# Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chaotic Map

#### 2.1. The Structure of Chaotic Maps

#### 2.2. Chaotic Performance of TDSCL

#### 2.2.1. Chaotic Trajectory

#### 2.2.2. Lyapunov Exponent

#### 2.2.3. Permutation Entropy

## 3. Image Encryption Algorithm

#### 3.1. Simultaneous Horizontal Confusion and Diffusion

- Step 1. Generate diffusion matrix ${S}_{1}$.

- Step 2. Set $i=1$.
- Step 3. Obtain begin index ${b}_{1}^{i}$ and circle shift the first row of the image $I(1,:)$ right by ${t}_{1}^{i}$ pixels

- Step 4. Horizontal diffusion.

- Step 5. Circle shift $I(i,:)$ horizontally by ${t}_{1}^{i}$ pixels.
- Step 6. Let $i=i+1$ and repeat steps 3 to 5 until all rows have been processed.

#### 3.2. Simultaneous Vertical Confusion and Diffusion

- Step 1. Generate diffusion matrix ${S}_{2}$.

- Step 2. Set $l=1$.
- Step 3. Generate index ${b}_{2}^{l}$ and circle shift the first column of the image $I(:,1)$ by ${t}_{2}^{l}$ pixels.

- Step 4. Vertical diffusion.

- Step 5. Circle shift $I(:,l)$ vertically by ${t}_{2}^{l}$ pixels.
- Step 6. Let $l=l+1$, and repeat steps 3 to 5 until all columns have been processed.

## 4. Experiment Results and Analysis

#### 4.1. Simulation Results

#### 4.2. Secret Key Space

#### 4.3. Statistical Analysis

#### 4.3.1. Correlation Coefficient Analysis

#### 4.3.2. Histogram Analysis

#### 4.4. Key Sensitivity Test

#### 4.5. Resistance Against Chosen-plain Text Attack

#### 4.6. Information Entropy

#### 4.7. Comparison with Other Methods

#### 4.8. Encryption Efficiency Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zhu, H.; Zhao, Y.; Song, Y. 2D Logistic-Modulated-Sine-Coupling-Logistic Chaotic Map for Image Encryption. IEEE Access
**2019**, 7, 14081–14098. [Google Scholar] [CrossRef] - Wu, J.; Liao, X.; Yang, B. Image encryption using 2D Hénon-Sine map and DNA approach. Signal Process.
**2018**, 153, 11–23. [Google Scholar] [CrossRef] - Pak, C.; Huang, L. A new color image encryption using combination of the 1D chaotic map. Signal Process.
**2017**, 138, 129–137. [Google Scholar] [CrossRef] - Hua, Z.; Jin, F.; Xu, B.; Huang, H. 2D Logistic-Sine-coupling map for image encryption. Signal Process.
**2018**, 149, 148–161. [Google Scholar] [CrossRef] - Li, B.; Liao, X.; Jiang, Y. A novel image encryption scheme based on logistic map and dynatomic modular curve. Multimed. Tools Appl.
**2018**, 77, 8911–8938. [Google Scholar] [CrossRef] - Zhou, Y.; Hua, Z.; Pun, C.M.; Chen, C.P. Cascade chaotic system with applications. IEEE Trans. Cybern.
**2014**, 45, 2001–2012. [Google Scholar] [CrossRef] [PubMed] - Xie, J.; Yang, C.; Xie, Q.; Tian, L. An encryption algorithm based on transformed logistic map. In Proceedings of the 2009 International Conference on Networks Security, Wireless Communications and Trusted Computing, Wuhan, China, 25–26 April 2009; Volume 2, pp. 111–114. [Google Scholar]
- Wu, X.; Zhu, B.; Hu, Y.; Ran, Y. A novel colour image encryption scheme using rectangular transform-enhanced chaotic tent maps. IEEE Access
**2017**, 5, 6429–6436. [Google Scholar] [CrossRef] [Green Version] - Cai, S.; Huang, L.; Chen, X.; Xiong, X. A Symmetric Plaintext-Related Color Image Encryption System Based on Bit Permutation. Entropy
**2018**, 20, 282. [Google Scholar] [CrossRef] [Green Version] - Zhu, C.; Wang, G.; Sun, K. Improved cryptanalysis and enhancements of an image encryption scheme using combined 1D chaotic maps. Entropy
**2018**, 20, 843. [Google Scholar] [CrossRef] [Green Version] - Pak, C.; An, K.; Jang, P.; Kim, J.; Kim, S. A novel bit-level color image encryption using improved 1D chaotic map. Multimed. Tools Appl.
**2019**, 78, 12027–12042. [Google Scholar] [CrossRef] - Wang, X.; Qin, X.; Liu, C. Color image encryption algorithm based on customized globally coupled map lattices. Multimed. Tools Appl.
**2019**, 78, 6191–6209. [Google Scholar] [CrossRef] - Wang, H.; Xiao, D.; Chen, X.; Huang, H. Cryptanalysis and enhancements of image encryption using combination of the 1D chaotic map. Signal Process.
**2018**, 144, 444–452. [Google Scholar] [CrossRef] - Wang, X.; Zhu, X.; Zhang, Y. An image encryption algorithm based on Josephus traversing and mixed chaotic map. IEEE Access
**2018**, 6, 23733–23746. [Google Scholar] [CrossRef] - Li, S.; Ding, W.; Yin, B.; Zhang, T.; Ma, Y. A novel delay linear coupling logistics map model for color image encryption. Entropy
**2018**, 20, 463. [Google Scholar] [CrossRef] [Green Version] - Nkandeu, Y.P.K.; Tiedeu, A. An image encryption algorithm based on substitution technique and chaos mixing. Multimed. Tools Appl.
**2019**, 78, 10013–10034. [Google Scholar] [CrossRef] - Hua, Z.; Xu, B.; Jin, F.; Huang, H. Image encryption using josephus problem and filtering diffusion. IEEE Access
**2019**, 7, 8660–8674. [Google Scholar] [CrossRef] - Huang, L.; Cai, S.; Xiong, X.; Xiao, M. On symmetric color image encryption system with permutation- diffusion simultaneous operation. Opt. Lasers Eng.
**2019**, 115, 7–20. [Google Scholar] [CrossRef] - Ur Rehman, A.; Liao, X. A novel robust dual diffusion/confusion encryption technique for color image based on Chaos, DNA and SHA-2. Multimed. Tools Appl.
**2019**, 78, 2105–2133. [Google Scholar] [CrossRef] - Li, S.; Yin, B.; Ding, W.; Zhang, T.; Ma, Y. A Nonlinearly Modulated Logistic Map with Delay for Image Encryption. Electronics
**2018**, 7, 326. [Google Scholar] [CrossRef] [Green Version] - Li, M.; Lu, D.; Xiang, Y.; Zhang, Y.; Ren, H. Cryptanalysis and improvement in a chaotic image cipher using two-round permutation and diffusion. Nonlinear Dyn.
**2019**, 96, 31–47. [Google Scholar] [CrossRef] - Shan, L.; Qiang, H.; Li, J.; Wang, Z.Q. Chaotic optimization algorithm based on Tent map. Control Decis.
**2005**, 20, 179–182. [Google Scholar] - Li, C.; Lin, D.; Lü, J.; Hao, F. Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE MultiMedia
**2018**, 25, 46–56. [Google Scholar] [CrossRef] [Green Version] - Pareek, N.K.; Patidar, V.; Sud, K.K. Image encryption using chaotic logistic map. Image Vis. Comput.
**2006**, 24, 926–934. [Google Scholar] [CrossRef] - Paar, V.; Buljan, H. Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion. Phys. Rev. E Stat. Phys. Plasmas Fluids Related Interdisciplinary Top.
**2000**, 62, 4869–4872. [Google Scholar] [CrossRef] [Green Version] - Amigo, J.M.; Kocarev, L.; Szczepanski, J. Discrete Lyapunov Exponent and Resistance to Differential Cryptanalysis. IEEE Trans. Circuits Syst. II Express Briefs
**2007**, 54, 882–886. [Google Scholar] [CrossRef] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] - Ahmad, J.; Khan, M.A.; Hwang, S.O.; Khan, J.S. A compression sensing and noise-tolerant image encryption scheme based on chaotic maps and orthogonal matrices. Neural Comput. Appl.
**2017**, 28, 953–967. [Google Scholar] [CrossRef] - Wu, Y.; Noonan, J.P.; Agaian, S. NPCR and UACI randomness tests for image encryption. Cyber J. Multidisciplinary J. Sci. Technol. J. Sel. Areas Telecommun. JSAT
**2011**, 1, 31–38. [Google Scholar] - Wu, Y.; Zhou, Y.; Saveriades, G.; Agaian, S.; Noonan, J.P.; Natarajan, P. Local Shannon entropy measure with statistical tests for image randomness. Inf. Sci.
**2013**, 222, 323–342. [Google Scholar] [CrossRef] [Green Version] - Enayatifar, R.; Abdullah, A.H.; Isnin, I.F.; Altameem, A.; Lee, M. Image encryption using a synchronous permutation-diffusion technique. Opt. Lasers Eng.
**2017**, 90, 146–154. [Google Scholar] [CrossRef] - Niyat, A.Y.; Moattar, M.H.; Torshiz, M.N. Color image encryption based on hybrid hyper-chaotic system and cellular automata. Opt. Lasers Eng.
**2017**, 90, 225–237. [Google Scholar] [CrossRef] - Farajallah, M. Chaos-Based Crypto and Joint Crypto-Compression Systems for Images and Videos. Ph.D. Thesis, Universite de Nantes, Nantes, France, 2015. [Google Scholar]

**Figure 3.**Trajectories for (

**a**) TDSCL with $\mu $ = 1, (

**b**) delay and linearly coupled logistic chaotic map (DLCL) with $\mu $ = 1, and (

**c**) logistic-modulated sine-coupling logistic chaotic map (LSMCL) with $\theta =0.75$.

**Figure 7.**Simulation results of the proposed image encryption algorithm: (

**a**,

**d**) original images, (

**b**,

**e**) encrypted images, and (

**c**,

**f**) decrypted images.

**Figure 8.**Adjacent pixels correlation analysis: the correlation between two horizontal, vertical, and diagonal pixels in (

**a**–

**c**) a plain image and (

**d**–

**f**) an encrypted image.

**Figure 9.**Histograms of (

**a**,

**b**) Lena and the encrypted image and (

**c**,

**d**) Pepper and the encrypted image.

Color Image | Horizontal | Vertical | Diagonal | |
---|---|---|---|---|

4.2.01.tiff | original | 0.9723 | 0.9843 | 0.9602 |

encrypted | 0.0001 | 0.0013 | 0.0040 | |

4.2.02.tiff | original | 0.9347 | 0.9413 | 0.8860 |

encrypted | −0.0032 | −0.0044 | 0.0011 | |

4.2.03.tiff | original | 0.8736 | 0.8261 | 0.7843 |

encrypted | 0.0075 | −0.0012 | −0.0014 | |

4.2.04.tiff | original | 0.9456 | 0.9727 | 0.9213 |

encrypted | −0.0040 | 0.0042 | 0.0063 | |

4.2.05.tiff | original | 0.9364 | 0.9302 | 0.8819 |

encrypted | 0.0007 | 0.0022 | −0.0007 | |

4.2.06.tiff | original | 0.9581 | 0.9564 | 0.9282 |

encrypted | 0.0049 | −0.0002 | −0.0029 | |

4.2.07.tiff | original | 0.9634 | 0.9704 | 0.9363 |

encrypted | −0.0043 | −0.0004 | −0.0008 |

**Table 2.**The number of pixel change rate (NPCR) and the unified average changing intensity (UACI) of different images for key sensitivity.

Image | NPCR | UACI |
---|---|---|

4.2.01.tiff | 0.9959 | 0.3354 |

4.2.02.tiff | 0.9960 | 0.3340 |

4.2.03.tiff | 0.9958 | 0.3345 |

4.2.04.tiff | 0.9958 | 0.3349 |

4.2.05.tiff | 0.9962 | 0.3354 |

4.2.06.tiff | 0.9962 | 0.3357 |

4.2.07.tiff | 0.9960 | 0.3356 |

Image | NPCR | UACI |
---|---|---|

4.2.01.tiff | 0.9961 | 0.3348 |

4.2.02.tiff | 0.9961 | 0.3345 |

4.2.03.tiff | 0.9961 | 0.3346 |

4.2.04.tiff | 0.9961 | 0.3352 |

4.2.05.tiff | 0.9961 | 0.3348 |

4.2.06.tiff | 0.9961 | 0.3350 |

4.2.07.tiff | 0.9961 | 0.3351 |

Image | Entropy |
---|---|

4.2.01.tiff | 7.9969 |

4.2.02.tiff | 7.9973 |

4.2.03.tiff | 7.9973 |

4.2.04.tiff | 7.9972 |

4.2.05.tiff | 7.9971 |

4.2.06.tiff | 7.9973 |

4.2.07.tiff | 7.9971 |

Paper | Correlation | NPCR | UACI | Entropy | ||
---|---|---|---|---|---|---|

Horizontal | Vertical | Diagonal | ||||

Paper [31] | 0.0062 | 0.0074 | 0.0009 | 0.9942 | 0.3352 | 7.9974 |

Paper [32] | 0.0054 | 0.0089 | 0.0021 | 0.9965 | 0.3351 | 7.9970 |

Paper [2] | 0.0028 | 0.0041 | 0.0010 | 0.9962 | 0.3363 | 7.9970 |

Proposed with one iteration | 0.0001 | −0.0007 | −0.0025 | 0.9961 | 0.3344 | 7.9971 |

Proposed with two iteration | 0.0007 | −0.0022 | −0.0007 | 0.9961 | 0.3346 | 7.9977 |

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**MDPI and ACS Style**

Zhang, G.; Ding, W.; Li, L.
Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map. *Symmetry* **2020**, *12*, 355.
https://doi.org/10.3390/sym12030355

**AMA Style**

Zhang G, Ding W, Li L.
Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map. *Symmetry*. 2020; 12(3):355.
https://doi.org/10.3390/sym12030355

**Chicago/Turabian Style**

Zhang, Guidong, Weikang Ding, and Lian Li.
2020. "Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map" *Symmetry* 12, no. 3: 355.
https://doi.org/10.3390/sym12030355