# Chaotic Color Image Encryption Scheme Using Deoxyribonucleic Acid (DNA) Coding Calculations and Arithmetic over the Galois Field

Authors: Lilian Huang; Shiming Wang; Jianhong Xiang; Yi Sun
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper proposes a chaotic color image encryption scheme based on DNA-coding calculations and arithmetic over the Galois field. Firstly, three modified one-dimensional (1D) chaotic maps with larger key space and better chaotic characteristics are presented. The experimental results show that their chaotic intervals are not only expanded to0,15, but their average largest Lyapunov Exponent reaches 10. They are utilized as initial keys. Secondly, DNA coding and calculations are applied in order to add more permutation of the cryptosystem. Ultimately, the numeration over the Galois field ensures the effect for the diffusion of pixels. The simulation analysis shows that the encryption scheme proposed in this paper has good encryption effect, and the numerical results verify that it has higher security than some of the latest cryptosystems.


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## Body

## 1. Introduction

Information security is an important issue in information communication nowadays. With the advancement of information technology, plenty of digital contents are stored and transmitted in various forms. Therefore, it becomes more and more significant to improve the security of them. As the main information carrier, digital images play an important role in the medical and military fields. Hence, the safety of image transmission has received extensive attention. Nevertheless, since the images have large amounts of data, high redundancy and strong correlation of adjacent pixels, conventional encryption technologies such as DES, AES, and RSA fail to satisfy the security of encryption.Chaotic systems are ergodic, dispersive, and highly sensitive to initial conditions [1–4]. They have a lot of similarities with cryptography. As a consequence, the chaos-based encryption methods have become a main branch of the cryptosystem. Fridrich first applied chaotic maps to image encryption algorithms, who used two-dimensional (2D) Baker map and Cat map for pixel position transformation in 1997 [5]. After that, many researchers encrypted images based on 1D and multidimensional (MD) chaotic maps [6–10]. Among them, the MD chaotic maps are widely used in image encryption due to their relatively intricate structures and parameters [11–14]. However, these traits increase the complexity of computation and the difficulty of implementation. In comparison, although the 1D chaotic maps have uneven distribution and discontinuous range, they have simpler structures. Thus, they are more convenient to be handled and implemented. In recent years, some efforts have been devoted to addressing the weaknesses of 1D chaotic systems and proposing encryption schemes. Zhou et al. presented a new system structure and encrypted images with random pixel insertion [15]. However, Dhall et al. pointed out that this cryptosystem can be broken by the differential attack [16]. Hua et al. introduced a Sine-Logistic modulation map to efficiently change the image pixel positions in [17]. Pak and Huang also proposed a new 1D chaotic system to determine pixels permutation and diffusion positions [18]. But the algorithms in [17, 18] are vulnerable to chosen plaintext attack. Hence, we propose three improved 1D chaotic maps with better dynamic complexity for our encryption scheme. The simulation results exhibit that the new 1D chaotic maps have larger chaotic ranges and more uniform output sequences, which makes them more suitable for image encryption.DNA-based image encryption methods mainly apply the principle of DNA cryptography. Due to huge parallelism, inherent information density, and fairly low power consumption, DNA-based image encryption methods have been rapidly developed. In [19], Zhang et al. proposed an image encryption scheme with DNA sequence addition operation and two Logistic maps. However, the Logistic map has been declared that it does not have outstanding randomness to achieve the desired encryption effect [15]. Hence, Li et al. changed the location of pixels and pixel values with the Arnold map before DNA encoding [20]. Since the key streams are independent of the plain images, Yong proved that this scheme can be cracked by the chosen plaintext attack and known plaintext attack [21]. The researchers also considered the application of chaotic systems with excellent chaotic dynamic properties to improve the robustness of the cryptosystems. In [22], Zhen et al. introduced an encryption method based on spatiotemporal chaotic systems and Logistic map. Moreover, the method also utilized DNA coding to promote the efficiency of image confusion and diffusion. Although the authors have claimed that the cryptosystem in [22] can resist a variety of attacks, Xin et al. still found two flaws that make the encryption fail under chosen plaintext attack [23].Motivated by the abovementioned discussions, we propose a new chaotic color image encryption scheme using DNA coding calculations and arithmetic over the Galois field. Particularly, the 1D chaotic maps applied in this paper are derived from the classical Logistic map, Sine map, and Chebyshev map. Besides, the contributions and innovations of this article are summarized as follows. (1) We propose improved 1D chaotic maps that are more appropriate for image encryption. Not only do they have better chaotic properties but they are also easier to be operated on hardware and software. (2) We design the calculation modes of the encoded DNA matrixes to be randomly decided by the chaotic sequences which are updated by plain images. In this case, the cryptosystem is more robust. (3) In order to prevent the danger of being cracked by utilizing DNA operations only, we add multiplication arithmetic over the Galois field GF(17) to our scheme. Moreover, the generation of the lookup table will not increase the time consumption.The arrangement of this paper is as follows. Section2 introduces the basic theories involved in this paper. Section 3 displays the improved 1D chaotic maps with their performance analysis. Section 4 proposes our image encryption and decryption schemes. Section 5 gives the experimental simulation results and security analysis. The Section 6 draws the conclusion.


## 2. Related Work

### 2.1. DNA Coding and Calculation

DNA coding is a concept derived from biology. A DNA sequence consists of four nucleotides. They are adenine (A), guanine (G), cytosine (C), and thymine (T). According to the principle of complementary base pairing, adenine (A) pairs with thymine (T) and cytosine (C) pairs with guanine (G) [24]. As shown in Table 1, there are eight DNA coding methods that satisfy the pairing rules [25]. In the binary system, 0 and 1 are also complementary. Thus, each 8-bit pixel value can be decomposed into four “2-bit” values, and the four values can be encoded with a certain coding rule to obtain a DNA sequence. For instance, if the value of a pixel is 177, it is “10110001” in binary encoding. One can get the DNA sequence “CTAG” by the DNA coding rule 2 in Table 1. And using disparate decoding rules will obtain diverse pixel values. Hence, when the sequence “CTAG” is decoded by the DNA decoding rule 5 in Table 1, the binary result “00100111” is obtained, and the corresponding decimal number is 39.Table 1
8 DNA coding and decoding rules.
1
2
3
4
5
6
7
8
A
00
00
11
11
01
10
01
10
T
11
11
00
00
10
01
10
01
C
10
01
10
01
00
00
11
11
G
01
10
01
10
11
11
00
00In addition to encoding pixel values, DNA sequences can also perform algebraic calculations. Because the operation modes of DNA are based on traditional arithmetic operations in binary, the eight DNA coding rules correspond to eight kinds of DNA addition, DNA subtraction, and DNA exclusive or (XOR) rules [26]. If we arbitrarily select the DNA coding rule 1 in Table 1, then the corresponding DNA addition, subtraction, and XOR modes are shown in Tables 2–4.Table 2
DNA addition.
+
A
T
G
C
A
A
T
G
C
T
T
C
A
G
G
G
A
C
T
C
C
G
T
ATable 3
DNA subtraction.
−
A
T
G
C
A
A
G
T
C
T
T
A
C
G
G
G
C
A
T
C
C
T
G
ATable 4
DNA XOR operation.
⊕
A
T
G
C
A
A
T
G
C
T
T
A
C
G
G
G
C
A
T
C
C
G
T
A


### 2.2. Multiplication over the Galois Field

The French mathematician Galois invented the Galois field, and the operations of addition, subtraction, multiplication, and division over the Galois field can be performed. Compared with the addition and subtraction over the Galois field, the multiplication and division are more complicated. That is, the diffusion effect on pixels is more prominent. In order to speed up the calculation, we need to construct a lookup table in advance by(1)0:16T×0:16mod17,where ⋅T denotes transposition of the sequence. As for the GF(17) multiplication, a 8-bit pixel value is divided into the upper four bits and the lower four bits whose value ranges are 0,15. The bits are further converted to 1,16 when the lookup table is established, and the results are exhibited in Table 5.Table 5
Multiplication lookup table on the GF(17).
×
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2
0
2
4
6
8
10
12
14
16
1
3
5
7
9
11
13
15
3
0
3
6
9
12
15
1
4
7
10
13
16
2
5
8
11
14
4
0
4
8
12
16
3
7
11
15
2
6
10
14
1
5
9
13
5
0
5
10
15
3
8
13
1
6
11
16
4
9
14
2
7
12
6
0
6
12
1
7
13
2
8
14
3
9
15
4
10
16
5
11
7
0
7
14
4
11
1
8
15
5
12
2
9
16
6
13
3
10
8
0
8
16
7
15
6
14
5
13
4
12
3
11
2
10
1
9
9
0
9
1
10
2
11
3
12
4
13
5
14
6
15
7
16
8
10
0
10
3
13
6
16
9
2
12
5
15
8
1
11
4
14
7
11
0
11
5
16
10
4
15
9
3
14
8
2
13
7
1
12
6
12
0
12
7
2
14
9
4
16
11
6
1
13
8
3
15
10
5
13
0
13
9
5
1
14
10
6
2
15
11
7
3
16
12
8
4
14
0
14
11
8
5
2
16
13
10
7
4
1
15
12
9
6
3
15
0
15
13
11
9
7
5
3
1
16
14
12
10
8
6
4
2
16
0
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1


## 2.1. DNA Coding and Calculation

DNA coding is a concept derived from biology. A DNA sequence consists of four nucleotides. They are adenine (A), guanine (G), cytosine (C), and thymine (T). According to the principle of complementary base pairing, adenine (A) pairs with thymine (T) and cytosine (C) pairs with guanine (G) [24]. As shown in Table 1, there are eight DNA coding methods that satisfy the pairing rules [25]. In the binary system, 0 and 1 are also complementary. Thus, each 8-bit pixel value can be decomposed into four “2-bit” values, and the four values can be encoded with a certain coding rule to obtain a DNA sequence. For instance, if the value of a pixel is 177, it is “10110001” in binary encoding. One can get the DNA sequence “CTAG” by the DNA coding rule 2 in Table 1. And using disparate decoding rules will obtain diverse pixel values. Hence, when the sequence “CTAG” is decoded by the DNA decoding rule 5 in Table 1, the binary result “00100111” is obtained, and the corresponding decimal number is 39.Table 1
8 DNA coding and decoding rules.
1
2
3
4
5
6
7
8
A
00
00
11
11
01
10
01
10
T
11
11
00
00
10
01
10
01
C
10
01
10
01
00
00
11
11
G
01
10
01
10
11
11
00
00In addition to encoding pixel values, DNA sequences can also perform algebraic calculations. Because the operation modes of DNA are based on traditional arithmetic operations in binary, the eight DNA coding rules correspond to eight kinds of DNA addition, DNA subtraction, and DNA exclusive or (XOR) rules [26]. If we arbitrarily select the DNA coding rule 1 in Table 1, then the corresponding DNA addition, subtraction, and XOR modes are shown in Tables 2–4.Table 2
DNA addition.
+
A
T
G
C
A
A
T
G
C
T
T
C
A
G
G
G
A
C
T
C
C
G
T
ATable 3
DNA subtraction.
−
A
T
G
C
A
A
G
T
C
T
T
A
C
G
G
G
C
A
T
C
C
T
G
ATable 4
DNA XOR operation.
⊕
A
T
G
C
A
A
T
G
C
T
T
A
C
G
G
G
C
A
T
C
C
G
T
A


## 2.2. Multiplication over the Galois Field

The French mathematician Galois invented the Galois field, and the operations of addition, subtraction, multiplication, and division over the Galois field can be performed. Compared with the addition and subtraction over the Galois field, the multiplication and division are more complicated. That is, the diffusion effect on pixels is more prominent. In order to speed up the calculation, we need to construct a lookup table in advance by(1)0:16T×0:16mod17,where ⋅T denotes transposition of the sequence. As for the GF(17) multiplication, a 8-bit pixel value is divided into the upper four bits and the lower four bits whose value ranges are 0,15. The bits are further converted to 1,16 when the lookup table is established, and the results are exhibited in Table 5.Table 5
Multiplication lookup table on the GF(17).
×
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2
0
2
4
6
8
10
12
14
16
1
3
5
7
9
11
13
15
3
0
3
6
9
12
15
1
4
7
10
13
16
2
5
8
11
14
4
0
4
8
12
16
3
7
11
15
2
6
10
14
1
5
9
13
5
0
5
10
15
3
8
13
1
6
11
16
4
9
14
2
7
12
6
0
6
12
1
7
13
2
8
14
3
9
15
4
10
16
5
11
7
0
7
14
4
11
1
8
15
5
12
2
9
16
6
13
3
10
8
0
8
16
7
15
6
14
5
13
4
12
3
11
2
10
1
9
9
0
9
1
10
2
11
3
12
4
13
5
14
6
15
7
16
8
10
0
10
3
13
6
16
9
2
12
5
15
8
1
11
4
14
7
11
0
11
5
16
10
4
15
9
3
14
8
2
13
7
1
12
6
12
0
12
7
2
14
9
4
16
11
6
1
13
8
3
15
10
5
13
0
13
9
5
1
14
10
6
2
15
11
7
3
16
12
8
4
14
0
14
11
8
5
2
16
13
10
7
4
1
15
12
9
6
3
15
0
15
13
11
9
7
5
3
1
16
14
12
10
8
6
4
2
16
0
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1


## 3. Modification of 1D Chaotic Maps

With only one variable and simple structures, 1D chaotic maps have low implementation costs. Hence, they are suitable for efficient image encryption. Nevertheless, existing 1D chaotic maps still have some defects. In this section, we will introduce and analyze three conventional 1D chaotic maps, and further present three modified 1D chaotic maps.


### 3.1. Three Classical 1D Chaotic Maps

The first classical chaotic map is the Logistic map, which is also called the insect mouth model. Its definition is as follows:(2)Xk+1=FLμ,Xk=μXk1−Xk,where parameter μ∈0,4, and Xk is the output sequence. The bifurcation diagram and Lyapunov Exponent of the Logistic map are shown in Figures 1(a) and 2(a), respectively. It can be seen from the bifurcation diagram that when the parameter is 3, the Logistic map appears 2 bifurcations from the steady-state solution, and it does not enter chaos until μ is close to 4. The chaotic region of the Logistic map is narrow and a blank window will appear. Only when the Lyapunov exponent exhibits a positive state, the chaotic map has excellent chaotic property. However, the Lyapunov exponents of the Logistic map are negative when μ<3.57. Figure 3(a) shows the chaotic sequence distribution with the initial value of 0.1, the bifurcation parameter of 4, and the iterations of 10000. Most values are close to 0 and 1. This indicates that the distribution of the sequence generated by the Logistic map is uneven. Hence, the application range of the Logistic map is tiny.Figure 1
The bifurcation diagrams of the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)Figure 2
The Lyapunov Exponents of the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)Figure 3
The distribution of the chaotic sequences for the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)The Sine map has similar chaotic properties with the Logistic map [17], which is described by (3)Xk+1=FSr,Xk=rsinπ×Xk,where parameter r∈0,1. The bifurcation analysis of the Sine map is exhibited in Figure 1(b), and the map has chaotic characteristic when r is close to 1. It can be seen in Figure 2(b) that a portion of the Lyapunov exponents for the Sine map are positive and others of them are negative. Figure 3(b) shows that the chaotic sequence distribution of the Sine map is similar to the Logistic map when the bifurcation parameter is set to 1 and the other parameters are unchanged. It is also not uniform. Thus, there are a number of security risks when using the Sine map for encryption.The Chebyshev map is also a traditional 1D chaotic map [27], and it can be expressed as follows:(4)Xk+1=FCu,Xk=cosu×arccosXk,where parameter u∈N. It can be seen from the bifurcation diagram in Figure 1(c) that the map does not have chaos property when u<1, and it has obvious blank areas within 1,2. Only when u is greater than 2, the map actually enters a chaotic state. Figure 2(c) displays that a portion of the Lyapunov exponents for the Chebyshev map are negative. Figure 3(c) shows the sequence distribution of the same parameters set as the Logistic map, and a great many values are close to 0 and 1. It also proves that the application range of the Chebyshev map is tiny and limited.


### 3.2. Three Modified 1D Chaotic Maps

In this section, the 1D chaotic maps mentioned above are combined to put forward three modified 1D chaotic maps. The purpose is to rectify the shortcomings of the classical 1D chaotic maps analyzed in the previous section. It can be found that the novel 1D chaotic maps have better chaotic properties than the original chaotic maps.The first map is the SLM (Sine-Logistic map), which can be defined by the following formula:(5)Xk+1=fn×FSr1,Xk+FL2r1,Xk−fn×FSr1,Xk+FL2r1,Xk,where fn=2n with 10≤n≤20, which is used to balance the entire function. The symbol ⌊⋅⌋ represents the integer function that make the element return the nearest integer towards minus infinity. It is utilized to control the chaotic sequence within the range of 0,1. After the improvement, the range of bifurcation parameter r1 is increased to 0,15, and x0 is the initial value of the output sequence. It can be seen from the bifurcation diagram in Figure 1(d) that there is no obvious blank area in the entire chaotic region. In other words, the improved SLM expands the original chaotic range. Moreover, Figure 2(d) displays that the Lyapunov exponents of the SLM are all positive within 0,15, and Figure 3(d) shows that its chaotic sequence is uniformly distributed when the parameter set is the same as the Logistic map.The second map is the CLM (Chebyshev-Logistic map). Its formula can be described as(6)Xk+1=fn×FCr2,Xk+FL2r2,Xk−fn×FCr2,Xk+FL2r2,Xk,where fn has the same effect as it in the SLM and control parameter r2∈0,15. From the bifurcation diagram in Figure 1(e) and Lyapunov exponent in Figure 2(e) of the CLM, it can be found that the CLM has superior chaotic behavior, which is similar to the SLM. Besides, Figure 3(e) shows that the distribution of its chaotic sequence is semblable to the SLM.The third map is the SCM (Sine-Chebyshev map). The combination equation is as follows:(7)Xk+1=fn×FSr3,Xk+FC2r3,Xk−fn×FSr3,Xk+FC2r3,Xk,where fn is the same as it in equation (6) and control parameter r3∈0,15. The bifurcation diagram and Lyapunov exponent of the SCM are revealed in Figures 1(f) and 2(f), respectively. Moreover, the distribution of the output sequence is displayed in Figure 3(f). Its chaos property is similar with the SLM and CLM, which is also improved a lot.Table6 lists the comparison of characteristics between our maps and other improved 1D chaotic maps. It can be seen that the maps proposed in this paper have better chaotic behaviors.Table 6
Chaotic characters of improved 1D chaotic maps.
Chaotic maps
Range of bifurcation parameters
Average maximum Lyapunov exponent
The proposed chaotic maps
0,15
10
Ref. [15]
0,4
0.7
Ref. [18]
0,10
2
Ref. [28]
−
4,4
1


## 3.1. Three Classical 1D Chaotic Maps

The first classical chaotic map is the Logistic map, which is also called the insect mouth model. Its definition is as follows:(2)Xk+1=FLμ,Xk=μXk1−Xk,where parameter μ∈0,4, and Xk is the output sequence. The bifurcation diagram and Lyapunov Exponent of the Logistic map are shown in Figures 1(a) and 2(a), respectively. It can be seen from the bifurcation diagram that when the parameter is 3, the Logistic map appears 2 bifurcations from the steady-state solution, and it does not enter chaos until μ is close to 4. The chaotic region of the Logistic map is narrow and a blank window will appear. Only when the Lyapunov exponent exhibits a positive state, the chaotic map has excellent chaotic property. However, the Lyapunov exponents of the Logistic map are negative when μ<3.57. Figure 3(a) shows the chaotic sequence distribution with the initial value of 0.1, the bifurcation parameter of 4, and the iterations of 10000. Most values are close to 0 and 1. This indicates that the distribution of the sequence generated by the Logistic map is uneven. Hence, the application range of the Logistic map is tiny.Figure 1
The bifurcation diagrams of the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)Figure 2
The Lyapunov Exponents of the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)Figure 3
The distribution of the chaotic sequences for the (a) Logistic map; (b) Sine map; (c) Chebyshev map; (d) SLM; (e) CLM; (f) SCM.
(a)
(b)
(c)
(d)
(e)
(f)The Sine map has similar chaotic properties with the Logistic map [17], which is described by (3)Xk+1=FSr,Xk=rsinπ×Xk,where parameter r∈0,1. The bifurcation analysis of the Sine map is exhibited in Figure 1(b), and the map has chaotic characteristic when r is close to 1. It can be seen in Figure 2(b) that a portion of the Lyapunov exponents for the Sine map are positive and others of them are negative. Figure 3(b) shows that the chaotic sequence distribution of the Sine map is similar to the Logistic map when the bifurcation parameter is set to 1 and the other parameters are unchanged. It is also not uniform. Thus, there are a number of security risks when using the Sine map for encryption.The Chebyshev map is also a traditional 1D chaotic map [27], and it can be expressed as follows:(4)Xk+1=FCu,Xk=cosu×arccosXk,where parameter u∈N. It can be seen from the bifurcation diagram in Figure 1(c) that the map does not have chaos property when u<1, and it has obvious blank areas within 1,2. Only when u is greater than 2, the map actually enters a chaotic state. Figure 2(c) displays that a portion of the Lyapunov exponents for the Chebyshev map are negative. Figure 3(c) shows the sequence distribution of the same parameters set as the Logistic map, and a great many values are close to 0 and 1. It also proves that the application range of the Chebyshev map is tiny and limited.


## 3.2. Three Modified 1D Chaotic Maps

In this section, the 1D chaotic maps mentioned above are combined to put forward three modified 1D chaotic maps. The purpose is to rectify the shortcomings of the classical 1D chaotic maps analyzed in the previous section. It can be found that the novel 1D chaotic maps have better chaotic properties than the original chaotic maps.The first map is the SLM (Sine-Logistic map), which can be defined by the following formula:(5)Xk+1=fn×FSr1,Xk+FL2r1,Xk−fn×FSr1,Xk+FL2r1,Xk,where fn=2n with 10≤n≤20, which is used to balance the entire function. The symbol ⌊⋅⌋ represents the integer function that make the element return the nearest integer towards minus infinity. It is utilized to control the chaotic sequence within the range of 0,1. After the improvement, the range of bifurcation parameter r1 is increased to 0,15, and x0 is the initial value of the output sequence. It can be seen from the bifurcation diagram in Figure 1(d) that there is no obvious blank area in the entire chaotic region. In other words, the improved SLM expands the original chaotic range. Moreover, Figure 2(d) displays that the Lyapunov exponents of the SLM are all positive within 0,15, and Figure 3(d) shows that its chaotic sequence is uniformly distributed when the parameter set is the same as the Logistic map.The second map is the CLM (Chebyshev-Logistic map). Its formula can be described as(6)Xk+1=fn×FCr2,Xk+FL2r2,Xk−fn×FCr2,Xk+FL2r2,Xk,where fn has the same effect as it in the SLM and control parameter r2∈0,15. From the bifurcation diagram in Figure 1(e) and Lyapunov exponent in Figure 2(e) of the CLM, it can be found that the CLM has superior chaotic behavior, which is similar to the SLM. Besides, Figure 3(e) shows that the distribution of its chaotic sequence is semblable to the SLM.The third map is the SCM (Sine-Chebyshev map). The combination equation is as follows:(7)Xk+1=fn×FSr3,Xk+FC2r3,Xk−fn×FSr3,Xk+FC2r3,Xk,where fn is the same as it in equation (6) and control parameter r3∈0,15. The bifurcation diagram and Lyapunov exponent of the SCM are revealed in Figures 1(f) and 2(f), respectively. Moreover, the distribution of the output sequence is displayed in Figure 3(f). Its chaos property is similar with the SLM and CLM, which is also improved a lot.Table6 lists the comparison of characteristics between our maps and other improved 1D chaotic maps. It can be seen that the maps proposed in this paper have better chaotic behaviors.Table 6
Chaotic characters of improved 1D chaotic maps.
Chaotic maps
Range of bifurcation parameters
Average maximum Lyapunov exponent
The proposed chaotic maps
0,15
10
Ref. [15]
0,4
0.7
Ref. [18]
0,10
2
Ref. [28]
−
4,4
1


## 4. The Proposed Image Encryption and Decryption Scheme

### 4.1. Encryption Scheme

Input: color plain imageP of size m×n, the security keys which are composed of the bifurcation parameters r1, r2, and r3 and three initial values X10,X20,andX30 of the SLM, CLM, and SCM.
Output: the color cipher image with the same size.Step 1: decompose the imageP into three matrixes R, G, and B. Then, update the three initial values with bit-planes recombination according to the following formula:
(8)
X
10
′
=
X
10
+
∑
R
odd
x
,
y
255
×
n
2
,
X
20
′
=
X
20
+
∑
G
odd
x
,
y
5
×
m
2
×
n
2
,
X
30
′
=
X
30
+
∑
B
odd
x
,
y
5
×
m
2
×
n
2
,
(9)
X
1
1
i
i
=
1
m
×
n
=
X
1
1
1
,
X
1
1
2
,
…
,
X
1
1
m
×
n
,
X
2
1
i
i
=
1
m
×
n
=
X
2
1
1
,
X
2
1
2
,
…
,
X
2
1
m
×
n
,
X
3
1
i
i
=
1
m
×
n
=
X
3
1
1
,
X
3
1
2
,
…
,
X
3
1
m
×
n
.
X
1
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
1
2
m
×
n
+
1
,
X
1
2
m
×
n
+
2
,
…
,
X
1
2
2
×
m
×
n
,
X
2
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
2
2
m
×
n
+
1
,
X
2
2
m
×
n
+
2
,
…
,
X
2
2
2
×
m
×
n
,
X
3
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
3
2
m
×
n
+
1
,
X
3
2
m
×
n
+
2
,
…
,
X
3
2
2
×
m
×
n
.
(10)
X
˜
1
1
i
=
X
1
1
i
+
50
×
10
12
mod
m
×
n
+
1
,
X
˜
2
1
i
=
X
2
1
i
+
100
×
10
12
mod
m
×
n
+
1
,
X
˜
3
1
i
=
X
3
1
i
+
500
×
10
12
mod
m
×
n
+
1
.
(11)
V
1
i
=
V
1
X
˜
1
1
i
,
V
2
i
=
V
2
X
˜
2
1
i
,
V
3
i
=
V
3
X
˜
3
1
i
.
(12)
X
^
1
1
i
=
fix
X
1
1
i
×
10
4
mod
256
,
X
^
2
1
i
=
fix
X
2
1
i
×
10
4
mod
256
,
X
^
3
1
i
=
fix
X
3
1
i
×
10
4
mod
256
.
(13)
X
^
1
2
i
=
fix
X
1
1
i
×
2
16
mod
256
,
X
^
2
2
i
=
fix
X
2
1
i
×
2
16
mod
256
,
X
^
3
2
i
=
fix
X
3
1
i
×
2
16
mod
256
.
(14)
X
¯
1
1
i
=
X
1
1
i
×
10
4
mod
3
,
(15)
E
D
R
,
D
G
,
D
B
=
C
D
R
,
D
G
,
D
B
+
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
0
,
C
D
R
,
D
G
,
D
B
−
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
1
,
C
D
R
,
D
G
,
D
B
⊕
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
2.
(16)
X
⌢
2
1
i
=
X
2
1
i
×
10
4
mod
8
+
1.
(17)
C
E
R
i
=
C
E
R
i
−
1
×
X
^
1
1
i
×
C
2
R
i
,
C
E
G
i
=
C
E
G
i
−
1
×
X
^
2
1
i
×
C
2
G
i
,
C
E
B
i
=
C
E
B
i
−
1
×
X
^
3
1
i
×
C
2
B
i
.
(18)
C
R
i
=
C
R
i
−
1
×
X
^
1
2
i
×
C
E
R
i
,
C
G
i
=
C
G
i
−
1
×
X
^
2
2
i
×
C
E
G
i
,
C
B
i
=
C
B
i
−
1
×
X
^
3
2
i
×
C
E
B
i
.where∑R,G,Bodd denotes sum of the odd bit-planes about components R, G, and B.
Step 2: iterate the SLM, CLM, and SCM2×m×n+500 times with X10′,X20′,andX30′, respectively. Discard the first 500 elements to eliminate the effects of transient processes. Then, two sets of new sequences are formed, which can be expressed as follows:
Step 3: convert the componentsR, G, and B into three 1D vectors V1, V2, and V3, and then perform pixel-level scrambling on them according to equations (10) and (11):
Then, restore the three vectors to the matrixesC1R, C1G, and C1B.
Step 4: generate two new sets of sequences by equations (12) and (13) After that, reshape the sequences in equation (12) to the matrixes M1, M2, and M3 of size m×n:
wherefix⋅ indicates the function rounds the element to the nearest integer towards minus infinity.
Step 5: fill each matrix into a square with zero elements and separate each matrix into parts of sizes×s, respectively. Arbitrarily, select the DNA coding rule 1 in Table 1 to encode each part of C1R, C1G, C1B, M1, M2, and M3 from steps 3 and 4. Then, six DNA matrixes named C1DR, C1DG, C1DB, MD1, MD2, and MD3 are obtained. After this, utilize equation (14) to execute DNA calculation on these matrixes:
where⌈⋅⌉ denotes that the element returns the smallest integer in the infinite direction, and the elements in X¯11ii=1m×n are 0, 1, and 2. Hence, the corresponding DNA calculations are addition, subtraction, and XOR operations, which can be defined as follows:
Step 6: utilize equation (16) to determine the decoding rules of equation (15). The DNA matrixes are decoded to binary matrixes and further transformed into decimal numbers to get three encrypted components C2R, C2G, and C2B:
Step 7: implement multiplication on the GF(17) with Table5 according to equations (17) and (18):
It should be noted that the matrixes need to be converted into the sequences for multiplication, and the three componentsCR, CG, and CB are merged to form the final encrypted image C. The flowchart of our encryption scheme is revealed in Figure 4.


### 4.2. Decryption Scheme

The decryption procedure is the inverse process of the encryption procedure. Thus, the steps presented in the previous section ought to proceed in the reverse order. However, the only distinction is that step 5 needs to apply equation (19) to determine the DNA calculation mode when decrypting:(19)EDR,DG,DB=CDR,DG,DB−MD1,D2,D3,ifX¯1i=0,CDR,DG,DB+MD1,D2,D3,ifX¯1i=1,CDR,DG,DB⊕MD1,D2,D3,ifX¯1i=2.Figure 4
Flowchart of the encryption scheme.


## 4.1. Encryption Scheme

Input: color plain imageP of size m×n, the security keys which are composed of the bifurcation parameters r1, r2, and r3 and three initial values X10,X20,andX30 of the SLM, CLM, and SCM.
Output: the color cipher image with the same size.Step 1: decompose the imageP into three matrixes R, G, and B. Then, update the three initial values with bit-planes recombination according to the following formula:
(8)
X
10
′
=
X
10
+
∑
R
odd
x
,
y
255
×
n
2
,
X
20
′
=
X
20
+
∑
G
odd
x
,
y
5
×
m
2
×
n
2
,
X
30
′
=
X
30
+
∑
B
odd
x
,
y
5
×
m
2
×
n
2
,
(9)
X
1
1
i
i
=
1
m
×
n
=
X
1
1
1
,
X
1
1
2
,
…
,
X
1
1
m
×
n
,
X
2
1
i
i
=
1
m
×
n
=
X
2
1
1
,
X
2
1
2
,
…
,
X
2
1
m
×
n
,
X
3
1
i
i
=
1
m
×
n
=
X
3
1
1
,
X
3
1
2
,
…
,
X
3
1
m
×
n
.
X
1
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
1
2
m
×
n
+
1
,
X
1
2
m
×
n
+
2
,
…
,
X
1
2
2
×
m
×
n
,
X
2
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
2
2
m
×
n
+
1
,
X
2
2
m
×
n
+
2
,
…
,
X
2
2
2
×
m
×
n
,
X
3
2
i
i
=
m
×
n
+
1
2
×
m
×
n
=
X
3
2
m
×
n
+
1
,
X
3
2
m
×
n
+
2
,
…
,
X
3
2
2
×
m
×
n
.
(10)
X
˜
1
1
i
=
X
1
1
i
+
50
×
10
12
mod
m
×
n
+
1
,
X
˜
2
1
i
=
X
2
1
i
+
100
×
10
12
mod
m
×
n
+
1
,
X
˜
3
1
i
=
X
3
1
i
+
500
×
10
12
mod
m
×
n
+
1
.
(11)
V
1
i
=
V
1
X
˜
1
1
i
,
V
2
i
=
V
2
X
˜
2
1
i
,
V
3
i
=
V
3
X
˜
3
1
i
.
(12)
X
^
1
1
i
=
fix
X
1
1
i
×
10
4
mod
256
,
X
^
2
1
i
=
fix
X
2
1
i
×
10
4
mod
256
,
X
^
3
1
i
=
fix
X
3
1
i
×
10
4
mod
256
.
(13)
X
^
1
2
i
=
fix
X
1
1
i
×
2
16
mod
256
,
X
^
2
2
i
=
fix
X
2
1
i
×
2
16
mod
256
,
X
^
3
2
i
=
fix
X
3
1
i
×
2
16
mod
256
.
(14)
X
¯
1
1
i
=
X
1
1
i
×
10
4
mod
3
,
(15)
E
D
R
,
D
G
,
D
B
=
C
D
R
,
D
G
,
D
B
+
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
0
,
C
D
R
,
D
G
,
D
B
−
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
1
,
C
D
R
,
D
G
,
D
B
⊕
M
D
1
,
D
2
,
D
3
,
if
X
¯
1
i
=
2.
(16)
X
⌢
2
1
i
=
X
2
1
i
×
10
4
mod
8
+
1.
(17)
C
E
R
i
=
C
E
R
i
−
1
×
X
^
1
1
i
×
C
2
R
i
,
C
E
G
i
=
C
E
G
i
−
1
×
X
^
2
1
i
×
C
2
G
i
,
C
E
B
i
=
C
E
B
i
−
1
×
X
^
3
1
i
×
C
2
B
i
.
(18)
C
R
i
=
C
R
i
−
1
×
X
^
1
2
i
×
C
E
R
i
,
C
G
i
=
C
G
i
−
1
×
X
^
2
2
i
×
C
E
G
i
,
C
B
i
=
C
B
i
−
1
×
X
^
3
2
i
×
C
E
B
i
.where∑R,G,Bodd denotes sum of the odd bit-planes about components R, G, and B.
Step 2: iterate the SLM, CLM, and SCM2×m×n+500 times with X10′,X20′,andX30′, respectively. Discard the first 500 elements to eliminate the effects of transient processes. Then, two sets of new sequences are formed, which can be expressed as follows:
Step 3: convert the componentsR, G, and B into three 1D vectors V1, V2, and V3, and then perform pixel-level scrambling on them according to equations (10) and (11):
Then, restore the three vectors to the matrixesC1R, C1G, and C1B.
Step 4: generate two new sets of sequences by equations (12) and (13) After that, reshape the sequences in equation (12) to the matrixes M1, M2, and M3 of size m×n:
wherefix⋅ indicates the function rounds the element to the nearest integer towards minus infinity.
Step 5: fill each matrix into a square with zero elements and separate each matrix into parts of sizes×s, respectively. Arbitrarily, select the DNA coding rule 1 in Table 1 to encode each part of C1R, C1G, C1B, M1, M2, and M3 from steps 3 and 4. Then, six DNA matrixes named C1DR, C1DG, C1DB, MD1, MD2, and MD3 are obtained. After this, utilize equation (14) to execute DNA calculation on these matrixes:
where⌈⋅⌉ denotes that the element returns the smallest integer in the infinite direction, and the elements in X¯11ii=1m×n are 0, 1, and 2. Hence, the corresponding DNA calculations are addition, subtraction, and XOR operations, which can be defined as follows:
Step 6: utilize equation (16) to determine the decoding rules of equation (15). The DNA matrixes are decoded to binary matrixes and further transformed into decimal numbers to get three encrypted components C2R, C2G, and C2B:
Step 7: implement multiplication on the GF(17) with Table5 according to equations (17) and (18):
It should be noted that the matrixes need to be converted into the sequences for multiplication, and the three componentsCR, CG, and CB are merged to form the final encrypted image C. The flowchart of our encryption scheme is revealed in Figure 4.


## 4.2. Decryption Scheme

The decryption procedure is the inverse process of the encryption procedure. Thus, the steps presented in the previous section ought to proceed in the reverse order. However, the only distinction is that step 5 needs to apply equation (19) to determine the DNA calculation mode when decrypting:(19)EDR,DG,DB=CDR,DG,DB−MD1,D2,D3,ifX¯1i=0,CDR,DG,DB+MD1,D2,D3,ifX¯1i=1,CDR,DG,DB⊕MD1,D2,D3,ifX¯1i=2.Figure 4
Flowchart of the encryption scheme.


## 5. Experimental Results and Security Analysis

The experiment is implemented by MATLAB R2016b on a PC with an Intel Core i5, 3.4 GHz CPU, 8 GB memory, and the encryption results for three color images “Female”, “Peppers,” and “Mandrill” of size256×256 are displayed in Figure 5. The control parameters and initial values are set as follows: r1=14.99120306595001, r2=14.985633002586235,r3=14.978965662236302, X10=0.654321563325991,X20=0.563214562356231, and X30=0.456326656565231. We specify these six elements as the key set K0. Visually, the scheme designed in this paper has good performance. The ciphered images are noise-like ones, and the decrypted images are almost identical with the plain images.Figure 5
Encryption results: (a) original images; (b) encrypted images; (c) decrypted images.
(a)
(b)
(c)


### 5.1. Differential Attack Analysis

In general, the intensity of sensitivity to the plain image can determine the ability of the algorithm to resist differential attack. The measured indicators are the number of pixels change rate (NPCR) and the unified average changing intensity (UACI), which can be defined as follows:(20)NPCRR,G,B=1M×N∑i=1M∑j=1NER,G,Bi,j×100%,UACIR,G,B=1M×N∑i=1M∑j=1NCR,G,Bi,j−CR,G,B′i,j255×100%,where M and N are the width and height of the image and C and C′ are two cipher images whose plain images have only one different pixel. Besides, ER,G,Bi,j is used to determine the distinction between two ciphered images, which can be calculated by(21)ER,G,Bi,j=0,CR,G,Bi,j=CR,G,B′i,j,1,CR,G,Bi,j≠CR,G,B′i,j.Nevertheless, Zhang [31] pointed out that these two indexes cannot precisely measure the difference between two images. He proposed block average changing intensity (BACI) to quantitatively analyze the antidifferential attack characteristic. It is a method of block calculation about subtraction images, and each small block can be defined by (22)Bi=bi1bi2bi3bi4.Moreover, the mean for the absolute values of differences between arbitrary two elements can be defined as(23)mi=16bi1−bi2+bi1−bi3+bi1−bi4+bi2−bi3+bi2−bi4+bi3−bi4.Then, the BACI can be calculated by(24)BACIR,G,B=1M−1N−1∑i=1M−1N−1mi255.Thus, the NPCR, UACI, and BACI are used together to analyze the sensitive property of our encryption scheme, and the results are revealed in Table7. It can be found that the measured values of six different color images are very close to the theoretical values 99.6094%, 33.4635%, and 26.7712%, respectively. In particular, Table 7 also compares the NPCR and UACI scores on the Lena, Mandrill, and Peppers images using our scheme with algorithms in [29, 30]. This further demonstrates that our cryptosystem has excellent ability to resist differential attack.Table 7
Plaintext sensitivity test for different color images.
Algorithms
Images
Channels
NPCR (%)
UACI (%)
BACI (%)
The proposed scheme
Female
Red
99.6109
33.5025
26.8215
Green
99.6039
33.4360
26.7813
Blue
99.6149
33.4430
26.7754
Peppers
Red
99.6005
33.4494
26.7600
Green
99.6278
33.4537
26.7353
Blue
99.6188
33.4529
26.7651
Mandrill
Red
99.5941
33.4530
26.7913
Green
99.6199
33.4472
26.7685
Blue
99.6091
33.4596
26.7405
Lena256×256
Red
99.6109
33.4783
26.7618
Green
99.6213
33.4503
26.7713
Blue
99.6004
33.4640
26.7794
Tree256×256
Red
99.6062
33.4911
26.7929
Green
99.5996
33.4628
26.7709
Blue
99.6191
33.4522
26.7487
Couple256×256
Red
99.6149
33.4617
26.7908
Green
99.6083
33.4563
26.7618
Blue
99.6114
33.4533
26.7512
Ref. [29]
Lena256×256
Red
99.5800
33.2700
—
Green
99.5600
33.3600
—
Blue
99.6400
33.5000
—
Mandrill256×256
Red
99.6200
33.4700
—
Green
99.6000
33.4800
—
Blue
99.6000
33.4500
—
Peppers512×512
Red
99.6000
33.3900
—
Green
99.6000
33.4600
—
Blue
99.6100
33.4000
—
Ref. [30]
Lena256×256
Red
99.6317
33.6783
—
Green
99.6205
33.7999
—
Blue
99.6211
33.6200
—
Mandrill256×256
Red
99.6199
33.6484
—
Green
99.6250
33.5908
—
Blue
99.6273
33.6749
—
Peppers256×256
Red
99.6202
33.6602
—
Green
99.6192
33.6575
—
Blue
99.6224
33.7314
—


### 5.2. Key Space

As mentioned earlier in this section, the secret keys for our scheme consist of three control parametersr1,r2,r3 and three initial values X10,X20,X30, which are all double-precision real numbers. In this paper, we use the 64-bit double-precision format in [32] to calculate our key space, and the total key space of our scheme can reach 10156=1090≈2299. In addition, Table 8 exhibits the comparison of key space between our scheme and other chaos-based encryption algorithms. It can be seen that the key space of our method is large enough to resist brute-force attack.Table 8
Comparison of the key space between our scheme and other chaos-based encryption algorithms.
Encryption algorithms
Key space
The proposed scheme
10
15
6
=
10
90
≈
2
299
Ref. [8]
2
106
Ref. [15]
10
84
≈
2
279
Ref. [18]
2
138
Ref. [33]
2
298
Ref. [34]
>
2
208


### 5.3. Key Sensitivity

The key sensitivity refers to the degree of variation for the corresponding ciphered image when the initial key alters slightly. Make a minor change toX10 while keeping other keys unchanged. For instance, X10′=X10+10−15. This produces a new key set K1. It is applied to encrypt the plain images in Figure 5(a), and the ciphered images in Figure 6(a) are displayed. The pixel-by-pixel differences between Figures 5(b) and 6(a) are revealed in Figure 6(b), which proves that their encrypted images are completely different. Moreover, K0 and K1 are utilized to decrypt the ciphered images in Figure 5(b), respectively. The decrypted images in Figure 6(c) can be acquired with the correct key set K0, and a minor change to the security keys will cause the failure of decryption, as shown in Figure 6(d).Figure 6
Key sensitivity test: (a) encrypted images with the key setK1; (b) the pixel-by-pixel difference between Figures 5(b) and 6(a); (c) decrypted images from Figure 5(b) with the correct key set K0; (d) decrypted images from Figure 5(b) with the wrong key set K1.
(a)
(b)
(c)
(d)In addition, Table9 lists the average NPCR and UACI results for the three channels of the key sensitivity test for the plain images in Figure 5(a). It can be found that a slight change in each key will result in the change rate of more than 99.5% for the ciphered images. Their corresponding average NPCR and UACI results are 99.6124% and 33.4714%, which are closer to ideal than the values (NPCR = 99.6389% and UACI = 33.4189) in [35]. Thus, the proposed scheme has high key sensitivity.Table 9
NPCR and UACI results for key sensitivity test.
Modified keys
Average NPCR (%)
Average UACI (%)
Female
Peppers
Mandrill
Female
Peppers
Mandrill
X
10
+
10
−
15
99.6132
99.6228
99.6164
33.4596
33.4677
33.4828
X
20
+
10
−
15
99.6088
99.6098
99.6062
33.4641
33.4606
33.4656
X
30
+
10
−
15
99.6112
99.6143
99.6124
33.4613
33.4860
33.4734
r
1
+
10
−
15
99.6191
99.6115
99.6072
33.4762
33.4866
33.4674
r
2
+
10
−
15
99.6124
99.6104
99.6120
33.4805
33.4678
33.4741
r
3
+
10
−
15
99.6109
99.6156
99.6089
33.4691
33.4764
33.4665


### 5.4. Histogram Analysis

The image histogram is an important feature statistic of the image. A great encryption algorithm theoretically makes the histogram of the encrypted image evenly distributed. This prevents the adversary from getting any useful information. Figure7 exhibits the histograms of the plain and ciphered Female, Peppers, and Mandrill images. It can be seen that the histograms of the ciphered images are fairly flat, and they are completely different from the histograms of the plain images.Figure 7
Histograms of the (a) plain images in Figure5(a); (b) ciphered images in Figure 5(b).
(a)
(b)Besides, the analysis of variance is a quantitative measure for the properties of histogram. The smaller the calculated variance value, the higher the uniformity of the image [36]. The specific calculation formula is as follows:(25)varianceY=1n2∑i=1n∑j=1nyi−yj22,where yi and yj indicate the number of pixels corresponding to the gray values i and j, Y represents the vector set of the histogram and Y=y1,y2,…,y256, and n denotes the total number of gray values. Table 10 exhibits the variances of the histograms of the color plain and ciphered images in Figure 5. It can be discovered that the histogram variance values of the encrypted images are much smaller than those of the original images. Moreover, Table 11 displays the comparison of the mean histogram variance values of the three channels for the Lena image and the calculated variance results in [26, 36–38]. Obviously, the histogram variance values of our scheme are smaller, that is, the histograms of the ciphered images have better uniformity.Table 10
Histogram variance analysis of plain and ciphered images.
Images
Plain images
Ciphered images
Red
Green
Blue
Red
Green
Blue
Female
168960
157950
158040
233.4375
263.3125
267.3516
Peppers
54484
64353
106880
244.1172
258.2734
223.9297
Mandrill
20770
30655
16857
220.3125
240.2109
270.2656Table 11
Histogram variance analysis of original Lena image and encrypted Lena image.
Images
Algorithms
Variance
Original Lena image (512×512)
6390323
Encrypted Lena image
The proposed scheme
953.7656
Ref. [26]
977.02
Ref. [36]
974.8
Ref. [37]
1077.3
Ref. [38]
1209.4Then, we utilize the chi-square test to validate the uniformity for the pixel values distribution of the ciphered images [36]. Similar to the variances of the histograms, the smaller the result of the chi-square test, the better the uniformity of the ciphered image. Its definition is as follows:(26)χ2=∑i=0255fi−pi2pi,where fi indicates the actual frequency of the pixel value i and pi represents the predicted frequency of the pixel value i. The predicted frequency pi can be calculated by(27)pi=m×n256,where m×n is the size of the image. Theoretically, the chi-square statistic with a significant level of 0.01 is 310.4574, while the chi-square statistic with a significant level of 0.05 is 293.2478. Obviously, the values of the chi-square test for six ciphered images in Table 12 are all smaller than the previous two theoretical values. Thus, it further illustrates the effectiveness of our scheme.Table 12
Chi-square test of some ciphered images.
Images
Components
Results
Red
Green
Blue
Female
233.4375
253.3125
247.3516
Pass
Peppers
244.1172
238.2734
223.9297
Pass
Mandrill
220.3125
240.2109
240.2656
Pass
Couple (256×256)
234.2500
246.0313
233.3906
Pass
Tree (256×256)
222.5781
246.3047
235.5313
Pass
Lena (512×512)
245.8965
230.9863
252.1680
Pass
Airplane (512×512)
240.7422
257.8555
235.8145
Pass


### 5.5. Correlation of Two Adjacent Pixels

As for plain images, there are strong correlations between their adjacent pixels. Thus, the purpose of encryption is to weaken this property. In the experiment, we randomly select 4000 pairs of adjacent pixels from the original images and corresponding encrypted images. Then, compute the correlation coefficients in horizontal, vertical, and diagonal directions as follows:(28)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,COVx,y=1N∑i=1Nxi−Exyi−Ey,ρxy=COVx,yDxDy,where Ex,Dx, and COVx,y indicate the mean value, variance, and covariance, respectively. Figure 8 plots the correlations between adjacent pixels of the plain Peppers image, and Figure 9 shows the correlation coefficients of the corresponding ciphered image. It can be seen from the comparison that the encryption makes the correlation between the pixels of the image significantly whittled. Moreover, Table 13 provides the correlation coefficients for the adjacent pixels of six color images with the size of 256×256 in three directions. It can be discovered that the results of the ciphered images are close to 0. Furthermore, Table 14 reveals that our encryption scheme has superior performance compared to some recent encryption algorithms.Figure 8
Correlation coefficients of plain Peppers image in all directions: (a) R channel; (b) G channel; (c) B channel.
(a)
(b)
(c)Figure 9
Correlation coefficients of ciphered Peppers image in all directions: (a) R channel; (b) G channel; (c) B channel.
(a)
(b)
(c)Table 13
Average correlation coefficients of three channels for plain and ciphered images.
Images
Plain images
Ciphered images
Horizontal
Vertical
Diagonal
Horizontal
Vertical
Diagonal
Female
0.9747
0.9621
0.9511
0.0078
0.0021
0.0004
Peppers
0.9647
0.9701
0.9602
0.0004
−0.0035
−0.0021
Mandrill
0.9188
0.8530
0.8534
−0.0031
0.0048
0.0005
Tree
0.9406
0.9551
0.9157
−0.0012
−0.0046
0.0006
Couple
0.9131
0.9011
0.9375
0.0074
0.0035
0.0013
House
0.9328
0.9676
0.9114
−0.0027
0.0047
0.0012Table 14
Correlation coefficients of original Lena image and encrypted Lena image.
Images
Algorithms
Directions
Horizontal
Vertical
Diagonal
Original Lena image256×256
0.9731
0.9881
0.9661
Encrypted Lena image
The proposed scheme
−0.0034
0.0021
−0.0003
Ref. [14]
0.0056
0.0065
−0.0073
Ref. [19]
0.0036
0.0023
0.0039
Ref. [33]
−0.0068
−0.0054
0.0010
Ref. [39]
0.0040
0.0011
0.0008
Ref. [40]
0.0059
−0.0042
0.0180
Ref. [41]
−0.0168
0.0445
−0.0022
Ref. [42]
−0.0003
−0.0013
−0.0066


### 5.6. Information Entropy Analysis

Information entropy is expressed as the probability of discrete random events, which is used to measure the randomness of a system. It can be defined as(29)Hs=−∑i=02n−1Psilog2Psi,where Psi denotes the probability that the gray value si may appear. For a grayscale image with data field of 0,255, the maximum value of information entropy is 8. Therefore, once the calculated value is pretty close to 8, it proves that the proposed algorithm is quite safe. We have computed the information entropy of six different color images with the size of 256×256 and their ciphered versions in Table 15. From the table, the information entropy results of ciphered images are approaching the ideal value 8. In addition, Table 16 exhibits the information entropy of the original Lena image and encrypted Lena image by utilizing our scheme and some other encryption algorithms. It can be clearly seen that our cryptosystem is closer to the desired state and has better randomness.Table 15
Information entropy analysis of plain and ciphered images.
Images
Plain images
Ciphered images
Red
Green
Blue
Red
Green
Blue
Female
6.4200
6.4457
6.3807
7.9974
7.9971
7.9971
Peppers
7.3449
7.5607
7.1003
7.9973
7.9972
7.9975
Mandrill
7.7255
7.5618
7.8031
7.9976
7.9974
7.9970
Tree
7.2104
7.4136
6.9207
7.9976
7.9971
7.9970
Couple
6.2499
5.9642
5.9309
7.9972
7.9973
7.9972
House
6.4311
6.5389
6.2320
7.9975
7.9968
7.9973Table 16
Comparison of the information entropy of the original Lena image and encrypted Lena image.
Information entropy
Red
Green
Blue
Original Lena image (256×256)
7.3140
7.6394
7.0506
The proposed algorithm
7.9971
7.9972
7.9972
Ref. [39]
7.9278
7.9744
7.9705
Ref. [41]
7.9897
7.9877
7.9896
Ref. [43]
7.9988
7.9967
7.9990
Ref. [44]
7.9791
7.9802
7.9827


### 5.7. Noise Attack Analysis

In practical applications, noise interference is inevitable. An outstanding encryption algorithm has the ability to resist noise attack. In our experiment, the encrypted Mandrill image in Figure5(b) is contaminated by Salt & Pepper noise, Gaussian noise, and speckle noise with different densities, respectively. The simulation results are exhibited in Figures 10–12. It can be observed that although the noise is increasing, the decrypted images can still be discerned. Moreover, we also tested the average NPCR and UACI values for the three channels of the original Mandrill image and deciphered Mandrill image under these different noises. The results are listed in Tables 17–19. It can be found that all the NPCR values are less than 99%, and all the UACI values are less than 20%. Hence, the proposed method is strongly robust against disparate noises.Figure 10
Experiment of Salt & Pepper noise attacks: (a) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.05; (b) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.1; (c) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.2; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Experiment of Gaussian noise attacks: (a) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0001; (b) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0003; (c) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0005; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Figure 12
Experiment of speckle noise attacks: (a) encrypted Mandrill image attacked by speckle noise with density 0.00005; (b) encrypted Mandrill image attacked by speckle noise with density 0.0001; (c) encrypted Mandrill image attacked by speckle noise with density 0.0005; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Table 17
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under Salt & Pepper noise.
Images
Density
Average NPCR
Average UACI
Figures5(a) and 10(d)
0.05
26.0681
7.7270
Figures5(a) and 10(e)
0.1
46.2418
13.6364
Figures5(a) and 10(f)
0.2
62.4237
18.5213Table 18
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under Gaussian noise.
Images
Mean
Variance
Average NPCR
Average UACI
Figures5(a) and 11(d)
0
0.0001
98.4116
8.8154
Figures5(a) and 11(e)
0
0.0003
98.7732
11.7899
Figures5(a) and 11(f)
0
0.0005
98.9624
12.7496Table 19
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under speckle noise.
Images
Density
Average NPCR
Average UACI
Figures5(a) and 12(d)
0.00005
94.5068
8.5631
Figures5(a) and 12(e)
0.0001
95.5780
11.3172
Figures5(a) and 12(f)
0.0005
97.4060
19.3016


### 5.8. Cropping Attack Analysis

Encrypted images are subject to cropping attack during transmission and may be partially damaged. Nevertheless, digital images allow a certain extent of distortion on the transmission channel. As long as the information to be conveyed by the image can be discriminated visually, it proves that the encryption algorithm has excellent anticropping attack capability. Figure13 displays ciphered House image of size 256×256 with data cuts in different sizes, and their corresponding deciphered images. Still, most of the pictorial information is available from the decrypted images. This shows that the proposed scheme can effectively resist cropping attack.Figure 13
Test of image cropping attacks: (a) original House image; (b) encrypted House image; (c) encrypted House image with a 30 × 90 data cut; (d) decrypted House image of (c); (e) encrypted House image with a 50 × 80 data cut; (f) decrypted House image of (e); (g) encrypted House image with a 70 × 70 data cut; (h) decrypted House image of (g); (i) encrypted House image with a 90 × 60 data cut; (j) decrypted House image of (i); (k) encrypted House image with a 110 × 50 data cut; (l) decrypted House image of (k).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)


### 5.9. Known Plaintext Attack and Chosen Plaintext Attack Analysis

Since the cryptosystems whose key streams are unrelated to the plain images are vulnerable to chosen plaintext attack and known plaintext attack [21], we design the initial keys of our scheme to be updated by the bit-planes recombination of the plain images. Under the circumstances, we can guarantee that different images are encrypted by diverse key streams, and the attackers cannot obtain serviceable information by selecting certain special images. Thus, our scheme can effectively resist the known plaintext attack and chosen plaintext attack.


### 5.10. MSE and PSNR Analysis

Mean square error (MSE) is a relatively straightforward method to measure the average error. When evaluating an encryption algorithm, verification is performed by comparing the MSE results between plain images and ciphered images, while also comparing the MSE results between plain images and deciphered images [37]. In addition, the larger the MSE value between plain images and their ciphers, the better the encryption effect. The definitions of MSE are as follows:(30)MSEPC=1M×N∑i=1M∑j=1NPi,j−Ci,j2,MSEPD=1M×N∑i=1M∑j=1NPi,j−Di,j2,where Pi,j denotes plain images, Ci,j represents ciphered images, and Di,j means deciphered images.Peak signal-to-noise ratio (PSNR) is utilized to appraise the distortion of images. In contrast to MSE, the smaller the PSNR value between plain images and their ciphers, the greater the difference. The definitions of PSNR are as follows:(31)PSNRPC=20log10LMSEPC,PSNRPD=20log10LMSEPD,where L is the maximum gray value. Table 20 shows the MSE and PSNR results between some plain images and their ciphered images. Obviously, the MSE results are quite large, and the PSNR values are pretty small. Moreover, the average MSE and PSNR values for the three channels of the Lena image with the size of 512×512 are 10116.3 and 8.0935, respectively. They are better than the test results (MSE=9081.2 and PSNR=8.2203) of the Lena image in [37]. In theory, the decrypted image is identical to the original image, thus the MSE value is 0 and the PSNR value is infinite. It can be found from Table 21 that the values of the test are consistent with the theoretical situation.Table 20
MSE and PSNR values between plain and ciphered images.
Images
MSE
PSNR
Red
Green
Blue
Red
Green
Blue
Female256×256
10081
12935
13536
8.0959
7.0131
6.8160
Peppers256×256
8087.8
11309
11211
9.0525
7.5967
7.6344
Mandrill256×256
8698.6
7871.9
9603.6
8.7363
9.1700
8.3065
Lena256×256
10791
9139.3
1067.2
7.8004
8.5217
7.8483
Lena512×512
10683
9035.8
10630
7.8440
8.5711
7.8655
Tree256×256
8801.6
11387
9718.1
8.6852
7.5667
8.2550
Couple256×256
14005
16095
16246
6.6680
6.0638
6.0233Table 21
MSE and PSNR values between plain and deciphered images.
Images
MSE
PSNR
Red
Green
Blue
Red
Green
Blue
Female256×256
0
0
0
∞
∞
∞
Peppers256×256
0
0
0
∞
∞
∞
Mandrill256×256
0
0
0
∞
∞
∞
Lena256×256
0
0
0
∞
∞
∞
Lena512×512
0
0
0
∞
∞
∞
Tree256×256
0
0
0
∞
∞
∞
Couple256×256
0
0
0
∞
∞
∞


### 5.11. Time Complexity Analysis

In order to analyze the time cost, we elaborate on the computational complexity. In the proposed scheme, the computational cost is relevant to the encryption steps. First, the iteration of the SLM, CLM, and SCM will produce the time complexity ofΟ3×m×n. Then, the time complexity of pixel-level scrambling is also Ο3×m×n. Subsequently, the time complexity of the sequence conversions is Ο6×m×n. After DNA coding, the time complexity of the DNA calculation determined by the sequence generated by the SLM is Ο3×m×n. Analogously, the time complexity of decoding DNA matrixes is Ο3×m×n. Ultimately, the time complexity of multiplication over the Galois field is Ο3×m×n. Therefore, the total time complexity of our scheme is Ο6×m×n. Furthermore, it can be seen from the time complexity results in Table 22 that although our scheme has higher time complexity than the encryption algorithms in [36], its computational complexity is lower than that of the algorithms in [45] and [46]. Thus, it can be proved that our scheme is effective.Table 22
Comparison of the time complexity of different algorithms.
Algorithms
Time complexity
The proposed scheme
6
mn
Ref. [36]
4 mn
Ref. [45]
8
mn
Ref. [46]
24
mn


## 5.1. Differential Attack Analysis

In general, the intensity of sensitivity to the plain image can determine the ability of the algorithm to resist differential attack. The measured indicators are the number of pixels change rate (NPCR) and the unified average changing intensity (UACI), which can be defined as follows:(20)NPCRR,G,B=1M×N∑i=1M∑j=1NER,G,Bi,j×100%,UACIR,G,B=1M×N∑i=1M∑j=1NCR,G,Bi,j−CR,G,B′i,j255×100%,where M and N are the width and height of the image and C and C′ are two cipher images whose plain images have only one different pixel. Besides, ER,G,Bi,j is used to determine the distinction between two ciphered images, which can be calculated by(21)ER,G,Bi,j=0,CR,G,Bi,j=CR,G,B′i,j,1,CR,G,Bi,j≠CR,G,B′i,j.Nevertheless, Zhang [31] pointed out that these two indexes cannot precisely measure the difference between two images. He proposed block average changing intensity (BACI) to quantitatively analyze the antidifferential attack characteristic. It is a method of block calculation about subtraction images, and each small block can be defined by (22)Bi=bi1bi2bi3bi4.Moreover, the mean for the absolute values of differences between arbitrary two elements can be defined as(23)mi=16bi1−bi2+bi1−bi3+bi1−bi4+bi2−bi3+bi2−bi4+bi3−bi4.Then, the BACI can be calculated by(24)BACIR,G,B=1M−1N−1∑i=1M−1N−1mi255.Thus, the NPCR, UACI, and BACI are used together to analyze the sensitive property of our encryption scheme, and the results are revealed in Table7. It can be found that the measured values of six different color images are very close to the theoretical values 99.6094%, 33.4635%, and 26.7712%, respectively. In particular, Table 7 also compares the NPCR and UACI scores on the Lena, Mandrill, and Peppers images using our scheme with algorithms in [29, 30]. This further demonstrates that our cryptosystem has excellent ability to resist differential attack.Table 7
Plaintext sensitivity test for different color images.
Algorithms
Images
Channels
NPCR (%)
UACI (%)
BACI (%)
The proposed scheme
Female
Red
99.6109
33.5025
26.8215
Green
99.6039
33.4360
26.7813
Blue
99.6149
33.4430
26.7754
Peppers
Red
99.6005
33.4494
26.7600
Green
99.6278
33.4537
26.7353
Blue
99.6188
33.4529
26.7651
Mandrill
Red
99.5941
33.4530
26.7913
Green
99.6199
33.4472
26.7685
Blue
99.6091
33.4596
26.7405
Lena256×256
Red
99.6109
33.4783
26.7618
Green
99.6213
33.4503
26.7713
Blue
99.6004
33.4640
26.7794
Tree256×256
Red
99.6062
33.4911
26.7929
Green
99.5996
33.4628
26.7709
Blue
99.6191
33.4522
26.7487
Couple256×256
Red
99.6149
33.4617
26.7908
Green
99.6083
33.4563
26.7618
Blue
99.6114
33.4533
26.7512
Ref. [29]
Lena256×256
Red
99.5800
33.2700
—
Green
99.5600
33.3600
—
Blue
99.6400
33.5000
—
Mandrill256×256
Red
99.6200
33.4700
—
Green
99.6000
33.4800
—
Blue
99.6000
33.4500
—
Peppers512×512
Red
99.6000
33.3900
—
Green
99.6000
33.4600
—
Blue
99.6100
33.4000
—
Ref. [30]
Lena256×256
Red
99.6317
33.6783
—
Green
99.6205
33.7999
—
Blue
99.6211
33.6200
—
Mandrill256×256
Red
99.6199
33.6484
—
Green
99.6250
33.5908
—
Blue
99.6273
33.6749
—
Peppers256×256
Red
99.6202
33.6602
—
Green
99.6192
33.6575
—
Blue
99.6224
33.7314
—


## 5.2. Key Space

As mentioned earlier in this section, the secret keys for our scheme consist of three control parametersr1,r2,r3 and three initial values X10,X20,X30, which are all double-precision real numbers. In this paper, we use the 64-bit double-precision format in [32] to calculate our key space, and the total key space of our scheme can reach 10156=1090≈2299. In addition, Table 8 exhibits the comparison of key space between our scheme and other chaos-based encryption algorithms. It can be seen that the key space of our method is large enough to resist brute-force attack.Table 8
Comparison of the key space between our scheme and other chaos-based encryption algorithms.
Encryption algorithms
Key space
The proposed scheme
10
15
6
=
10
90
≈
2
299
Ref. [8]
2
106
Ref. [15]
10
84
≈
2
279
Ref. [18]
2
138
Ref. [33]
2
298
Ref. [34]
>
2
208


## 5.3. Key Sensitivity

The key sensitivity refers to the degree of variation for the corresponding ciphered image when the initial key alters slightly. Make a minor change toX10 while keeping other keys unchanged. For instance, X10′=X10+10−15. This produces a new key set K1. It is applied to encrypt the plain images in Figure 5(a), and the ciphered images in Figure 6(a) are displayed. The pixel-by-pixel differences between Figures 5(b) and 6(a) are revealed in Figure 6(b), which proves that their encrypted images are completely different. Moreover, K0 and K1 are utilized to decrypt the ciphered images in Figure 5(b), respectively. The decrypted images in Figure 6(c) can be acquired with the correct key set K0, and a minor change to the security keys will cause the failure of decryption, as shown in Figure 6(d).Figure 6
Key sensitivity test: (a) encrypted images with the key setK1; (b) the pixel-by-pixel difference between Figures 5(b) and 6(a); (c) decrypted images from Figure 5(b) with the correct key set K0; (d) decrypted images from Figure 5(b) with the wrong key set K1.
(a)
(b)
(c)
(d)In addition, Table9 lists the average NPCR and UACI results for the three channels of the key sensitivity test for the plain images in Figure 5(a). It can be found that a slight change in each key will result in the change rate of more than 99.5% for the ciphered images. Their corresponding average NPCR and UACI results are 99.6124% and 33.4714%, which are closer to ideal than the values (NPCR = 99.6389% and UACI = 33.4189) in [35]. Thus, the proposed scheme has high key sensitivity.Table 9
NPCR and UACI results for key sensitivity test.
Modified keys
Average NPCR (%)
Average UACI (%)
Female
Peppers
Mandrill
Female
Peppers
Mandrill
X
10
+
10
−
15
99.6132
99.6228
99.6164
33.4596
33.4677
33.4828
X
20
+
10
−
15
99.6088
99.6098
99.6062
33.4641
33.4606
33.4656
X
30
+
10
−
15
99.6112
99.6143
99.6124
33.4613
33.4860
33.4734
r
1
+
10
−
15
99.6191
99.6115
99.6072
33.4762
33.4866
33.4674
r
2
+
10
−
15
99.6124
99.6104
99.6120
33.4805
33.4678
33.4741
r
3
+
10
−
15
99.6109
99.6156
99.6089
33.4691
33.4764
33.4665


## 5.4. Histogram Analysis

The image histogram is an important feature statistic of the image. A great encryption algorithm theoretically makes the histogram of the encrypted image evenly distributed. This prevents the adversary from getting any useful information. Figure7 exhibits the histograms of the plain and ciphered Female, Peppers, and Mandrill images. It can be seen that the histograms of the ciphered images are fairly flat, and they are completely different from the histograms of the plain images.Figure 7
Histograms of the (a) plain images in Figure5(a); (b) ciphered images in Figure 5(b).
(a)
(b)Besides, the analysis of variance is a quantitative measure for the properties of histogram. The smaller the calculated variance value, the higher the uniformity of the image [36]. The specific calculation formula is as follows:(25)varianceY=1n2∑i=1n∑j=1nyi−yj22,where yi and yj indicate the number of pixels corresponding to the gray values i and j, Y represents the vector set of the histogram and Y=y1,y2,…,y256, and n denotes the total number of gray values. Table 10 exhibits the variances of the histograms of the color plain and ciphered images in Figure 5. It can be discovered that the histogram variance values of the encrypted images are much smaller than those of the original images. Moreover, Table 11 displays the comparison of the mean histogram variance values of the three channels for the Lena image and the calculated variance results in [26, 36–38]. Obviously, the histogram variance values of our scheme are smaller, that is, the histograms of the ciphered images have better uniformity.Table 10
Histogram variance analysis of plain and ciphered images.
Images
Plain images
Ciphered images
Red
Green
Blue
Red
Green
Blue
Female
168960
157950
158040
233.4375
263.3125
267.3516
Peppers
54484
64353
106880
244.1172
258.2734
223.9297
Mandrill
20770
30655
16857
220.3125
240.2109
270.2656Table 11
Histogram variance analysis of original Lena image and encrypted Lena image.
Images
Algorithms
Variance
Original Lena image (512×512)
6390323
Encrypted Lena image
The proposed scheme
953.7656
Ref. [26]
977.02
Ref. [36]
974.8
Ref. [37]
1077.3
Ref. [38]
1209.4Then, we utilize the chi-square test to validate the uniformity for the pixel values distribution of the ciphered images [36]. Similar to the variances of the histograms, the smaller the result of the chi-square test, the better the uniformity of the ciphered image. Its definition is as follows:(26)χ2=∑i=0255fi−pi2pi,where fi indicates the actual frequency of the pixel value i and pi represents the predicted frequency of the pixel value i. The predicted frequency pi can be calculated by(27)pi=m×n256,where m×n is the size of the image. Theoretically, the chi-square statistic with a significant level of 0.01 is 310.4574, while the chi-square statistic with a significant level of 0.05 is 293.2478. Obviously, the values of the chi-square test for six ciphered images in Table 12 are all smaller than the previous two theoretical values. Thus, it further illustrates the effectiveness of our scheme.Table 12
Chi-square test of some ciphered images.
Images
Components
Results
Red
Green
Blue
Female
233.4375
253.3125
247.3516
Pass
Peppers
244.1172
238.2734
223.9297
Pass
Mandrill
220.3125
240.2109
240.2656
Pass
Couple (256×256)
234.2500
246.0313
233.3906
Pass
Tree (256×256)
222.5781
246.3047
235.5313
Pass
Lena (512×512)
245.8965
230.9863
252.1680
Pass
Airplane (512×512)
240.7422
257.8555
235.8145
Pass


## 5.5. Correlation of Two Adjacent Pixels

As for plain images, there are strong correlations between their adjacent pixels. Thus, the purpose of encryption is to weaken this property. In the experiment, we randomly select 4000 pairs of adjacent pixels from the original images and corresponding encrypted images. Then, compute the correlation coefficients in horizontal, vertical, and diagonal directions as follows:(28)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,COVx,y=1N∑i=1Nxi−Exyi−Ey,ρxy=COVx,yDxDy,where Ex,Dx, and COVx,y indicate the mean value, variance, and covariance, respectively. Figure 8 plots the correlations between adjacent pixels of the plain Peppers image, and Figure 9 shows the correlation coefficients of the corresponding ciphered image. It can be seen from the comparison that the encryption makes the correlation between the pixels of the image significantly whittled. Moreover, Table 13 provides the correlation coefficients for the adjacent pixels of six color images with the size of 256×256 in three directions. It can be discovered that the results of the ciphered images are close to 0. Furthermore, Table 14 reveals that our encryption scheme has superior performance compared to some recent encryption algorithms.Figure 8
Correlation coefficients of plain Peppers image in all directions: (a) R channel; (b) G channel; (c) B channel.
(a)
(b)
(c)Figure 9
Correlation coefficients of ciphered Peppers image in all directions: (a) R channel; (b) G channel; (c) B channel.
(a)
(b)
(c)Table 13
Average correlation coefficients of three channels for plain and ciphered images.
Images
Plain images
Ciphered images
Horizontal
Vertical
Diagonal
Horizontal
Vertical
Diagonal
Female
0.9747
0.9621
0.9511
0.0078
0.0021
0.0004
Peppers
0.9647
0.9701
0.9602
0.0004
−0.0035
−0.0021
Mandrill
0.9188
0.8530
0.8534
−0.0031
0.0048
0.0005
Tree
0.9406
0.9551
0.9157
−0.0012
−0.0046
0.0006
Couple
0.9131
0.9011
0.9375
0.0074
0.0035
0.0013
House
0.9328
0.9676
0.9114
−0.0027
0.0047
0.0012Table 14
Correlation coefficients of original Lena image and encrypted Lena image.
Images
Algorithms
Directions
Horizontal
Vertical
Diagonal
Original Lena image256×256
0.9731
0.9881
0.9661
Encrypted Lena image
The proposed scheme
−0.0034
0.0021
−0.0003
Ref. [14]
0.0056
0.0065
−0.0073
Ref. [19]
0.0036
0.0023
0.0039
Ref. [33]
−0.0068
−0.0054
0.0010
Ref. [39]
0.0040
0.0011
0.0008
Ref. [40]
0.0059
−0.0042
0.0180
Ref. [41]
−0.0168
0.0445
−0.0022
Ref. [42]
−0.0003
−0.0013
−0.0066


## 5.6. Information Entropy Analysis

Information entropy is expressed as the probability of discrete random events, which is used to measure the randomness of a system. It can be defined as(29)Hs=−∑i=02n−1Psilog2Psi,where Psi denotes the probability that the gray value si may appear. For a grayscale image with data field of 0,255, the maximum value of information entropy is 8. Therefore, once the calculated value is pretty close to 8, it proves that the proposed algorithm is quite safe. We have computed the information entropy of six different color images with the size of 256×256 and their ciphered versions in Table 15. From the table, the information entropy results of ciphered images are approaching the ideal value 8. In addition, Table 16 exhibits the information entropy of the original Lena image and encrypted Lena image by utilizing our scheme and some other encryption algorithms. It can be clearly seen that our cryptosystem is closer to the desired state and has better randomness.Table 15
Information entropy analysis of plain and ciphered images.
Images
Plain images
Ciphered images
Red
Green
Blue
Red
Green
Blue
Female
6.4200
6.4457
6.3807
7.9974
7.9971
7.9971
Peppers
7.3449
7.5607
7.1003
7.9973
7.9972
7.9975
Mandrill
7.7255
7.5618
7.8031
7.9976
7.9974
7.9970
Tree
7.2104
7.4136
6.9207
7.9976
7.9971
7.9970
Couple
6.2499
5.9642
5.9309
7.9972
7.9973
7.9972
House
6.4311
6.5389
6.2320
7.9975
7.9968
7.9973Table 16
Comparison of the information entropy of the original Lena image and encrypted Lena image.
Information entropy
Red
Green
Blue
Original Lena image (256×256)
7.3140
7.6394
7.0506
The proposed algorithm
7.9971
7.9972
7.9972
Ref. [39]
7.9278
7.9744
7.9705
Ref. [41]
7.9897
7.9877
7.9896
Ref. [43]
7.9988
7.9967
7.9990
Ref. [44]
7.9791
7.9802
7.9827


## 5.7. Noise Attack Analysis

In practical applications, noise interference is inevitable. An outstanding encryption algorithm has the ability to resist noise attack. In our experiment, the encrypted Mandrill image in Figure5(b) is contaminated by Salt & Pepper noise, Gaussian noise, and speckle noise with different densities, respectively. The simulation results are exhibited in Figures 10–12. It can be observed that although the noise is increasing, the decrypted images can still be discerned. Moreover, we also tested the average NPCR and UACI values for the three channels of the original Mandrill image and deciphered Mandrill image under these different noises. The results are listed in Tables 17–19. It can be found that all the NPCR values are less than 99%, and all the UACI values are less than 20%. Hence, the proposed method is strongly robust against disparate noises.Figure 10
Experiment of Salt & Pepper noise attacks: (a) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.05; (b) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.1; (c) encrypted Mandrill image attacked by Pepper & Salt noise with density 0.2; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Experiment of Gaussian noise attacks: (a) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0001; (b) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0003; (c) encrypted Mandrill image attacked by Gaussian noise with mean value 0 and variance value 0.0005; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Figure 12
Experiment of speckle noise attacks: (a) encrypted Mandrill image attacked by speckle noise with density 0.00005; (b) encrypted Mandrill image attacked by speckle noise with density 0.0001; (c) encrypted Mandrill image attacked by speckle noise with density 0.0005; (d) decrypted Mandrill image of a; (e) decrypted Mandrill image of b; (f) decrypted Mandrill image of c.
(a)
(b)
(c)
(d)
(e)
(f)Table 17
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under Salt & Pepper noise.
Images
Density
Average NPCR
Average UACI
Figures5(a) and 10(d)
0.05
26.0681
7.7270
Figures5(a) and 10(e)
0.1
46.2418
13.6364
Figures5(a) and 10(f)
0.2
62.4237
18.5213Table 18
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under Gaussian noise.
Images
Mean
Variance
Average NPCR
Average UACI
Figures5(a) and 11(d)
0
0.0001
98.4116
8.8154
Figures5(a) and 11(e)
0
0.0003
98.7732
11.7899
Figures5(a) and 11(f)
0
0.0005
98.9624
12.7496Table 19
NPCR and UACI values for the original Mandrill image and deciphered Mandrill image under speckle noise.
Images
Density
Average NPCR
Average UACI
Figures5(a) and 12(d)
0.00005
94.5068
8.5631
Figures5(a) and 12(e)
0.0001
95.5780
11.3172
Figures5(a) and 12(f)
0.0005
97.4060
19.3016


## 5.8. Cropping Attack Analysis

Encrypted images are subject to cropping attack during transmission and may be partially damaged. Nevertheless, digital images allow a certain extent of distortion on the transmission channel. As long as the information to be conveyed by the image can be discriminated visually, it proves that the encryption algorithm has excellent anticropping attack capability. Figure13 displays ciphered House image of size 256×256 with data cuts in different sizes, and their corresponding deciphered images. Still, most of the pictorial information is available from the decrypted images. This shows that the proposed scheme can effectively resist cropping attack.Figure 13
Test of image cropping attacks: (a) original House image; (b) encrypted House image; (c) encrypted House image with a 30 × 90 data cut; (d) decrypted House image of (c); (e) encrypted House image with a 50 × 80 data cut; (f) decrypted House image of (e); (g) encrypted House image with a 70 × 70 data cut; (h) decrypted House image of (g); (i) encrypted House image with a 90 × 60 data cut; (j) decrypted House image of (i); (k) encrypted House image with a 110 × 50 data cut; (l) decrypted House image of (k).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)


## 5.9. Known Plaintext Attack and Chosen Plaintext Attack Analysis

Since the cryptosystems whose key streams are unrelated to the plain images are vulnerable to chosen plaintext attack and known plaintext attack [21], we design the initial keys of our scheme to be updated by the bit-planes recombination of the plain images. Under the circumstances, we can guarantee that different images are encrypted by diverse key streams, and the attackers cannot obtain serviceable information by selecting certain special images. Thus, our scheme can effectively resist the known plaintext attack and chosen plaintext attack.


## 5.10. MSE and PSNR Analysis

Mean square error (MSE) is a relatively straightforward method to measure the average error. When evaluating an encryption algorithm, verification is performed by comparing the MSE results between plain images and ciphered images, while also comparing the MSE results between plain images and deciphered images [37]. In addition, the larger the MSE value between plain images and their ciphers, the better the encryption effect. The definitions of MSE are as follows:(30)MSEPC=1M×N∑i=1M∑j=1NPi,j−Ci,j2,MSEPD=1M×N∑i=1M∑j=1NPi,j−Di,j2,where Pi,j denotes plain images, Ci,j represents ciphered images, and Di,j means deciphered images.Peak signal-to-noise ratio (PSNR) is utilized to appraise the distortion of images. In contrast to MSE, the smaller the PSNR value between plain images and their ciphers, the greater the difference. The definitions of PSNR are as follows:(31)PSNRPC=20log10LMSEPC,PSNRPD=20log10LMSEPD,where L is the maximum gray value. Table 20 shows the MSE and PSNR results between some plain images and their ciphered images. Obviously, the MSE results are quite large, and the PSNR values are pretty small. Moreover, the average MSE and PSNR values for the three channels of the Lena image with the size of 512×512 are 10116.3 and 8.0935, respectively. They are better than the test results (MSE=9081.2 and PSNR=8.2203) of the Lena image in [37]. In theory, the decrypted image is identical to the original image, thus the MSE value is 0 and the PSNR value is infinite. It can be found from Table 21 that the values of the test are consistent with the theoretical situation.Table 20
MSE and PSNR values between plain and ciphered images.
Images
MSE
PSNR
Red
Green
Blue
Red
Green
Blue
Female256×256
10081
12935
13536
8.0959
7.0131
6.8160
Peppers256×256
8087.8
11309
11211
9.0525
7.5967
7.6344
Mandrill256×256
8698.6
7871.9
9603.6
8.7363
9.1700
8.3065
Lena256×256
10791
9139.3
1067.2
7.8004
8.5217
7.8483
Lena512×512
10683
9035.8
10630
7.8440
8.5711
7.8655
Tree256×256
8801.6
11387
9718.1
8.6852
7.5667
8.2550
Couple256×256
14005
16095
16246
6.6680
6.0638
6.0233Table 21
MSE and PSNR values between plain and deciphered images.
Images
MSE
PSNR
Red
Green
Blue
Red
Green
Blue
Female256×256
0
0
0
∞
∞
∞
Peppers256×256
0
0
0
∞
∞
∞
Mandrill256×256
0
0
0
∞
∞
∞
Lena256×256
0
0
0
∞
∞
∞
Lena512×512
0
0
0
∞
∞
∞
Tree256×256
0
0
0
∞
∞
∞
Couple256×256
0
0
0
∞
∞
∞


## 5.11. Time Complexity Analysis

In order to analyze the time cost, we elaborate on the computational complexity. In the proposed scheme, the computational cost is relevant to the encryption steps. First, the iteration of the SLM, CLM, and SCM will produce the time complexity ofΟ3×m×n. Then, the time complexity of pixel-level scrambling is also Ο3×m×n. Subsequently, the time complexity of the sequence conversions is Ο6×m×n. After DNA coding, the time complexity of the DNA calculation determined by the sequence generated by the SLM is Ο3×m×n. Analogously, the time complexity of decoding DNA matrixes is Ο3×m×n. Ultimately, the time complexity of multiplication over the Galois field is Ο3×m×n. Therefore, the total time complexity of our scheme is Ο6×m×n. Furthermore, it can be seen from the time complexity results in Table 22 that although our scheme has higher time complexity than the encryption algorithms in [36], its computational complexity is lower than that of the algorithms in [45] and [46]. Thus, it can be proved that our scheme is effective.Table 22
Comparison of the time complexity of different algorithms.
Algorithms
Time complexity
The proposed scheme
6
mn
Ref. [36]
4 mn
Ref. [45]
8
mn
Ref. [46]
24
mn


## 6. Conclusions

In this paper, we introduce a novel chaotic color image encryption scheme based on DNA coding calculations and arithmetic over the Galois field. Firstly, three 1D chaotic maps with better chaotic properties are obtained by improving the classical 1D chaotic maps, and we use them as the secret keys for the cryptosystem. Meanwhile, the application of plain images to update the initial values protects our scheme from the threat of chosen plaintext attack and known plaintext attack. In order to increase the degree of diffusion, the scheme also adds the coding calculations of DNA sequences and multiplication over the Galois fields GF(17). At last, the simulation results verify that the proposed algorithm has excellent performance. The future work is to research the fast encryption scheme based on chaos, which can be applied in real-time communication scenarios such as telemedicine.



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*Source: 3965281-2020-03-09.xml*