# A Novel Image Encryption Scheme Using the Composite Discrete Chaotic System

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## Abstract

**:**

## 1. Introduction

## 2. Composite Discrete Chaotic System

#### 2.1. Two-Dimensional Composite Discrete Chaotic System

**Definition 1.**

**Theorem 2.**

**Proof of Theorem 2.**

#### 2.2. Sensitivity Analysis of the Initial Values and Control Parameters in CDCS

**Step 1:**- Get 6 different CDCS by setting $k=2,3,4,5,6,7$, respectively. Then, we obtain 6 chaotic sequences with the initial value ${x}_{0}=0.65382364$ by Equations (3) and (4), respectively. If the output $|{y}_{i}|$ is larger than 1, then we let ${y}_{i}=\frac{\sqrt{k-1}-{y}_{i}}{\sqrt{k-1}}$ in Equation (3) and ${y}_{i}=\frac{\sqrt{k-1}-1+{y}_{i}}{\sqrt{k-1}-1}$ in Equation (4), and if the output ${y}_{i}<0$, then ${y}_{i}=\left|{y}_{i}\right|$. Next, we turn them into 6 binary sequence by $y=\lceil x\ast {10}^{9}\rceil \phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}2$.
**Step 2:**- Extend the plain gray image matrix to an 1-dimensional integer sequence, and transform the integer sequence into a binary sequence.
**Step 3:**- Do exclusive OR for the binary sequence with the 6 chaotic binary sequences, respectively, then get 6 diffused binary sequences
**Step 4:**

#### 2.3. Trajectory

#### 2.4. Gottwald and Melbourne Test

## 3. The Proposed Scheme

#### 3.1. Secret Key Generation

Algorithm 1. The generation of the secret key |

Input: Random key K with length of 505 bits |

Output: Secret key $(A,B)$ used in the proposed algorithm. |

1: $i=1$; |

2: for $j=0:1:10$; |

3: ${x}_{j}=({\sum}_{t=0}^{24}K[i+t]\times {2}^{t})/{2}^{25}$; |

4: $i=i+25$; |

5: end for |

6: for $j=0:1:10$; |

7: ${k}_{j}=({\sum}_{t=0}^{9}K[i+t]\times {2}^{t})/{2}^{10}$; |

8: $i=i+10$; |

9: end for |

10: for $j=0:1:10$; |

11: ${s}_{j}=({\sum}_{t=0}^{9}K[i+t]\times {2}^{t})/{2}^{10}$; |

12: $i=i+10$; |

13: end for |

14: $a=({\sum}_{t=0}^{9}K[496+t]\times {2}^{t})/{2}^{10}$; |

15: ${k}_{0}^{\prime}=({k}_{0}\times mod\phantom{\rule{4pt}{0ex}}1)+3$; |

16: for $j=0:1:10$ |

17: ${x}_{j}^{\prime}=({x}_{j}+a\times {s}_{j})mod\phantom{\rule{4pt}{0ex}}1$ |

18: end for |

19: for $j=1:1:10$ |

20: ${k}_{j}^{\prime}=(round(({k}_{j}+a\times {s}_{j})mod\phantom{\rule{4pt}{0ex}}1\times {10}^{9})mod\phantom{\rule{4pt}{0ex}}32$; |

21: end for |

#### 3.2. Encryption Process

#### 3.2.1. Bit-Level Permutation Stage

**Step 1:**- Extend the plain image gray value matrix ${({a}_{ij})}_{h\times w}$ to a binary sequence: $E=\{{E}_{1},{E}_{2},\cdots ,{E}_{h\ast 8w}\}$. Then, turn E into 8 different bit planes: ${B}_{1},{B}_{2},\cdots ,{B}_{8}$ by the following rules: ${B}_{1}=\left\{{E}_{i}\right|i\equiv 1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{2}=\left\{{E}_{i}\right|i\equiv 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{3}=\left\{{E}_{i}\right|i\equiv 3\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{4}=\left\{{E}_{i}\right|i\equiv 4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{5}=\left\{{E}_{i}\right|i\equiv 5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{6}=\left\{{E}_{i}\right|i\equiv 6\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{7}=\left\{{E}_{i}\right|i\equiv 7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${B}_{8}=\left\{{E}_{i}\right|i\equiv 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$.
**Step 2:**- Get 8 chaotic sequences of size $w\ast h$ by Equations (3) and (4) with 8 pairs parameters (${x}_{0}$, k), and denote them as ${F}_{i}=\{{F}_{i1},{F}_{i2},\cdots ,{F}_{iw\ast h}\},i=1,\cdots ,8$. If the output $|{y}_{i}|$ is larger than 1, we let ${y}_{i}=\frac{\sqrt{k-1}-{y}_{i}}{\sqrt{k-1}}$ in Equation (3) and ${y}_{i}=\frac{\sqrt{k-1}-1+{y}_{i}}{\sqrt{k-1}-1}$ in Equation (4), and if the output ${y}_{i}<0$, then ${y}_{i}=\left|{y}_{i}\right|$. Then, sort ${F}_{i}$ in ascending order and get 8 index order sequences ${I}_{i}=\{{I}_{i1},{I}_{i2},\cdots ,{I}_{ih\ast w}\},i=1,2,\cdots ,8$.
**Step 3:**- Permutate the binary sequence ${B}_{i}$ by ${I}_{i}$ in the following way to get a shuffled binary sequence ${T}_{i}=\left\{{T}_{ij}\right|j=1,2,\cdots ,h\ast w\}$:$$\begin{array}{c}\hfill {T}_{ij}={B}_{{I}_{ij}},i=1,\cdots ,8,j=1,\cdots ,h\ast w\end{array}$$
**Step 4:**- Rearrange the 8 permutated binary sequences ${T}_{i},i=1,2,\cdots ,8$ into the permutated sequence J with size $h\ast 8w$ in the following way:$$J=\{{T}_{11},{T}_{21},\cdots ,{T}_{81},{T}_{12},{T}_{22},\cdots ,{T}_{82},\cdots ,{T}_{1h\ast w},{T}_{2h\ast w},\cdots ,{T}_{8h\ast w}\}$$
**Step 5:**- Divide the intermediate binary sequence J into $h\ast w$ blocks: ${K}_{1}=\{{J}_{1},{J}_{2},\cdots ,{J}_{8}\}$, ${K}_{2}=\{{J}_{9},{J}_{10},\cdots ,{J}_{16}\}$, ⋯, ${K}_{(h\ast w)}=\{{J}_{h\ast 8w-7},{J}_{h\ast 8w-6},\cdots ,{J}_{h\ast 8w}\}$, then change each block into an integer, and get the permutated integer sequence $Int=\{Int(1),Int(2),\cdots ,Int(h\ast w)\}$, and reshape to the permutated image.

#### 3.2.2. Pixel-Level Diffusion Stage

**Step 1:****Step 2:**- For each $Int(k)\in Int,k=1,2,\cdots ,h\ast w$, do the following operations:$$\{\begin{array}{l}te{m}_{2}=Int(k)\oplus te{m}_{1}\oplus ran{d}_{2}(k)\\ C(k)=(ran{d}_{1}(te{m}_{0})+te{m}_{2}),\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}256\\ te{m}_{0}=k;te{m}_{1}=C(k)\end{array}$$
**Step 3:**- Reshape the encrypted integer sequence C back to the 2-dimensional gray value matrix of size $h\ast w$ to form the finally encrypted image.

#### 3.3. Decryption Process

#### 3.3.1. Pixel-Level Diffusion Decryption Stage

**Step 1:**- For the encrypted image, turn it into an integer sequence: $C=\{C(1),C(2),\cdots ,C(h\ast w)\}$.
**Step 2:**- Obtain 2 CDCS chaotic sequences again by Equations (3) and (4) with the same parameters used in the encryption procedure, respectively. Then they are quantified by $y=\lceil x\ast {10}^{9}\rceil \phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}256$ and named $ran{d}_{1}^{\prime}$, $ran{d}_{2}^{\prime}$, respectively.
**Step 3:**- Let the initial value $te{m}_{0}^{\prime}=h\ast w-1,te{m}_{1}^{\prime}=C(h\ast w-1)$. For each $C(k)\in C,k=h\ast \phantom{\rule{3.33333pt}{0ex}}w,h\ast \phantom{\rule{3.33333pt}{0ex}}(w-1),\cdots ,2$, do the following operations to get the permutated sequence $In{t}^{\prime}$:$$\{\begin{array}{l}te{m}_{2}^{\prime}=(C(k)-ran{d}_{1}^{\prime}(te{m}_{0}^{\prime})),(\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}256),\\ In{t}^{\prime}(k)=te{m}_{2}^{\prime}\oplus te{m}_{1}^{\prime}\oplus ran{d}_{2}^{\prime}(k-1),\\ te{m}_{0}^{\prime}=k-1;te{m}_{1}^{\prime}=C(k-1).\end{array}$$

#### 3.3.2. Bit-Level Permutation Decryption Stage

**Step 1:**- Extend the decrypted sequence $In{t}^{\prime}$ to a binary sequence $G=\{{g}_{1},{g}_{2},\cdots ,{g}_{h\ast 8w}\}$, where $h\ast w$ is the length of $In{t}^{\prime}$, respectively. Then, turn binary sequence G into 8 different bit planes ${T}_{1},{T}_{2},\cdots ,{T}_{8}$ by the following rules: ${T}_{1}=\left\{{g}_{i}\right|i\equiv 1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{2}=\left\{{g}_{i}\right|i\equiv 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{3}=\left\{{g}_{i}\right|i\equiv 3\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{4}=\left\{{g}_{i}\right|i\equiv 4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{5}=\left\{{g}_{i}\right|i\equiv 5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{6}=\left\{{g}_{i}\right|i\equiv 6\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{7}=\left\{{g}_{i}\right|i\equiv 7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$, ${T}_{8}=\left\{{g}_{i}\right|i\equiv 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}8,i=1,2,\cdots ,h\ast 8w\}$.
**Step 2:**- Get the 8 chaotic sequences of size $w\times h$ again by Equations (3) and (4) with the 8 pairs same parameters (${x}_{0}$, k) used in the encryption procedure, and denote them as ${H}_{i}=\{{H}_{i1},{H}_{i2},\cdots ,{H}_{iw\ast h}\},i=1,\cdots ,8$. If the output $|{y}_{i}|$ is larger than 1, we let ${y}_{i}=\frac{\sqrt{k-1}-{y}_{i}}{\sqrt{k-1}}$ in Equation (3) and ${y}_{i}=\frac{\sqrt{k-1}-1+{y}_{i}}{\sqrt{k-1}-1}$ in Equation (4), and if the output ${y}_{i}<0$, then ${y}_{i}=\left|{y}_{i}\right|$. Then, sort ${H}_{i}$ in ascending order and get 8 index order sequences ${I}_{i}^{\prime}=\{{I}_{i1}^{\prime},{I}_{i2}^{\prime},\cdots ,{I}_{ih\ast w}^{\prime}\},i=1,2,\cdots ,8$.
**Step 3:**- Permutate the binary sequence ${T}_{i}$ by ${I}_{i}^{\prime}$, in the following way to obtain the original binary sequence ${B}_{i}^{\prime}$:$$\begin{array}{c}\hfill {B}_{{I}_{ij}^{\prime}}^{\prime}={T}_{ij}=,i=1,\cdots ,8,j=1,\cdots ,h\ast w\end{array}$$
**Step 4:**- Rearrange the 8 permutated binary sequences ${B}_{1}^{\prime},\cdots ,{B}_{8}^{\prime}$ into the permutated sequence P of size $h\ast 8w$ in the following way: $Q=\{{B}_{1}^{\prime}(1),{B}_{2}^{\prime}(1),\cdots ,{B}_{8}^{\prime}(1)$, ${B}_{1}^{\prime}(2),{B}_{2}^{\prime}(2),\cdots ,{B}_{8}^{\prime}(2),\cdots ,$ ${B}_{1}^{\prime}(h\ast w),{B}_{2}^{\prime}(h\ast w),\cdots ,{B}_{8}^{\prime}(h\ast w)\}$.
**Step 5:**- Divide the intermediate binary sequence Q into $(h\ast w)$ blocks: ${Y}_{1}=\{{B}_{1}^{\prime},{B}_{2}^{\prime},\cdots ,{B}_{8}^{\prime}\}$, ${Y}_{2}=\{{B}_{9}^{\prime},{B}_{10}^{\prime},\cdots ,{B}_{16}^{\prime}\}$, ⋯, ${Y}_{h\ast w}=\{{B}_{h\ast 8w-7}^{\prime},{B}_{h\ast 8w-6}^{\prime},\cdots ,{B}_{h\ast 8w}\}$, then turn each block into an integer, and get the decrypted integer sequence $P=\{P(1),P(2),\cdots ,P(h\ast w)\}$.
**Step 6:**- Reshape the decrypted integer sequence P back to the 2-dimensional gray value matrix of size $h\ast w$ to get the finally decrypted image.

## 4. Simulation Results and Security Analyses

#### 4.1. Gray and Color Image Encryption

#### 4.2. Key Size Analysis

#### 4.3. The Chi-Square Test Analysis of Cipher Image

#### 4.4. Correlation Analysis

#### 4.5. Information Entropy Analysis

#### 4.6. Local Shannon Entropy Analysis

#### 4.7. Key Sensitivity Analysis

#### 4.7.1. Encrypted Key Sensitivity Analysis

- $Ke{y}_{1}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{2}$:
- A = (0.34556789,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{3}$:
- A = (0.34556788, 0.13456791, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{4}$:
- A = (0.34556788,0.13456790, 0.24567982, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{5}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567931, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{6}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345680, 0.53456794, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{7}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456795, 0.64567958, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{8}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567959, 0.76456793, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10).
- $Ke{y}_{9}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456794, 0.86456797,0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10)
- $Ke{y}_{10}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456798, 0.754712846, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10)
- $Ke{y}_{11}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797, 0.754712847, 0.567889322), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10)
- $Ke{y}_{12}$:
- A = (0.34556788,0.13456790, 0.24567981, 0.34567932, 0.42345679, 0.53456794, 0.64567958, 0.76456793, 0.86456797, 0.754712846, 0.567889323), B = (π, 8, 7, 6, 5, 4, 3, 2, 13,11,10)

#### 4.7.2. Decrypted Key Sensitivity Analysis

#### 4.8. Chosen/Known Plaintext Attacks Analysis

#### 4.9. Differential Attack Analysis

#### 4.10. Randomness Analysis of CDCS

#### 4.11. Speed Performance

#### 4.12. Robustness of the Proposed Algorithm in Noise and Data Loss

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The diffused images by different control parameter k of CDCS. (

**a**) Lena image; (

**b**) k = 2; (

**c**) k = 3; (

**d**) k = 4; (

**e**) k = 5; (

**f**) k = 6; (

**g**) k = 7.

**Figure 2.**Trajectories of different chaotic maps. (

**a**) CDCS; (

**b**) single discrete dynamic chaotic system Equation (3); (

**c**) Logistic map.

**Figure 4.**The histogram distribution analysis. (

**a**) Plain image; (

**b**) The histogram distribution of (a); (

**c**) Permutated image; (

**d**) The histogram distribution of (c); (

**e**) Cipher image; (

**f**) The histogram distribution of (e).

**Figure 5.**Color image encryption effect. (

**a**) House image; (

**b**) The corresponding cipher image of (a); (

**c**) Peppers image; (

**d**) The corresponding cipher of (c).

**Figure 6.**The correlation analyses of the proposed algorithms. (

**a**) Horizontal direction of plain image lena; (

**b**) Horizontal direction of cipher image; (

**c**) Vertical direction of plain image; (

**d**) Vertical direction of cipher image; (

**e**) Diagonal direction of plain image; (

**f**) Diagonal direction of cipher image.

**Figure 7.**Encrypted key sensitivity analysis.

**(a)**$Ke{y}_{1}$;

**(b)**$Ke{y}_{2}$;

**(c)**$Ke{y}_{3}$;

**(d)**$Ke{y}_{4}$;

**(e)**$Ke{y}_{5}$;

**(f)**$Ke{y}_{6}$;

**(g)**$Ke{y}_{7}$;

**(h)**$Ke{y}_{8}$;

**(i)**$Ke{y}_{9}$;

**(j)**$Ke{y}_{10}$;

**(k)**$Ke{y}_{11}$;

**(l)**$Ke{y}_{12}$.

**Figure 8.**Decrypted key sensitivity analysis. (

**a**) $Ke{y}_{1}$; (

**b**) $Ke{y}_{2}$; (

**c**) $Ke{y}_{3}$; (

**d**) $Ke{y}_{4}$; (

**e**) $Ke{y}_{5}$; (

**f**) $Ke{y}_{6}$; (

**g**) $Ke{y}_{7}$; (

**h**) $Ke{y}_{8}$; (

**i**) $Ke{y}_{9}$; (

**j**) $Ke{y}_{10}$; (

**k**) $Ke{y}_{11}$; (

**l**) $Ke{y}_{12}$.

**Figure 11.**Robustness analysis results. (

**a**) 3.6% data loss with a black square; (

**b**) 3.6% data modification with a square; (

**c**) 5% Salt and Pepper noise; (

**d**) 60.12% data loss with a white square; (

**e**) 94.23% data loss with a black square.

Chaotic Sequence | x_{0} = 0.65382364 | x_{0} = 0.65382365 |
---|---|---|

${y}_{2001}$ | 0.6916087935 | 0.5510774642 |

${y}_{2002}$ | 0.6190457067 | 0.3196168464 |

${y}_{2003}$ | 0.4879461173 | 0.3993617501 |

${y}_{2004}$ | 0.8447332446 | 0.5513615043 |

${y}_{2005}$ | 0.8303411884 | 0.3205043035 |

${y}_{2006}$ | 0.8128237059 | 0.4008410954 |

${y}_{2007}$ | 0.7909787683 | 0.5546711225 |

${y}_{2008}$ | 0.7250674640 | 0.3306693894 |

${y}_{2009}$ | 0.6974535701 | 0.4180539361 |

${y}_{2010}$ | 0.6709209552 | 0.595164073 |

Control Parameter | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 |
---|---|---|---|---|---|---|

k = 2 | 0 | 0.99607849 | 0.99603271 | 0.99586487 | 0.99604797 | 0.99598694 |

k = 3 | 0.99607849 | 0 | 0.99621582 | 0.99627686 | 0.99639893 | 0.9962616 |

k = 4 | 0.99603271 | 0.99621582 | 0 | 0.99629211 | 0.99645996 | 0.9962616 |

k = 5 | 0.99586487 | 0.99627686 | 0.99629211 | 0 | 0.99615479 | 0.99568176 |

k = 6 | 0.99604797 | 0.99639893 | 0.99645996 | 0.99615479 | 0 | 0.99591064 |

k = 7 | 0.99598694 | 0.9962616 | 0.9962616 | 0.99568176 | 0.99591064 | 0 |

Initial Value x_{0} | Control Parameters k | Test Results |
---|---|---|

0.12345678 | k = 2 | 0.99823000 |

0.12345679 | k = 2 | 0.99839183 |

0.12345677 | k = 2 | 0.99790976 |

0.12345676 | k = 2 | 0.99798716 |

0.21345678 | k= 3 | 0.99782871 |

0.21345678 | k = 4 | 0.99831404 |

0.21345678 | k = 5 | 0.99842179 |

0.21345678 | k = 6 | 0.99854142 |

0.321345678 | k = 7 | 0.99735072 |

Image Name | p-value |
---|---|

5.2.08 | 0.257198003 |

5.2.09 | 0.22534446 |

5.2.10 | 0.220229861 |

7.1.01 | 0.200104753 |

7.1.02 | 0.115958523 |

7.1.03 | 0.478716242 |

7.1.04 | 0.477272383 |

7.1.05 | 0.536532281 |

7.1.06 | 0.681580846 |

7.1.07 | 0.492913738 |

7.1.08 | 0.919338576 |

7.1.09 | 0.101262279 |

boat.512 | 0.631836356 |

elaine | 0.52057636 |

lena | 0.224053673 |

goldhill | 0.883876476 |

peppers | 0.13530398 |

baboon | 0.763747416 |

house(R) | 0.099225363 |

house(G) | 0.412077954 |

house(B) | 0.285299456 |

**Table 5.**Correlation coefficients analysis. (Numbers in bold face means the crresoponding encryption scheme has the smallest correlation coefficients.)

Test Image | Direction | Plain Image | Proposed Scheme | Ref. [16] | Ref. [22] | Ref. [8] | Ref. [25] |
---|---|---|---|---|---|---|---|

horizontal | 0.98727175 | 0.00911870 | −0.01598859 | 0.00921633 | −0.03444986 | 0.0255741 | |

lena | vertical | 0.99060282 | −0.02799349 | −0.00928994 | 0.00438226 | 0.0162397 | −0.0060722 |

diagonal | 0.98232025 | −0.008005781 | 0.00955705 | 0.00984828 | 0.05472108 | 0.03712793 | |

horizontal | 0.98359562 | 0.00984443 | −0.01681935 | 0.02161067 | 0.04823954 | 0.00106722 | |

goldhill | vertical | 0.97498297 | 0.01843329 | 0.03527012 | 0.03929575 | −0.01953583 | 0.01852442 |

diagonal | 0.96849169 | 0.002687612 | 0.006798479 | 0.02047153 | −0.01473057 | 0.01321081 | |

horizontal | 0.98486792 | 0.06072394 | 0.0237943 | 0.02161067 | −0.00350778 | −0.0115003 | |

peppers | vertical | 0.97916019 | −0.00116499 | −0.011170982 | −0.037837 | −0.021423 | −0.00237434 |

diagonal | 0.97515696 | −0.00571419 | 0.00664635 | 0.02047153 | 0.02345089 | −0.00098435 | |

horizontal | 0.74954747 | −0.03856932 | −0.02999071 | 0.0161014 | 0.00289877 | −0.00547718 | |

house(R) | vertical | 0.80262072 | 0.002023 | −0.0154941 | −0.04053886 | 0.018131106 | −0.01215746 |

diagonal | 0.60609133 | 0.00207905 | −0.01487206 | −0.00227398 | 0.018180302 | −0.01803246 | |

horizontal | 0.76294284 | 0.00181239 | −0.0161565 | 0.0485065 | −0.00423018 | −0.03230305 | |

house(G) | vertical | 0.86429319 | −0.01279169 | −0.00183728 | 0.01576013 | 0.00157326 | 0.0002708 |

diagonal | 0.66868095 | 0.00241394 | 0.00156149 | −0.03474189 | -0.003761 | −0.00867577 | |

horizontal | 0.90852712 | −0.00921239 | 0.02068469 | 0.015610986 | −0.04302791 | 0.02820804 | |

house(B) | vertical | 0.9477393 | 0.0034053 | −0.02170484 | −0.00567228 | 0.0137757 | 0.0107359 |

diagonal | 0.86744223 | −0.0217718 | −0.0075945 | 0.01271975 | 0.00921989 | 0.03581673 |

Test Image | Plain Image | The Proposed Scheme | Ref. [16] | Ref. [22] | Ref. [8] | Ref. [25] |
---|---|---|---|---|---|---|

5.2.08 | 7.201008 | 7.9992989 | 7.9970096 | 7.9993075 | 7.999206 | 7.9993742 |

5.2.09 | 6.9939942 | 7.9992833 | 7.9968423 | 7.9992492 | 7.9991342 | 7.9992579 |

5.2.10 | 5.7055602 | 7.9992883 | 7.9969656 | 7.9993485 | 7.9992213 | 7.9993323 |

7.1.01 | 6.0274148 | 7.9993239 | 7.9972729 | 7.9993146 | 7.9991445 | 7.9993389 |

7.1.02 | 4.0044994 | 7.9993947 | 7.9931779 | 7.999289 | 7.9989956 | 7.9993285 |

7.1.03 | 5.49574 | 7.9992167 | 7.997577 | 7.9993951 | 7.9991431 | 7.9993135 |

7.1.04 | 6.1074181 | 7.9991862 | 7.9970146 | 7.9993017 | 7.999126 | 7.9993481 |

7.1.05 | 6.5631956 | 7.9992767 | 7.9969023 | 7.9993046 | 7.9991403 | 7.9993864 |

7.1.06 | 6.6952834 | 7.999398 | 7.9975578 | 7.999246 | 7.9992993 | 7.9992284 |

7.1.07 | 5.9915988 | 7.9992502 | 7.997237 | 7.9992476 | 7.9989706 | 7.9992728 |

7.1.08 | 5.053448 | 7.9992442 | 7.9967758 | 7.9993288 | 7.9989898 | 7.9991881 |

7.1.09 | 6.1898137 | 7.99938 | 7.9972559 | 7.9991956 | 7.9991552 | 7.9992166 |

boat.512 | 7.1913702 | 7.9993893 | 7.997026 | 7.99931 | 7.9991832 | 7.9993511 |

lena | 7.4455676 | 7.9993283 | 7.9973605 | 7.9993589 | 7.999155 | 7.9992604 |

goldhill | 7.4777796 | 7.9993354 | 7.9974798 | 7.9992933 | 7.9992657 | 7.999319 |

baboon | 7.3735278 | 7.9992275 | 7.9970364 | 7.9993183 | 7.9991787 | 7.9993072 |

peppers | 7.5714776 | 7.9992535 | 7.9974015 | 7.9991921 | 7.9992645 | 7.9992152 |

elaine | 7.4664262 | 7.9993301 | 7.9972385 | 7.9993498 | 7.9991789 | 7.9992656 |

house(R) | 7.415627 | 7.9992942 | 7.9970608 | 7.999344 | 7.9992396 | 7.9992674 |

house(G) | 7.2294792 | 7.9993214 | 7.9974495 | 7.9992617 | 7.9991642 | 7.9993073 |

house(B) | 7.4353838 | 7.9992996 | 7.9973264 | 7.9992621 | 7.999225 | 7.9992576 |

average | 6.6016959 | 7.9993039 | 7.9971971 | 7.9992913 | 7.9991954 | 7.9992817 |

**Table 7.**Local Shannon entropy analysis. (Numbers in bold face means the test image can pass the LSE test.)

Test Image | The Proposed Scheme | Ref. [16] | Ref. [22] | Ref. [8] | Ref. [25] |
---|---|---|---|---|---|

5.2.08 | 7.9028691 | 7.9055763 | 7.9053199 | 7.902356 | 7.9028432 |

5.2.09 | 7.9037385 | 7.9029891 | 7.900893 | 7.899853 | 7.9025761 |

5.2.10 | 7.9030217 | 7.9041229 | 7.9026793 | 7.902654 | 7.9016977 |

7.1.01 | 7.9031848 | 7.9031774 | 7.9031721 | 7.902634 | 7.9027515 |

7.1.02 | 7.9018403 | 7.8976268 | 7.9003936 | 7.901634 | 7.902448 |

7.1.03 | 7.9035924 | 7.9011942 | 7.901988 | 7.905423 | 7.9039657 |

7.1.04 | 7.902570 | 7.9060551 | 7.9023579 | 7.902125 | 7.9055074 |

7.1.05 | 7.9050477 | 7.9018336 | 7.9022384 | 7.883653 | 7.9044964 |

7.1.06 | 7.9025262 | 7.9058613 | 7.9008032 | 7.902356 | 7.9009599 |

7.1.07 | 7.9018694 | 7.9028083 | 7.9000806 | 7.902364 | 7.9044062 |

7.1.08 | 7.9031321 | 7.9028933 | 7.9032622 | 7.904456 | 7.9024535 |

7.1.09 | 7.9030009 | 7.8998789 | 7.9017465 | 7.90312 | 7.9025151 |

boat.512 | 7.9026992 | 7.9000555 | 7.9017958 | 7.901879 | 7.9009823 |

Elaine | 7.9009196 | 7.9006208 | 7.9046929 | 7.902989 | 7.9029109 |

Lena | 7.903462 | 7.902938 | 7.900975 | 7.904512 | 7.904671 |

Goldhill | 7.9025015 | 7.9009052 | 7.902251 | 7.9015092 | 7.9020145 |

peppers | 7.9024452 | 7.9016155 | 7.9040266 | 7.9053045 | 7.9007481 |

baboon | 7.9033626 | 7.9004801 | 7.9001366 | 7.902999 | 7.9013492 |

house(R) | 7.9019456 | 7.9007318 | 7.9029686 | 7.9010447 | 7.905035 |

house(G) | 7.9019228 | 7.904166 | 7.9023234 | 7.9058879 | 7.9033633 |

house(B) | 7.9026658 | 7.9014576 | 7.8998792 | 7.1993477 | 7.9046128 |

pass rate | 16/21 | 8/21 | 11/21 | 13/21 | 10/21 |

$\mathit{K}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{1}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{2}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{3}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{4}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{5}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{6}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{7}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{8}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{9}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{10}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{11}}$ | ${\mathit{K}\mathit{e}\mathit{y}}_{\mathbf{12}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

$Ke{y}_{1}$ | 0 | 0.99617 | 0.99600 | 0.99207 | 0.98381 | 0.96877 | 0.93839 | 0.97691 | 0.97549 | 0.97625 | 0.99577 | 0.99630 |

$Ke{y}_{2}$ | 0.99617 | 0 | 0.99631 | 0.99622 | 0.99599 | 0.99621 | 0.99612 | 0.99609 | 0.99616 | 0.99599 | 0.99588 | 0.99615 |

$Ke{y}_{3}$ | 0.99600 | 0.99631 | 0 | 0.99613 | 0.99635 | 0.99619 | 0.99614 | 0.99597 | 0.99611 | 0.99609 | 0.99611 | 0.99604 |

$Ke{y}_{4}$ | 0.99207 | 0.99622 | 0.99613 | 0 | 0.99246 | 0.99254 | 0.99178 | 0.99192 | 0.99210 | 0.99217 | 0.99598 | 0.99597 |

$Ke{y}_{5}$ | 0.98381 | 0.99599 | 0.99635 | 0.99246 | 0 | 0.98441 | 0.98413 | 0.98443 | 0.98447 | 0.98463 | 0.99611 | 0.99612 |

$Ke{y}_{6}$ | 0.96877 | 0.99621 | 0.99619 | 0.99254 | 0.98441 | 0 | 0.96814 | 0.96885 | 0.96847 | 0.96888 | 0.99614 | 0.99615 |

$Ke{y}_{7}$ | 0.93839 | 0.99612 | 0.99614 | 0.99178 | 0.98413 | 0.96814 | 0 | 0.93682 | 0.93850 | 0.93759 | 0.99593 | 0.99614 |

$Ke{y}_{8}$ | 0.97691 | 0.99609 | 0.99597 | 0.99192 | 0.98443 | 0.96885 | 0.93682 | 0 | 0.95173 | 0.90193 | 0.99608 | 0.99616 |

$Ke{y}_{9}$ | 0.97549 | 0.99616 | 0.99611 | 0.99210 | 0.98447 | 0.96847 | 0.93850 | 0.95173 | 0 | 0.95015 | 0.99604 | 0.99623 |

$Ke{y}_{10}$ | 0.97625 | 0.99599 | 0.99609 | 0.99217 | 0.98463 | 0.96888 | 0.93759 | 0.90193 | 0.95015 | 0 | 0.99617 | 0.99622 |

$Ke{y}_{11}$ | 0.99577 | 0.99588 | 0.99611 | 0.99598 | 0.99611 | 0.99614 | 0.99593 | 0.99608 | 0.99604 | 0.99617 | 0 | 0.99606 |

$Ke{y}_{12}$ | 0.99630 | 0.99615 | 0.99604 | 0.99597 | 0.99612 | 0.99615 | 0.99614 | 0.99616 | 0.99623 | 0.99622 | 0.99606 | 0 |

Round | The Proposed Scheme | Ref. [16] | Ref. [22] | Ref. [8] | Ref. [25] |
---|---|---|---|---|---|

1 | 0.6696014404 | 0.000015259 | 0.00654386 | 0.9960098267 | 0.9963431625 |

2 | 0.995967865 | 0.000015259 | 0.80495842 | 0.9960746765 | 0.9959527564 |

3 | 0.9961090088 | 0.000015259 | 0.99615466 | 0.9961242676 | 0.9965357538 |

4 | 0.9961585999 | 0.000015259 | 0.99595247 | 0.9961776733 | 0.9960346326 |

5 | 0.9961013794 | 0.000015259 | 0.99616793 | 0.9959716797 | 0.9961644326 |

Round | The Proposed Scheme | Ref. [16] | Ref. [22] | Ref. [8] | Ref. [25] |
---|---|---|---|---|---|

1 | 0.2630562577 | 0.000012087 | 0.00321365 | 0.3326689627 | 0.3360676146 |

2 | 0.3353189655 | 0.000012087 | 0.24366436 | 0.3348488303 | 0.335123463 |

3 | 0.3346790837 | 0.000012087 | 0.33401162 | 0.3351597805 | 0.3350163487 |

4 | 0.3342736338 | 0.000012087 | 0.33389708 | 0.3346462175 | 0.3344254165 |

5 | 0.3340085647 | 0.000012087 | 0.33441636 | 0.3348087535 | 0.3343425278 |

Test Name | p-value | Results |
---|---|---|

Frequency test | 0.5731 | Success |

Block Frequency test | 0.6825 | Success |

Cusum-Forward test | 0.9293 | Success |

Cusum-Reverse test | 0.3514 | Success |

Runs test | 0.5536 | Success |

Long Runs test of Ones | 0.6154 | Success |

Binary Matrix Rank Test | 0.7635 | Success |

Spectral DFT test | 0.4674 | Success |

Non-overlapping test Templates (m = 9, B = 000000001) | 0.8710 | Success |

Overlapping test Templates (m = 9) | 0.9241 | Success |

Maurer’s Universal test (L = 7, Q = 1280) | 0.3533 | Success |

Approximate Entropy test (m = 5) | 0.9987 | Success |

Random Excursions test (x = +1) | 0.2085 | Success |

Lempel Ziv compression test | 0.6784 | Success |

Linear complexity test | 0.2314 | Success |

Random Excursions Variant test (x = −1) | 0.5811 | Success |

Serial test (m = 5,$\nabla {\phi}_{m}^{2}$) | 0.8989 | Success |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhu, H.; Zhang, X.; Yu, H.; Zhao, C.; Zhu, Z.
A Novel Image Encryption Scheme Using the Composite Discrete Chaotic System. *Entropy* **2016**, *18*, 276.
https://doi.org/10.3390/e18080276

**AMA Style**

Zhu H, Zhang X, Yu H, Zhao C, Zhu Z.
A Novel Image Encryption Scheme Using the Composite Discrete Chaotic System. *Entropy*. 2016; 18(8):276.
https://doi.org/10.3390/e18080276

**Chicago/Turabian Style**

Zhu, Hegui, Xiangde Zhang, Hai Yu, Cheng Zhao, and Zhiliang Zhu.
2016. "A Novel Image Encryption Scheme Using the Composite Discrete Chaotic System" *Entropy* 18, no. 8: 276.
https://doi.org/10.3390/e18080276