# Image Encryption Scheme Based on Block Scrambling, Closed-Loop Diffusion, and DNA Molecular Mutation

Authors: Li-Hua Gong; Jin Du; Jing Wan; Nan-Run Zhou
Journal: Security and Communication Networks
 (2021)

---

## Abstract

A new image encryption scheme is proposed with a combination of block scrambling, closed-loop diffusion, and DNA molecular mutation. The new chaotic block scrambling mechanism is put forward to replace the traditional swapping rule by combining the rectangular-ambulatory-plane cyclic shift with the bidirectional random disorganization. The closed-loop diffusion strategy is designed to form a feedback system, which improves the anti-interference capacity of the algorithm. To further destroy the blocks characteristics and eliminate the correlations among adjacent blocks, two efficient methods of DNA molecular mutation are adopted in the mutation stage. Moreover, the proposed algorithm possesses a large key space and the keys are highly related with the plaintext image. Experimental results demonstrate that the suggested image encryption strategy is practicable and has strong ability against a variety of common attacks.


---

## Body

## 1. Introduction

Lots of images travel over the Internet due to the shareability and the openness of network transmission, which may threaten the security of private image information. To tackle this problem, various image encryption algorithms have been put forward successively [1–4]. The main purpose of these algorithms is to encrypt the serviceable image information into a noise-like one. For example, Li devised an image encryption algorithm with a chaotic tent map, since the key stream generated with the modified chaotic sequence was more suitable for encryption [5]. To achieve lower computing consumption and increase efficiency on scrambling, Wang provided a fast color image encryption with correlated logistic map [6]. Nevertheless, the key distribution of the low-dimensional chaos is nonuniform, and the generated sequence is also unstable. Thus, some high-dimensional chaotic systems emerged [7, 8] and a new hyperchaotic system was explored [9]. It is capable of generating chaotic attractors with multiring and multiwing, which enables it to have complex dynamic behavior. Besides, hyperchaotic systems have a wider chaotic range and better ergodicity, which are extremely profitable for image encryption. It might be an ideal strategy to adopt multiple chaotic systems on a single image encryption system [10–12]. To effectively apply the advantages of composite chaotic systems and inherit their randomness, it is necessary to design the frameworks of confusion and diffusion carefully.Scrambling could destroy the strong correlation among adjacent pixels. Quite a few image encryption algorithms sorted the random sequences generated from chaotic systems, and then the positions of the image pixels are rearranged with a group of index numbers. However, simply mapping the index into the ciphertext image pixels one by one might ignore some fixed points [13]. Furthermore, it has been pointed out that the periodicity and the inefficiency of traditional permutation architecture may not meet the security requirements [14–16]. Consequently, it is necessary to further study scrambling strategy. Shahna gave a permutation strategy on both pixel level and bit level, which makes the encryption algorithm complicated and realizes high-performance random permutation [17]. Xian constructed a fractal sorting matrix to perturb the elements in the plaintext image [18]. The disordered sorting matrix could be flexibly derived from an initial block to any size, and it was safer in image encryption. To complicate the scrambling process, Wang split the plaintext image into four sub-blocks and executed various degrees of Arnold transform [19]. In addition, the statistical law of pixel values could be changed by an effective diffusion algorithm [20, 21]. Gong adopted the chaotic sequences to XOR the compressed plaintext directly [22]. The image might be partially revealed because the decryption between unrelated pixels would not affect each other [23]. Thus, Mirzaei suggested a new diffusion method in the cryptosystem, where the plaintext image was split into 4 subimages. Each encrypted pixel of the previous subimage participated in the operation of the next subimage pixel [24]. To hide the statistical structure of plaintext more effectively, Sheela alternately employed the two-level diffusion operations and the pixels transform in image encryption system. In the first diffusion level, each pixel was processed by two points relative to its position in the chaos matrix, while the next level was handled by the former ciphered pixel [25]. Although the diffusion operations were more complex, a tiny alteration of the pixel could only affect the latter ones [14, 26]. Once the diffusion sequence was obtained, the ciphered pixels could be decrypted in a reverse order.There were also numerous image encryption algorithms combining chaos theory with DNA computing [27–32]. The kernel of these image encryption algorithms was to encode the image pixels with the DNA encoding rules and then to perform biological and algebraic manipulations on the encoded sequence. These manipulations include DNA addition and subtraction and base complementation rules. For instance, Jian encoded the plaintext image and the generated mask by the logistic map, and then they were added up with a DNA addition rule. Subsequently, a complement matrix was utilized to execute the base complementation rule, and the final encryption image was obtained with a DNA decoding rule [27]. It might be more beneficial to combine DNA with Chen’s hyperchaotic system in an image composite encryption mechanism [28]. Nevertheless, the DNA computing rules used in the image encryption algorithm were not well designed. Even if the algorithm was very sensitive, its security would still be questionable under the passive attack, and the cryptanalyses discussed in [29, 30] proved that the algorithm in [28] was vulnerable to the chosen-plaintext attack. There are two main security issues with generally used manipulations. One is that all DNA manipulations are based on the binary calculation, and the coding results might be easily predicted with four DNA bases. The other is that DNA coding rules are fixed, which is not conducive to the security of image encryption algorithms. Thus, Yu expatiated the deletion and insertion operations of DNA-based image encryption to update the computational pattern [31], and two images were regarded as keys to each other, which strengthens the image cryptosystem security. Yang explained three DNA mutation operations on the 12 layers of DNA molecules [32]. The improved Lorenz sequences were employed to operate the interlayer and intramolecular mutations, so that the final mutation results are random and unpredictable.In this paper, a new chaotic block scrambling mechanism will be investigated, which contains rectangular-ambulatory-plane cyclic shift and bidirectional random disorganization. The rings of each sub-block could be dynamically managed during cyclic shift, and pixels could be randomly selected and inserted during bidirectional random disorganization. In addition, the closed-loop diffusion strategy could form a feedback mechanism among the key block, the plaintext image, and the ciphertext image. To further alter the features of the ciphertext blocks, two kinds of DNA molecular mutation rules based on the theory of biological variation are adopted.The structure of the remaining parts in this paper is as follows: some fundamental tools are explained in Section2. In Section 3, the key generation, the block scrambling and the closed-loop diffusion algorithm, and the entire image encryption process are dwelled on. Section 4 provides simulation results and performance evaluations. A brief conclusion is drawn in Section 5.


## 2. Fundamental Knowledge

### 2.1. Affine Transform

Affine transform is a linear transform in the two-dimensional coordinates, which can extend/retract the image to any angle and direction. The general type of affine transform is(1)x′=a1x+a2y+b1,y′=a3x+a4y+b2,where x,y is the initial coordinate, and x′,y′ is the coordinate after transform. a1, a2, a3, .., b1, and b2 are the parameters of affine transform.


### 2.2. Chaotic Systems

2D logistic-sine-coupling map (2D-LSCM) was designed by combining logistic map with sine map to enhance the complexity of the chaotic behavior, where the control parameterδ belongs to 0,1 and the original position at xn,yn is updated to the new position at xn+1,yn+1,(2)xn+1=sinπ4δxn1−xn+1−δsinπyn,yn+1=sinπ4δyn1−yn+1−δsinπxn+1.2D logistic-adjusted-sine map (2D-LASM) is an integration of two 1D sine logistic modulation maps [33]. Its system parameter γ ranges from 0 to 1.(3)xn+1=sinγπxn1−xnyn+3,yn+1=sinγπyn1−ynxn+1+3.Henon map is a dynamical chaotic system in discrete-time, as described in equation (4). If the parameters λ1 and λ2 of Henon map are 1.4 and 0.3, respectively, it turns into a chaotic state,(4)xn+1=1−λ1xn2+yn,yn+1=λ2yn.


### 2.3. Hilbert Curve

As a space filling curve, Hilbert curve could be utilized as a scan tool to scan the whole points on the2n×2n plane through quartering continuously [34]. The scan path starts from the right bottom, via the right top and the left top and ends at the left bottom of each square. Hilbert curve is a good shuffling tool to obtain a scrambled image. Figure 1 shows the Hilbert curves, which are drawn in the blocks of size 22, 42 and 82, respectively.Figure 1
Hilbert curve of orders 1, 2, and 3, respectively. (a) Order 1. (b) Order 2. (c) Order 3.
(a)(b)(c)


### 2.4. DNA Encoding Rules

The complementary pairing rule between the four nucleobases in DNA is analogous to the complementation of 0 and 1 in the binary system. If each nucleobase is represented by a two-digit binary, there will be 24 kinds of encoding rules. Since the limitations of DNA complementation rules should be taken into account, only 8 of them are acceptable. The 8 encoding rules are recorded in Table1.Table 1
DNA encoding and decoding methods.
Rule1234567800AATTGGCC01CGCGTATA10GCGCATAT11TTAACCGG


### 2.5. Mutation in DNA Molecules

Two kinds of DNA variation rules based on gene mutation are investigated to destroy the statistical law of images. The first one is the dynamic point substitution according to the rules of base transversion and base transition. A random sequence is employed to construct the mutation environment and the rule is listed in Table2. The point substitution could only be performed when the random value is in the range from 0 to 3. The other DNA mutation is the cross mutation among adjacent chains, where the starting point of the exchange is random. Figure 2 illustrates the interchange pattern when the starting point is in the middle of the exchange chain.Table 2
Points substitution rule.
OperationBase transitionBase transversionFixedRandom sequence0123ElseAAGCAATCTTGTGGAGTGCTCACCFigure 2
Cross-mutation among DNA chains.


## 2.1. Affine Transform

Affine transform is a linear transform in the two-dimensional coordinates, which can extend/retract the image to any angle and direction. The general type of affine transform is(1)x′=a1x+a2y+b1,y′=a3x+a4y+b2,where x,y is the initial coordinate, and x′,y′ is the coordinate after transform. a1, a2, a3, .., b1, and b2 are the parameters of affine transform.


## 2.2. Chaotic Systems

2D logistic-sine-coupling map (2D-LSCM) was designed by combining logistic map with sine map to enhance the complexity of the chaotic behavior, where the control parameterδ belongs to 0,1 and the original position at xn,yn is updated to the new position at xn+1,yn+1,(2)xn+1=sinπ4δxn1−xn+1−δsinπyn,yn+1=sinπ4δyn1−yn+1−δsinπxn+1.2D logistic-adjusted-sine map (2D-LASM) is an integration of two 1D sine logistic modulation maps [33]. Its system parameter γ ranges from 0 to 1.(3)xn+1=sinγπxn1−xnyn+3,yn+1=sinγπyn1−ynxn+1+3.Henon map is a dynamical chaotic system in discrete-time, as described in equation (4). If the parameters λ1 and λ2 of Henon map are 1.4 and 0.3, respectively, it turns into a chaotic state,(4)xn+1=1−λ1xn2+yn,yn+1=λ2yn.


## 2.3. Hilbert Curve

As a space filling curve, Hilbert curve could be utilized as a scan tool to scan the whole points on the2n×2n plane through quartering continuously [34]. The scan path starts from the right bottom, via the right top and the left top and ends at the left bottom of each square. Hilbert curve is a good shuffling tool to obtain a scrambled image. Figure 1 shows the Hilbert curves, which are drawn in the blocks of size 22, 42 and 82, respectively.Figure 1
Hilbert curve of orders 1, 2, and 3, respectively. (a) Order 1. (b) Order 2. (c) Order 3.
(a)(b)(c)


## 2.4. DNA Encoding Rules

The complementary pairing rule between the four nucleobases in DNA is analogous to the complementation of 0 and 1 in the binary system. If each nucleobase is represented by a two-digit binary, there will be 24 kinds of encoding rules. Since the limitations of DNA complementation rules should be taken into account, only 8 of them are acceptable. The 8 encoding rules are recorded in Table1.Table 1
DNA encoding and decoding methods.
Rule1234567800AATTGGCC01CGCGTATA10GCGCATAT11TTAACCGG


## 2.5. Mutation in DNA Molecules

Two kinds of DNA variation rules based on gene mutation are investigated to destroy the statistical law of images. The first one is the dynamic point substitution according to the rules of base transversion and base transition. A random sequence is employed to construct the mutation environment and the rule is listed in Table2. The point substitution could only be performed when the random value is in the range from 0 to 3. The other DNA mutation is the cross mutation among adjacent chains, where the starting point of the exchange is random. Figure 2 illustrates the interchange pattern when the starting point is in the middle of the exchange chain.Table 2
Points substitution rule.
OperationBase transitionBase transversionFixedRandom sequence0123ElseAAGCAATCTTGTGGAGTGCTCACCFigure 2
Cross-mutation among DNA chains.


## 3. Proposed Image Encryption Scheme

### 3.1. Generation of Random Sequences

To possess an excellent capacity against the differential attack, the original values of chaotic systems are restricted with both the MD5 hash values and the parity quantization values of the plaintext image. The 32-bit hexadecimal key streamM generated from the MD5 hash function is divided into 16 groups, represented as ki, i=1,2,…,16,(5)ki=h2dM2i−1+1+M2i,where h2d⋅ converts a hexadecimal number into a decimal integer. The sum of ki is denoted as SUM=∑i=116ki. The parity property of the plaintext image is(6)u=floorabske−koe¯−o¯,where the numbers of even and odd integers in the plaintext image are represented as ke and ko, respectively; e¯ and o¯ are the mean values of the even and odd numbers, respectively. Then, the initial values are computed as(7)xj=u+k4j⊕k4j−2⊕u+k4j−1⊕k4j−3SUM,j=1,2,3,4,y1=∑j=16kj+uSUM,y2=∑j=16k2j+uSUM,where A⊕B means bitxorA,B. x1, x2, x3, x4, y1, and y2 are the initial values. The three chaotic systems are iterated with corresponding times and the former 1000 iteration elements are discarded to avert the so-called transient effect. The specific parameter values are collected in Table 3.Table 3
Parameter values of chaotic systems.
2D-LSCM2D-LASMHenon mapInitial valuesx1, x2x3, x4y1, y2System parametersδ=0.8μ=0.9λ1=1.4, λ2=0.3Resulting random sequencesXls, YlsXla, YlaHx, Hy


### 3.2. Block Scrambling and Closed-Loop Diffusion

The mechanisms of block scrambling and closed-loop diffusion are constructed to shuffle the preprocessed blocks and to diffuse the scrambled blocks in a linkage system. The whole process can be represented as(8)B=SDPH,Xls,Yls,Xla,Yla,where SD⋅ is the block scrambling and closed-loop diffusion function; Xls, Yls, Xla, and Yla are four random sequences; PH is the preprocessed plaintext block set and stored as a cell array of size n×n, where n is set to 16. The l−th sub-block in the cell array can be labeled as PHl. The flowchart of block scrambling and closed-loop diffusion is displayed in Figure 3. The specific process is given as follows:Step 1. Rectangular-ambulatory-plane cutting: the rings of each sub-block PHl are extracted and stored in OPHl. The x−th ring in the sub-block PHl can be represented as OxPHl, where x belongs to 1,2,…,8.Step 2. Four sequences r1, r2, r3, and r4 of length n2 are selected from Xls and the elements in r1, r2, r3, and r4 are all converted into integers in 0,255. The control sequence f of length 8n2 is updated from Xla. When the number in Xla is less than 0.5, it is updated to −1; otherwise, it is updated to 1.Step 3. The rectangular-ambulatory-plane cyclic shift operation is executed on each ring,(9)CxPHl=circshiftOxPHl,rx+1/2,f,x=1,3,5,7;circshiftOxPHl,xrx/2,f,x=2,4,6,8,wherecircshifta′,b′,c′ is a function for the bidirectional cyclic shift operation. a′ is a ring to be operated, b′ determines the shift times, and the ring OxPHl rotates clockwise when c′<0. CxPHl is the scrambled ring. After repeating the cyclic shift operation in each ring, a block set S is generated.Step 4. Yls and Yla are evenly divided into blocks, and then each sub-block of them is arranged in an ascending order to yield Yls′ and Yla′, respectively. The corresponding index block sets are D1 and D2.Step 5. Bidirectional random disorganization: each sub-block in S is scrambled by the double random block sets. The specific operation is(10)p′=dφD1l,p″=dφD2l,kp″Kl=sp′Sl,where1≤φ≤256; p′ and p″ correspond to the φ−th pixels in D1l and D2l, respectively; sp′Sl refers to the pixel extracted from the position p′ of Sl, and then all the extracted pixels are placed into a new block set K at a random position p″.Step 6. Two key block sets V1 and V2 in the range of 0,255 are generated with Yls′ and Yla′. The overall ciphertext sub-blocks in K are diffused by(11)gϕGl=kϕKl⊕kn2Kl⊕vϕV1l,ϕ=1,gϕ−1Gl⊕kϕKl⊕kϕ+1Kl⊕vϕV1l,ϕ=2,3,…,n2−1,gϕ−1Gl⊕kϕKl⊕vϕV1l,ϕ=n2,whereGl is the sub-block when all the pixels in the sub-block Kl are updated.Step 7. The diffusion operation between sub-blocks is subsequently executed. Then, the final ciphertext block set B is obtained after executing the closed-loop diffusion operation.(12)Bm=Gm⊕Gn2⊕V2m,m=1,Bm−1⊕Gm⊕Gm+1⊕V2m,m=2,3,…,n2−1,Bm−1⊕Gm⊕V2m,m=n2.Figure 3
Block scrambling and closed-loop diffusion.There exist some highlights in the above steps. First, the key blocks are associated with the plaintext image, and the proposed rectangular-ambulatory-plane cyclic shift takes full advantages of the chaotic sequences’ randomness and scrambles each block efficiently. Also, the double random blocks mapping rule could attain the pixels extraction and insertion randomly at the same time. Ultimately, the blocks are related to each other after being diffused. The whole confusion and diffusion operations can strengthen the security of the block encryption process.


### 3.3. Image Encryption Scheme Based on Block Scrambling, Closed-Loop Diffusion, and DNA Molecular Mutation

In the proposed image encryption scheme, a global scrambling tool and a scanning method are adopted first. Subsequently, the strategy described inSection 2.5 will be executed and the ciphertext image will be obtained after mutation operations. The encryption steps are detailed as follows.Step 1. The MD5 hash function and the parity formula are adopted on the plaintext image of size N×N to generate a key stream. The methods described in Section 3.1 are utilized to covert the key stream into several initial valves for the used chaotic systems. Six random sequences Xls, Yls, Xla, Yla, Hx, and Hy are generated according to the parameters in Table 3.Step 2. Global scrambling: affine transform is exploited to scramble the plaintext image P for N times and generate a preliminary scrambled image PA.Step 3.PA is evenly divided into n×n sub-blocks. These sub-blocks are treated as the points to be scanned, and then a shuffled block set PH is obtained with the Hilbert matrix.Step 4. Steps 1 to 7 in Section 3.2 are executed to accomplish block scrambling and closed-loop diffusion operations in the block set PH and acquire a ciphered block set B.Step 5. The key block h is randomly selected from D1 to extract two sub-blocks located at h2z−1 and h2z in B. Then, the extracted sub-blocks are converted into 8-bit blocks T1 and T2, respectively. They are subsequently spliced into Q,(13)Q2ω−1=T1ω,Q2ω=T2ω,where1≤ω≤8n2.Step 6. The binary sequence Q is encoded into a DNA matrix E1 of size n×8n, and the DNA encoding rule adopted in the DNA encoding operation is Rc=modh2z−1+h2z,8+1.Step 7. The random sequence f1 utilized to construct the environment for point substitution could be calculated with equation (14), where q∈1,4N2. Then, the DNA molecules of E1 are substituted by the DNA mutation rule listed in Table 2. After substitution, the DNA matrix is updated to ...(14)f1q=1,Hxq∈0,0.15,2,Hxq∈0.15,0.35,3,Hxq∈0.35,0.5,4,Hxq∈0.5,0.65,5,Hxq∈0.65,0.85,6,Hxq∈0.85,1.Step 8. The sequence f2 calculated with equation (15) is in the set 1,2,…,n−1 and is utilized to determine the starting point of cross-mutation,(15)f2q′=floormodH¯xq′+Hyq′×1014,15+1,1≤q′≤32N.The vertical adjacent DNA chains ofE2 would be exchanged randomly with f2. For instance, W1 and W2 are the adjacent columns in E2, and the molecules from points f2 to n are exchanged. This process is expressed as(16)W′=excW1f2:n,W2f2:n,whereexca,b is a function to exchange the values of a and b. After executing the cross-mutation operation on the whole adjacent columns in E2, the DNA matrix E3 is obtained.Step 9. The DNA block set E3 is converted into decimal numbers with the DNA decoding rule Rd=modh2z−1×h2z,8+1.Step 10. Steps 5 to 9 are repeated until all the sub-blocks are traversed, and then the final ciphered image C is acquired after splicing the whole ciphered blocks.The encryption image can be decrypted with the inverse process of the encryption algorithm, and the encryption and the decryption processes are summarized in Figure4.Figure 4
Encryption and decryption processes.


## 3.1. Generation of Random Sequences

To possess an excellent capacity against the differential attack, the original values of chaotic systems are restricted with both the MD5 hash values and the parity quantization values of the plaintext image. The 32-bit hexadecimal key streamM generated from the MD5 hash function is divided into 16 groups, represented as ki, i=1,2,…,16,(5)ki=h2dM2i−1+1+M2i,where h2d⋅ converts a hexadecimal number into a decimal integer. The sum of ki is denoted as SUM=∑i=116ki. The parity property of the plaintext image is(6)u=floorabske−koe¯−o¯,where the numbers of even and odd integers in the plaintext image are represented as ke and ko, respectively; e¯ and o¯ are the mean values of the even and odd numbers, respectively. Then, the initial values are computed as(7)xj=u+k4j⊕k4j−2⊕u+k4j−1⊕k4j−3SUM,j=1,2,3,4,y1=∑j=16kj+uSUM,y2=∑j=16k2j+uSUM,where A⊕B means bitxorA,B. x1, x2, x3, x4, y1, and y2 are the initial values. The three chaotic systems are iterated with corresponding times and the former 1000 iteration elements are discarded to avert the so-called transient effect. The specific parameter values are collected in Table 3.Table 3
Parameter values of chaotic systems.
2D-LSCM2D-LASMHenon mapInitial valuesx1, x2x3, x4y1, y2System parametersδ=0.8μ=0.9λ1=1.4, λ2=0.3Resulting random sequencesXls, YlsXla, YlaHx, Hy


## 3.2. Block Scrambling and Closed-Loop Diffusion

The mechanisms of block scrambling and closed-loop diffusion are constructed to shuffle the preprocessed blocks and to diffuse the scrambled blocks in a linkage system. The whole process can be represented as(8)B=SDPH,Xls,Yls,Xla,Yla,where SD⋅ is the block scrambling and closed-loop diffusion function; Xls, Yls, Xla, and Yla are four random sequences; PH is the preprocessed plaintext block set and stored as a cell array of size n×n, where n is set to 16. The l−th sub-block in the cell array can be labeled as PHl. The flowchart of block scrambling and closed-loop diffusion is displayed in Figure 3. The specific process is given as follows:Step 1. Rectangular-ambulatory-plane cutting: the rings of each sub-block PHl are extracted and stored in OPHl. The x−th ring in the sub-block PHl can be represented as OxPHl, where x belongs to 1,2,…,8.Step 2. Four sequences r1, r2, r3, and r4 of length n2 are selected from Xls and the elements in r1, r2, r3, and r4 are all converted into integers in 0,255. The control sequence f of length 8n2 is updated from Xla. When the number in Xla is less than 0.5, it is updated to −1; otherwise, it is updated to 1.Step 3. The rectangular-ambulatory-plane cyclic shift operation is executed on each ring,(9)CxPHl=circshiftOxPHl,rx+1/2,f,x=1,3,5,7;circshiftOxPHl,xrx/2,f,x=2,4,6,8,wherecircshifta′,b′,c′ is a function for the bidirectional cyclic shift operation. a′ is a ring to be operated, b′ determines the shift times, and the ring OxPHl rotates clockwise when c′<0. CxPHl is the scrambled ring. After repeating the cyclic shift operation in each ring, a block set S is generated.Step 4. Yls and Yla are evenly divided into blocks, and then each sub-block of them is arranged in an ascending order to yield Yls′ and Yla′, respectively. The corresponding index block sets are D1 and D2.Step 5. Bidirectional random disorganization: each sub-block in S is scrambled by the double random block sets. The specific operation is(10)p′=dφD1l,p″=dφD2l,kp″Kl=sp′Sl,where1≤φ≤256; p′ and p″ correspond to the φ−th pixels in D1l and D2l, respectively; sp′Sl refers to the pixel extracted from the position p′ of Sl, and then all the extracted pixels are placed into a new block set K at a random position p″.Step 6. Two key block sets V1 and V2 in the range of 0,255 are generated with Yls′ and Yla′. The overall ciphertext sub-blocks in K are diffused by(11)gϕGl=kϕKl⊕kn2Kl⊕vϕV1l,ϕ=1,gϕ−1Gl⊕kϕKl⊕kϕ+1Kl⊕vϕV1l,ϕ=2,3,…,n2−1,gϕ−1Gl⊕kϕKl⊕vϕV1l,ϕ=n2,whereGl is the sub-block when all the pixels in the sub-block Kl are updated.Step 7. The diffusion operation between sub-blocks is subsequently executed. Then, the final ciphertext block set B is obtained after executing the closed-loop diffusion operation.(12)Bm=Gm⊕Gn2⊕V2m,m=1,Bm−1⊕Gm⊕Gm+1⊕V2m,m=2,3,…,n2−1,Bm−1⊕Gm⊕V2m,m=n2.Figure 3
Block scrambling and closed-loop diffusion.There exist some highlights in the above steps. First, the key blocks are associated with the plaintext image, and the proposed rectangular-ambulatory-plane cyclic shift takes full advantages of the chaotic sequences’ randomness and scrambles each block efficiently. Also, the double random blocks mapping rule could attain the pixels extraction and insertion randomly at the same time. Ultimately, the blocks are related to each other after being diffused. The whole confusion and diffusion operations can strengthen the security of the block encryption process.


## 3.3. Image Encryption Scheme Based on Block Scrambling, Closed-Loop Diffusion, and DNA Molecular Mutation

In the proposed image encryption scheme, a global scrambling tool and a scanning method are adopted first. Subsequently, the strategy described inSection 2.5 will be executed and the ciphertext image will be obtained after mutation operations. The encryption steps are detailed as follows.Step 1. The MD5 hash function and the parity formula are adopted on the plaintext image of size N×N to generate a key stream. The methods described in Section 3.1 are utilized to covert the key stream into several initial valves for the used chaotic systems. Six random sequences Xls, Yls, Xla, Yla, Hx, and Hy are generated according to the parameters in Table 3.Step 2. Global scrambling: affine transform is exploited to scramble the plaintext image P for N times and generate a preliminary scrambled image PA.Step 3.PA is evenly divided into n×n sub-blocks. These sub-blocks are treated as the points to be scanned, and then a shuffled block set PH is obtained with the Hilbert matrix.Step 4. Steps 1 to 7 in Section 3.2 are executed to accomplish block scrambling and closed-loop diffusion operations in the block set PH and acquire a ciphered block set B.Step 5. The key block h is randomly selected from D1 to extract two sub-blocks located at h2z−1 and h2z in B. Then, the extracted sub-blocks are converted into 8-bit blocks T1 and T2, respectively. They are subsequently spliced into Q,(13)Q2ω−1=T1ω,Q2ω=T2ω,where1≤ω≤8n2.Step 6. The binary sequence Q is encoded into a DNA matrix E1 of size n×8n, and the DNA encoding rule adopted in the DNA encoding operation is Rc=modh2z−1+h2z,8+1.Step 7. The random sequence f1 utilized to construct the environment for point substitution could be calculated with equation (14), where q∈1,4N2. Then, the DNA molecules of E1 are substituted by the DNA mutation rule listed in Table 2. After substitution, the DNA matrix is updated to ...(14)f1q=1,Hxq∈0,0.15,2,Hxq∈0.15,0.35,3,Hxq∈0.35,0.5,4,Hxq∈0.5,0.65,5,Hxq∈0.65,0.85,6,Hxq∈0.85,1.Step 8. The sequence f2 calculated with equation (15) is in the set 1,2,…,n−1 and is utilized to determine the starting point of cross-mutation,(15)f2q′=floormodH¯xq′+Hyq′×1014,15+1,1≤q′≤32N.The vertical adjacent DNA chains ofE2 would be exchanged randomly with f2. For instance, W1 and W2 are the adjacent columns in E2, and the molecules from points f2 to n are exchanged. This process is expressed as(16)W′=excW1f2:n,W2f2:n,whereexca,b is a function to exchange the values of a and b. After executing the cross-mutation operation on the whole adjacent columns in E2, the DNA matrix E3 is obtained.Step 9. The DNA block set E3 is converted into decimal numbers with the DNA decoding rule Rd=modh2z−1×h2z,8+1.Step 10. Steps 5 to 9 are repeated until all the sub-blocks are traversed, and then the final ciphered image C is acquired after splicing the whole ciphered blocks.The encryption image can be decrypted with the inverse process of the encryption algorithm, and the encryption and the decryption processes are summarized in Figure4.Figure 4
Encryption and decryption processes.


## 4. Simulation Results and Performance Analyses

To validate the reliability and the security of the proposed image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation, a series of numerical experiments with test images of size256×256 are carried out in this section.


### 4.1. Encryption and Decryption Results and Quality Assessments

Figures5(a)–5(c) show the plaintext images “Bridge,” “Elaine,” and “Bird,” respectively. The corresponding encryption and decryption results are listed in Figures 5(d)–5(i). Visually, the ciphertext images reveal no information about the original ones, and Figures 5(g)–5(i) displayed that they can be decrypted intactly. To appraise the fidelity of the encryption and decryption images, the peak signal-to-noise ratio (PSNR) and the structural similarity index metric (SSIM) [35] are employed,(17)PSNR=10log10S1×S2×2552∑s1=1S1∑s2=1S2Q1s1,s2−Q2s1,s2,where Q1, Q2 denote two contrast images and S1, S2 represent the image dimensions,(18)SSIMm,m′=2μmμm′+ζ12σmm′+ζ2μm2+μm′2+ζ1σm2+σm′2+ζ2,where μm=1/ST∑s=1S∑s=1Tms,t and μm′=1/ST∑s=1S∑t=1Tm′s,t are the means for images m and m′, respectively, σm2 and σm′2 are the images’ variances, and σmm′2 is their covariance. Constants ζ1=v1L2 and ζ2=v2L2 can be obtained with v1=0.01 and v2=0.03 in the dynamic range L of an image. If the two images are nearly identical, the PSNR value would approach infinity and the SSIM value would approach 1. Thus, from the encryption results displayed in Table 4, the encryption images are severely disturbed, and the PSNR and the SSIM results for all the decryption images represent that there is no apparent data loss. In other words, the devised image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation could encrypt and decrypt images effectively.Figure 5
Experiment results: (a)–(c) the plaintext images “Bridge,” “Elaine,” and “Bird,” respectively; (d)–(f) the ciphertext images “Bridge,” “Elaine,” and “Bird,” respectively; (g)–(i) correct decryption images (d)–(f).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 4
Quality assessments for different encryption and decryption images.
BridgeElaineBirdPSNR (dB)SSIMPSNR (dB)SSIMPSNR (dB)SSIMEncryption image8.76820.01049.29250.00969.31120.0090Decryption image∞1∞1∞1


### 4.2. Statistical Analyses

#### 4.2.1. Histogram Analysis

The histograms of the plaintext images “Couple,” “Camera,” and “Peppers” are respectively presented in Figures6(b)–6(j), while the histograms of their corresponding ciphertext images are exhibited in Figures 6(d)–6(l). The histograms after encryption are smoother with no raised spikes. This benefits from the devised close-loop block diffusion scheme and the mutation operation, which can distribute the pixel values uniformly in the range from 0 to 255. To further verify the histogram homogeneity, the chi-square test is adopted and the corresponding results are recorded in Table 5.(19)χ2=∑M=0255ks−os2os,where ks is an observed frequency of the encryption image at level s and os represents the expected one. Theoretically, χ2 should not be more than 293.2478 when the probability is 5%. Conclusively, it is impractical for an attacker to obtain the corresponding plaintext images with the histogram analysis attack, since the histograms of all test encryption images are smooth and featureless.Figure 6
Histograms of several images: (b)–(j) the histograms for the plaintext images (a)–(i); (d)–(l) the histograms for the ciphertext images (c)–(k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 5
Chi-square results for ciphertext images.
ImageCoupleCameraPeppersχtest2226.6016244.3672240.0391χ2552293.2478293.2478293.2478DecisionAcceptedAcceptedAccepted


#### 4.2.2. Correlation Coefficients of Adjacent Pixels

The high correlation among the adjacent pixels of a ciphertext image would increase the risk of being cracked [13]. To inspect the correlation between plaintext and ciphertext, 10,000 pairs of pixels are arbitrarily selected. As displayed in Figure 7, the scatter plots of the plaintext image seem like a linear distribution and the adjacent pixels are highly correlated. After executing the proposed strategies of confusion, diffusion, and DNA mutation, the positions and the values of the pixels are altered randomly and adequately. Therefore, the pixels of the corresponding ciphertext image are almost evenly dispersed on the plane. It can be seen from Tables 6 and 7 that our scheme is more effective in reducing the correlation and can stand up to the statistical analysis attack.Figure 7
Correlation coefficients of the plaintext image “Couple” and the ciphertext one in three directions: (a) “Couple,” vertical direction (VD). (b) “Couple,” horizontal direction (HD). (c) “Couple,” diagonal direction (DD). (d) Ciphertext image, VD. (e) Ciphertext image, HD. (f) Ciphertext image, DD.
(a)(b)(c)(d)(e)(f)Table 6
Correlation coefficients of plaintext images and ciphertext ones.
Test imagePlaintext imageCiphertext imageHDVDDDHDVDDDBridge0.84910.88690.81870.00260.0029–0.0009Elaine0.95760.94830.9316–0.0010–0.0008–0.0036Bird0.99080.98110.97110.00000.00350.0000Couple0.94390.94240.9150–0.0015–0.0016–0.0004Camera0.96150.93210.90580.0002–0.00180.0003Peppers0.95350.94390.89880.00210.00140.0002Table 7
Correlation coefficients of image “Peppers” obtained by other encryption schemes.
DirectionRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeHD0.0057–0.00560.02340.0068–0.01600.0021VD0.0037–0.01620.0121–0.00220.03500.0014DD0.0043–0.02020.00050.0005–0.00970.0002


#### 4.2.3. Information Entropy

Shannon entropy is a commonly used indicator to evaluate the randomness,(20)Hv=∑i=0N−1pvlog1pvi,where pvi refers to the probability of the random gray value vi. However, compared with the local entropy [41], the global Shannon entropy is insufficient in evaluating the uniformity. The local entropy is the sample average value of the global Shannon entropy of 1936 pixels taken from 30 nonoverlapping image blocks, which is more accurate, consistent, and efficient. The results of global and local entropies with our image encryption scheme are exhibited in Table 8, which are very close to 8 bits. Based on the above analysis, the information entropy analysis attack on our proposed image encryption scheme is ineffective.Table 8
Global and local Shannon entropies.
ImagePlaintext image entropy (bit)Global Shannon entropy (bit)Local Shannon entropy (bit)Local entropy critical values (bit)hleft1∗0.05=7.9019hleft1∗0.01=7.9017hleft1∗0.001=7.9015hright1∗0.05=7.9030hleft1∗0.01=7.9032hleft1∗0.001=7.9034Couple7.30217.99867.9025PassPassPassCamera7.00977.99907.9026PassPassPassPeppers7.53277.99837.9023PassPassPass


### 4.3. Sensibility Analyses

#### 4.3.1. Key Sensitivity Analysis

To rate the key sensitivity, simulation experiments are executed under a slight alteration of the correct keys. In Figure8, the decryption images are noise-like ones and will not reveal any meaningful information. The accuracy of the key would seriously affect the image decryption. Therefore, the proposed encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation could withstand the brute-force attack successfully.Figure 8
Decryption image “Camera” with incorrect keys: (a)x1=0.3623×10−16, (b) x2=0.3281+10−16, (c) x3=0.3102+10−15, (d) x4=0.5263+10−15, (e) y1=0.7035+10−15, and (f) y2=0.5794+10−15.
(a)(b)(c)(d)(e)(f)


#### 4.3.2. Key Space Analysis

Exhaustive attack may threaten the cryptosystem security, but the key space expansion would make it harder to defeat the system. In the proposed algorithm, the initial valuesx1, x2, x3x4, y1, and y2 for chaotic systems are the main keys. These initial values are all specified within −1,1 and the simulation results demonstrate that the computational precision of the above key space are about 10−15 or 10−16. Totally, the key space of the devised image encryption scheme is about 1092, which is larger than the key space in [42]. Therefore, the brute-force attack is impracticable for the presented image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation.


#### 4.3.3. Differential Attack Analysis

The number of pixel change rate (NPCR) and the unified average changing intensity (UACI) are applied to assess the sensitivity of the image encryption systems and the differential attack resistance,(21)NPCR%=∑i=1M∑j=1NGi,jM×N×100,UACI%=∑i=1M∑j=1Ngi,j−g′i,j255×M×N×100,Gi,j=0,gi,j=g′i,j,1,gi,j≠g′i,j,where g and g′ are the ciphertext images of size M×N corresponding to the normal test image and the one pixel altered test image. The NPCR values and the UACI values of various test images are tabulated in Tables 9 and 10 compiles the results of other schemes with “Peppers.” The results are close to their ideal values, indicating that the pixel change has a big impact on the encryption results. It substantiates that the proposed image encryption strategy could resist the differential attack.Table 9
NPCR and UACI analyses of the proposed scheme.
ImageBridgeElaineBirdCoupleCameraPeppersNPCR99.626399.608299.610799.608699.612499.6285UACI33.424833.438733.401333.407733.417033.3803Table 10
NPCR and UACI results of other schemes with “Peppers.”
SchemeRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeNPCR99.586799.631999.474299.553899.622499.6285UACI36.687833.692333.181633.354133.542133.3803


### 4.4. Robustness Analyses under Attack

#### 4.4.1. Chosen-Plaintext Attack

In our proposed scheme, the key stream is highly connected with the plaintext. It is composed of the MD5 hash values and the quantization values of the plaintext image. In other words, the slight change of the plaintext will have a great impact on the entire image encryption system, and in the designed closed-loop diffusion algorithm, the ciphertext blocks are interconnected with the key blocks and the plaintext blocks. Hence, a linkage system is formed that not only invalidates differential attack but also invalidates the chosen-plaintext. Besides, the mutations performed between two random selected sub-blocks are nonlinear operations, which render an attacker incapable of obtaining the correct keys. In brief, the presented image encryption system is nonlinear and the entire encryption blocks are highly interconnected, which make it immune to chosen-plaintext attack.


#### 4.4.2. Gaussian Noise Attack

Assume the Gaussian noise attack is modeled as(22)E′=E+kNG,where E′ is the encryption image affected by noise and E is the normal encryption one, NG represents the white Gaussian noise with the standardized normal distribution, and k is the noise intensity. Figure 9 shows the simulation results when “Camera” is polluted by the white Gaussian noise of different intensities. Its primary information is still visible as the noise intensity increases, and the MSE curve in Figure 10 indicates that the proposed image encryption scheme is immune to the white Gaussian noise attack to a certain degree.Figure 9
Gaussian noise on “Camera” with different noise intensities: (a)k=1, (b) k=10, (c) k=20, and (d) k=30.
(a)(b)(c)(d)Figure 10
MSE curve with variation of noise intensity.


#### 4.4.3. Salt and Pepper Noise Attack

The encryption “Camera” is added with Salt and Pepper noise of different variances 0.01, 0.05, 0.07, and 0.1, respectively. The test results are shown in Figures11(a)–11(d), and it could verify that the proposed image encryption technology based on block scrambling, closed-loop diffusion, and DNA molecular mutation has a certain anti-interference ability under Salt and Pepper noise attack.Figure 11
Salt and Pepper noise attack on image “Camera” with different variances: (a) 0.01; (b) 0.05; (c) 0.07; (d) 0.1.
(a)(b)(c)(d)


#### 4.4.4. Occlusion Attack

Experiments with varying extents of data loss are executed on “Elaine,” and the decryption images after cropping in different positions or sizes are shown in Figures12(a)–12(d). The cropped pixels of the decryption images are replaced by 0. The devised operations, rectangular-ambulatory-plane cyclic shift and the bidirectional random disorganization, are controlled by the cascade chaotic systems. They would rearrange the image pixels stochastically, which makes the decryption results of the same size cropping in different positions similar. Visually, the encryption images in Figure 12 still contain general information as the cutting area increases. Thus, the proposed image encryption system has certain robustness against the cropping attack.Figure 12
(a), (c)) The encryption “Elaine” with 6.5% cropped; ((b), (d))the corresponding decryption ones; ((e), (g)) the encryption “Elaine” with 10% cropped; ((f), (h)) the corresponding decryption ones.
(a)(b)(c)(d)(e)(f)(g)(h)


### 4.5. Computation Complexity and Execution Time

The computation complexity of an image encryption algorithm will greatly affect the execution time. To this end, complexity analysis is carried out for the main encryption modules in our algorithm. The total complexity isO2N2 for the block scrambling and closed-loop diffusion module. Specifically, the rectangular-ambulatory-plane cyclic shift operation is executed in 8 rings and the complexity is O8N. The subsequent bidirectional random disorganization operation would traverse all of the image pixels, whose complexity is ON2. The complexities of the intrablock diffusion and the outer-block diffusion are ON2 and ON, respectively. The total complexity for the DNA mutation module is O17N2. The complexity of encoding and decoding process is O8N2. The complexities of the concatenation operations of binary sub-blocks and the conversion of the decimal matrices are O4N2 and ON2, respectively, and the complexity for the mutation processes in the sub-blocks is O4N2. The computation time of the algorithm is related to the program design, operating environment, and so on. The proposed image encryption algorithm is carried out under MATLAB (version R2016a) on the computer with 8 GB RAM and Windows 10. In our scheme, the operations of the block scrambling and closed-loop diffusion need one-round encryption only, which costs 0.200 s, and the DNA mutation process takes 14.147 s. Since the encoding and decoding rules for each selected sub-block are different, more iterations are involved, so it takes longer execution time. The complete encryption process needs 16.906 s, which is acceptable for most of the real-time image encryption schemes and also should be reduced for some cases.


## 4.1. Encryption and Decryption Results and Quality Assessments

Figures5(a)–5(c) show the plaintext images “Bridge,” “Elaine,” and “Bird,” respectively. The corresponding encryption and decryption results are listed in Figures 5(d)–5(i). Visually, the ciphertext images reveal no information about the original ones, and Figures 5(g)–5(i) displayed that they can be decrypted intactly. To appraise the fidelity of the encryption and decryption images, the peak signal-to-noise ratio (PSNR) and the structural similarity index metric (SSIM) [35] are employed,(17)PSNR=10log10S1×S2×2552∑s1=1S1∑s2=1S2Q1s1,s2−Q2s1,s2,where Q1, Q2 denote two contrast images and S1, S2 represent the image dimensions,(18)SSIMm,m′=2μmμm′+ζ12σmm′+ζ2μm2+μm′2+ζ1σm2+σm′2+ζ2,where μm=1/ST∑s=1S∑s=1Tms,t and μm′=1/ST∑s=1S∑t=1Tm′s,t are the means for images m and m′, respectively, σm2 and σm′2 are the images’ variances, and σmm′2 is their covariance. Constants ζ1=v1L2 and ζ2=v2L2 can be obtained with v1=0.01 and v2=0.03 in the dynamic range L of an image. If the two images are nearly identical, the PSNR value would approach infinity and the SSIM value would approach 1. Thus, from the encryption results displayed in Table 4, the encryption images are severely disturbed, and the PSNR and the SSIM results for all the decryption images represent that there is no apparent data loss. In other words, the devised image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation could encrypt and decrypt images effectively.Figure 5
Experiment results: (a)–(c) the plaintext images “Bridge,” “Elaine,” and “Bird,” respectively; (d)–(f) the ciphertext images “Bridge,” “Elaine,” and “Bird,” respectively; (g)–(i) correct decryption images (d)–(f).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 4
Quality assessments for different encryption and decryption images.
BridgeElaineBirdPSNR (dB)SSIMPSNR (dB)SSIMPSNR (dB)SSIMEncryption image8.76820.01049.29250.00969.31120.0090Decryption image∞1∞1∞1


## 4.2. Statistical Analyses

### 4.2.1. Histogram Analysis

The histograms of the plaintext images “Couple,” “Camera,” and “Peppers” are respectively presented in Figures6(b)–6(j), while the histograms of their corresponding ciphertext images are exhibited in Figures 6(d)–6(l). The histograms after encryption are smoother with no raised spikes. This benefits from the devised close-loop block diffusion scheme and the mutation operation, which can distribute the pixel values uniformly in the range from 0 to 255. To further verify the histogram homogeneity, the chi-square test is adopted and the corresponding results are recorded in Table 5.(19)χ2=∑M=0255ks−os2os,where ks is an observed frequency of the encryption image at level s and os represents the expected one. Theoretically, χ2 should not be more than 293.2478 when the probability is 5%. Conclusively, it is impractical for an attacker to obtain the corresponding plaintext images with the histogram analysis attack, since the histograms of all test encryption images are smooth and featureless.Figure 6
Histograms of several images: (b)–(j) the histograms for the plaintext images (a)–(i); (d)–(l) the histograms for the ciphertext images (c)–(k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 5
Chi-square results for ciphertext images.
ImageCoupleCameraPeppersχtest2226.6016244.3672240.0391χ2552293.2478293.2478293.2478DecisionAcceptedAcceptedAccepted


### 4.2.2. Correlation Coefficients of Adjacent Pixels

The high correlation among the adjacent pixels of a ciphertext image would increase the risk of being cracked [13]. To inspect the correlation between plaintext and ciphertext, 10,000 pairs of pixels are arbitrarily selected. As displayed in Figure 7, the scatter plots of the plaintext image seem like a linear distribution and the adjacent pixels are highly correlated. After executing the proposed strategies of confusion, diffusion, and DNA mutation, the positions and the values of the pixels are altered randomly and adequately. Therefore, the pixels of the corresponding ciphertext image are almost evenly dispersed on the plane. It can be seen from Tables 6 and 7 that our scheme is more effective in reducing the correlation and can stand up to the statistical analysis attack.Figure 7
Correlation coefficients of the plaintext image “Couple” and the ciphertext one in three directions: (a) “Couple,” vertical direction (VD). (b) “Couple,” horizontal direction (HD). (c) “Couple,” diagonal direction (DD). (d) Ciphertext image, VD. (e) Ciphertext image, HD. (f) Ciphertext image, DD.
(a)(b)(c)(d)(e)(f)Table 6
Correlation coefficients of plaintext images and ciphertext ones.
Test imagePlaintext imageCiphertext imageHDVDDDHDVDDDBridge0.84910.88690.81870.00260.0029–0.0009Elaine0.95760.94830.9316–0.0010–0.0008–0.0036Bird0.99080.98110.97110.00000.00350.0000Couple0.94390.94240.9150–0.0015–0.0016–0.0004Camera0.96150.93210.90580.0002–0.00180.0003Peppers0.95350.94390.89880.00210.00140.0002Table 7
Correlation coefficients of image “Peppers” obtained by other encryption schemes.
DirectionRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeHD0.0057–0.00560.02340.0068–0.01600.0021VD0.0037–0.01620.0121–0.00220.03500.0014DD0.0043–0.02020.00050.0005–0.00970.0002


### 4.2.3. Information Entropy

Shannon entropy is a commonly used indicator to evaluate the randomness,(20)Hv=∑i=0N−1pvlog1pvi,where pvi refers to the probability of the random gray value vi. However, compared with the local entropy [41], the global Shannon entropy is insufficient in evaluating the uniformity. The local entropy is the sample average value of the global Shannon entropy of 1936 pixels taken from 30 nonoverlapping image blocks, which is more accurate, consistent, and efficient. The results of global and local entropies with our image encryption scheme are exhibited in Table 8, which are very close to 8 bits. Based on the above analysis, the information entropy analysis attack on our proposed image encryption scheme is ineffective.Table 8
Global and local Shannon entropies.
ImagePlaintext image entropy (bit)Global Shannon entropy (bit)Local Shannon entropy (bit)Local entropy critical values (bit)hleft1∗0.05=7.9019hleft1∗0.01=7.9017hleft1∗0.001=7.9015hright1∗0.05=7.9030hleft1∗0.01=7.9032hleft1∗0.001=7.9034Couple7.30217.99867.9025PassPassPassCamera7.00977.99907.9026PassPassPassPeppers7.53277.99837.9023PassPassPass


## 4.2.1. Histogram Analysis

The histograms of the plaintext images “Couple,” “Camera,” and “Peppers” are respectively presented in Figures6(b)–6(j), while the histograms of their corresponding ciphertext images are exhibited in Figures 6(d)–6(l). The histograms after encryption are smoother with no raised spikes. This benefits from the devised close-loop block diffusion scheme and the mutation operation, which can distribute the pixel values uniformly in the range from 0 to 255. To further verify the histogram homogeneity, the chi-square test is adopted and the corresponding results are recorded in Table 5.(19)χ2=∑M=0255ks−os2os,where ks is an observed frequency of the encryption image at level s and os represents the expected one. Theoretically, χ2 should not be more than 293.2478 when the probability is 5%. Conclusively, it is impractical for an attacker to obtain the corresponding plaintext images with the histogram analysis attack, since the histograms of all test encryption images are smooth and featureless.Figure 6
Histograms of several images: (b)–(j) the histograms for the plaintext images (a)–(i); (d)–(l) the histograms for the ciphertext images (c)–(k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 5
Chi-square results for ciphertext images.
ImageCoupleCameraPeppersχtest2226.6016244.3672240.0391χ2552293.2478293.2478293.2478DecisionAcceptedAcceptedAccepted


## 4.2.2. Correlation Coefficients of Adjacent Pixels

The high correlation among the adjacent pixels of a ciphertext image would increase the risk of being cracked [13]. To inspect the correlation between plaintext and ciphertext, 10,000 pairs of pixels are arbitrarily selected. As displayed in Figure 7, the scatter plots of the plaintext image seem like a linear distribution and the adjacent pixels are highly correlated. After executing the proposed strategies of confusion, diffusion, and DNA mutation, the positions and the values of the pixels are altered randomly and adequately. Therefore, the pixels of the corresponding ciphertext image are almost evenly dispersed on the plane. It can be seen from Tables 6 and 7 that our scheme is more effective in reducing the correlation and can stand up to the statistical analysis attack.Figure 7
Correlation coefficients of the plaintext image “Couple” and the ciphertext one in three directions: (a) “Couple,” vertical direction (VD). (b) “Couple,” horizontal direction (HD). (c) “Couple,” diagonal direction (DD). (d) Ciphertext image, VD. (e) Ciphertext image, HD. (f) Ciphertext image, DD.
(a)(b)(c)(d)(e)(f)Table 6
Correlation coefficients of plaintext images and ciphertext ones.
Test imagePlaintext imageCiphertext imageHDVDDDHDVDDDBridge0.84910.88690.81870.00260.0029–0.0009Elaine0.95760.94830.9316–0.0010–0.0008–0.0036Bird0.99080.98110.97110.00000.00350.0000Couple0.94390.94240.9150–0.0015–0.0016–0.0004Camera0.96150.93210.90580.0002–0.00180.0003Peppers0.95350.94390.89880.00210.00140.0002Table 7
Correlation coefficients of image “Peppers” obtained by other encryption schemes.
DirectionRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeHD0.0057–0.00560.02340.0068–0.01600.0021VD0.0037–0.01620.0121–0.00220.03500.0014DD0.0043–0.02020.00050.0005–0.00970.0002


## 4.2.3. Information Entropy

Shannon entropy is a commonly used indicator to evaluate the randomness,(20)Hv=∑i=0N−1pvlog1pvi,where pvi refers to the probability of the random gray value vi. However, compared with the local entropy [41], the global Shannon entropy is insufficient in evaluating the uniformity. The local entropy is the sample average value of the global Shannon entropy of 1936 pixels taken from 30 nonoverlapping image blocks, which is more accurate, consistent, and efficient. The results of global and local entropies with our image encryption scheme are exhibited in Table 8, which are very close to 8 bits. Based on the above analysis, the information entropy analysis attack on our proposed image encryption scheme is ineffective.Table 8
Global and local Shannon entropies.
ImagePlaintext image entropy (bit)Global Shannon entropy (bit)Local Shannon entropy (bit)Local entropy critical values (bit)hleft1∗0.05=7.9019hleft1∗0.01=7.9017hleft1∗0.001=7.9015hright1∗0.05=7.9030hleft1∗0.01=7.9032hleft1∗0.001=7.9034Couple7.30217.99867.9025PassPassPassCamera7.00977.99907.9026PassPassPassPeppers7.53277.99837.9023PassPassPass


## 4.3. Sensibility Analyses

### 4.3.1. Key Sensitivity Analysis

To rate the key sensitivity, simulation experiments are executed under a slight alteration of the correct keys. In Figure8, the decryption images are noise-like ones and will not reveal any meaningful information. The accuracy of the key would seriously affect the image decryption. Therefore, the proposed encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation could withstand the brute-force attack successfully.Figure 8
Decryption image “Camera” with incorrect keys: (a)x1=0.3623×10−16, (b) x2=0.3281+10−16, (c) x3=0.3102+10−15, (d) x4=0.5263+10−15, (e) y1=0.7035+10−15, and (f) y2=0.5794+10−15.
(a)(b)(c)(d)(e)(f)


### 4.3.2. Key Space Analysis

Exhaustive attack may threaten the cryptosystem security, but the key space expansion would make it harder to defeat the system. In the proposed algorithm, the initial valuesx1, x2, x3x4, y1, and y2 for chaotic systems are the main keys. These initial values are all specified within −1,1 and the simulation results demonstrate that the computational precision of the above key space are about 10−15 or 10−16. Totally, the key space of the devised image encryption scheme is about 1092, which is larger than the key space in [42]. Therefore, the brute-force attack is impracticable for the presented image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation.


### 4.3.3. Differential Attack Analysis

The number of pixel change rate (NPCR) and the unified average changing intensity (UACI) are applied to assess the sensitivity of the image encryption systems and the differential attack resistance,(21)NPCR%=∑i=1M∑j=1NGi,jM×N×100,UACI%=∑i=1M∑j=1Ngi,j−g′i,j255×M×N×100,Gi,j=0,gi,j=g′i,j,1,gi,j≠g′i,j,where g and g′ are the ciphertext images of size M×N corresponding to the normal test image and the one pixel altered test image. The NPCR values and the UACI values of various test images are tabulated in Tables 9 and 10 compiles the results of other schemes with “Peppers.” The results are close to their ideal values, indicating that the pixel change has a big impact on the encryption results. It substantiates that the proposed image encryption strategy could resist the differential attack.Table 9
NPCR and UACI analyses of the proposed scheme.
ImageBridgeElaineBirdCoupleCameraPeppersNPCR99.626399.608299.610799.608699.612499.6285UACI33.424833.438733.401333.407733.417033.3803Table 10
NPCR and UACI results of other schemes with “Peppers.”
SchemeRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeNPCR99.586799.631999.474299.553899.622499.6285UACI36.687833.692333.181633.354133.542133.3803


## 4.3.1. Key Sensitivity Analysis

To rate the key sensitivity, simulation experiments are executed under a slight alteration of the correct keys. In Figure8, the decryption images are noise-like ones and will not reveal any meaningful information. The accuracy of the key would seriously affect the image decryption. Therefore, the proposed encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation could withstand the brute-force attack successfully.Figure 8
Decryption image “Camera” with incorrect keys: (a)x1=0.3623×10−16, (b) x2=0.3281+10−16, (c) x3=0.3102+10−15, (d) x4=0.5263+10−15, (e) y1=0.7035+10−15, and (f) y2=0.5794+10−15.
(a)(b)(c)(d)(e)(f)


## 4.3.2. Key Space Analysis

Exhaustive attack may threaten the cryptosystem security, but the key space expansion would make it harder to defeat the system. In the proposed algorithm, the initial valuesx1, x2, x3x4, y1, and y2 for chaotic systems are the main keys. These initial values are all specified within −1,1 and the simulation results demonstrate that the computational precision of the above key space are about 10−15 or 10−16. Totally, the key space of the devised image encryption scheme is about 1092, which is larger than the key space in [42]. Therefore, the brute-force attack is impracticable for the presented image encryption scheme based on block scrambling, closed-loop diffusion, and DNA molecular mutation.


## 4.3.3. Differential Attack Analysis

The number of pixel change rate (NPCR) and the unified average changing intensity (UACI) are applied to assess the sensitivity of the image encryption systems and the differential attack resistance,(21)NPCR%=∑i=1M∑j=1NGi,jM×N×100,UACI%=∑i=1M∑j=1Ngi,j−g′i,j255×M×N×100,Gi,j=0,gi,j=g′i,j,1,gi,j≠g′i,j,where g and g′ are the ciphertext images of size M×N corresponding to the normal test image and the one pixel altered test image. The NPCR values and the UACI values of various test images are tabulated in Tables 9 and 10 compiles the results of other schemes with “Peppers.” The results are close to their ideal values, indicating that the pixel change has a big impact on the encryption results. It substantiates that the proposed image encryption strategy could resist the differential attack.Table 9
NPCR and UACI analyses of the proposed scheme.
ImageBridgeElaineBirdCoupleCameraPeppersNPCR99.626399.608299.610799.608699.612499.6285UACI33.424833.438733.401333.407733.417033.3803Table 10
NPCR and UACI results of other schemes with “Peppers.”
SchemeRef. [36]Ref. [37]Ref. [38]Ref. [39]Ref. [40]Our schemeNPCR99.586799.631999.474299.553899.622499.6285UACI36.687833.692333.181633.354133.542133.3803


## 4.4. Robustness Analyses under Attack

### 4.4.1. Chosen-Plaintext Attack

In our proposed scheme, the key stream is highly connected with the plaintext. It is composed of the MD5 hash values and the quantization values of the plaintext image. In other words, the slight change of the plaintext will have a great impact on the entire image encryption system, and in the designed closed-loop diffusion algorithm, the ciphertext blocks are interconnected with the key blocks and the plaintext blocks. Hence, a linkage system is formed that not only invalidates differential attack but also invalidates the chosen-plaintext. Besides, the mutations performed between two random selected sub-blocks are nonlinear operations, which render an attacker incapable of obtaining the correct keys. In brief, the presented image encryption system is nonlinear and the entire encryption blocks are highly interconnected, which make it immune to chosen-plaintext attack.


### 4.4.2. Gaussian Noise Attack

Assume the Gaussian noise attack is modeled as(22)E′=E+kNG,where E′ is the encryption image affected by noise and E is the normal encryption one, NG represents the white Gaussian noise with the standardized normal distribution, and k is the noise intensity. Figure 9 shows the simulation results when “Camera” is polluted by the white Gaussian noise of different intensities. Its primary information is still visible as the noise intensity increases, and the MSE curve in Figure 10 indicates that the proposed image encryption scheme is immune to the white Gaussian noise attack to a certain degree.Figure 9
Gaussian noise on “Camera” with different noise intensities: (a)k=1, (b) k=10, (c) k=20, and (d) k=30.
(a)(b)(c)(d)Figure 10
MSE curve with variation of noise intensity.


### 4.4.3. Salt and Pepper Noise Attack

The encryption “Camera” is added with Salt and Pepper noise of different variances 0.01, 0.05, 0.07, and 0.1, respectively. The test results are shown in Figures11(a)–11(d), and it could verify that the proposed image encryption technology based on block scrambling, closed-loop diffusion, and DNA molecular mutation has a certain anti-interference ability under Salt and Pepper noise attack.Figure 11
Salt and Pepper noise attack on image “Camera” with different variances: (a) 0.01; (b) 0.05; (c) 0.07; (d) 0.1.
(a)(b)(c)(d)


### 4.4.4. Occlusion Attack

Experiments with varying extents of data loss are executed on “Elaine,” and the decryption images after cropping in different positions or sizes are shown in Figures12(a)–12(d). The cropped pixels of the decryption images are replaced by 0. The devised operations, rectangular-ambulatory-plane cyclic shift and the bidirectional random disorganization, are controlled by the cascade chaotic systems. They would rearrange the image pixels stochastically, which makes the decryption results of the same size cropping in different positions similar. Visually, the encryption images in Figure 12 still contain general information as the cutting area increases. Thus, the proposed image encryption system has certain robustness against the cropping attack.Figure 12
(a), (c)) The encryption “Elaine” with 6.5% cropped; ((b), (d))the corresponding decryption ones; ((e), (g)) the encryption “Elaine” with 10% cropped; ((f), (h)) the corresponding decryption ones.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.4.1. Chosen-Plaintext Attack

In our proposed scheme, the key stream is highly connected with the plaintext. It is composed of the MD5 hash values and the quantization values of the plaintext image. In other words, the slight change of the plaintext will have a great impact on the entire image encryption system, and in the designed closed-loop diffusion algorithm, the ciphertext blocks are interconnected with the key blocks and the plaintext blocks. Hence, a linkage system is formed that not only invalidates differential attack but also invalidates the chosen-plaintext. Besides, the mutations performed between two random selected sub-blocks are nonlinear operations, which render an attacker incapable of obtaining the correct keys. In brief, the presented image encryption system is nonlinear and the entire encryption blocks are highly interconnected, which make it immune to chosen-plaintext attack.


## 4.4.2. Gaussian Noise Attack

Assume the Gaussian noise attack is modeled as(22)E′=E+kNG,where E′ is the encryption image affected by noise and E is the normal encryption one, NG represents the white Gaussian noise with the standardized normal distribution, and k is the noise intensity. Figure 9 shows the simulation results when “Camera” is polluted by the white Gaussian noise of different intensities. Its primary information is still visible as the noise intensity increases, and the MSE curve in Figure 10 indicates that the proposed image encryption scheme is immune to the white Gaussian noise attack to a certain degree.Figure 9
Gaussian noise on “Camera” with different noise intensities: (a)k=1, (b) k=10, (c) k=20, and (d) k=30.
(a)(b)(c)(d)Figure 10
MSE curve with variation of noise intensity.


## 4.4.3. Salt and Pepper Noise Attack

The encryption “Camera” is added with Salt and Pepper noise of different variances 0.01, 0.05, 0.07, and 0.1, respectively. The test results are shown in Figures11(a)–11(d), and it could verify that the proposed image encryption technology based on block scrambling, closed-loop diffusion, and DNA molecular mutation has a certain anti-interference ability under Salt and Pepper noise attack.Figure 11
Salt and Pepper noise attack on image “Camera” with different variances: (a) 0.01; (b) 0.05; (c) 0.07; (d) 0.1.
(a)(b)(c)(d)


## 4.4.4. Occlusion Attack

Experiments with varying extents of data loss are executed on “Elaine,” and the decryption images after cropping in different positions or sizes are shown in Figures12(a)–12(d). The cropped pixels of the decryption images are replaced by 0. The devised operations, rectangular-ambulatory-plane cyclic shift and the bidirectional random disorganization, are controlled by the cascade chaotic systems. They would rearrange the image pixels stochastically, which makes the decryption results of the same size cropping in different positions similar. Visually, the encryption images in Figure 12 still contain general information as the cutting area increases. Thus, the proposed image encryption system has certain robustness against the cropping attack.Figure 12
(a), (c)) The encryption “Elaine” with 6.5% cropped; ((b), (d))the corresponding decryption ones; ((e), (g)) the encryption “Elaine” with 10% cropped; ((f), (h)) the corresponding decryption ones.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.5. Computation Complexity and Execution Time

The computation complexity of an image encryption algorithm will greatly affect the execution time. To this end, complexity analysis is carried out for the main encryption modules in our algorithm. The total complexity isO2N2 for the block scrambling and closed-loop diffusion module. Specifically, the rectangular-ambulatory-plane cyclic shift operation is executed in 8 rings and the complexity is O8N. The subsequent bidirectional random disorganization operation would traverse all of the image pixels, whose complexity is ON2. The complexities of the intrablock diffusion and the outer-block diffusion are ON2 and ON, respectively. The total complexity for the DNA mutation module is O17N2. The complexity of encoding and decoding process is O8N2. The complexities of the concatenation operations of binary sub-blocks and the conversion of the decimal matrices are O4N2 and ON2, respectively, and the complexity for the mutation processes in the sub-blocks is O4N2. The computation time of the algorithm is related to the program design, operating environment, and so on. The proposed image encryption algorithm is carried out under MATLAB (version R2016a) on the computer with 8 GB RAM and Windows 10. In our scheme, the operations of the block scrambling and closed-loop diffusion need one-round encryption only, which costs 0.200 s, and the DNA mutation process takes 14.147 s. Since the encoding and decoding rules for each selected sub-block are different, more iterations are involved, so it takes longer execution time. The complete encryption process needs 16.906 s, which is acceptable for most of the real-time image encryption schemes and also should be reduced for some cases.


## 5. Conclusion

A novel secure image encryption scheme with block scrambling, closed-loop diffusion, and the strategy of DNA molecular mutation is presented. During the block scrambling phase, the rectangular-ambulatory-plane cyclic shift and the bidirectional random disorganization are utilized to scramble the preprocessed blocks completely, and the results of the correlation coefficients demonstrate that our scheme inherits the randomness of chaotic sequences more effectively. The feedback mechanism in the diffusion phase is constructed with a new closed-loop block diffusion strategy, which improves the ability against the differential attack. The final encryption image is obtained with the DNA molecular mutation operations. Different types of simulation experiments and theoretical analyses demonstrate that the proposed image encryption scheme has strong reliability and high security.



---

*Source: 6627005-2021-02-22.xml*