# A Simple Chaotic Map-Based Image Encryption System Using Both Plaintext Related Permutation and Diffusion

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## Abstract

**:**

## 1. Introduction

- (1)
- The proposed encryption system can be used to encrypt color images or gray images of any size. Some algorithms [5,8,9,10,15,29,31,33,35] mentioned above are developed to encrypt gray images. If these algorithms are used to encrypt R, G and B channels of original color image and then transform three encrypted gray imges into encrypted color image, the encryption system has less plaintext sensitivity because three channels of original color image are encrypted separately and do not have interaction in the encryption process. Furthermore, some other algorithms [7,10,11] are suitable for encrypting the original square image.
- (2)
- As mentioned above, most plaintext related image encryption schemes used only plaintext related confusion operation [28,29,30,31,32] or only plaintext related diffusion operation [33,34,35,36,37] related to plaintext inadequately. For security purposes, in our scheme, both permutation operation and diffusion operation are related to plain images, which can achieve high plaintext sensitivity to chosen/known plaintext attack efficiently.
- (3)
- Different from most chaotic based image cryptosystems in [4,5,6,7,9,10,12,16,19,28,29,31,36], in which the permutation–diffusion operation is performed several times to obtain the desired security level, the plaintext related permutation and diffusion in our scheme is only performed a single time in the entire encryption process.
- (4)
- Complete simulations are given and the simulation results prove an excellent performance in security and efficiency.

## 2. Related Work

#### 2.1. Extended Arnold Map

#### 2.2. Chebyshev Map

## 3. Algorithm of Image Encryption

#### 3.1. Secret Key Formulation

#### 3.2. Encryption Process

#### 3.2.1. Permutation Stage

#### 3.2.2. Diffusion Stage

#### 3.3. Decryption Process

## 4. Experimental Results and Security Analysis

#### 4.1. Security Key Space

#### 4.2. Histogram Analysis

#### 4.3. Correlation Analysis

#### 4.4. Sensitivity Analysis

#### 4.5. Known and Chosen Plaintext Analysis

#### 4.6. Robustness against Noise and Occlusion Attacks

#### 4.7. Information Entropy

#### 4.8. Encryption Speed Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The Bifurcation diagram and Lyapunov Exponent diagram of the Chebyshev map. (

**a**) Bifurcation diagram; (

**b**) Lyapunov Exponent diagram.

**Figure 4.**Encryption and decryption result of two images. (

**a**) The plain-image and corresponding histograms; (

**b**) The cipher images and corresponding histograms; (

**c**) The decrypted images and corresponding histograms; (

**d**) The gray original image and encrypted image and they corresponding histograms.

**Figure 5.**Correlation of R channel in standard “Lena” image. (

**a**) Correlation of R channel in plain image; (

**b**) Correlation of R channel in encrypted image.

**Figure 6.**Key sensitivity tests: (

**a**,

**e**): plain-image and its histogram; (

**b**,

**f**): $E1$ and its histogram; (

**c**,

**g**): $E2$ and its histograms; (

**d**,

**h**): $|E1-E2|$ and its histogram; (

**i**): decrypted image of $E1$ using key set ${k}_{ey}\left(1\right)$ ; (

**j**–

**n**): decrypted image of $E1$ using security key set ${k}_{ey}\left(2\right)$, ${k}_{ey}\left(3\right)$, ${k}_{ey}\left(4\right)$, ${k}_{ey}\left(5\right)$ or ${k}_{ey}\left(6\right)$.

**Figure 7.**Encryption result of special images: (

**a**) black image; (

**b**) the encrypted black image; (

**c**) white image; (

**d**) the encrypted white image.

**Figure 8.**Robustness against noise results: (

**a**,

**e**): encrypted image and its decrypted image; (

**b**,

**f**): encrypted image added with salt and pepper noise with 0.1 density and corresponding decrypted image; (

**c**,

**g**): encrypted image added with salt and pepper noise with 0.2 density and its decrypted image; (

**d**,

**h**): encrypted image added with salt and pepper noise with 0.3 density and its decrypted image.

**Figure 9.**Robustness analysis results. (

**a**) The color cipher images of lena with different percentages data loss; (

**b**) The corresponding decrypted images of (

**a**); (

**c**) The gray cipher images of lena with different percentages data loss; (

**d**) The corresponding decrypted images of (

**c**).

Cryptosystems | Attacked by | Attack Approaches |
---|---|---|

Zhang et al. (2013) [4] | Hoang et al. (2018) [21] | chosen ciphertext |

Zhou et al. (2015) [6] | Chen et al. (2017) [22] | Differential |

Zhang et al. (2016) [7] | Wu et al. (2018) [23] | chosen plaintext |

Huang et al. (2012) [9] | Wang et al. (2014) [24] | chosen plaintext |

Chen et al. (2015) [10] | Hu et al. (2017) [25] | chosen plaintext and ciphertext |

Liu et al. (2016) [14] | Zhang et al. (2017) [26] | chosen plaintext |

Pak et al. (2017) [18] | Wang et al. (2018) [27] | chosen plaintext |

Image | Lena | Baboon | Flower | Fruits | Yacht | Girl | Flowers |
---|---|---|---|---|---|---|---|

Variance | 898.25 | 1017.29 | 1095.71 | 899.87 | 917.07 | 1413.86 | 783.00 |

**Table 3.**Average correlation coefficients of the original images and the cipher-images in four directions.

Image | Original-Image | Encrypted-Image | |||||||
---|---|---|---|---|---|---|---|---|---|

V | H | D | A | V | H | D | A | ||

Lena | R | 0.9798 | 0.9893 | 0.9777 | 0.9697 | 0.0003 | 0.0040 | 0.0013 | 0.0021 |

G | 0.9689 | 0.9824 | 0.9653 | 0.9554 | −0.0018 | 0.0005 | 0.0002 | 0.0009 | |

B | 0.9325 | 0.9574 | 0.9253 | 0.9181 | 0.0019 | −0.0093 | 0.0002 | 0.0005 | |

baboon | R | 0.9231 | 0.8660 | 0.8519 | 0.8543 | 0.0002 | 0.0019 | −0.0005 | 0.0022 |

G | 0.8654 | 0.7650 | 0.7249 | 0.7348 | −0.0019 | −0.0046 | 0.0040 | 0.0010 | |

B | 0.9072 | 0.8808 | 0.8424 | 0.8398 | −0.0039 | −0.0062 | 0.0013 | 0.0020 | |

fruits | R | 0.9936 | 0.9928 | 0.9897 | 0.9868 | −0.0022 | −0.0021 | −0.0027 | 0.0010 |

G | 0.9855 | 0.9848 | 0.9783 | 0.9694 | 0.0069 | 0.0073 | 0.0021 | −0.0016 | |

B | 0.9265 | 0.9192 | 0.8809 | 0.8531 | 0.0007 | 0.0088 | −0.0003 | −0.0010 | |

flowers | R | 0.9718 | 0.9719 | 0.9504 | 0.9551 | 0.0045 | −0.0002 | −0.0005 | 0.0007 |

G | 0.9510 | 0.9497 | 0.9123 | 0.9218 | 0.0047 | −0.0015 | 0.0026 | 0.0022 | |

B | 0.9527 | 0.9527 | 0.9178 | 0.9256 | 0.0004 | −0.0032 | 0.0019 | −0.0008 |

**Table 4.**Correlation coefficients comparison for different encryption algorithms (R channel of Lena).

Direction | Original Image | Our Scheme | Ref. [11] | Ref. [18] | Ref. [28] | Ref. [32] | Ref. [36] |
---|---|---|---|---|---|---|---|

Horizontal | 0.9853 | 0.0003 | 0.0013 | −0.0038 | −0.0031 | 0.0046 | 0.0005 |

Vertical | 0.9753 | 0.0040 | 0.0034 | −0.0026 | 0.0025 | −0.0028 | −0.0070 |

Diagonal | 0.9734 | 0.0013 | 0.0072 | 0.0017 | −0.0001 | 0.0014 | 0.0006 |

Image | NPCR (99.6094) | UACI (33.4635) | |||||
---|---|---|---|---|---|---|---|

R | G | B | R | G | B | ||

Lena | $ke{y}_{\_x0}$ | 99.6089 | 99.6089 | 99.6085 | 33.4589 | 33.4598 | 33.4624 |

${K}_{1}$ | 99.6092 | 99.6075 | 99.6095 | 33.4615 | 33.4676 | 33.4624 | |

${K}_{2}$ | 99.6082 | 99.6101 | 99.6087 | 33.4626 | 33.4670 | 33.4686 | |

${K}_{3}$ | 99.6087 | 99.6094 | 99.6098 | 33.4609 | 33.4623 | 33.4628 | |

${N}_{0}$ | 99.6090 | 99.6091 | 99.6090 | 33.4662 | 33.4619 | 33.4684 | |

baboon | $ke{y}_{\_x0}$ | 99.6086 | 99.6096 | 99.6106 | 33.4639 | 33.4621 | 33.4641 |

${K}_{1}$ | 99.6107 | 99.6089 | 99.6093 | 33.4641 | 33.4651 | 33.4696 | |

${K}_{2}$ | 99.6083 | 99.6103 | 99.6093 | 33.4685 | 33.4653 | 33.4620 | |

${K}_{3}$ | 99.6087 | 99.6112 | 99.6087 | 33.4616 | 33.4636 | 33.4593 | |

${N}_{0}$ | 99.6096 | 99.6090 | 99.6087 | 33.4686 | 33.4645 | 33.4662 |

Image | NPCR (99.6094) | UACI (33.4635) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

Lena | 99.6091 | 99.6099 | 99.6090 | 33.4678 | 33.4577 | 33.4608 |

baboon | 99.6111 | 99.6097 | 99.6094 | 33.4617 | 33.4680 | 33.4617 |

fruits | 99.6091 | 99.6081 | 99.6091 | 33.4631 | 33.4663 | 33.4593 |

Girl | 99.6090 | 99.6100 | 99.6095 | 33.4627 | 33.4597 | 33.4588 |

Flower | 99.6113 | 99.6098 | 99.6100 | 33.4603 | 33.4666 | 33.4588 |

Yacht | 99.6092 | 99.6095 | 99.6099 | 33.4613 | 33.4666 | 33.4651 |

Lena in Ref. [11] | 99.6892 | 99.6943 | 99.6922 | 33.3256 | 33.3324 | 33.3313 |

Lena in Ref. [18] | 99.6552 | 99.6277 | 99.5882 | 33.4846 | 33.4132 | 33.3441 |

Lena in Ref. [28] | 99.6917 | 99.6887 | 99.6704 | 33.5418 | 33.5327 | 33.5164 |

Lena in Ref. [32] | 99.6086 | 99.6083 | 99.6104 | 33.4709 | 33.4683 | 33.4682 |

Theoretically NPCR Critical Value [42] | |||

${N}_{0.05}^{*}=99.5893$ | ${N}_{0.01}^{*}=99.5810$ | ${N}_{0.001}^{*}=99.5717$ | |

Tested Image Size | NPCR Test Results | ||

0.05-level | 0.01-level | 0.001-level | |

512 by 512 | 4/4 | 4/4 | 4/4 |

256 by 256 | 4/4 | 4/4 | 4/4 |

Theoretically UACI Critical Value [42] | |||

${u}_{0.05}^{*-}=33.3730$ ${u}_{0.05}^{*+}=33.5541$ | ${u}_{0.01}^{*-}=33.3445$ ${u}_{0.01}^{*+}=33.5826$ | ${u}_{0.001}^{*-}=33.3115$ ${u}_{0.001}^{*+}=33.6156$ | |

Tested Image Size | UACI Test Results | ||

0.05-level | c0.01-level | 0.001-level | |

512 by 512 | 4/4 | 4/4 | 4/4 |

256 by 256 | 4/4 | 4/4 | 4/4 |

Original Image | NPCR (99.6094) | UACI (33.4635) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

All-black | 99.6121 | 99.6102 | 99.6105 | 33.5744 | 33.5200 | 33.3917 |

All-white | 99.6075 | 99.6100 | 99.6100 | 33.4578 | 33.4683 | 33.4555 |

${P}_{1}$ | 99.5284 | 99.4976 | 99.4469 | 33.4466 | 33.6076 | 33.3642 |

${P}_{2}$ | 99.6104 | 99.6089 | 99.6113 | 33.4254 | 33.5033 | 33.4453 |

Noise Attacks or Date Loss | MSE (Proposed) | PSNR (Proposed) | MSE (Ref. [35]) | PSNR (Ref. [35]) |
---|---|---|---|---|

Salt & peppers noise (density 0.05) | 734.3922 | 19.4715 | 869.8890 | 18.7362 |

Salt & pepper noise (density 0.1) | 1465.4644 | 16.4711 | 1829.6416 | 15.5071 |

(100:220,110:230)=0 | 1729.6835 | 15.7511 | 2894.6596 | 13.5148 |

(90:110,:)=0 | 658.2202 | 19.9471 | 1073.0810 | 17.8245 |

(100:240,:)=0 | 4335.8104 | 11.7601 | 6813.5770 | 9.7971 |

(:,100:120)=0 | 632.5614 | 20.1198 | 946.0235 | 18.3718 |

Image | Plain-Image | Encrypted Image | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

Lena | 7.2531 | 7.5952 | 6.9686 | 7.9993 | 7.9994 | 7.9994 |

baboon | 7.7067 | 7.4753 | 7.7522 | 7.9993 | 7.9993 | 7.9993 |

fruits | 7.5172 | 7.3230 | 6.7785 | 7.9991 | 7.9993 | 7.9991 |

flower | 7.4428 | 7.4062 | 7.3371 | 7.9993 | 7.9994 | 7.9994 |

Girl | 7.4346 | 7.2354 | 7.0578 | 7.9996 | 7.9995 | 7.9995 |

Yacht | 7.6071 | 7.4062 | 7.3371 | 7.9993 | 7.9993 | 7.9991 |

Lena in Ref. [11] | 7.2531 | 7.5952 | 6.9686 | 7.9996 | 7.9997 | 7.9997 |

Lena in Ref. [28] | 7.2531 | 7.5952 | 6.9686 | 7.9972 | 7.9972 | 7.9976 |

Lena in Ref. [32] | 7.2531 | 7.5952 | 6.9686 | 7.9992 | 7.9993 | 7.9994 |

© 2018 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**

Huang, L.; Cai, S.; Xiao, M.; Xiong, X. A Simple Chaotic Map-Based Image Encryption System Using Both Plaintext Related Permutation and Diffusion. *Entropy* **2018**, *20*, 535.
https://doi.org/10.3390/e20070535

**AMA Style**

Huang L, Cai S, Xiao M, Xiong X. A Simple Chaotic Map-Based Image Encryption System Using Both Plaintext Related Permutation and Diffusion. *Entropy*. 2018; 20(7):535.
https://doi.org/10.3390/e20070535

**Chicago/Turabian Style**

Huang, Linqing, Shuting Cai, Mingqing Xiao, and Xiaoming Xiong. 2018. "A Simple Chaotic Map-Based Image Encryption System Using Both Plaintext Related Permutation and Diffusion" *Entropy* 20, no. 7: 535.
https://doi.org/10.3390/e20070535