# Joint Lossless Image Compression and Encryption Scheme Based on CALIC and Hyperchaotic System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{(4)}quantum wavelet transform (DQWT). Li et al. [36] proposed a quantum image compression-encryption scheme based on quantum discrete cosine transform. However, they perform lossy compression.

## 2. Research on Encryption Locations for CALIC

#### 2.1. Brief Description of CALIC

#### 2.1.1. Continuous-Tone Mode

- Gradient-adjusted prediction (GAP)

_{h}and d

_{v}are estimates of the gradients of the intensity function near the pixel I[i,j] in the horizontal and vertical directions. According to Equation (2), the current pixel value can be predicted, and the prediction algorithm is shown in Algorithm 1 [45].

Algorithm 1 Pixel gradient prediction algorithm |

Input Original pixels IOutput The predicted value Î (a gradient-adjusted prediction Î of I)1. if (d _{v} − d_{h} > 80){sharp horizontal edge} $\widehat{I}[i,j]={I}_{W}$2. else (d _{v} − d_{h} < −80){sharp vertical edge} $\widehat{I}[i,j]={I}_{N}$3. else { 4. $\widehat{I}[i,j]=({I}_{W}+{I}_{N})/2+({I}_{NE}-{I}_{NW})/4$ 5. if (d _{v} − d_{h} > 32){ horizontal edge} $\widehat{I}[i,j]=(\widehat{I}[i,j]+{I}_{W})/2$6. else if (d _{v} − d_{h} > 8){weak horizontal edge} $\widehat{I}[i,j]=(3\widehat{I}[i,j]+{I}_{W})/4$7. else if (d _{v} − d_{h} < −32){vertical edge} $\widehat{I}[i,j]=(\widehat{I}[i,j]+{I}_{N})/2$8. else if (d _{v} − d_{h} < −8){weak vertical edge} $\widehat{I}[i,j]=(3\widehat{I}[i,j]+{I}_{N})/4$9. } |

- 2.
- The predicted values of pixels are used for coding context selection and quantization. Coding context selection and quantization

_{w}is the previous prediction error, ${e}_{w}=I[i-1,j]-\widehat{I}[i-1,j],a=b=1,c=2$. Quantize $\Delta $ to L levels. L = 8 is found to be sufficient in practice. Denote the $\Delta $ quantizer by Q, i.e., $Q:\Delta \to \{0,1,2,...,7\}$. The quantization criterion is to minimize conditional entropy of errors. According to the definition of entropy, such that [45]

- 3.
- Context modeling and prediction errors

- Formation and quantization of contexts

- Estimation of error magnitude within a context

- Error feedback and sign flipping

- 4.
- Entropy Coding

#### 2.1.2. Binary Mode

_{1}= I

_{W}, and let other values, if any, be s

_{2}. A ternary code T used to describe three states of I[i,j] is defined in Equation (11) [45].

#### 2.2. Feasibility Analysis on Image Encryption Based on CALIC

#### 2.2.1. Feasibility Analysis of Plaintext Image in Encoder

#### 2.2.2. Feasibility Analysis of the Predicted Values of Pixels

#### 2.2.3. Feasibility Analysis of the Final Predicted Errors

#### 2.2.4. Feasibility Analysis of Two Lines of Pixel Values

#### 2.2.5. Feasibility Analysis of Compressed File after Entropy Coding

## 3. Design of Lossless Image Compression and Encryption Scheme

#### 3.1. Design of Pseudo-Random Sequence Generator

#### 3.1.1. Design of Hyperchaotic System

- 1.
- The necessary condition for the hyperchaotic system based on the Lorenz chaotic system is in differential equations:

- 2.
- Add the nonlinear term. Add a new variable and its first-order dynamic differential equation and introduce its nonlinear term and increase the dimension of the system.
- 3.
- Lyapunov function V of differential equations satisfies:

#### 3.1.2. Performance Analysis of Hyperchaotic System

- 1.
- Lyapunov exponent of Hyperchaotic system

- 2.
- Dissipativity of hyperchaotic system

- 3.
- Stability and periodicity of equilibrium for hyperchaotic system

#### 3.1.3. Pseudo-Random Sequence Generation

#### 3.2. Encryption Algorithm on Predicted Values and Final Predicted Errors

- 1.
- Encryption based on CBC mode

- 2.
- Encryption based on XOR mode

#### 3.3. Design of Image Pixels and Entropy Coding Encryption

- 1.
- Scrambling with the two-dimensional baker map

- 2.
- Pixel diffusion with table lookup method

## 4. Performance Analysis of Lossless Image Compression and Encryption Scheme

#### 4.1. Compression Performance of Lossless Image Compression and Encryption Scheme

- 1.
- Compression ratio analysis

- 2.
- Compression efficiency analysis

#### 4.2. Security Analysis of Lossless Image Compression and Encryption Scheme

- 1.
- Key Space

^{320}. In addition, for a random matrix used in pseudo-random sequence generator, there are 8! = 40,320 non repetitive permutations. Five permutations are randomly selected, and there are ${A}_{\mathrm{40,320}}^{5}$ selections in total.

^{64}.

^{384}, together with ${A}_{\mathrm{40,320}}^{5}$ random matrix space, which is strong enough to resist the brute force attack.

- 2.
- Fault tolerance of algorithm

- 3.
- Key sensitivity analysis

- 4.
- Statistical analysis

- 5.
- Correlation Analysis

_{j}and y

_{j}represent the gray values of two adjacent pixels and N represents the number of pixels in the sample. The maximum absolute value of the correlation coefficient is 1, and the minimum absolute value is 0. The lower the correlation coefficient is, the lower the correlation of the image pixels. A good encryption algorithm should remove correlation of the image pixels to improve the resistance against an attack.

- 6.
- Information entropy analysis

_{i}) denotes the probability of symbol x

_{i}. Taking 8 bytes as a unit, if the probability of every symbol in accordance with a uniform distribution is 1/8, the entropy should be 8. A good encryption scheme should make the entropy approach 8. The entropies of the encryption file and predicted errors are shown in Table 11.

- 7.
- Analysis of image processing attack

- 8.
- Security analysis of pseudo-random sequence

- Entropy test

_{i}= C

_{j}

^{3}, j = log

_{2}

^{i}. C

_{i}

^{m}is frequency of N overlapping blocks.

_{0}when $\tau =0$ and the value of sequence in the box of i

_{d}when $\tau =0$. The K entropy is defined as:

- Autocorrelation test

_{1}and l

_{2}represent the two pseudo-random sequences, respectively. A is the number of the same bit in l

_{1}, D is the number of the same bit in l

_{2}and N represents the total length of the key stream sequences.

_{1}and l

_{2}are the same sequence, $\psi $ is called autocorrelation. The best state of $\psi $ is close to a horizontal line. If the test result is a horizontal line close to 0, it shows that the test sequence has a good randomness.

- Balance test

- Sequence distribution test

- NIST SP800-22 tests

^{6}bits are tested. Test results are shown in Table 14.

- 9.
- Time complexity analysis

#### 4.3. Performance Comparison with Other Schemes

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Chaotic attractors and projection on different planes. (

**a**) Projection on Y-Z plane; (

**b**) projection on X-Y plane; (

**c**) projection on X-Z plane; (

**d**) projection on X-Y-Z plane.

**Figure 9.**Histogram of plaintext and cipher-image of different images. (

**a**) Lena plain image histogram; (

**b**) Lena cipher-image histogram; (

**c**) Barbara plain image histogram; (

**d**) Barbara cipher-image histogram; (

**e**) Baboon plain image histogram; (

**f**) Baboon cipher-image histogram; (

**g**) Boat plain image histogram; (

**h**) Boat cipher-image histogram; (

**i**) Peppers plain image histogram; (

**j**) Peppers cipher-image histogram.

**Figure 10.**The distribution of plaintext and cipher-text of different images. (

**a**) Lena plain image distribution; (

**b**) Lena cipher-image distribution; (

**c**) Barbara plain image distribution; (

**d**) Barbara cipher-image distribution; (

**e**) Baboon plain image distribution; (

**f**) Barbara cipher-image distribution; (

**g**) Boat plain image distribution; (

**h**) Boat cipher-image distribution; (

**i**) Peppers plain image distribution; (

**j**) Peppers cipher-image distribution.

Encryption Location | The Feasibility of Encryption | Impact on Compression Ratio | Implementation of Encryption |
---|---|---|---|

Plaintext image | √ | Significant impact | × |

The predicted values of pixels | √ | Depend on encryption strength | √ |

The final prediction error | √ | Depend on encryption strength | √ |

Two lines of pixel values | √ | No impact | √ |

Compressed file after entropy coding | √ | No impact | √ |

Function | Lyapunov1 | Lyapounov2 | Lyapounov3 | Lyapounov4 |
---|---|---|---|---|

Proposed system | 2.404 | 0.302 | 0.00 | −17.534 |

Lorenz system | 1.497 | 0.00 | −22.46 | —— |

Rossler system | 0.112 | 0.019 | 0 | −25.188 |

Ref. [12] | 0.456 | 0.219 | 0 | −15.060 |

Ref. [13] | 2.253019 | 1.406374 | 0.054342 | −38.339706 |

Ref. [15] | 0.81 | 0.31 | 0 | −24.11 |

Ref. [46] | 0.5697 | 0.0453 | 0 | −12.6078 |

Equilibrium | Eigenvalues 1 | Eigenvalues 2 | Eigenvalues 3 | Eigenvalues 4 |
---|---|---|---|---|

(0, 0, 0, 0) | −34.68 | 22.94 | 0.38 | −3.5 |

(−37.87, −9.15, −7.48, 46.16) | −62.38 | 0.29 | 23.56 + 38.04i | 23.56–38.04i |

(36.50,8.82,7.17,46.11) | −58.89 | 0.29 | 21.87 + 38.88i | 21.87–38.88i |

Operation Number | Encryption Operation |
---|---|

0 | ${P}_{i}=~{P}_{i}$ (invert by bit) |

1 | ${P}_{i}=rollerleft({P}_{i},{K}_{i})$ (rotate left ${K}_{i}$ bits of P) |

2 | ${P}_{i}=rollerright({P}_{i},{K}_{i})$ (rotate right ${K}_{i}$ bits of P) |

3 | ${P}_{i}={P}_{i}\oplus {K}_{i}$ (XOR ${K}_{i}$) |

4 | ${P}_{i}=rollerleft({P}_{i},{K}_{i})\oplus 123$ (rotate left ${K}_{i}$ bits of P and do XOR operation with constant) |

5 | ${P}_{i}=rollerleft({P}_{i},{K}_{i})\oplus 35$ (rotate right ${K}_{i}$ bits of P and do XOR operation with constant) |

Image | Lena | Baboon | Barbara | Boat | Peppers |
---|---|---|---|---|---|

MSE | 0 | 0 | 0 | 0 | 0 |

Encryption Mode | Lena | Baboon | Barbara | Boat | Peppers |
---|---|---|---|---|---|

No encryption | 4.2812 | 6.0000 | 4.6563 | 4.3438 | 4.5625 |

Encryption with XOR mode | 4.7186 | 6.3438 | 5.5625 | 4.5625 | 4.8438 |

Encryption with CBC mode | 6.5938 | 7.0938 | 6.9688 | 6.4375 | 6.4375 |

Time(s) | Lena | Barbara | Baboon | Boat | Peppers |
---|---|---|---|---|---|

Compression time | 0.218 | 0.234 | 0.265 | 0.218 | 0.218 |

Encryption time | 0.078 | 0.078 | 0.062 | 0.078 | 0.063 |

Total run time | 0.296 | 0.312 | 0.327 | 0.296 | 0.281 |

Encryption time/Total run time | 26.35% | 25% | 18.96% | 26.35% | 22.42% |

Encryption Location | Predicted Values | Predicted Errors | Image Pixels | Entropy Coding |
---|---|---|---|---|

Lena | × | × | √ | × |

Barbara | × | × | √ | × |

Baboon | × | × | √ | × |

Boat | × | × | √ | × |

Peppers | × | × | √ | × |

Image Name | Lena | Barbara | Baboon | Boat | Peppers |
---|---|---|---|---|---|

Change rate of cipher text | 0.49944 | 0.49963 | 0.49975 | 0.49993 | 0.49967 |

Image | Horizontal Direction | Vertical Direction | Diagonal Direction | |
---|---|---|---|---|

Lena | Plain image | 0.9722821 | 0.9852186 | 0.9608765 |

Cipher-image | 0.0116133 | 0.0125545 | 0.0218114 | |

Barbara | Plain image | 0.8612652 | 0.9595282 | 0.8468656 |

Cipher-image | 0.0122084 | 0.0079887 | 0.0200846 | |

Baboon | Plain image | 0.8689049 | 0.7629869 | 0.7403367 |

Cipher-image | 0.0071669 | 0.0115003 | 0.0181280 | |

Boat | Plain image | 0.9635128 | 0.9789292 | 0.9478472 |

Cipher-image | 0.0063547 | 0.0098399 | 0.0221604 | |

Peppers | Plain image | 0.9793075 | 0.9827495 | 0.9685861 |

Cipher-image | 0.0110749 | 0.0073561 | 0.0194759 |

Lena | Barbara | Baboon | Boat | Peppers | ||
---|---|---|---|---|---|---|

Information entropy | Original image | 7.4483 | 7.4664 | 7.3579 | 7.1237 | 7.5714 |

Encryption file | 7.9989 | 7.9901 | 7.9987 | 7.9989 | 7.9988 | |

Final prediction error | 4.2812 | 6.0000 | 4.6563 | 4.3438 | 4.5625 |

Test Item | Lena Cipher-image | Mean Filter | Median Filter | Fuzzy Contrast Enhancement Filter | Wiener Filter |
---|---|---|---|---|---|

PSNR | 5.22 | 6.43 | 5.56 | 4.18 | 6.74 |

MSSIM | 0.011 | 0.019 | 0.008 | 0.008 | 0.020 |

**Table 13.**Comparison about approximate entropy, information entropy and K entropy of pseudo-random sequence.

Sequence Length N | Test Item | Approximate Entropy | Information Entropy | K Entropy |
---|---|---|---|---|

800 | Logistic | 0.6932 | 2.8266 | 0.6950 |

Proposed algorithm | 0.6963 | 2.8261 | 0.6980 | |

1000 | Logistic | 0.6946 | 2.8174 | 0.6958 |

Proposed algorithm | 0.6962 | 2.8364 | 0.6976 | |

2000 | Logistic | 0.6928 | 2.8210 | 0.6935 |

Proposed algorithm | 0.6942 | 2.8297 | 0.6949 |

Statistical Test | p-Value | Proportion |
---|---|---|

Frequency | 0.494392 | 0.9800 |

Block Frequency | 0.574903 | 0.9900 |

Cumulative Sums | 0.955835 | 0.9900 |

Runs | 0.779188 | 0.9900 |

Longest Run | 0.319084 | 0.9800 |

Rank | 0.574903 | 0.9900 |

FFT | 0.102625 | 0.9900 |

NonOverlappingTemplate | 0.484732 | 0.9800 |

Overlapping Template | 0.470723 | 1.0000 |

Universal | 0.759756 | 0.9900 |

Approximate Entropy | 0.437274 | 0.9800 |

Random Excursions | 0.657933 | 0.9921 |

Random Excursions Variant | 0.383827 | 0.9932 |

Serial | 0.987896 | 0.9900 |

Linear Complexity | 0.249284 | 0.9700 |

Operation Item | Time Complexity |
---|---|

CALIC compression algorithm | O(n) |

GAP predicted pixel encryption | O(n) |

Predicted errors encryption | O(n) |

Image pixels encryption | O(n) |

Entropy encoding encryption | O(n) |

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## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Tong, X.; Wang, Z.; Chen, P.
Joint Lossless Image Compression and Encryption Scheme Based on CALIC and Hyperchaotic System. *Entropy* **2021**, *23*, 1096.
https://doi.org/10.3390/e23081096

**AMA Style**

Zhang M, Tong X, Wang Z, Chen P.
Joint Lossless Image Compression and Encryption Scheme Based on CALIC and Hyperchaotic System. *Entropy*. 2021; 23(8):1096.
https://doi.org/10.3390/e23081096

**Chicago/Turabian Style**

Zhang, Miao, Xiaojun Tong, Zhu Wang, and Penghui Chen.
2021. "Joint Lossless Image Compression and Encryption Scheme Based on CALIC and Hyperchaotic System" *Entropy* 23, no. 8: 1096.
https://doi.org/10.3390/e23081096