Adaptive Color Image Encryption Scheme Based on Multiple Distinct Chaotic Maps and DNA Computing

Abstract

In this paper, we propose an adaptive encryption scheme for color images using Multiple Distinct Chaotic Maps (MDCM) and DNA computing. We have chosen three distinct chaotic maps, including a 2D-Henon map, a Tent map, and a Logistic map, to separately encrypt the red, green, and blue channels of the original image. The proposed scheme adaptively modifies the parameters of the maps, utilizing various statistical characteristics such as mean, variance, and median of the image to be encrypted. Thus, whenever there is a change in the plain image, the secret keys also change. This makes the proposed scheme robust against the chosen and known plaintext attacks. DNA encoding has also been used to add another layer of security. The experimental analysis of the proposed scheme shows that the average value of entropy is approximately eight, the Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are 99.61% and 33%, respectively, and correlation coefficients close to zero, making the scheme not only reliable but also resilient against many attacks. Moreover, the use of low-dimensional maps reduces the computational costs of the scheme to a large extent.

1. Introduction

With the increase in digitization, the need for and importance of information security also increase. To transmit information with ease, security is the ultimate target of digital communication. Nowadays, it has become an important issue to ensure secure communication for the defense department, commercial, and medical fields [1,2]. Digital information is available in various formats such as images, videos, and audio. This information can be accessed on different social and web platforms. One of the most common carriers of digital information is images, and thus, they require significant protection against various attacks that could otherwise reveal crucial information and put the security of the system in danger.

Image encryption is an important and prominent method that ensures the security of images. Numerous standard encryption methods such as International Data Encryption Algorithm (IDEA), Rivest Shamir Adleman (RSA), Advanced Encryption Standard (AES), and Data Encryption Standard (DES) exist, but these are used mainly for encryption of textual data and are not suitable for digital images. The reason is that these standard methods of encryption comprise many rounds and operations. Moreover, there are some inherent features of digital images, such as data redundancy, bulk data capacity, and strongly correlated adjacent pixels, as a result of which the traditional ciphers require high computational costs, thus, making them inappropriate for use in real-time image encryption [3,4,5].

The trend nowadays involves designing encryption algorithms that are based on chaos. The reason for this is that chaos has various fundamental characteristics related to cryptography [6,7,8,9,10,11]. Among various features of chaos, its high sensitivity to the preliminary conditions and control parameters, computational efficiency, and non-periodicity are the important properties that make chaos-based systems a preferable option for creating a physical security system [12,13,14]. Several efficient chaos-based image encryption schemes have been presented [15,16,17,18,19,20,21]. However, some of the schemes are insecure, such as the ones given in [22,23].

In recent years, because of the various striking characteristics of DNA computing, such as massive parallelism, high speed, very large storage capability, and ultra-low power consumption, it is now being used in the field of cryptography. Many researchers are now combining chaos with DNA computing to create further improvement in the security and efficiency of the cryptosystem. Many such algorithms have been proposed in [24,25,26,27,28,29,30,31,32,33,34].

However, some of the reported algorithms, such as [24,25,28,31,33], have certain drawbacks, i.e., they are secret keys plain-image independent, making them vulnerable to chosen and known plain text attacks. In some methods, such as [27,30,31,32], either high-dimensional maps are being used or some transform operations are involved, thus increasing the computational complexity of the schemes in terms of power, time, and other resources. These limitations make the existing systems less suitable for critical information exchange, reducing their adoptability for several security-related applications.

Taking into consideration the above drawbacks, we propose an adaptive encryption scheme for color images with the following original key contributions:

• The proposed scheme uses a logistic map, a tent map, and a 2D Henon map. Each chaotic map separately encrypts the red channel, green channel, and blue channel, respectively.
• The use of low-dimensional maps ensures that the proposed algorithm has better computational efficiency. At the same time, the scheme performs better than some recently proposed state-of-the-art image encryption schemes.
• Adaptive encryption helps to determine various preliminary conditions and control variables of the chaotic maps by making the secret keys plain image dependent. So, every time the plain image is changed, different secret keys will be generated. This makes the scheme robust against the chosen and known plaintext attacks.
• Further enhancement in the efficiency of the scheme is provided by involving DNA computation in the diffusion phase.

The rest of the paper is structured by presenting the related work in Section 2 and the preliminaries in Section 3. The proposed scheme is introduced in Section 4. The analysis of the scheme is carried out and various results are shown in Section 5. In the end, Section 6 gives the conclusion of this work.

2. Related Work

Numerous image-encryption algorithms based on chaos have been introduced. Khan and Masood [35] presented a chaos-based encryption technique for color images that involves multiple discrete dynamical maps. To carry out diffusion and confusion, the scheme makes use of several 1D and 2D maps. Farah et al. [27] used chaos theory, fractional Fourier transform, and DNA operations to propose an optical image encryption scheme. A DNA matrix was obtained upon the transformation of the plain image by generating random phase masks utilizing the Lorenz map. Then, fractional Fourier transform was implemented thrice on the matrix. Wu et al. [11] proposed a 2D Discrete Wavelet Transform (DWT) and 6D hyperchaotic system-based color image encryption algorithm. Two-dimensional DWT is employed to divide the original image into four image sub-bands and then a keystream is used to permute the sub-bands. Kang and Guo [36] presented a spatiotemporal chaotic system and DNA encoding-based color image encryption technique. Firstly, three DNA matrices are obtained from the plain image based on DNA coding rules. Then, a mixed linear non-linear coupled map lattice (MLNCML) system is used to generate a scrambling matrix that is used to perform permutation on the combined DNA matrix.

Wang et al. [25] presented a Coupled Map Lattice (CML) and DNA sequence operations-based color-image encryption scheme. A matrix is constructed using the three-color plain image components. CML is used to accomplish confusion on the matrix of pixels and the rules of DNA encoding and decoding are applied in the process of permutation. Rehman et al. [26] proposed an encryption algorithm for color images based on the Secure Hash Algorithm (SHA-2) and chaos theory, employing exclusive-OR (XOR) and complementary rules of DNA.

Valandar et al. [37] put forward a three-dimensional chaotic map-based fast encryption scheme for color images. RGB channels are bit-XORed with the three numbers produced by the map. The image is divided into 4 × 4 parts with each part consisting of 16 × 16 blocks, and then different keys of the presented map are used to permute the blocks. Elshamy et al. [38] presented an encryption scheme for color images using chaos. A hybrid encryption technique utilizing three distinct chaos maps: Baker, Arnold, and Henon, each applied to a single channel of Red-Green-Blue is used in the model. Three separate keys are used in unlocking encryptions, thus enhancing information security. Alghafis et al. [39] put forward an efficient color image encryption technique based on chaotic and DNA sequencing. This scheme uses a chaotic system consisting of a logistic map, a Henon map, and a Lorenz system to generate random sequences. In addition, after confusion and diffusion, an operation of DNA fusion is implemented on the DNA image.

Zheng and Liu [40] proposed an encryption algorithm based on an improved 2D logistic sine chaotic map (2D-LSMM). The input of the sine map is controlled using logistic map. DNA coding and operation rules are decided by 2D-LSMM chaotic sequences. Zang et al. [41] introduced a 1D discrete chaotic system. By parameter adjustment, the chaotic system is distributed uniformly. Finally, based on the uniformly distributed discrete chaotic system and DNA encoding, an image encryption algorithm is put forward. Plaintext is used to determine the DNA coding and decoding rules.

Broumandnia [42] presented an image encryption algorithm based on the Galois field in chaotic maps. The algorithm mainly consists of diffusion and permutation. In the diffusion stage, the overlapping rows and columns of the image pixels are mixed by using matrix multiplication operation in GF (256). In the permutation stage, a 2D or 3D chaotic map is used to change the pixel positions. Nardo et al. [43] proposed an image encryption scheme using finite-precision error. Two natural different interval extensions are used to implement a chaotic system to obtain the error. Zhang and Yan [44] proposed an RNA and pixel depth-based adaptive chaotic image encryption algorithm. The secret keys are generated by simultaneously using the hash value of a plain image and the current time to achieve one-image, one-key, and one-time pad. The pixel depth plays an integral role in the entire process of key generation, scrambling, and diffusion.

Among the above-mentioned algorithms, those reported in [25,35,37,38] have shortcomings such as the secret key being independent of the plain image, thereby making the algorithm insecure against chosen and known attacks. Few of them, such as [11,27], involve either high dimensional maps or transform operations as well; thus, the overall computational cost is high and such systems are not suitable for applications, needing a fast response.

The proposed encryption algorithm resolves the above issues by using a tent map, a logistic map, and a Henon map. The logistic map is utilized to encrypt the red channel, the tent map encrypts the green channel, and the Henon map is used to encrypt the blue channel. All the preliminary conditions and controlling parameters of the maps are adaptively modified by mean, variance, and median of the plain image to make the keys plain-image dependent to avoid any known and chosen attacks. At the same time, low-dimensional maps make the overall algorithm computationally less complex. The efficiency and security of the algorithm are further improved using DNA computation.

3. Preliminaries

Chaos is used in the study of dynamic systems. Because of various properties of chaos, such as being highly sensitive to preliminary conditions and other control variables, it has found its use in cryptography. Pseudo-Random Number (PRN) sequences or key streams are generated using chaotic maps. In the proposed scheme, three discrete chaotic maps are utilized to generate the key streams.

3.1. Logistic Map

This is the most commonly used chaotic map and, being one-dimensional, it is among the simplest maps as well. Biologist Robert May first introduced this map in 1976 [45].

Mathematically, it is expressed by Equation (1) given below:

$$x\left(k\right)=\mu \times x\left(k-1\right)\times \left(1-x\left(k-1\right)\right)$$

(1)

where the iteration index is k, µ is the control parameter, and ∈ (0, 4), x(0) is the preliminary condition, and ∈ (0, 1). To exhibit chaotic behavior, µ ∈ (3.5699456, 4).

3.2. Tent Map

The tent map is also a 1D map like the logistic map, and is piecewise chaotic. Equation (2) represents the mathematical expression of the tent map:

$$\begin{gathered} x\left(n\right)=f\left(x\left(n\right),r\right)=r\times x\left(n-1\right), 0<x\left(n-1\right)\le 0.5 \\ r\times \left(1-x\left(n-1\right)\right), 0.5<x\left(n-1\right)\le 1 \end{gathered}$$

(2)

Here, the iteration index is n, r ∈ [0, 2] is the control parameter, but to display chaotic behavior, r ∈ [1, 2] or closer to 2. x(n − 1) is the initial condition of the map, and ∈ [0, 1].

3.3. Henon Map

It is a 2D discrete chaotic map and maps a point ($x_0,y_0$) to another point. Equations (3) and (4) express the Henon map equations:

$$x_{n+1}=1-a{x}_n+y_n$$

(3)

$$y_{n+1}=b{x}_n$$

(4)

where, $x_{0 }\mathrm{and} y_0$are the preliminary conditions, and a and b are the control parameters. To produce chaotic behavior, the usual values of (a, b) $\in$ (1.4, 0.3).

3.4. DNA Coding

DNA structure comprises double helical strands, which in turn are composed of simpler units called nucleotides [46]. The four bases or nucleotides are Adenine (A), Thymine (T), Guanine (G), and Cytosine (C). Adenine and Thymine are complements of each other. Similarly, Guanine and Cytosine are each other’s complements. In binary numbers, zero and one are complements. Similarly, 00 and 11 and 01 and 10 are also complements. So, if these four DNA bases are used to encode 00, 11, 01, and 11, there will be a total of 24 combinations, out of which only eight coding combinations satisfy Watson–Crick complementary rule. These eight coding rules are given in

Table 1. Eight kinds of DNA map rules.

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

.

3.5. DNA Computing

The first experiment on DNA computing was performed by Adleman in 1991, and thus, a new stage in the information era began [47]. Based on the rules of DNA encoding and decoding, certain operations can be carried out on the DNA sequences such as addition, subtraction, XOR, etc., given in

Table 2. Rule 3 for DNA addition.

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

,

Table 3. Rule 3 for DNA subtraction.

−	A	T	G	C
A	G	C	A	T
T	A	G	T	C
G	C	T	G	A
C	T	A	C	G

and

Table 4. Rule 3 for DNA XOR.

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

, respectively, as per rule 3.

4. Proposed Scheme

In this section, a novel adaptive color image encryption scheme based on MDCM and DNA computation is presented. The block diagram of the proposed encryption scheme is presented in Figure 1, comprising various phases.

4.1. The Encryption Algorithm

The various phases in encryption algorithm are discussed in the following steps:

• Step 1: In the proposed scheme, the first step is the key generation phase. In this phase, three chaotic maps are used: a tent map, a logistic map, and a Henon map. The preliminary conditions and control variables of all three maps are dynamically controlled using statistical plain image characteristics such as the mean, variance, and median. This makes the secret keys dependent on the plain image so that any change in the image may be reflected in the output as well. In this scheme, a random $8\times 8$ plain image pixel block is chosen. The arithmetic means, variance, and median of this pixel block are determined and then normalized. The normalized mean is employed to obtain the starting conditions of the logistic map and tent map and to obtain control variable ‘b’ of the Henon map. Likewise, the use of normalized variance is made to obtain the control parameters of the logistic map and tent map, and control variable ‘a’ of the Henon map. The two starting conditions of the Henon map are obtained using the normalized median of the block.
• Step 2: In this step, the permutation process is carried out. Permutation involves changing the pixel positions to reduce the correlation among the neighboring pixels in the plain image. The original color image with size M × N × 3 is initially taken as the input and then split into red, green, and blue channels, each reshaped to a size of $M\times N\times 1$. The preliminary condition and control parameter values are given to the logistic map and it is iterated $M\times N$ times to generate a PRN sequence, which is then employed to permute the red channel of the image. Similarly, on giving the values of the preliminary condition and control variable, the tent map is iterated $M\times N$ times and a PRN sequence is generated. This sequence is used in scrambling the green channel of the image. Finally, the Henon map is iterated $M\times N$ times to generate two PRN sequences. One of these sequences is used in the permutation of the blue channel of the image. So, in the permutation phase, three permuted images are obtained.
• Step 3: This step involves the DNA-encoding phase. In this phase, the three permuted images obtained are encoded into three DNA sequences $DN{A}_{seq1}$, $DN{A}_{seq2}$, and $DN{A}_{seq3}$ according to a DNA-encoding rule, each with a size of $4\times M\times N$. In the proposed algorithm, the encoding is performed as per DNA rule 3. After giving different values of preliminary conditions and control variables, the logistic and tent map are iterated again $M\times N$ times to generate two new PRN sequences. These two sequences, along with the other sequence from the Henon map, are also encoded into DNA sequences as per the same rule 3 to get three more DNA sequences: $DN{A}_{seq4}$, $DN{A}_{seq5}, \mathrm{and} DN{A}_{seq6}$, respectively.
• Step 4: This step involves the substitution process of the suggested algorithm. In any encryption algorithm, substitution is of great significance and is incorporated in changing or modifying the pixel values. In this phase, DNA computation is carried out on the six DNA sequences obtained so far. The $DN{A}_{seq1}$ is added with the $DN{A}_{seq4}$ as per DNA addition rule 3. Likewise, the $DN{A}_{seq2}$ is added with the $DN{A}_{seq5}$ and finally $DN{A}_{seq3 }$is added with $DN{A}_{seq6}$as per the same rule 3. Thus, at the end of the substitution phase, we get three DNA sequences.
• Step 5: This step involves the DNA decoding process. In this phase, each of the three DNA sequences obtained in the substitution phase is decoded into a binary stream according to the DNA decoding rule 3 and then the binary stream is converted into decimal form. After reshaping the decimal sequence, each element is XORed with the elements preceding that index in the sequence to finally get the three ciphered channels. The concatenation of these channels gives the final encrypted image.

4.2. The Decryption Algorithm

The algorithm for decryption is similar to that of the encryption and involves the same steps in reverse. After decrypting all three channels, they are concatenated to give the final decrypted image.

5. Experimental Results and Security Analysis

In this section, experimental results of the suggested scheme are presented, and performance analysis is carried out. The analysis is given in terms of key sensitivity, keyspace, various statistical parameters such as histogram, information entropy, correlation coefficients, differential attack analysis, etc. In the proposed algorithm, various standard RGB images were used as plain images. Figure 2 shows Lena of $256\times 256\times 3$ as the plain image and its three channels red, green, and blue after decomposition, along with their corresponding encrypted and decrypted versions as well.

Various other standard RGB images and their encrypted and decrypted versions are displayed in Figure 3.

5.1. Keyspace Analysis

Keyspace means all the digital space that can be employed as the encryption/decryption key. For an effective encryption scheme, its value should be as large as possible, at least equal to $2^{100}$ (≈${10}^{30}$), to withstand the brute-force attacks [1]. The presented scheme makes use of three chaotic maps. There are two parameters: one initial condition and one control parameter in both the logistic map and tent map. Both maps are used twice in the algorithm, making a total of eight parameters or keys. Additionally, a 2D-Henon map is used once. It has four parameters: two control variables and two preliminary conditions. If each key has precision up to ${10}^{-10},$the keyspace is (${10}^{10}{)}^{12}={\left(10\right)}^{120} \left(=2^{400}\right).$ Additionally, one rule is used out of eight DNA-encoding rules and the encoding process is carried out six times. Likewise, the DNA addition and decoding operations are also performed thrice each; therefore, the total keyspace of the proposed algorithm = ($2^{400}$) × ($2^3{)}^{12}$=$2^{436}$, which is much larger than the required value; thus, this scheme could very well resist the attacks of brute force.

5.2. Key Sensitivity Analysis

There should be an extreme sensitivity in a cryptosystem concerning even the slightest change in any of the keys used. For this, key-sensitivity analysis is performed by first encrypting the image using a set of right keys and then decrypting it after bringing a very minute change in one of the keys used in the algorithm. Under such a scenario, the decrypted image should not reveal any sort of information about the plain image, depicting the high sensitivity of the algorithm to the keys, and should withstand any such attack from the adversary. To test the sensitivity of the suggested algorithm to the secret key, image Lena is firstly encrypted by a set of correct keys and then decrypted by insignificantly changing one of the keys. Here, the value of the normalized variance used in obtaining the various parameters or secret keys is changed by ${10}^{-10}.$ Figure 4 shows the key-sensitivity analysis of the scheme.

Figure 4d clearly shows that the proposed algorithm is extremely sensitive to the secret keys after changing the value of one of the keys, i.e., increasing the normalized variance by ${10}^{-10}$.

5.3. Statistical Attack Analysis

This section analyzes various statistical attacks carried out on the proposed system. The results in terms of histogram analysis, entropy analysis, and fidelity analysis are discussed as follows.

5.3.1. Histogram Analysis

A histogram represents the occurrences of pixels in digital images. It should be quite uniform after an encryption algorithm is applied to an image to deceive the attacker. The uniformity of encrypted images is one aspect to conceal the actual information of digital images. Figure 5 shows the histogram analysis of image Lena. One can quite obviously observe that the histogram of RGB channels after encryption is very much uniform, and thus, the information about plain image remains hidden.

5.3.2. Information Entropy Analysis

The haphazardness in the plain image information content is determined by information entropy. For an eight-bit image, its ideal value is eight. It is represented using the expression given in Equation (5).

$$H\left(S\right)={\displaystyle \sum }_{i=0}^{2^{N-1}}P\left(S_i\right)\times log(1/P(S_i))$$

(5)

where N is the bit length of the value of a pixel and $P\left(S_i\right)$ gives the probability of symbol $S_i$. The entropy of some standard RGB images obtained using the proposed algorithm is given in

Table 5. Information entropy analysis.

Images	Proposed		[33]		[35]		[39]
	Entropy Value, H(S)	%Age = H(S)/8 × 100	Entropy Value, H(S)	%Age = H(S)/8 × 100	Entropy Value, H(S)	%Age = H(S)/8 × 100	Entropy Value, H(S)	%Age = H(S)/8 × 100
Lena (256 × 256 × 3)	R = 7.9973	99.966	R = 7.9892	99.865	R = 7.9973	99.966	R = 7.9976	99.970
	G = 7.9972	99.965	G = 7.9902	99.877	G = 7.9972	99.965	G = 7.9975	99.968
	B = 7.9974	99.967	B = 7.9896	99.870	B = 7.9975	99.968	B = 7.9974	99.967
Baboon (256 × 256 × 3)	R = 7.9972	99.965	-	-	R = 7.9972	99.965	R = 7.9972	99.965
	G = 7.9970	99.962			G = 7.9970	99.962	G = 7.9972	99.965
	B = 7.9973	99.966			B = 7.9977	99.971	B = 7.9972	99.965
Peppers (256 × 256 × 3)	R = 7.9974	99.967	-	-	-	-	R = 7.9967	99.958
	G = 7.9971	99.963					G = 7.9970	99.962
	B = 7.9972	99.965					B = 7.9973	99.966
Peppers (256 × 256 × 3)	R = 7.9992	99.990	-	-	R = 7.9993	99.991	-	-
	G = 7.9993	99.991			G = 7.9992	99.990
	B = 7.9992	99.990			B = 7.9993	99.991

. The results are also compared with other schemes, and it is visible that the results are better in the proposed scheme, being near enough to the ideal value of eight. So, there is a very low probability of information leakage making these images resistant against statistical attacks.

5.3.3. Correlation Coefficient Analysis

There is a very high correlation, which also means a large similarity between the adjacent plain image pixels along with different directions. In a digital image, its value lies in the range of −1 and 1. One aspect of an encryption scheme is to break this correlation among the adjacent pixels and bring its value closer to zero. Such a scheme is said to be a cryptographically secure one. Mathematically, it can be defined as Equation (6):

$$r_{xy}=cov\left(x,y\right)/\sqrt{V\left(x\right)\times V\left(y\right)}$$

(6)

where, $cov\left(x,y\right)=1/N{\displaystyle \sum }_{i=1}^N\left(x_{i-}E\left(x\right)\right)\times (y_i-E\left(y\right))$

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

$$\left(x\right)=1/N{\displaystyle \sum }_{i=1}^N\left(x_i\right)$$

where the number of adjacent pixel pairs is given by N, the adjacent pixel pair has gray values of x and y, E(x) denotes the mean, the variance is denoted by V(x), and covariance by $Cov\left(x,y\right)$.

Table 6. Correlation coefficient analysis.

Images	Proposed	[35]	[39]
Lena (256 × 256 × 3)	CHR = 0.0018	CHR = 0.0017	CHR = 0.0003
	CHG = −0.0032	CHG = 0.0011	CHG = 0.001
	CHB = 0.0022	CHB = −0.0030	CHB = −0.0009
	CVR = 0.0028	CVR = −0.0004	CVR = 0.003
	CVG = 0.0286	CVG = 0.0076	CVG = −0.004
	CVB = 0.1074	CVB = 0.0050	CVB = −0.0008
	CDR = 0.0016	CDR = 0.0049	CDR = 0.0008
	CDG = 0.0022	CDG = −0.0002	CDG = 0.002
	CDB = −0.00075	CDB = 0.0049	CDB = 0.002
Baboon (256 × 256 × 3)	CHR = −0.0037	CHR = −0.0007	CHR = 0.0005
	CHG = 0.0010	CHG = 0.0057	CHG = −0.00003
	CHB = 0.0091	CHB = 0.0056	CHB = 0.005
	CVR = −0.1196	CVR = 0.0023	CVR = 0.002
	CVG = −0.0889	CVG = 0.0043	CVG = 0.005
	CVB = 0.0313	CVB = 0.0002	CVB = 0.0009
	CDR = −0.0043	CDR = −0.0077	CDR = 0.006
	CDG = 0.00059	CDG = −0.0002	CDG = 0.005
	CDB = 0.0070	CDB = −0.0040	CDB = −0.004
Peppers (256 × 256 × 3)	CHR = −0.0027	-	CHR = 0.003
	CHG = 0.00023		CHG = −0.009
	CHB = −0.00084		CHB = −0.003
	CVR = −0.0174		CVR = −0.001
	CVG = 0.0105		CVG = −0.004
	CVB = −0.0732		CVB = −0.0002
	CDR = 0.0022		CDR = 0.006
	CDG = −0.0017		CDG = −0.0002
	CDB = −0.0029		CDB = −0.0008
Peppers (512 × 512 × 3)	CHR = −0.0015	CH = 0.0008 CV = 0.0013 CD = 0.0011	-
	CHG = −0.0011
	CHB = 0.000349
	CVR = −0.0210
	CVG = 0.0111
	CVB = −0.1088
	CDR = −0.00082
	CDG = 0.00023
	CDB = 0.0037

displays values of correlation coefficients of different encrypted images along with horizontal (CHR, CHG, CHB), vertical (CVR, CVG, CVB), and diagonal (CDR, CDG, CDB) directions. The values are closer to zero, indicating the encrypted images to be uncorrelated, which also suggests that the suggested scheme is secure enough against statistical attacks.

5.4. Peak Signal to Noise Ratio (PSNR)

PSNR is an attribute that helps to check the quality of an image-encryption technique. It measures and indicates the changes in the value of plain-image pixels and that of encrypted images. For a secure encryption technique, its value should be low enough, from seven to nine. PSNR can be expressed mathematically by Equation (7).

$$\mathrm{PSNR}=10\times log{\left(P\right)}^2/MSE$$

(7)

where P denotes the peak pixel intensity for an eight-bit image, MSE denotes the mean square error and is given in Equation (8).

$$MSE=\frac{1}{M\times N}{\displaystyle \sum }_{i-1}^M{\displaystyle \sum }_{j=1}^N{\left(O\left(i,j\right)-E\left(i,j\right)\right)}^2$$

(8)

where O is the original image and E is the encrypted image.

5.5. Structural Similarity (SSIM)

SSIM measures the similarity between the plain image and the encrypted image. It indicates the amount of structural information modified in the plain image. Its value should be low enough, closer to 0. SSIM can be calculated using Equation (9).

$$\mathrm{SSIM}\left(o,e\right)=\frac{\left(2{\mu }_o{\mu }_e+C_1\right)\left(2{\sigma }_{oe}+C_2\right)}{\left({\mu }_o{}^2{\mu }_e{}^2+C_1\right)\left({\sigma }_o{}^2{\sigma }_e{}^2+C_2\right)}$$

(9)

where ${\mu }_o$ = mean of the original image, ${\mu }_e$ = mean of the encrypted image, ${\sigma }_o{}^2$ = original image variance, ${\sigma }_e{}^2$= encrypted image variance, and ${\sigma }_{oe}$ = covariance between original and encrypted images. $C_1$ = ${(h_{1}L)}^2$, $C_2$ = ${(h_{2}L)}^2$, L = 255, $h_1$= 0.01 and $h_2$ = 0.03.

Table 7. SSIM and PSNR analysis.

Images	Proposed	[39]	[35]
Lena (256 × 256 × 3)	SSIMR = 0.0101	SSIMR = 0.0091	-
	SSIMG = 0.0089	SSIMG = 0.0061	-
	SSIMB = 0.0112	SSIMB = 0.0087	-
	PSNRR = 8.3348	PSNRR = 8.7544	PSNRR = 7.7930
	PSNRG = 8.5570	PSNRG = 8.4328	PSNRG = 7.7739
	PSNRB = 10.4662	PSNRB = 8.0809	PSNRB = 7.7363
Baboon (256 × 256 × 3)	SSIMR = 0.0111	SSIMR = 0.0103	-
	SSIMG = 0.0102	SSIMG = 0.0094	-
	SSIMB = 0.0094	SSIMB = 0.0105	-
	PSNRR = 9.1395	PSNRR = 8.9018	PSNRR = 7.7432
	PSNRG = 9.4347	PSNRG = 9.5258	PSNRG = 7.7427
	PSNRB = 8.6421	PSNRB = 8.6235	PSNRB = 7.7482
Peppers (256 × 256 × 3)	SSIMR = 0.0118	SSIMR = 0.0093	-
	SSIMG = 0.0092	SSIMG = 0.0070
	SSIMB = 0.0083	SSIMB = 0.0074
	PSNRR = 9.4346	PSNRR = 8.2465
	PSNRG = 7.7963	PSNRG = 7.4135
	PSNRB = 8.2885	PSNRB = 7.3602
Peppers (512 × 512 × 3)	SSIMR = 0.0118	-	-
	SSIMG = 0.0085		-
	SSIMB = 0.0071		-
	PSNRR = 9.4433		PSNRR = 7.7618
	PSNRG = 7.6337		PSNRG = 7.7478
	PSNRB = 8.1225		PSNRB = 7.7635

gives the SSIM and PSNR values for various RGB images obtained using the presented scheme. Both SSIM and PSNR values are low enough to suggest that there is very little similarity between the plain and the encrypted image and high noise between the two images, respectively.

5.6. Differential Attack Analysis

An algorithm for encryption of images should be such that even if a slight change occurs in the plain image, its cipher image should also change, i.e., the sensitivity of the algorithm concerning the plain image should be high. This property makes an algorithm robust against differential attacks. In such attacks, the adversary slightly changes a plain image. He then encrypts the image before and post the change, and then analyzes the distribution of these images to determine any kind of statistical pattern. Two parameters are employed to determine resistance against such attacks: Number of Pixels Changing Rate (NPCR) and Unified Average Changing Intensity (UACI).

The absolute pixels whose value varies in differential attacks is given by NPCR, whereas the focus of UACI is on the averaged difference between the two encrypted images. NPCR and UACI can be determined using Equations (10) and (11), respectively.

$$\mathrm{NPCR}\left(C_{1,}{C}_2\right)=\frac{1}{M\ast N}{\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^{N}D\left(i,j\right)\times 100\%$$

(10)

$$\mathrm{UACI}\left(C_{1,}{C}_2\right)=\frac{1}{M\ast N}{\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^N\frac{\left|C_1\left(i,j\right)-C_2\left(i,j\right)\right|}{255}\times 100\%$$

(11)

$$D\left(i,j\right)=\{\begin{array}{c}0, if C_1\left(i,j\right)=C_2\left(i,j\right)\\ 1,if C_1\left(i,j\right)\ne C_2\left(i,j\right)\end{array}$$

(12)

where the dimensions of the original image are given by M and N. Cipher images before and post the change in the value of the pixel at location $\left(i,j\right)$ are given by $C_1\left(i,j\right)$ and $C_2\left(i,j\right)$, respectively. Ideally, the respective NPCR and UACI values should be equal to 99.61% and 33.46%, [48]. In the proposed scheme, various RGB images are taken, and one-bit change is introduced in the plain images. UACI and NPCR are evaluated, and their values are given in

Table 8. NPCR and UACI analysis.

Images	Proposed	[39]	[35]	[33]	[32]
Lena (256 × 256 × 3)	N = 99.61 U = 32.95	N = 99.57 U = 33.49	N = 99.59 U = 33.37	N = 99.61 U = 32.20	N = 99.61 U = 30.41
Baboon (256 × 256 × 3)	N = 99.62 U = 33.05	N = 99.60 U = 33.53	N = 99.60 U = 33.46	-	N = 99.62 U = 29.78
Peppers (256 × 256 × 3)	N = 99.60 U = 33.50	N = 99.52 U = 33.50	-	-	N = 99.61 U = 32.19
Peppers (512 × 512 × 3)	N = 99.61 U = 33.50	-	N = 99.60 U = 33.41	-	-

along with a comparison to other schemes.

The results obtained from the proposed scheme outperform the results of those compared with it and are found to be close to their ideal values. It implies that the presented scheme is extremely sensitive to the plain image and can resist the differential attacks very well.

5.7. Robustness Analysis

It is inevitable in real-world applications that some noise gets added to the information or it suffers some losses during transmission over a communication channel. Robustness is the ability of an encryption algorithm to recover the original image from a cropped or noisy cipher image.

5.7.1. Noise Attack Analysis

To check the robustness of the proposed scheme against noise attacks, we added salt and pepper noise of different densities to the encrypted image of Lena, as shown in Figure 6. The noise densities used are 0.01, 0.05, and 0.1 in Figure 6a–c, respectively. The corresponding decrypted images are shown in Figure 6d–f. It is evident from the figure that despite the cipher images being noisy, the corresponding decrypted images are still perceivable. This shows the robustness of the proposed scheme.

5.7.2. Cropping Attack Analysis

In order to test the robustness of the proposed scheme against cropping attack, we used different cropping levels such as 1/8, 1/4, and 1/2, as depicted in Figure 7a–c, respectively. We find out in Figure 7d–f that the corresponding decrypted images can still be recognized. Thus, the proposed algorithm also succeeds in resisting the cropping attack.

One of the most important indexes to measure the quality of encryption is PSNR. A low PSNR value signifies good encryption quality, and a high value signifies better perceptual security. We evaluate the robustness of the proposed algorithm against salt-and-pepper-noise attack and cropping attack in terms of PSNR and MSE, defined in Equations (7) and (8), respectively. The results obtained for the image Lena are presented in

Table 9. Noise attack analysis.

Salt and Pepper Noise Density	PSNR			MSE
	R	G	B	R	G	B
0.01	25.0491	25.6354	26.4249	203.3168	177.6411	148.1111
0.05	17.8247	18.6317	19.7586	1873.0	891.1748	687.4135
0.1	15.0764	15.7996	16.8090	2022.7	1710.5	1355.8

(noise attack analysis) and

Table 10. Cropping attack analysis.

Cropping Level	PSNR			MSE
	R	G	B	R	G	B
1/8	15.7048	17.4616	18.0581	1748.2	1166.6	1016.9
1/4	12.7580	14.4784	15.0746	3445.8	2318.7	2021.2
1/2	9.7916	11.5753	12.0974	6822.1	4524.2	4011.8

(cropping attack analysis).

From Table 9 and Table 10, it is observed that both PSNR and MSE have good values, which indicate that the proposed scheme has good perceptual security and is robust against noise and cropping attacks.

5.8. Computational Complexity Analysis

The chaotic sequence generation, permutation operations, and diffusion operations in any chaos-based image encryption algorithm constitute its computational costs. The computational complexity is described by the number of times each operation is repeated in various stages based on plain image pixels M $\times$ N.

The key generation phase in the proposed algorithm involves generating the sequences using a logistic map, a tent map, and a Henon map, with each sequence equal to the number of image pixels. Both the logistic map and tent map are iterated twice to generate two sequences and the Henon map is iterated only once. So, the complexity is O(5 × M × N).

Table 11. Computational complexity of the proposed scheme.

Operations	Number of Times Repeated
Permutation	3 × (M × N)
DNA encoding_1	3 × 4 × (M × N)
DNA encoding_2	3 × 4 × (M × N)
DNA addition	3 × 4 × (M × N)
DNA decoding	3 × 4 × (M × N)
XOR	3 × (M × N)

shows the number of times various other operations are repeated in the proposed algorithm.

On aggregating the complexities of all these operations, the total computational complexity of the proposed scheme is O(59 × M × N), which is much less than the complexity of other state-of-the-art schemes given in

Table 12. Comparison based on computational complexity.

Scheme	Computational Complexity
Proposed	O(59 × M × N)
[49]	O(168 × M × N)
[50]	O(69 × M × N)
[51]	O(124 × M × N)
[52]	O(579 × M × N)

.

Table 12 shows that the proposed scheme is computationally efficient compared to other related schemes.

5.9. Speed Analysis

The swiftness of an algorithm is a critical feature of a good cryptosystem. Speed analysis of the suggested algorithm is carried out by measuring the encryption/decryption time for various standard color images. The scheme implementation was performed using MATLAB 2019a (version 9.6, Mathworks, Natick, MA, USA) on the operating system Windows 10 with processor Intel^® core™ i7-8565U CPU @ 1.8GHZ and 8GB RAM. The speed analysis of the suggested scheme is given in

Table 13. Speed analysis by measuring the encryption and decryption time in seconds.

Image	Average per Channel Encryption Time (s)	Average per Channel Decryption Time (s)
Lena (256 × 256 × 3)	R = 1.3847 G = 1.5412 B = 1.5293	R = 0.9633 G = 0.9485 B = 0.9887
Lena (512 × 512 × 3)	R = 4.9086 G = 5.0535 B = 5.0043	R = 3.3672 G = 3.3942 B = 3.4466

.

6. Conclusions

Recently, chaotic systems have found their application in various fields such as cryptography and digital communication. In this paper, an adaptive color-image encryption scheme based on MDCM and DNA computing is presented. In the proposed scheme, three chaotic maps are used to perform encryption of the three channels separately. A logistic map encrypts the red channel, a Tent map encrypts the green channel, and a 2D-Henon map is used to encrypt the blue channel. We have tried other scenarios as well, wherein these maps are exchanged with each other to encrypt the different channels. However, in every such situation, the values of various parameters for performance evaluation start degrading. We only get optimal values of these parameters by adopting the proposed encryption method. The use of low-dimensional maps to encrypt the images makes the scheme computationally less complex. The preliminary conditions and control variables of all these maps are controlled using statistical plain-image characteristics such as mean, variance, and median, making the secret keys plain-image dependent, and thus the scheme is robust against chosen and known plaintext attacks. In the diffusion phase, DNA addition operation followed by XOR operation is carried out to obtain the encrypted channels that, upon concatenating, give the final encrypted image. Various experimental results are obtained and are found to be closer to their ideal values. NPCR and UACI values come out to be more than 99.61% and 33%, respectively. Correlation coefficient values are approaching zero and the entropy value is approximately equal to the ideal value of eight. Thus, the experimental analysis shows that the suggested scheme is secure enough against many attacks such as statistical attacks and differential attacks. The proposed scheme also shows extreme sensitivity to the keys and has a huge keyspace that makes it secure against brute-force attacks. It also gives a good structural similarity index, making it safe and reliable for image encryption.