# A Novel Five-Dimensional Three-Leaf Chaotic Attractor and Its Application in Image Encryption

^{1}

^{2}

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^{*}

## Abstract

**:**

^{150}that had strong key sensitivity. It effectively resisted the attacks of statistical analysis and gray value analysis, and had a good encryption effect on the encryption of digital images.

## 1. Introduction

## 2. New Five-Dimensional Chaotic System

#### 2.1. Dissipative Analysis

_{0}e

^{−(a+c+e)t}. Clearly, the volume element V

_{0}shrinks to the volume V

_{0}e

^{−(a+c+e)t}at moment t. Now consider when $t\to \infty $. Each volume element that contains the system trajectory shrinks to 0 at an exponential rate a + c + e = −9.

#### 2.2. Balance Point Analysis

#### 2.3. Time Series Chart

#### 2.4. Phase Diagram Analysis

#### 2.5. Bifurcation Diagram

#### 2.6. Power Spectrum Analysis

## 3. Circuit Design and Experimental Results

## 4. Related Information

#### 4.1. Convolution Operation

**Step 1:**Extend matrix $A$ to an $(n+2)\times (n+2)$ matrix with 0.

**Step2:**Obtain matrix $C={({c}_{ij})}_{nn}$ using a convolution operation between matrix $A$ and convolution kernel $h$, where

#### 4.2. “Same OR” Operation

## 5. Algorithm Descriptions

#### 5.1. Encryption Algorithm Description

**Step 1:**Input a grayscale image $A$, initial value of the chaotic system ${y}_{0}=[0.6,0.1,0.2,0.5,0.4]$, and the step size $L=0.01$.

**Step 2:**Find the total iteration time $T=(250+P\times P)\times L$, where $P=\mathrm{max}(M,N)$.

**Step 3:**Call the ode45 function, iterate system (2), and generate five chaotic sequences.

**Step 4:**The five chaotic sequences are treated separately as follows.

**Step 5:**Given a convolution kernel of $3\times 3$.

**Step 6:**For the five $M\times N$ matrices $A1(:,:,i)$, $i=1,2,3,4,5$ obtained in step 4, apply the convolution operation with the given convolution kernel to obtain five $M\times N$ matrices $Y(:,:,i)$, $i=1,2,3,4,5$.

**Step 7:**Divide grayscale image $A$ into five regions starting from the center area in the proportion $l=M:N$. Input the value of ${m}_{1}$ using ${n}_{1}={m}_{1}\xf7l$, and obtain n

_{1}. The formulas for m

_{i}and n

_{i}are as follows.

_{1}× n

_{1}, T2 = m

_{2}× n

_{2}, T3 = m

_{3}× n

_{3}, T4 = m

_{4}× n

_{4}, T5 = M × N as shown in Figure 10.

**Step 8:**Divide the region $H1$ in matrix $Y(:,:,1)$ according to the starting point and size of $T1$ in grayscale image $A$, as shown in Figure 11.

**Step 9:**Convert matrix $T1$ and matrix $H1$ sums of one row and $\left({m}_{1}\times {n}_{1}\right)$ column matrix $T11$ and $H11$, respectively. $H11$ is treated as follows.

**Step 10:**Combine $T11$ and $HH$, and perform a bitwise “$XOR$” operation to obtain matrix $B1$.

**Step 11:**Process $H11$ as follows to obtain $M1$ and $M2$.

**Step 12:**Convert matrix $B1$ into binary matrix $F1$.

**Step 13:**Disturb each line of binary numbers in each row of $F1$, and then obtain matrix $C1$.

**Step 14:**Disturb all column elements of $C1$, and then obtain $C2$. The disturbance formula is as follows.

**Step 15:**Convert binary number matrix C2 to decimal number matrix D1.

**Step 17:**The area $H2$ is taken out according to $T2$ in the starting point and size in $A$ from $Y(:,:,2)$, as shown in Figure 13.

**Step 18:**Convert matrix $T2$ and matrix $H2$ sums of one row and $\left({m}_{2}\times {n}_{2}\right)$ column matrix $T22$ and $H22$, respectively. $H22$ is treated as follows.

**Step 19:**Combine $T22$ and $HH1$, and perform a bitwise “same OR” operation to obtain matrix $B2$

**Step 20:**Process $H22$ to obtain $M3$ and $M4$ as follows.

**Step 21:**Convert matrix $B2$ into binary matrix $F2$.

**Step 22:**Scramble each row of binary numbers in $F2$ to obtain matrix $C3$. The scrambling formula is as follows.

**Step 23:**Disturb all the column elements of $C3$ to obtain $C4$. The scrambling formula is as follows.

**Step 24:**Convert binary number matrix $C4$ to decimal number matrix $D2$.

**Step26:**According to $T3$ from the starting point and size in the $A$, take the area $H3$ from $Y(:,:,3)$, as shown in Figure 15.

**Step 27:**Convert the matrices $T3$ and $H3$ into matrices $T33$ and $H33$ with one row and $\left({m}_{3}\times {n}_{3}\right)$ columns, respectively, and process $H33$ as follows.

**Step 28:**Combine $T33$ and $HH2$, and perform a bitwise “same OR” operation to obtain matrix $B3$.

**Step 29:**Process $H33$ as follows to obtain $M5$ and $M6$.

**Step 30:**Convert matrix $B3$ into binary matrix $F3$.

**Step 31:**Scramble the binary numbers in each row of $F3$ to obtain matrix $C5$. The scrambling formula is as follows.

**Step 32:**Disturb all the column elements of $C5$ to obtain $C6$. The scrambling formula is as follows.

**Step 33:**Convert binary number matrix $C6$ to decimal number matrix $D3$.

**Step 35:**Take region $H4$ from $Y(:,:,4)$ according to the starting point and size of $T4$ in $A$, as shown in Figure 17.

**Step 36:**Convert matrices $T4$ and $H4$ into matrices $T44$ and $H44$ with one row and $\left({m}_{4}\times {n}_{4}\right)$ columns, respectively, and process $H44$ as follows.

**Step 37:**Combine $T44$ and $HH3$, and perform a bitwise “XOR” operation to obtain matrix $B4$.

**Step 38:**Process $H44$ to obtain $M7$ and $M8$ as follows.

**Step 39:**Convert matrix $B4$ into binary matrix $F4$.

**Step 40:**Scramble the binary numbers in each row in $F4$ to obtain matrix $C7$. The scrambling formula is as follows.

**Step 41:**Disturb all the column elements of $C7$ to obtain $C8$. The scrambling formula is as follows.

**Step42:**Convert binary number matrix $C8$ to decimal number matrix $D4$.

**Step 44:**Convert matrix $T5$ and $Y(:,:,5)$ to matrix $T55$ and $H55$ with one row and $\left(M\times N\right)$ columns, respectively, and $H55$ is treated as follows.

**Step 45:**Combine $T55$ and $HH4$, and perform a bitwise “same OR” operation to obtain matrix $B5$.

**Step 46:**Process $H55$ as follows to obtain $M9$ and $M10$.

**Step 47:**Convert matrix $B5$ into binary matrix $F5$.

**Step 48:**Scramble the binary numbers in each row of $F5$ to obtain matrix $C9$. The scrambling formula is as follows.

**Step 49:**Disturb all the column elements of $C9$. The scrambling formula is as follows.

**Step 50:**Convert binary number matrix $C10$ to decimal number matrix $D5$, and $D5$ is the final encrypted image. The result is shown in Figure 19.

#### 5.2. Decryption Algorithm Description

**Step 1:**Input the initial value of the chaotic system ${y}_{0}=[0.6,0.1,0.2,0.5,0.4]$ and step size $L=$ $0.02$, and find the total iteration time $T=(250+P\times P)\times L$, where $P=\mathrm{max}(M,N)$.

**Step 2:**Call the ode45 function, iterate system (2), and generate five chaotic sequences.

**Step 3:**The five chaotic sequences are treated as follows.

**Step 4:**Regarding the matrix $A1(:,:,i)$, $i=1,2,3,4,5$ of $M\times N$, the five obtained in the previous step convolution operation with the given convolution kernel of the encryption algorithm include five matrix $Y1(:,:,i)$, $i=1,2,3,4,5$ of $M\times N$ are obtained.

**Step 5:**Divide the encrypted image $D5$ of $M\times N$ into five regions in proportion to the middle region, with $l=M:N,$ $E1=({m}_{1}\times {n}_{2}),E2({m}_{2}\times {n}_{2}),E3({m}_{3}\times {n}_{3}),E4({m}_{4}\times {n}_{4}),E5(M\times N)$, as shown in Figure 20.

**Step6:**Restore $E5,E4,E3,E2,E1$ in turn to obtain decrypted image $A$.

## 6. Experimental Results and Analysis

#### 6.1. Experiment Platform

#### 6.2. Experimental Result

#### 6.3. Key Space Analysis

#### 6.4. Convolution Nuclear Sensitivity Analysis

^{11}and performed convolution operations with the convolution kernel of $3\times 3$. The convolution kernel of $3\times 3$ in this algorithm was $c=[1,2,3;4,5,6;7,8,9]$. In the decryption process, when any of the parameters in the convolution kernel were slightly changed, the original image could not be successfully decrypted. When any parameter in the convolution kernel changed slightly with ${10}^{-15}$, the decrypted image was blurred, but the outline could be seen, as shown in Figure 22c. When any of the parameters in the convolution kernel was slightly changed with ${10}^{-14}$, the decrypted image could not substantially display the plaintext image information, as shown in Figure 22d. When any parameter in the convolution kernel changed slightly with ${10}^{-13}$, the plaintext image information could not be solved at all, as shown in Figure 22e. Taking the Lena image as an example, we made a slight change to the parameters of the second row and the second column of the convolution kernel: ${c}_{1}=[1,2,3;4,5.000000000000001,6;7,8,9]$, ${c}_{2}=[1,2,3;4,5.00000000000001,6;7,8,9]$, ${c}_{3}=[1,2,3;4,$ $5.0000000000001,6;7,8,9]$. The plaintext images, ciphertext images, and the corresponding decrypted images of ${c}_{1}$, ${c}_{2}$, and ${c}_{3}$ are shown in Figure 22.

#### 6.5. Key Sensitivity Analysis

#### 6.6. Information Entropy Analysis

#### 6.7. Histogram Analysis

#### 6.8. Histogram Statistics

#### 6.9. Correlation Analysis of Adjacent Pixels

#### 6.10. Robustness Analysis

#### 6.10.1. Quality Metrics Analysis

#### 6.10.2. Occlusion Attack Analysis

#### 6.10.3. Noise Attack Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time series diagram (

**a**) $x-t$ time series, (

**b**) $y-t$ time series, (

**c**) $z-t$ time series, (

**d**) $w-t$ time series, (

**e**) $v-t$ time series.

**Figure 2.**3D phase diagram (

**a**) $x-y-z$ 3D map, (

**b**) $x-y-w$ 3D map, (

**c**) $x-y-v$ 3D map, (

**d**) $x-z-w$ 3D map, (

**e**) $x-z-v$ 3D map, (

**f**) $x-w-v$ 3D map, (

**g**) $y-z-w$ 3D map, (

**h**) $y-z-v$ 3D map, (

**i**) $y-w-v$ 3D map, (

**j**) $z-w-v$ 3D map.

**Figure 3.**Plane phase diagram (

**a**) $x-y$ flat, (

**b**) $x-z$ flat, (

**c**) $x-w$ flat, (

**d**) $x-v$ flat, (

**e**) $y-z$ flat, (

**f**) $y-w$ flat, (

**g**) $y-v$ flat, (

**h**) $z-w$ flat, (

**i**) $z-v$ flat, and (

**j**) $w-v$ flat.

**Figure 6.**System power spectrum: (

**a**) power spectrum of the $x$ sequence, (

**b**) power spectrum of the $y$ sequence, (

**c**) power spectrum of the $w$ sequence, and (

**d**) power spectrum of the $v$ sequence.

**Figure 8.**Experimental results for the circuit (

**a**) $x-w$ plane, (

**b**) $w-v$ plane, and (

**c**) $w-v$ plane.

**Figure 21.**Encrypted/decrypted experimental results: (

**a**) Lena original image, (

**b**) Lena encrypted image, (

**c**) Lena decrypted image, (

**d**) boat original image, (

**e**) boat encrypted image, (

**f**) boat decrypted image, (

**g**) leaf original image, (

**h**) leaf encrypted image, and (

**i**) leaf decrypted image.

**Figure 22.**${c}_{1},{c}_{2},{c}_{3}$ decryption map: (

**a**) Lena plaintext image, (

**b**) $c$ corresponding decrypted image, (

**c**) ${c}_{1}$ corresponding decrypted image, (

**d**) ${c}_{2}$ corresponding decrypted image, and (

**e**) ${c}_{3}$ corresponding decrypted image.

**Figure 23.**Convolution nuclear sensitivity experiment analysis chart: (

**a**) Plaintext image, (

**b**) Ciphertext ${C}_{0}$ (convolution kernel ${c}_{0}$), (

**c**) Ciphertext ${C}_{1}$ (convolution kernel ${c}_{1}$), (

**d**) ${C}_{0}$ correct decryption result, (

**e**) ${C}_{0}$ error decryption result using ${c}_{1}$, and (

**f**) ${C}_{1}$ error decryption result using ${c}_{0}$.

**Figure 24.**Key sensitivity experiment analysis chart: (

**a**) plaintext image, (

**b**) ciphertext ${Y}_{0}$ (key is ${y}_{0}$), (

**c**) ciphertext ${Y}_{1}$ (key is ${y}_{1}$), (

**d**) ${Y}_{0}$ Correct decryption result, (

**e**) error decryption result ${Y}_{0}$ using ${y}_{1}$, and (

**f**) error decryption result ${Y}_{1}$ using ${y}_{0}$.

**Figure 25.**Histogram of the plaintext image and ciphertext image: (

**a**) histogram of Lena plaintext, (

**b**) histogram of Lena ciphertext, (

**c**) histogram of baboon plaintext, (

**d**) histogram of baboon ciphertext, (

**e**) histogram of the clear text of the boat, and (

**f**) histogram of the boat ciphertext.

**Figure 26.**Correlation analysis of the three directions before and after Lena image encryption: (

**a**) and (

**b**) horizontally adjacent, (

**c**) and (

**d**) vertically adjacent, (

**e**) and (

**f**) diagonally adjacent.

**Figure 27.**The results of occlusion attack: (

**a**) encrypted with 12.5% occlusion, (

**b**) encrypted with 25% occlusion, (

**c**) encrypted with 50% occlusion, (

**d**) decrypted with 12.5% occlusion, (

**e**) decrypted with 25% occlusion, and (

**f**) decrypted with 50% occlusion.

**Figure 28.**The results of noise attack analysis: (

**a**) noise with 10 of intensity, (

**b**) noise with 15 of intensity, and (

**c**) noise with 20 of intensity.

A(Input) | B(Input) | F(Result) |
---|---|---|

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

Lena Image | $\mathbf{Kernel}\text{}{\mathit{c}}_{1}$ | $\mathbf{Kernel}\text{}{\mathit{c}}_{2}$ | $\mathbf{Kernel}\text{}{\mathit{c}}_{3}$ | $\mathbf{Kernel}\text{}{\mathit{c}}_{4}$ | $\mathbf{Kernel}\text{}{\mathit{c}}_{5}$ | |
---|---|---|---|---|---|---|

Index | Pixel Change Rate (NPCR) | 0.9716 | 0.9962 | 0.9962 | 0.9963 | 0.996 |

Normalized Mean Change Intensity (UACI) | 0.2271 | 0.3341 | 0.3335 | 0.3352 | 0.3346 |

Images | Lena | Baboon | Boat | Peppers | ||||
---|---|---|---|---|---|---|---|---|

Index | Pixel Change Rate (NPCR) | Normalized Mean Change Intensity (UACI) | NPCR | UACI | NPCR | UACI | NPCR | UACI |

Test value | 0.9961 | 0.3356 | 0.9962 | 0.3344 | 0.996 | 0.3363 | 0.9962 | 0.3357 |

Reference [4] | 0.9961 | 0.3346 | - | - | - | - | - | - |

Reference [14] | 0.9961 | 0.3346 | - | - | - | - | 0.9962 | 0.3341 |

Reference [17] | 0.9952 | 0.3359 | 0.991 | 0.3325 | 0.9925 | 0.3339 | 0.985 | 0.3295 |

Image | Lena | Baboon | Boat | Peppers | Couple |
---|---|---|---|---|---|

Original Image | 7.4832 | 7.3713 | 7.1267 | 7.5715 | 7.2369 |

Encrypted Image | 7.9978 | 7.9977 | 7.997 | 7.9974 | 7.9974 |

Image | Original Image Information Entropy | Encrypted Image Information Entropy | |||||
---|---|---|---|---|---|---|---|

Algorithm | Reference [4] | Reference [11] | Reference [12] | Reference [13] | Reference [17] | ||

Lena | 7.4832 | 7.9978 | 7.9967 | 7.9972 | 7.9900 | 7.9959 | 7.9975 |

Image | Global Entropy | Local Entropy No. of Blocks = 20 (Block Size = 44 × 44) | Local Entropy Critical Values | ||
---|---|---|---|---|---|

h^{l*0.05}left = 7.9019h ^{l*0.05}right = 7.9030 | h^{l*0.01}left = 7.9017; h ^{l*0.01}right = 7.9032 | h^{l*0.001}left = 7.9015l h ^{l*0.001}right = 7.9034 | |||

Lena | 7.9978 | 7.9028 | Pass | Pass | Pass |

Baboon | 7.9977 | 7.9023 | Pass | Pass | Pass |

Boat | 7.997 | 7.9027 | Pass | Pass | Pass |

Couple | 7.9974 | 7.9022 | Pass | Pass | Pass |

**Table 7.**Histogram statistics with the variance and standard deviation of plain and encrypted images.

Plain Image | Scale | $\alpha $ | $\beta $ |

Lena $256\times 256$ | gray | 37,963 | 195 |

Reference [22] (lena $256\times 256$) | gray | 38,451 | 196 |

Boat $256\times 256$ | gray | 103,380 | 321.5 |

Baboon $256\times 256$ | gray | 58,542 | 241.9 |

Couple $256\times 256$ | gray | 79,457 | 281.9 |

Encrypted Image | Scale | ||

Lena $256\times 256$ | gray | 230 | 15 |

Reference [22] (lena $256\times 256$) | gray | 414 | 20 |

Boat $256\times 256$ | gray | 250 | 15.8 |

Baboon $256\times 256$ | gray | 260 | 16.1 |

Couple $256\times 256$ | gray | 242 | 15.6 |

Images | Horizontal Correlation Coefficient | Vertical Correlation Coefficient | Diagonal Direction Correlation Coefficient | |||
---|---|---|---|---|---|---|

Clear Image | Ciphertext Image | Clear Image | Ciphertext Image | Clear Image | Ciphertext Image | |

Lena | 0.971 | 0.012 | 0.9402 | 0.002 | 0.9121 | −0.0083 |

Baboon | 0.8343 | −0.0109 | 0.8712 | 0.0043 | 0.794 | −0.0074 |

Boat | 0.9574 | −0.0076 | 0.9533 | −0.0137 | 0.915 | 0.0036 |

© 2020 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**

Wang, T.; Song, L.; Wang, M.; Chen, S.; Zhuang, Z.
A Novel Five-Dimensional Three-Leaf Chaotic Attractor and Its Application in Image Encryption. *Entropy* **2020**, *22*, 243.
https://doi.org/10.3390/e22020243

**AMA Style**

Wang T, Song L, Wang M, Chen S, Zhuang Z.
A Novel Five-Dimensional Three-Leaf Chaotic Attractor and Its Application in Image Encryption. *Entropy*. 2020; 22(2):243.
https://doi.org/10.3390/e22020243

**Chicago/Turabian Style**

Wang, Tao, Liwen Song, Minghui Wang, Shiqiang Chen, and Zhiben Zhuang.
2020. "A Novel Five-Dimensional Three-Leaf Chaotic Attractor and Its Application in Image Encryption" *Entropy* 22, no. 2: 243.
https://doi.org/10.3390/e22020243