Fast Color Image Encryption Algorithm Based on DNA Coding and Multi-Chaotic Systems

Abstract

At present, there is a growing emphasis on safeguarding image data, yet conventional encryption methods are full of numerous limitations. In order to tackle the limitations of conventional color image encryption methodologies, such as inefficiency and insufficient security, this paper designs an expedited encryption method for color images that uses DNA coding in conjunction with multiple chaotic systems. The encryption algorithm proposed in this paper is based on three-dimensional permutation, global scrambling, one-dimensional diffusion and DNA coding. First of all, the encryption algorithm uses three-dimensional permutation algorithms to scramble the image, which disrupts the high correlation among the image pixels. Second, the RSA algorithm and the SHA-256 hashing algorithm are utilized to derive the starting value necessary for the chaotic system to produce the key. Third, the image is encrypted by using global scrambling and one-dimensional diffusion. Finally, DNA coding rules are used to perform DNA computing. The experimental results indicate that the encryption scheme exhibits a relatively weak inter-pixel correlation, uniform histogram distribution, and an information entropy value approaching eight. This shows that the proposed algorithm is able to protect the image safely and efficiently.

1. Introduction

In the digital age, digital images play an indispensable role as the main visual communication medium on the Internet and in all areas of life.With the relentless expansion in the scale and velocity of multimedia data dissemination in public networks, the imperative of safeguarding data integrity, authenticity, and confidentiality has escalated to a pivotal concern. The issue of how to encrypt images in a secure and efficient manner and effectively protect the information from being stolen has become a concern in the field of information security.

Over the past many years, numerous encryption algorithms have emerged [1,2], offering diverse approaches to safeguarding image security. There are generally three ways to protect image security. The first method is to embed private data into images [3,4]. This encryption method primarily involves disguising unnecessary information in digital media as important information to avoid guessing from attackers [5]. The second method is to insert invisible watermarks into the image [6]. The third method converts the image into ciphertext for encryption. Compared with the first and second methods, the third method can ensure higher security of the image. The commonly used image encryption algorithms are DES [7] and AES [8]. However, the encryption of images is difficult owing to the enormous volume, the high inherent data redundancy of images, and the strong correlation between neighboring pixels [9].

Chaotic systems have demonstrated considerable strength on account of their remarkable sensitivity to initial values. Minor alterations in these can lead to significantly different outcomes, rendering chaotic signals elusive to interception and prediction. This characteristic has made chaotic systems a popular choice in image encryption applications in recent years [10,11]. For one-dimensional chaotic systems, due to the possibility of periodic behavior in some trajectories and their relatively simple structure, using them for image encryption may lead to performance degradation, small key space, and fragility to various enumeration attacks. Recognizing these shortcomings, in the past few decades, many scholars have strived to design more chaotic systems with richer behaviors for more secure encryption [12,13]. For instance, Liu et al. [12] used novel parametric variable chaotic mapping to scramble planar images, which could effectively resist spatial reconstruction attacks by using parameter-varying chaotic mappings. Yin et al. [13] proposed an image encryption algorithm based on a two-dimensional dual discrete quadratic chaotic map, which improved the security of image encryption by generating highly random pseudo-random numbers and a feedback key mechanism. However, because of the inability of low-dimensional chaotic systems to accurately predict the behavior of certain complex systems and limited application scope, more and more scholars have shifted their focus to designing high-dimensional chaotic systems [14,15,16]. Tong et al. [14] utilized a 4D chaotic system to address issues of low complexity and slow image encryption speeds characteristic of low-dimensional chaotic systems. Zhu et al. [15] used compressed sensing and a new four-dimensional chaotic system to encrypt images, which enhanced the unpredictability and security of the encryption process by utilizing a four-dimensional chaotic system to generate a key stream. Xu et al. [16] applied a new four-dimensional hyperchaotic system in the encryption algorithm and obtained satisfactory encryption results.

Nowadays, there are more and more ways to attack images, and encryption schemes using only chaotic systems no longer satisfy the needs of more demanding encryptions. Consequently, integrating chaotic systems with innovative techniques is essential to enhance the image protection. Inspired by DNA coding, the field of bioimage coding is being used by scientists to address potential security issues and effectively counter emerging threats. DNA coding enhances the protection of data by converting image data into DNA sequences to create a more secure encryption scheme. Combining DNA coding technology with hyper-chaotic systems can bring more advantages to encryption schemes, enhancing their security and performance. Therefore, conventional attacks can be well resisted by using DNA theory in encryption algorithms [17]. DNA coding can convert data into the form of bases and manipulate the data through the form of bases, a form that can greatly increase the difficulty for attackers to attack and thus effectively protect the image. As a result, a significant portion of scholars have turned to utilizing chaotic systems and DNA-based encoding techniques for the purpose of images encryption [18,19,20,21]. Li et al. [18] used DNA encoding and a six-dimensional hyperchaotic system to encrypt images. The outcomes of the experiments demonstrated that the encryption method possesses high resistance against both geometric and truncation attacks. Yu et al. [19] encrypted images through used spatiotemporal chaos and DNA coding, which utilized DNA manipulation to encrypt two images simultaneously in confusion and diffusion, thus improving encryption speed and security. Kumar et al. [20] utilized a new chaotic mapping and DNA-based diffusion for encryption by applying DNA manipulation in the diffusion phase, which effectively enhanced the obfuscation and randomization of the image. Patro et al. [21] designed a scheme that incorporates hyper-chaos and DNA encoding operations, enhancing the strength of encryption and decryption.

Reordering the images throughout the encryption process can significantly enhance the algorithm’s security. Common transformation algorithms include Zigzag, Fisher–Yates shuffle, etc. Zigzag transformation is a common permutation operation [22,23] that is often used in encryption algorithms. In Zigzag transformation, the plaintext is arranged in a zigzag pattern and then read according to specific rules to form the ciphertext. Guo et al. [24] employed an enhanced reverse Zigzag transformation loop traversal algorithm to encrypt images. The algorithm proposed by Wang et al. [25] utilized the Fisher–Yates algorithm to generate chaotic sequences, thereby enhancing the randomness of encrypted images. Naim et al. [26] used the Fisher–Yates shuffle algorithm to rearrange the rows and columns of images, thereby effectively encrypting the images. The above mentioned transformation algorithms can provide some protection for images encryption and is suitable for basic encryption needs. However, it is not a high-strength encryption method and is vulnerable to attack such as statistical and frequency analysis. Therefore, during the encryption process, this paper utilizes a condition based on the sum of the row and column indices to reorder the images.

The encryption schemes generated by a chaotic system may have some security risks because a single or simply structured chaotic map may be less accurate and less secure [27]. Moreover, conventional chaotic maps suffer from drawbacks like cyclic and limited key space [28], which makes it difficult to resist advanced attacks. Therefore, the encryption scheme proposed in this paper includes multiple chaotic systems, and different sequences are generated by different chaotic systems to encrypt the image. Compared with the traditional encryption methods, the proposed scheme in this paper will go through multiple stages during encryption, which can make the data more secure during transmission and thus increase the difficulty of cracking. By testing the images at different scales separately, the experimental results are satisfactory. Therefore, it is evident that we propose a scheme that significantly improves the encryption speed and anti-attack capability of images, enhancing bandwidth utilization and saving storage space.

The contributions made by this article are listed below:

(1)

To make the arrangement pattern of the images unpredictable, this paper uses the parity of the sum of index values as a condition and applies different permutation rules accordingly.

(2)

The image information added to the hash algorithm, generating the initial conditions required for chaotic mapping and applying to the entire encryption process.

(3)

The encryption method necessitates the integration of several chaotic systems. In parallel, this paper produces distinct sequences from various chaotic systems to perform scrambling and diffusion operations, respectively.

(4)

This paper proposes new scrambling and diffusion algorithms to encrypt color images. After the scrambling and diffusion operations are completed, the image is then subjected to a DNA coding operation. After a round of encryption operations, the image exhibits a very high level of complexity.

(5)

After analyzing and comparing with other encryption algorithms, it can be seen that the encryption algorithm proposed in this paper shows excellent encryption performance. Furthermore, the encryption rate can be guaranteed at the same time while simultaneously ensuring the protection of the image.

The rest of the paper is structured as follows. In Section 2, the theoretical framework is described in detail. Section 3 outlines the process of encryption for the proposed scheme. Section 4 displays the outcomes of the experiments and contrasts these with the previously proposed algorithm. Eventually, Section 5 summarizes the conclusions of the paper.

2. Methods

2.1. FHCCS

Contrasting with the simpler low-dimensional chaotic systems, the more complex high-dimensional hyperchaotic systems exhibit a greater number of key spaces, a greater number of control parameters and a more pronounced chaotic behavior [29]. In [30], a paradigm is introduced for creating 2n + 1 dimensional minimalist Hamiltonian Conservative Chaotic Systems (HCCSs). Expanding upon this paradigm, a novel five-dimensional HCCS (FHCCS) is formulated, which incorporates the least possible number of elements while exhibiting intricate dynamical properties. Its expression is as follows:

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\\ {\dot{x}}_4\\ {\dot{x}}_5\end{array}\right]=\left[\begin{array}{ccccc}0& 0& 0& 0& asin\left(\pi x_2\right)\\ 0& 0& 0& b{x}_1& 0\\ 0& 0& 0& c& 0\\ 0& -b{x}_1& -c& 0& 0\\ -asin\left(\pi x_2\right)& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\end{array}\right]=\left[\begin{array}{c}a{x}_{5}sin\left(\pi x_2\right)\\ b{x}_1{x}_4\\ c{x}_4\\ -c{x}_3-b{x}_1{x}_2\\ -a{x}_{1}sin\left(\pi x_2\right)\end{array}\right]$$

(1)

In [30], the bifurcation behavior of parameter a is investigated with parameters b = 35 and c = 25, and initial values [1,1,1,1,1]. The results exhibit abundant chaotic behavior.

2.2. MOTDCM

To elevate the intricacy of the encryption methodology, A mixed one-and two- dimensional chaotic map (MOTDCM) [31] is used to generate sequences which are used to diffuse the images. MOTDCM consists of a two-dimensional logistic map and a modified one-dimensional Feigenbaum transcendenta map. It retains the same structure as the two-dimensional logistic map while incorporates the modified one-dimensional Feigenbaum transcendental map. The mathematical description of the MOTDCM system is shown as follows:

$$\left\{\begin{array}{c}{x}_{n+1}=mod(4\ast \alpha \ast x_n\ast (1-x_n)+{\gamma }_1\ast y_n^2,1),\hfill \\ y_{n+1}=mod(12\ast \beta \ast sin(\pi \ast y_n)\ast (1-3\ast sin(\pi \ast y_n)+{\gamma }_2\ast x_n^2,1),\hfill \end{array}\right.$$

(2)

where $\alpha$∈[0, 6], $\beta$∈[0, 6], ${\gamma }_1$∈[0, 2] and ${\gamma }_2$∈[0, 2]. When $\beta$ > 0.1, the system exhibits chaotic behavior, and chaotic performance with the increase in $\beta$. Here, we take $\alpha$ = 6, $\beta$ = 5, ${\gamma }_1$ = 1, ${\gamma }_2$ = 1. Figure 1 illustrates that the MOTDCM system’s trajectory plot is distributed uniformly across the entire range, indicating the presence of favorable chaotic characteristics.

2.3. Three-Dimensional Permutation

The initial position for the standard Zigzag transformation is the first position and is often easier to crack by attacking in the opposite direction as shown in Figure 2. This paper proposes a three-dimensional permutation that effectively addresses this drawback. Before the permutation begins, the matrix is first divided into four blocks. Afterwards, the user needs to manually input an index value, and each subsequent traversal requires using this index value for computation to determine the next permutation position. During the permutation process, reordering each position requires the sum of the current position index as a condition, ensuring that odd sums and even sums are permuted in different ways. If the position of the first pixel to be replaced in the encryption process is (1,1), it contains a pixel value with three channels. First of all, the sum of the index values is needed to determine which substitution method should be used. If the sum of the index values is even, then the operation is performed according to the even-sum substitution method, and then a new pixel position is obtained, which also contains the pixel values of the three channels. Finally, the value at position (1,1) is replaced with the value of the newly obtained pixel. The implementation diagram of the three-dimensional permutation is shown in Figure 3. The specific implementation of the three-dimensional permutation is shown in Equation (3):

$$\left\{\begin{array}{c}U(i,j,:)=CU(A_1,B_1),ifmod(i,j)==0,\hfill \\ U(i,j,:)=CU(A_2,B_2),ifmod(i,j)==1,\hfill \end{array}\right.$$

(3)

where $CU$ is the image obtained after partitioning. U is the new image obtained after permutation. Row_index and Col_index are the index values input by the encryptor. $A_1$ = $\left(mod\right(size(CU,1)-i+row$_$index,size(CU,1))$, $B_1$ = $\left(mod\right(size(CU,2)-j+col$_$index,size(CU,2))+1,:)$, $A_2$ = $\left(mod\right(i+row$_$index,size(CU,1))+1)$, and $B_2$ = $\left(mod\right(j+col$_$index,size(CU,2))+1,:)$.

2.4. Basic Theory of DNA

The DNA coding is composed of four bases—adenine (A), thymine (T), cytosine (C), and guanine (G). Among them, A pairs with T, and C pairs with G [20]. This forms the basis for the genetic information encoded in DNA. Base A complements base T, and base C complements base G. Since computers only have binary zeros and ones, and they are complementary, that is, 0 and 1 complement each other, 01 and 10 complement each other, and 00 and 11 complement each other. By specifying this, A, T, G, and C are coded as 00, 11, 01, and 10, respectively. They have a total of 8 groups that satisfy the complementarity rule as illustrated in in

Table 1. DNA coding rules.

Rule	1	2	3	4	5	6	7	8
A	00	00	01	01	10	10	11	11
G	11	11	10	10	01	01	00	00
C	01	10	00	11	00	11	01	10
T	10	01	11	00	11	00	10	01

.

If the pixel value of the image is 180, the first step in the encoding process is to convert it to the binary code “10110100”. The above tables can correspond to 8 different encoding rules, namely, $TGCA$, $CGTA$, $GTAC$, $GCAT$, $ATGC$, $ACGT$, $TACG$, $CATG$. We use rule 1 and obtain a DNA code of $TGCA$. If rule 5 is used to decode it, the binary number “11010010” is obtained, which is converted into a decimal number of 210. It can be seen that the use of DNA coding can significantly alter the original values. Following the encoding of values into DNA strands, the security can be significantly bolstered by performing arithmetic manipulations like adding and subtracting on the images [32]. The DNA-encoding implementation flow is depicted in Figure 4. DNA addition, subtraction, XOR and XNOR operations are shown in

Table 2. DNA addition Operation.

+	A	T	C	G
A	A	T	C	G
T	T	C	G	A
C	C	G	A	T
G	G	A	T	C

,

Table 3. DNA subtractive operation.

-	A	T	C	G
A	A	G	C	T
T	T	A	G	C
C	C	T	A	G
G	G	C	T	A

,

Table 4. DNA XOR operation.

⊕	A	T	C	G
A	A	G	A	T
T	G	C	T	A
C	A	T	C	G
G	T	A	G	C

and

Table 5. DNA XNOR operation.

⊙	A	T	C	G
A	A	T	C	G
T	T	A	G	C
C	C	G	A	T
G	G	C	T	A

, respectively.

2.5. Key Generation Theory

This paper employs the SHA-256 algorithm to generate the initial value of the chaotic map. To make the encrypted images more secure, the information of images is used to generate the key. Therefore, the initial value required by MOTDCM is obtained by the SHA-256 algorithm and images information. In addition, the initial values of FHCCS are obtained differently from MOTDCM. The encryption algorithm employs the RSA and SHA-256 hashing algorithm to produce the essential parameters for the FHCCS. The SHA-256 algorithm provides hash calculations, and the RSA algorithm offers asymmetric encryption, digital signatures, and key exchange, generating keys that possess higher security and functionality.

2.6. Global Scrambling

This paper uses FHCCS to generate a chaotic sequence S. The sequence S is generated by a modulo operation to keep all the values in an interval, after which the obtained values are rounded to obtain a sequence of integers. Finally, S is divided into two sequences $E_1$ and $E_2$. The sequences $E_1$ and $E_2$ are then arranged according to their values, and two new sequences, $sort\_E_1$ and $sort\_E_2$, are obtained. Then, according to $E_1$ and $E_2$, and $sort\_E_1$ and $sort\_E_2$, we can obtain its index sequences $index\_E_1$ and $index\_E_2$. The final index values are used to scramble the image as depicted in Figure 5. In the scrambling process, $index\_E_1$ and $index\_E_2$ are taken as coordinates that need to be exchanged. Through multiple exchanges, the original values of the matrix can be disrupted on a large scale to achieve the scrambling effect.

2.7. One-Dimensional Diffusion

After performing global scrambling on the images, the algorithm applies one-dimensional diffusion that can enhance the encryption effect of the encrypted images. One-dimensional diffusion involves converting the matrix into a one-dimensional array and operating on it with a sequence, which complets the diffusion of the encrypted images. The values obtained after the diffusion depend on both the sequence and the previously diffused values. Performing diffusion at the one-dimensional level increases the complexity of the images. A schematic diagram of one-dimensional diffusion is shown in Figure 6. The specific implementation of one-dimensional diffusion is shown in Equation (4):

$$C\left(i\right)=\left\{\begin{array}{c}C\left(i\right)\oplus C\left(N\right)\oplus C(N-1)\oplus mod\left(M2\left(i\right)\ast {32}^{32},255\right),i=1,\hfill \\ C\left(i\right)\oplus C(i-1)\oplus C\left(N\right)\oplus mod\left(M2\left(i\right)\ast {32}^{32},255\right),i=2,\hfill \\ C\left(i\right)\oplus C(i-1)\oplus C(i-2)\oplus mod\left(M2\left(i\right)\ast {32}^{32},255\right),i>2.\hfill \end{array}\right.$$

(4)

The information of the pixels at each position is diffused throughout the images after the one-dimensional diffusion is performed.

3. Algorithm Flow for Images Encryption and Images Decryption

In this section, we propose a color image encryption algorithm by incorporating three-dimensional permutation, global scrambling, one-dimensional diffusion, and DNA coding theories. First of all, three-dimensional permutation operations are performed on three dimensions of the color images. Subsequently, the MOTDCM system is utilized to produce DNA coding and operation rules, key matrices, and sequences for diffusion. Then, the chaotic sequences generated by FHCCS and MOTDCM are utilized for scrambling and diffusion for the image, respectively. Ultimately, the obtained images undergo DNA coding for the purpose of achieving comprehensive encryption. The detailed process can be visualized in the flowchart depicted in Figure 7.

3.1. Encryption Algorithm

3.1.1. Three-Dimensional Permutation

Image Q is loaded, and Q is divided into four blocks. Next, we perform three-dimensional permutation on the images.

3.1.2. Key Generation

To enhance the intricacy of the key, the SHA-256 hashing algorithm combined with image information is incorporated into the key generation process. First, a 32-bit key is obtained using the SHA-256 algorithm and image information. Then, the 32-bit key is divided into $V_1$, $V_2$, and $V_3$…$V_{32}$. Secondly, using $V_1$, $V_2$, and $V_3$…$V_{32}$, we perform 8 rounds and shifts, and obtain a key K. The variables $K_1$, $K_2$, and $K_3$…$K_8$ required by MOTDCM are obtained by K. The calculation rules are shown in Equation (5):

$$\left\{\begin{array}{c}{K}_1={\displaystyle \frac{1+5\ast K_1}{2^{64}}},\hfill \\ K_2={\displaystyle \frac{K_1+5\ast K_2}{2^{64}}},\hfill \\ K_3={\displaystyle \frac{K_2+5\ast K_3}{2^{64}}},\hfill \\ K_4={\displaystyle \frac{K_3+5\ast K_4}{2^{64}}},\hfill \\ K_5={\displaystyle \frac{K_4+5\ast K_5}{2^{64}}},\hfill \\ K_6={\displaystyle \frac{K_5+5\ast K_6}{2^{64}}},\hfill \\ K_7={\displaystyle \frac{K_6+5\ast K_7}{2^{64}}},\hfill \\ K_8={\displaystyle \frac{K_7+5\ast K_8}{2^{64}}}.\hfill \end{array}\right.$$

(5)

The initial values of the MOTDCM system $A_1$ and $A_2$…$A_6$ are calculated by combining the initial variable and the embedded image information with the value obtained by the SHA-256 algorithm. The generation rules of six MOTDCM initial values are shown in Equation (6). We take the initial variables ${\overline{A}}_1=4, {\overline{A}}_2=3, {\overline{A}}_3=2, {\overline{A}}_4=2, \mathrm{and} {\overline{A}}_5=0.4, {\overline{A}}_6=0.3$:

$$\left\{\begin{array}{c}{A}_1={\overline{A}}_1+{\displaystyle \frac{mod\left((K_1+K_2+K_3+K_4)\ast 2^{10},255\right)}{256}},\hfill \\ A_2={\overline{A}}_2+{\displaystyle \frac{mod\left((K_1+K_2+K_5+K_6)\ast 2^{10},255\right)}{256}},\hfill \\ A_3={\overline{A}}_3+{\displaystyle \frac{mod\left((K_1+K_2+K_7+K_8)\ast 2^{10},255\right)}{256}},\hfill \\ A_4={\overline{A}}_4+{\displaystyle \frac{mod\left(K_3+K_4+K_5+K_6\right)\ast 2^{10},255)}{256}},\hfill \\ A_5={\overline{A}}_5+{\displaystyle \frac{mod\left((K_3+K_4+K_7+K_8)\ast 2^{10},255\right)}{256}},\hfill \\ A_6={\overline{A}}_6+{\displaystyle \frac{mod\left((K_5+K_6+K_7+K_8)\ast 2^{10},255\right)}{256}}.\hfill \end{array}\right.$$

(6)

The FHCCS also uses Equations (5) and (6) to generate the eight required variable values. In addition, the value obtained by the RSA algorithm is also added to the initial variable of the FHCCS. At the same time, the generation of RSA key values also requires the use of the variable value of the FHCCS. The obtained values are substituted into the RSA algorithm to calculate the public and private keys. The public key obtains the ciphertext, while the private key decrypts it. The result is the encrypted ciphertext. Finally, the key value generated by RSA is obtained. The key value generated by the RSA algorithm replaces some of the key values generated by using SHA-256 and the images information, which can make the encryption algorithm more secure.

3.1.3. Generating Chaotic Sequences for Scrambling and Diffusion

Step 1: Input image Q and obtain the initial values required by the MOTDCM ($A_1$, $A_2$, $A_3$, $A_4$, $A_5$, and $A_6$) through SHA-256.

Step 2: Substituting the initial value K = ($A_1$, $A_2$, $A_3$, $A_4$, $A_5$, and $A_6$) into the MOTDCM yields a chaotic sequence. To further increase the chaos of the sequence, we discard the first w digits of the sequence, and finally obtain chaotic sequences $M_1$ and $M_2$, which are used to obtain a key matrix for DNA computation.

Step 3: The initial values ($B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$, $B_7$, and $B_8$) required by the FHCCS system are obtained using the SHA-256 algorithm and the RSA algorithm.

Step 4: The initial values obtained through the RSA algorithm are K1 = ($Z_1$, $Z_2$, $Z_3$, $Z_4$, and $Z_5$). Some variables are replaced with values obtained using RSA, resulting in the final initial value K = ($Z_1$, $Z_2$, $Z_3$, $Z_4$, $Z_5$, $B_6$, $B_7$, and $B_8$) used for generating the chaotic sequence.

Step 5: In order to guarantee that the encryption algorithm is more secure, this paper uses different keys to generate chaotic sequences that are scrambled and diffused. Moreover, the key is generated one at a time during encryption. Substituting the initial value K = ($Z_1$, $Z_2$, $Z_3$, $Z_4$, $Z_5$, $B_6$, $B_7$, $B_8$) into FHCCS generates a chaotic sequence S for scrambling. The chaotic sequences obtained from MOTDCM are $M_1$ and $M_2$. $M_2$ is used to generate the sequence for diffusion operations on the image.

3.1.4. Diffusion and Scrambling

The encryption algorithm employs the principles of diffusion and scrambling. In the scrambling phase, the image is scrambled using the sequence S, thus obtaining the encrypted image $C_1$. Subsequently, an image diffusion process is executed with the $M_2$ sequence applied, resulting in the production of the encrypted image $C_2$.

3.1.5. DNA Coding

The key matrix Q1 is obtained from the sequence $M_1$. The sequences ($X,Y,Z,H$) for DNA operations and DNA coding rules are obtained from $M_2$. X represents the coding rule of the Q. Y is the coding rule of the Q1. Z represents DNA coding operations. H represents the DNA decoding operation. Finally, Q is subjected to DNA operation with Q1 to finalize the encryption process.

3.2. Decryption Algorithm

For the decryption of an image, the exact opposite of encryption must be performed, otherwise, the encryption will not be completed. In addition, the processing of the DNA coding should also be reversed in the decryption process, and only by performing the complete opposite of the encryption process can the correct decryption images be obtained.

Step 1: During decryption, the DNA encoding performed must be the opposite of the encryption process, followed by executing the inverse DNA computation operation.

Step 2: After completing the DNA operation, the one-dimensional diffusion operation of the inverse placement is performed. In the encryption phase, the image is diffused from front to back. Therefore, the same diffusion logic is used in the decryption phase, but the diffusion process needs to be performed from back to front.

Step 3: The scramble process also requires the opposite approach. An inverse global scrambling operation is performed on the encrypted image according to the resulting sequence. In encryption, the scrambling starts with the first element and swaps elements row by row and column by column. For decryption, the last element is used as the starting point, and the elements are also exchanged row by row and column by column in the reverse direction.

Step 4: Divide the image into four pieces. Then, each piece is individually performed in reverse three-dimensional permutation. Thus, the decrypted image is obtained.

4. Simulation Results and Safety Tests

This paper uses Matlab R2023b to run our proposed algorithm. This experiment is run by using a computer with Windows 11 operating system, Intel(R) Core i5-13400F, 2.50 GHz and 8.00 GB RAM. All the pictures used in the experiment are 512*512 color pictures.

4.1. Histogram Analysis

In this experiment, a total of four images are selected, namely, Airplane, Baboon, Lake and Pepper.

The graphical representation known as a histogram provides a potent method to illustrate the spread of shades of gray throughout the complete images. Figure 8 illustrates that the histogram of the original images presents uneven distribution, containing a significant amount of image information. However, the histogram of the encrypted images shows a steady state. As such, it invalidates statistical attacks.

4.2. Correlation Analysis

The lower the correlation between neighboring pixels of an image, the higher the security of the algorithm.

In an effort to assess the image’s resistance against attacks, 5000 pixel pairs are randomly picked from the horizontal, vertical, and diagonal orientations of the image. The specific formulas are provided in Equations (7)–(10):

$$r_{xy}={\displaystyle \frac{cov(x,y)}{\sqrt{D\left(x\right)D\left(y\right)}}},$$

(7)

$$Cov(x,y)={\displaystyle \frac{1}{N}}\sum _{i=1}^N(x_i-E\left(x\right))(y_i-E\left(y\right)),$$

(8)

$$E\left(x\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{x}_i,$$

(9)

$$D\left(x\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(x_i-E\left(x\right)\right)}^2,$$

(10)

where N denotes the total count of chosen pixel pairs. $r_{xy}$ is the correlation coefficient. The average value of the selected pixels is denoted by $E\left(x\right)$. $D\left(x\right)$ is the variance, and $Cov(x,y)$ represents the correlation function.

Table 6. Correlation coefficients of different images.

Image			Airplane	Peppers	Lake	Baboon
Plain	Level	R	0.97355	0.96754	0.95455	0.92224
images		G	0.95908	0.98136	0.97123	0.85976
		B	0.96599	0.96947	0.96937	0.90562
	vertical	R	0.95516	0.96546	0.95414	0.86959
		G	0.9674	0.98217	0.96703	0.7521
		B	0.92997	0.96471	0.97158	0.88098
	Diagonal line	R	0.93559	0.95997	0.94264	0.85806
		G	0.93373	0.97091	0.95186	0.74225
		B	0.91332	0.94999	0.95201	0.83879
Encrypt	Level	R	−0.0091464	−0.00013964	−0.0039296	−0.000264
images		G	0.00055585	0.033616	0.0048301	−0.0006278
		B	−0.0071193	−0.0017792	0.0051603	−0.0015394
	vertical	R	0.0071193	−0.020117	0.0075364	−0.0003714
		G	−0.00071544	−0.0045737	−0.003334	0.00036302
		B	0.010181	0.0084629	−0.040256	0.00082484
	Diagonal line	R	-0.010503	-0.006991	0.014811	−0.000269
		G	−0.0060571	−0.001500	0.0009328	−0.0015874
		B	−0.0082853	0.024363	−0.010184	−0.00016499

demonstrates the correlation coefficients before and after image encryption. It can be seen that the correlation coefficients obtained after encryption are considerably lower. Meanwhile, a comparison is conducted with other algorithms as shown in

Table 7. Comparison of correlation coefficients of Baboon images.

		Proposed	Ref. [32]	Ref. [33]	Ref. [34]	Ref. [35]
Level	R	−0.000264	−0.0056	−0.0005	−0.002389	0.006319
	G	−0.0006278	−0.0171	0.0021	−0.001452	0.002034
	B	−0.0015394	−0.0034	0.0024	0.000427	−0.004609
vertical	R	−0.0003714	−0.0235	−0.0023	0.000689	−0.008340
	G	0.00036302	0.0052	−0.0005	0.000863	0.016949
	B	0.00082484	−0.0071	−0.0019	−0.001932	−0.005533
Diagonal line	R	−0.000269	−0.0015	0.0001	0.000492	−0.000352
	G	−0.0015874	−0.0092	0.0033	−0.000382	−0.000142
	B	−0.00016499	0.0216	0.0016	0.001381	−0.007744

. It is evident that the correlation coefficient of the encrypted image tends to 0, which indicates that the correlation between the pixels is destroyed.

4.3. Differential Attack Analysis

NPCR and UACI can be used as metrics to measure the strength and security of encryption algorithms [32]. The ideal values for NPCR and UACI are 99.6094% and 33.4635%, respectively. The mathematical expressions for NRCR and UACI are provided in Equations (11) and (12):

$$NPCR={\displaystyle \frac{1}{CG}}\sum _{i=1}^C\sum _{j=1}^{G}D(i,j)\times 100\%,$$

(11)

$$UACI={\displaystyle \frac{1}{CG}}\left[\sum _{i=1}^C\sum _{j=1}^G|E(i,j)-E^{\prime }(i,j)|/255\right]\times 100\%,$$

(12)

where the height and width of the image are denoted by C and G, respectively. $E(i,j)$ denotes the encryption of the image at a pixel location prior to modification, whereas $E^{\prime }$$(i,j)$ signifies the encrypted form of the image post-pixel alteration.

Table 8. Experimental results for NPCR and UACI.

		NPCR(%)			UACI(%)
Images	R	G	B	R	G	B
Airplane	99.6185	99.6125	99.6113	33.4845	33.5153	33.5251
Peppers	99.6231	99.6098	99.6147	33.5705	33.4751	33.5267
Lake	99.6173	99.6372	99.6254	33.5043	33.4647	33.4767
Baboon	99.6197	99.6181	99.6250	33.4954	33.4996	33.5360
Ref. [1]	99.6071	99.6087	99.6026	33.4355	33.4355	33.4358
Ref. [9]	99.6151	99.6071	99.6021	33.4718	33.4148	33.4378

shows NPCR, UACI, and the values used for comparison. It indicates that the obtained results are better than most of the literature values, suggesting that our algorithm can effectively protect the image data while maintaining their original quality and information integrity.

4.4. Information Entropy Analysis

The degree of randomness within an encrypted image is assessed through the measure of information entropy. An algorithm exhibiting elevated information entropy levels is deemed to possess enhanced security. Its expression is (13)

$$H\left(K\right)=-\sum _{j=1}^{N}P\left(k_j\right)lo{g}_{2}P\left(k_j\right),$$

(13)

where $K_j$ represents the value of each level. As evidenced by

Table 9. Experimental results of information entropy results.

Image	Before Encryption				After Encryption
	image	R	G	B	R	G	B	image
Airplane	6.6639	6.7178	6.7990	6.2138	7.9992	7.9993	7.9993	7.9998
Peppers	7.6698	7.3388	7.4963	7.0583	7.9993	7.9993	7.9993	7.9998
Lake	7.7622	7.3124	7.6429	7.2136	7.9993	7.9994	7.9993	7.9998
Baboon	7.7624	7.7067	7.4744	7.7522	7.9992	7.9993	7.9993	7.9998

, the encrypted image’s information entropy is always close to 8, indicating a good encryption effect. From

Table 10. Comparison of information entropy in Baboon images.

	Original	R	G	B	R	G	B	Result
Baboon	7.7624	7.7067	7.4744	7.7522	7.9992	7.9993	7.9993	7.9998
Ref. [35]	-	7.3650	7.3975	7.3115	7.9992	7.9993	7.9991	7.9992
Ref. [36]	7.6444	-	-	-	-	-	-	7.9994
Ref. [37]	7.3585	-	-	-	-	-	-	7.9993

, it is evident that the encryption method we propose surpasses the performance of other competing encryption methods.

4.5. Noise Attack Analysis

A robust encryption algorithm must possess the capacity to withstand noise attacks. Figure 9 and Figure 10 respectively demonstrate that the decrypted images of encrypted images are subjected to noise attacks of varying intensities. The experimental results indicate that even under high levels of noise interference, the main information can still be recognized. This demonstrates excellent resistance to noise attacks.

4.6. Shear Attack Analysis

During the images storage and transfer phase, the loss of information may happen. A good encryption algorithm allows for the recovery of as much original image information as possible in the case of partial information loss. Experimenting with Baboon image, in this test, the image is intentionally subjected to information loss before decryption and then decrypted. Figure 11 demonstrates that even in the event of 50% loss of information, the decrypted image is still capable of recovering a substantial portion of the original image.

4.7. Key Space

An algorithm with a key space greater than $2^{100}$ is generally considered to be resistant to brute force attacks. In this paper, part 1 and part 2 together form the key space of this paper. Part 1 is the core key space of the hashing function, encompassing a magnitude of $2^{128}$. Part 2 pertains to the variable parameters within the chaotic system, with a precision level of ${10}^{-15}$, resulting in a key space of ${10}^{120}$ for this part. Consequently, the combined key space amounts to $2^{128}$ × ${10}^{120}$ ≈ $2^{526}$, far exceeding the value of $2^{100}$. Hence, it can be considered that this algorithm has excellent resistance against exhaustive attacks.

4.8. Encryption Speed Analysis

The evaluation of encryption algorithms requires a dual approach, which is composed of security and encryption speed. The encryption efficiency of an encryption algorithm is also one of the important indexes to measure whether an encryption algorithm is excellent or not.

Table 11. Encryption efficiency with different schemes.

	Proposed	Ref. [38]	Ref. [39]	Ref. [40]	Running on Poor Hardware
Encryption time (s)	2.6840	3.1143	10.8785	6.1855	2.8913

demonstrates the comparison of encryption time for different encryption methods. The outcome reveals that the encryption method presented in this paper outperforms the encryption methods found in the literature in terms of efficiency. Concurrently, in order to thoroughly assess the performance of the encryption method introduced, this section conducts an analysis of its encryption capabilities on a computer with poor hardware conditions. The experimental results obtained are almost indistinguishable from those obtained on computers with excellent hardware conditions.

4.9. Key Sensitivity Analysis

A robust encryption method must exhibit extreme responsiveness to its key. To verify the key sensitivity, we make a minor adjustment to the value of one initial key in the algorithm while keeping other parameters constant during the experiment. This section tests the sensitivity of both correct and incorrect keys separately. We obtain secret key1 and key2, where both key matrices consist of eight digits. Then, we let key1(1) + 0.0005 and key2(1) + 0.0005, while keeping the others unchanged. Figure 12 demonstrates that an incorrect key renders the decryption process ineffective.

5. Conclusions

This paper presents an expedited encryption method which combines several chaotic systems and DNA coding techniques. In this encryption algorithm, the initial value required for MOTDCM is obtained using the SHA-256 algorithm and the image information, whereas the acquisition of the initial value of FHCCS requires the addition of the RSA algorithm on top of the SHA-256 algorithm and the information of the image itself. In the encryption stage, the pixel correlation of the R, G, B channels is firstly destroyed by using three-dimensional permutation, thus effectively cutting off the interrelationship between the data. Then MOTDCM is used to generate the sequence used to perform the diffusion operation, the encryption rules for DNA coding, and the key matrix for the DNA operations. Next, FHCCS is used to generate the sequence for the scrambling operation on the image. During the scramble operation, the index values of the sequences are obtained by sorting, after which the coordinates consisting of these index values are used to globally scramble the image. After the scrambling is completed, a one-dimensional diffusion operation is performed on the images. Finally, according to the encoding rules, the image is first encoded using DNA. Then, based on the encoding calculation rules, DNA arithmetic operations are performed to complete the final encryption process and obtain the final encrypted image. Upon conducting a comparison of various encryption methodologies, it is evident that the information entropy of the encryption scheme designed in this study approaches eight. Additionally, the obtained values of NPCR and UACI surpass those of alternative methods, while the scheme also ensures both speed and security encryption processes. Moreover, upon conducting a thorough analysis of the prevalent assault methods including histogram analysis, correlation analysis, defense against clipping tactics, and safeguarding against noise interference, it is obvious that the proposed encryption approach exhibits exceptional efficiency and robust security features. In practical applications, on the precondition of ensuring the encryption effect, we consider compressing the image to be processed before encryption to reduce resource waste and improve encryption speed.