control parameter.

4.1.2 PWLCM map

The PWLCM sequence is a piecewise linear sequence defined by Eq. (2):

$t_n=f\left(t_{n-1}\right)= \begin{cases}\left.t_0 \in\right] 0 ; 1[ & h \in] 0,5 ; 1[ \\ \frac{t_{n-1}}{h} & \text { if } 0 \leq t_{n-1} \leq h \\ \frac{t_{n-1}-h}{0.5-h} \quad & \text { if } h \leq t_{n-1} \leq 0.5 \\ f\left(1-t_{n-1}\right) \text { else }\end{cases}$           (2)

The parameters $\left(t_0\right)$ and $(h)$ represent, respectively, the initial state and its control parameter.

4.2 Sub keys construction

For the encryption and decryption processes smooth execution, this technique requires the construction of several pseudo-random vectors to establish an algorithm capable of addressing any known attack. These vectors come in various types.

4.2.1 Binary control vectors

Our algorithm requires the presence of two chaotic vectors $\left(V B_1\right)$ and $\left(V B_2\right)$ with coefficients in $\left(G_2\right)$. These vectors are generated by Algorithm 1.

These two vectors will be used to control any image transformation operation and make precise decisions.

These two vectors will be used to control any image transformation operation and make precise decisions.

4.2.2 Confusion / Diffusion pseudorandom vectors

Our technique requires the generation of multiple random vectors for confusion and diffusion. These vectors are:

·(VC₁), (VC₂) and (VC₃) of coefficients in ring (Z/256Z) for confusion at pixel level;

·(CO) in (Z⁄4Z) for nucleotide value computation;

·(VE) in (Z⁄16Z) for reading in the nucleotide table (indicate the table row);

·(VD) in (Z⁄64Z) for searching in the RNA table (TR).

These vectors are generated by the following algorithm:

It is noted that due to the high sensitivity of the chaotic maps used to initial conditions, the slightest disturbance in these chaotic sequences results in different pseudo-random vectors.

In line with Kirchhoff's guidelines, our simulations involve the generation of encryption parameters and pseudo-random vectors (including confusion/diffusion vectors and control vectors) using algorithms 1 and 2.

4.3 Original image preparation

Any original image must undergo modeling before entering the operating block to undergo a future encryption process and must go through the following steps:

4.3.1 Original image vectoring

After extracting the three-color channels $(R G B)$ and converting them into vectors $(V r),(V g),(V b)$, each of dimension $(1, \mathrm{~nm})$, a concatenation operation controlled by the decision vector $\left(V B_1\right)$ is performed to generate a vector $(\mathrm{VX})=\left(\mathrm{VX}_1, \mathrm{VX}_2, \ldots, \mathrm{VX}_{\mathrm{n}_{\mathrm{nm}}}\right)$ of dimension $(1,3 \mathrm{~nm})$, as specified by Algorithm 3 below.

This algorithm can be seen in Figure 1.

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Figure 1. Original image vectoring diagram

The completion of this encryption step involves the computation of an initialization value closely linked to the original image and pseudo-random vectors.

4.3.2 Confusion / Diffusion process

The initialization value (GI) involves the modification of the seed pixel's content, thus initiating the confusion/diffusion process. This value is calculated according to Algorithm 4 below.

This algorithm can be represented by (Figure 2).

A first slight transformation of the vector (VX) into the vector (X) controlled by the vector (VB1) using diffusion and confusion operations, is given by the following algorithm:

This algorithm can be illustrated by the Figure 3.

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Figure 2. Initialization value computation diagram

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Figure 3. Confusion and diffusion process diagram

To increase the temporal attack complexity of our system, we proceed with a second round acting at the DNA level and then at the RNA level through well-defined algebraic operations.

4.3.3 DNA coding

Definition. DNA, or deoxyribonucleic acid, is the molecule carrying the genetic code of an individual. This genetic information is present in all cells of the human body. The genetic code is represented by a sequence of four nucleotides: Adenine, Cytidine, Guanine, and Thymine, symbolized respectively by the letters A, C, G, and T as depicted in Table 1.

Table 1. Initial natural values of DNA nucleotides

4.3.4 Pixels to nucleotides transformation

A pixel is converted into 4 nucleotides, each represented by two bits. This transition involves the following steps:

Transition to (Z⁄4Z). The output vector (X), with coefficients in (Z⁄256Z), is converted into a vector (Y) over (Z⁄16Z), and then into a vector (Z) over (Z⁄4Z) of size (1, 12 nm). This phase is illustrated by the diagram below (Figure 4):

This transition can be described by the following algorithm:

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Figure 4. Transition to (Z⁄4Z)

Figure 5 give an example of such operation.

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Figure 5. Transition table to (Z⁄4Z)

Transition to nucleotides. The transition from (Z⁄4Z) to nucleotides is a simple notation change, it can be defined by the following steps:

Transition from (Z⁄4Z) to DNA. The transition from (Z⁄4Z) to nucleotides is a simple notation change, whose pseudo-random selection of values for the components DNA of the table (AN) of size (8;4) is defined by:

The 1st row of the table ($A N$)  is defined by the natural nucleotides’ 'A'=0, 'C'=1, 'G'=2, and 'T'=3.

·Any row of rank (i)$>$1 is obtained by shifting the previous row (i-1) by a pseudo-random step defined by the vector (VC₁) or (VC₂) depending on the values of the control vector (VB₁).

Table construction (AN). The nucleotide table (AN) is generated by the following algorithm:

An example of DNA table creation is depicted in Figure 6.

6.png

Figure 6. Example of DNA table creation

Nucleotides assignment. The vector (Z) will be transcribed into vector (XN), whose components are nucleotides read from the table (AN) based on the values of the control vector (VB₂):

Note:

A simple perturbation to the secret key of the system results in the generation of a new DNA table (AN), and consequently, a new vector (XN). This table may have a pseudo-random dimension computed from the used chaotic maps.

The transition from the initial image to DNA notation is described in the following Figure 7.

Example:

Nucleotides Complementary. Each nucleotide has a unique complementary counterpart in RNA duplication. Naturally, the complements of nucleotides are defined in the following table:

In our work, the complement of a nucleotide will be determined by a pseudo-random table (CN) of size (16;4) generated through the following steps:

·1st line is constructed from natural complementariness.

·The line j>1 is the shifting of the line j-1 of rank VC₂ (j) or VC₃ (j) across the VB₁ (j) control value.

The construction of the nucleotide complement (CN) table is described by the algorithm below.

This algorithm can be represented by Figure 8 below.

7.png

Figure 7. Transition to DNA

8-1.png

8-2.png

8-3.png

Figure 8. Construction of table (CN)

The vector (XC) of complementary nucleotides to vector (XN) is determined by the following algorithm:

The complementary strand forms the double helix of the DNA strand (Figure 9).

Figure 10 depict an example of vector (XC) transformation to vector (XN).

We observe that the location and values of the complements are closely related to the table (CN) and are taken in a pseudorandom manner.

DNA pseudorandom vector construction. We observe that the location and values of the complements are closely linked to the table (CN) and taken in a pseudo-random manner. A pseudo-random vector (BD) with DNA coefficients is constructed to introduce confusion with vectors (XN) and (XC) under the control of (VB3). This construction is determined by the algorithm below:

Figure 11 depict an example of Algorithm 10 above.

9.png

Figure 9. (XN) pseudorandom vector generation DNA

10.png

Figure 10. Example of vector (XC) conversion to vector (XN)

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Figure 11. Example of DNA pseudorandom vector generation

Genetic algebraic operator. An algebraic operator (⨂) between DNA values will be established under the control of the following table (AD). This operator is depicted in (Figure 12) below.

12.png

Figure 12. Genetic operator

The algebraic operator $(\otimes)$ employed to link two nucleotides is a reversible, commutative operator that provides a group structure to facilitate transformations on DNA. Figure 13 give an example of such operation.

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Figure 13. Algebraic operator $(\otimes)$ application example

DNA genetic crossover. The vectors (XN) and (XC) undergo genetic crossover with the vector (BD) to generate a new vector (XB) with DNA components, under the control of the table (TB) of size (12 nm; 3).

Construction crossover of table (TB):

·The first column obtained by sorting in ascending order the 12nm values of (VC₁) vector;

·The second column obtained by sorting in ascending order the 12nm values of (VC₂) vector;

·The third column is the (VB₁) vector.

Description of the role of the table (TB):

·The 1st column indicates the index of the vector (XN) or the vector (XC). Crossed with the following vector (BD);

·The second column gives the index of the position in the vector (XB);

·The third column indicates the choice of the vector (XN) or (XC) crossed with the vector (BD) through a genetic crossover table (AD).

·This genetic crossover at the DNA level is described by the following algorithm:

This passage is defined by the following diagram (Figure 14).

An example of such transformation is depicted in Figure 15.

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Figure 14. DNA crossover operation

15.png

Figure 15. Genetic DNA crossover transformation

Transition to RNA. RNA is the concatenation of three nucleotides read from a gene for the formation of a protein. For this, the vector (XB) is subdivided into three sub-vectors (XB₁), (XB₂), and (XB₃) each of size (1; 4 nm). The transition from DNA to RNA requires the construction of the table (TR) of size (4 nm; 5) defined by the following process:

·The first column is the arrangement (PR1), which is obtained by roughly sorting the first 4 nm values of (VC₁) in ascending order;

·The second column is the arrangement (PR2), which is obtained by roughly sorting the first 4nm values of (VC₂) in ascending order;

·The third column is the arrangement (PR3), which is obtained by roughly sorting the first 4 nm values of (VC₃) in ascending order;

·The fourth column is the 4 nm vector values (CO);

·The fifth column is the vector (VE).

Construction of (TR) table. These table construction steps (TR) are defined by:

Description of the table (EN):

·The first column gives the index of nucleotides in the sub vector (XB₁);

·The 2nd column gives the index of nucleotides in the sub vector (XB₂);

·The 3rd column gives the index of nucleotides in the sub vector (XB₃);

·The 4th column gives the index of the nucleotides that should be transcribed in complementary;

·The 5th column gives the row in the table (CN) for selecting the nucleotide value.

Figure 16 depict an example of (TR) table transition.

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Figure 16. Transition table (TR) example

The conversion of nucleotides into RNA is given by the following algorithm:

Figure 17 depict an example of transition from nucleotides to codons.

17-1.png

17-2.png

Figure 17. Transition from nucleotides to codons example

Algebraic operation $(\boxtimes)$ on RNA. Using the operator table $(A D)$, it is possible to combine two RNA through the following genetic operation: $(x y z) \boxtimes(u v w)=(x \otimes u)(y \otimes v)(z \otimes w)$.

On the other hand, the algebraic operator $(\boxtimes)$ used to link two codons is a reversible, commutative operator applied to RNA, dependent on $(\otimes)$. However, it is noteworthy that in the case of RNA, only one nucleotide at a time undergoes complementarity under the control of the control vector (CR).

An example of algebraic operator ( $\boxtimes)$ application is depicted in Figure 18.

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Figure 18. example of algebraic operator $(\boxtimes)$ application

We notice that the algebraic operation defined on the RNA is not commutative, which increases the complexity of attacks on our cryptographic system.

RNA crossover.

Construction of the reference RNA table

In a first step, a reference RNA table (TA) of size (2; 64) is constructed by the following process:

The 1st row is given by the following algorithm:

The second line includes codons whose numerical value is associated with the first line generated by the algorithm below:

This algorithm can be explained by an example depicted in Figure 19 below.

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Figure 19. Construction of the (TA) table example

The construction of the reference table (TA) is closely related to the nucleotide table (TR).

Generation of the pseudo-random vector (CR) with coefficient (RNA)

A vector (CR) with pseudo-random component RNA taking values from the reference table (TA) is constructed by the following process:

Construction of the (CR) vector is explained by an example depicted in Figure 20 below.

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Figure 20. Construction of the (CR) vector

The vector (XS) obtained through genetic crossover between the chaotic vector (CR) and the vector (XR) at the RNA level is given by the following algorithm:

The vector (XS) represents the RNA-coefficient encrypted image using our approach. Figure 21 represents an example of the (XS) vector construction.

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Figure 21. Construction of the (XS) vector example

Example:

Reconstruction of the encrypted image

The reconstruction of the encrypted image from the vector (XS) with RNA coefficient requires going through the following steps:

·Transition to DNA

The transition from RNA to DNA notation is ensured by the reference table (RT).

Example:

The transition from vector (XS) to vector (XD) in DNA is determined by the following algorithm:

·Transition to (Z⁄4Z)

We assume: ‘A’=0, ‘C’=1, ‘G’=2 et ‘T’=3

The vector (XD) will be converted to $(X T)$  with a coefficient of (Z⁄4Z):

The vector (XT) will be converted to degree level to (XF) by the following process:

An example of transition to (Z⁄256Z) is depicted in Figure 22 below.

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Figure 22. Transition to (Z⁄256Z) example

4.4 Decryption of the encrypted image

Our algorithm is characterized by a symmetric encryption system, involving the use of an identical key during the decryption process and the same encryption parameters. As our approach employs diffusion functions, the decryption procedure must start with the last encrypted block by applying the inverse functions of the corresponding encryption functions. This decryption process involves the following sequences:

·Encrypted image loading;

·Encryption key generation;

·Encoding of the encrypted image in RNA phase;

·Encoding of the encrypted image in DNA phase;

·Reverse genetic crossover;

·Transition to (Z⁄256Z);

·Confusion section;

·Construction of the original image.

These steps are illustrated by the diagram (Figure 23).

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Figure 23. Decryption scheme

To assess how well the algorithm performs under differential attacks, metrics such as the Number of Pixel Changes Rate (NPCR), Unified Average Change Intensity (UACI), and the avalanche effect are employed.

6.1 NPCR constant

The NPCR is determined by Eq. (6).

$N P C R=\left(\frac{1}{3 n m} \sum_{i, j=1}^{n m} D(i, j)\right) \cdot 100$                      (6)

where,

$D(i, j)=\left\{\begin{array}{lll}1 & \text { if } & C_1(i, j) \neq C_2(i, j) \\ 0 & \text { if } & C_1(i, j)=C_2(i, j)\end{array}\right.$             

6.2 UACI constant

The analysis of the mathematical constant $U A C I$ of the image is given by the Eq. (7).

$U A C I=\left(\frac{1}{255 n m} \sum_{i, j=1}^{n m} A B S\left(C_1(i, j)-C_2(i, j)\right)\right) \cdot 100$             (7)

The data in Table 6 presents the measures of NPCR and UACI for two images (Im1, Im2), showing that NPCR ≥ 99.6 and UACI≥33.4. These values clearly indicate that the performance of the NPCR of the proposed encryption algorithm was comparable to that of Alawida et al. [28] and superior to that of Manzoor et al. [36], Alawida et al. [37], Teseleanu [38], Jarjar et al. [39], Ye and Huang [40], and Chen et al. [41], while the UACI value was comparable to those of others.

6.3 Robustness against occlusion attacks

In the field of image processing, the parameters of Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) are frequently used to assess the quality of encryption. These metrics are the most commonly employed criteria for evaluating the quality of two images within a cryptographic system. PSNR measures the similarity between two images and is the complement of MSE. The Mean Squared Error (MSE) for the original, decrypted, and encrypted images can be computed using the following formula (8):

$M S E=\frac{1}{(n m)^2} \sum_{i, j=1}^{n m}|\operatorname{Imo}(i, j)-\operatorname{Imc}(i, j)|^2$              (8)

In the context of image processing) "Imo" and "Imc" respectively represent the original and encrypted images. MSE corresponds to the mean squared error. (n) denotes the number of rows in the original image, and (m) represents the number of columns in the image. PSNR is evaluated in decibels and is inversely proportional to the mean squared error. It is determined by Eq. (9).

$P S N R=10 * \log _{10}\left(\frac{\left(2^L-1\right)^2}{M S E}\right)(d B)$              (9)

where, L=8 represents the bit depth of the specific image.

A greater PSNR signifies reduced differences between the original and decrypted images as represented in Table 7. When the original and encrypted images are identical, their PSNR becomes infinite.

Table 6. Comparison of (NPCR) and (UACI)

Table 7. (PSNR) (dB) between the original image, the encrypted image, and the decrypted image

I would like to express my gratitude to all the authors who have contributed, directly or indirectly, to the completion of this article. I would like to emphasize that our research group is not supported by any public or private organization.