# Homomorphic Encryption-Based Robust Reversible Watermarking for 3D Model

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## Abstract

**:**

## 1. Introduction

## 2. Paillier Cryptosystem

- Key Generation

- Encryption

- Decryption

- Lemma One

- Homomorphic Multiplication

- Modular Multiplication Inverse (MMI)

## 3. The Proposed Method

#### 3.1. Preprocessing

#### 3.2. Patch Dividing and Patch Encryption

#### 3.2.1. Patch Dividing

#### 3.2.2. Patch Encryption

#### 3.3. Watermark Embedding

#### 3.3.1. Three Direction Values Calculation of Each Patch

#### 3.3.2. Constructing the Mapping Table

#### 3.3.3. Constructing the Symmetrical Direction Histogram

#### 3.3.4. Embedding Watermark by Histogram Shifting

#### 3.4. Watermark Extraction

#### 3.4.1. Extracting Watermark in an Encrypted Domain and Restore the Original Encrypted Model

#### 3.4.2. Extracting Watermark in Decrypted Model

## 4. Experimental Results and Discussion

#### 4.1. The Value of $\beta $

#### 4.2. The Value of $t$

#### 4.3. Feasibility of the Watermarking

#### 4.4. Robustness Analysis

#### 4.4.1. Robustness Against Translation Attacks

#### 4.4.2. Robustness Against Scaling Attacks

#### 4.4.3. Robustness to Gaussian Noise Attacks

#### 4.5. Compared with the Existing Watermark Method in an Encrypted Domain

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The patch with four vertices. (

**a**) $M(p)$ correspond to the vertex. (

**b**) The encrypted coordinate.

**Figure 5.**The blue line represents relationship between the maximum direction value and the number of vertices of the patch, and the red line is the fitted curve of the blue line.

**Figure 8.**The watermarked histogram. After embedding the watermark, the original direction histogram can be divided into 0-bit area and 1-bit area. The 0-bit area and 1-bit area are separated by the robust interval of size $T({N}_{l})$.

**Figure 10.**The effect of $\beta $ on the distortion of decrypted model and the bit error rate of the extracted watermark. (

**a**) $\beta $ is related to signal-to-noise ratio ($SNR$ ). (

**b**) $\beta $ is related to bit error rate (BER).

**Figure 12.**Experiment with 3D model ‘devil’ (

**a**) The original model; (

**b**) the encrypted model; (

**c**) the watermarked model; (

**d**) the decrypted model. After decryption, the $SNR$ was 30.93. (

**e**) The restored model. After restoration, the $SNR$ approached infinity. (

**f**) The bit error rate after watermark extraction.

**Figure 13.**Five watermarked 3D models. (

**a**) The watermarked “Fairy”; (

**b**) the watermarked “Boss”; (

**c**) the watermarked “Solider”; (

**d**) the watermarked “Thing”; (

**e**) the watermarked “Lord”; (

**f**) $SNR$ of the five watermarked models.

Model | t | SNR | Gaussian | Translation | Scaling | ||||
---|---|---|---|---|---|---|---|---|---|

(0.005) | (0.01) | (0.02) | 0.8 | 1.2 | 1.5 | ||||

Fairy | 40 | 30.96 | 1.75% | 2.63% | 6.15% | 1 | 0.21 | 0.073 | 0.18 |

Boss | 40 | 30.96 | 1.24% | 1.52% | 6.26% | 1 | 0.17 | 0.064 | 0.19 |

Solider | 35 | 31.44 | 1.98% | 2.67% | 5.98% | 1 | 0.13 | 0.053 | 0.15 |

Devil | 30 | 31.44 | 2.06% | 2.41% | 6.74% | 1 | 0.16 | 0.069 | 0.19 |

Thing | 30 | 31.44 | 1.68% | 2.47% | 6.45% | 1 | 0.18 | 0.057 | 0.21 |

Lord | 25 | 32.1 | 2.28% | 3.24% | 8.64% | 1 | 0.23 | 0.071 | 0.24 |

**Table 2.**Compared to the method of Jiang [1].

Capability | Robustness | Security | SNR of Decrypted Model | SNR of Restored Model | BER | |
---|---|---|---|---|---|---|

The proposed method | 0.396 | yes | high | 30.08 | +∞ | 0 |

Method of Jiang [1] | 0.365 | no | low | 5.35 | 31.97 | 4.22% |

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**MDPI and ACS Style**

Li, L.; Wang, S.; Zhang, S.; Luo, T.; Chang, C.-C.
Homomorphic Encryption-Based Robust Reversible Watermarking for 3D Model. *Symmetry* **2020**, *12*, 347.
https://doi.org/10.3390/sym12030347

**AMA Style**

Li L, Wang S, Zhang S, Luo T, Chang C-C.
Homomorphic Encryption-Based Robust Reversible Watermarking for 3D Model. *Symmetry*. 2020; 12(3):347.
https://doi.org/10.3390/sym12030347

**Chicago/Turabian Style**

Li, Li, Shengxian Wang, Shanqing Zhang, Ting Luo, and Ching-Chun Chang.
2020. "Homomorphic Encryption-Based Robust Reversible Watermarking for 3D Model" *Symmetry* 12, no. 3: 347.
https://doi.org/10.3390/sym12030347