# A Novel Chaos-Based Color Image Encryption Scheme Using Bit-Level Permutation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{8}permutation.

## 2. Proposed Scheme

#### 2.1. Encryption Scheme

**PI**.

**Red**,

**Green,**and

**Blue**are color components.

**PI**is an M × 3N matrix.

**PI**, and denote the summation as

**SUM**.

**k1**,

**k2**,…,

**k9**are the inputted keys (similarly hereinafter).

**T**. Take the first M elements of

**T**to generate a new sequence

**X1**(Equation (7)). Then, take the repeated numbers away from

**X1**and append the absent elements to the end for generating the rows’ substitute control sequence

**X**. In the same way, take the M + 1 to M + 3N elements to generate a new sequence

**Y1**(Equation (8)). Then, take the repeated numbers away from

**Y1**and append the absent elements to the end for generating the columns’ substitute control sequence

**Y**.

**X**sequence controlling as in Equation (9).

**Y**sequence controlling as in Equation (10).

**XS**,

**YS,**and

**ZS**, respectively. Here, the transitional state—some states before the chaos system goes stable—may cause the bad randomness seen in the generation sequences for the cryptosystem.

**KEY**into 3N vectors with M elements, and denote them as$\mathit{K}{\mathit{V}}_{\mathit{i}}$. Then, partition the matrix generated in Step 15 into column vectors, and denote them as$\mathit{E}{\mathit{C}}_{\mathit{i}}$, where$i=1,2,\dots ,3N$.

#### 2.2. Decryption Scheme

**D**.

**Red**,

**Green**, and

**Blue**are color components.

**D**is an M × 3N matrix.

**XS**,

**YS,**and

**ZS**, respectively.

**KEY**into 3N vectors with M elements, and denote them as$\mathit{K}{\mathit{V}}_{\mathit{i}}$. Then, partition the matrix

**D**generated from Step 2 into column vectors, and denote them as$\mathit{D}{\mathit{C}}_{\mathit{i}}$, where$i=1,2,\dots ,3N$.

**DP**generated in the previous step, and denote the summation as

**SUM**.

**T**. Take the first M elements of

**T**to generate a new sequence

**X1**as in Equation (47). Then, take the repeated numbers away from

**X1**, and append the absent elements to the end for generating the rows’ substitute control sequence

**X**. In the same way, take the M + 1 to M + 3N elements to generate a new sequence

**Y1**as in Equation (48). Then, take the repeated numbers from

**Y1**and append the absent elements to the end for generating the columns’ substitute control sequence

**Y**.

**DP**generated in Step 13 into column vectors, and denote them as $\mathit{C}{\mathit{C}}_{\mathit{t}}$, where t = 1, 2, …, 3N. Substitute the column vectors under

**Y**sequence controlling as in Equation (49).

**X**sequence controlling as in Equation (50).

## 3. Simulation and Security Analysis

#### 3.1. Key Space Analysis

#### 3.2. Differential Attack

#### 3.3. Statistical Analysis

#### 3.3.1. Histogram Analysis

#### 3.3.2. Correlation Coefficient

#### 3.4. Key Sensitivity Analysis

#### 3.5. Information Entropy Analysis

#### 3.6. Speed Analysis and Comparisons

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Attractors of the Rucklidge system. (

**a**) x–y plane, (

**b**) x–z plane, (

**c**) y–z plane, and (

**d**) x–y–z plane.

**Figure 4.**Encryption results. (

**a**–

**f**) The plain images of ‘pepper’, ‘whole black’, ‘whole white’, ‘Lena’, ‘baboon’, and ‘airplane’. (

**g**–

**l**) The corresponding cipher images. (

**m**–

**r**) The corresponding recovery images by decryption.

**Figure 5.**Histograms. (

**a**–

**f**) The plain images of pepper, whole black, whole white, Lena, baboon, and airplane. (

**g**–

**l**) The corresponding histograms of those plain images. (

**m**–

**r**) The corresponding cipher images generated from encryption. (

**s**–

**x**) The corresponding histograms of those cipher images.

**Figure 6.**Correlation distributions. (

**a**–

**d**) The data of pepper, Lena, baboon, and airplane, respectively. Column (1) is plain images. Columns (2)–(4) are the distributions of the plain images. Column (5) is the corresponding cipher image. Columns (6)–(8) are the distributions of cipher images. Columns (2) and (6) are the red component, columns (3) and (7) are the green component, and columns (4) and (8) are the blue component.

**Figure 7.**Key sensitivity analysis. (

**a**) Original image, (

**b**) cipher image C1, (

**c**) cipher image C2, (

**d**) cipher image C3, (

**e**) image of

**|**C1-C2

**|**, and (

**f**) image of

**|**C1–C3

**|**.

**Table 1.**The results of number of pixels change rate (NPCR) and unified average changing intensity (UACI).

Images | Red | Green | Blue | |||
---|---|---|---|---|---|---|

NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | |

Lena | 99.6052 | 33.4025 | 99.6120 | 33.4428 | 99.6303 | 33.5029 |

Baboon | 99.6174 | 33.3923 | 99.6117 | 33.3923 | 99.6166 | 33.5534 |

Pepper | 99.6098 | 33.4131 | 99.6106 | 33.4946 | 99.6322 | 33.4067 |

Airplane | 99.6170 | 33.4704 | 99.6136 | 33.4304 | 99.6075 | 33.5222 |

[30] | 99.6097 | 33.5012 | 99.6218 | 33.4414 | 99.5947 | 33.4535 |

[31] | 99.6012 | 33.4459 | 99.6002 | 33.4129 | 99.6174 | 33.4681 |

[32] | 99.61 | 31.12 | 99.61 | 30.23 | 99.61 | 29.74 |

Images | Horizontal | Vertical | Diagonal | ||||||
---|---|---|---|---|---|---|---|---|---|

Red | Green | Blue | Red | Green | Blue | Red | Green | Blue | |

Lena | −0.0022 | 0.0057 | 0.00007 | 0.0009 | −0.0041 | 0.00004 | 0.0013 | 0.0017 | 0.0104 |

Baboon | 0.0049 | −0.0026 | 0.0068 | 0.0006 | −0.0100 | 0.0038 | 0.0021 | 0.0147 | −0.0040 |

Pepper | 0.0038 | 0.0014 | −0.0094 | −0.0026 | 0.0038 | 0.0035 | −0.0187 | 0.0058 | 0.0035 |

Airplane | 0.0006 | 0.0023 | 0.0009 | 0.0013 | 0.0010 | −0.0017 | 0.0016 | −0.0057 | 0.0083 |

[21] | 0.0112 | −0.0039 | 0.0373 | 0.0118 | −0.0156 | 0.0153 | −0.0095 | 0.0193 | 0.0373 |

[30] | 0.0013 | 0.0032 | 0.0020 | 0.0047 | −0.0005 | 0.0094 | 0.00232 | 0.0048 | 0.0040 |

[31] | 0.0127 | −0.0338 | 0.0221 | −0.0584 | −0.0029 | 0.0196 | −0.0633 | −0.0558 | −0.0167 |

[32] | −0.0164 | −0.0071 | 0.0053 | 0.0630 | −0.02911 | 0.0015 | −0.0460 | −0.0371 | −0.0115 |

Ciphers | Red | Green | Blue | |||
---|---|---|---|---|---|---|

NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | |

C1 and C2 | 99.6029 | 33.4676 | 99.5995 | 33.5168 | 99.6029 | 33.5368 |

C1 and C3 | 99.6223 | 33.4970 | 99.6120 | 33.5514 | 99.6075 | 33.4839 |

Color Channels | Lena | Baboon | Pepper | Airplane | Ref. [21] | Ref. [30] | Ref. [31] | Ref. [8] |
---|---|---|---|---|---|---|---|---|

Red | 7.999279 | 7.999238 | 7.999353 | 7.999306 | 7.90302 | 7.99171 | 7.9949 | 7.9913 |

Green | 7.999264 | 7.999280 | 7.999222 | 7.999357 | 7.90238 | 7.99121 | 7.9945 | 7.9949 |

Blue | 7.999353 | 7.999299 | 7.999283 | 7.999357 | 7.90157 | 7.99117 | 7.9941 | 7.9889 |

Image Size (Pixels) | Image Size (MB) | Mean Time (s) | Mean Speed (MB/s) |
---|---|---|---|

256 × 256 | 0.187 | 0.0820 | 2.280 |

512 × 512 | 0.750 | 0.3239 | 2.315 |

1024 × 1024 | 3.000 | 1.2878 | 2.329 |

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**MDPI and ACS Style**

Li, Z.; Peng, C.; Tan, W.; Li, L.
A Novel Chaos-Based Color Image Encryption Scheme Using Bit-Level Permutation. *Symmetry* **2020**, *12*, 1497.
https://doi.org/10.3390/sym12091497

**AMA Style**

Li Z, Peng C, Tan W, Li L.
A Novel Chaos-Based Color Image Encryption Scheme Using Bit-Level Permutation. *Symmetry*. 2020; 12(9):1497.
https://doi.org/10.3390/sym12091497

**Chicago/Turabian Style**

Li, Zhen, Changgen Peng, Weijie Tan, and Liangrong Li.
2020. "A Novel Chaos-Based Color Image Encryption Scheme Using Bit-Level Permutation" *Symmetry* 12, no. 9: 1497.
https://doi.org/10.3390/sym12091497