This paper proposes a robust and reversible watermarking scheme for the encrypted image by using Paillier cryptosystem. In the proposed method, the original image is divided into a number of non-overlapping blocks sized by 8 × 8 and Paillier cryptosystem is applied to encrypt the pixels in each block. Firstly, a data hider can calculate the statistical values of encrypted blocks by employing modular multiplicative inverse (MMI) method and looking for a mapping table. Then a watermark sequence can be embedded into the encrypted image by shifting the histogram of the statistical values. On the receiver side, the shifted histogram can be obtained from both the encrypted image and the decrypted image. Furthermore, the embedded watermark can be extracted from the shifted histogram. The encrypted original image can be restored by employing inverse operations of histogram shifting. This is followed by a decryption operation to restore the original image. In the proposed method, the hidden bits can still be extracted correctly under some typical content-preserving operations, such as JPEG/JPEG2000 compression and additive Gaussian noise. Compared with the previous reversible watermarking methods in plaintext domain, the proposed method has satisfactory performance in image quality and robustness. Experimental results have shown the validity of the proposed method.

本文利用Paillier密码系统提出了一种鲁棒且可逆的加密图像水印方案。在所提出的方法中，原始图像被分成多个大小为 8 × 8 的非重叠块，并应用 Paillier 密码系统对每个块中的像素进行加密。首先，数据隐藏器可以通过采用模块化乘法逆（MMI）方法并查找映射表来计算加密块的统计值。然后可以通过移动统计值的直方图将水印序列嵌入到加密图像中。在接收方，可以从加密图像和解密图像中获得移位直方图。此外，可以从移位的直方图中提取嵌入的水印。可以通过直方图移位的逆运算来恢复加密的原始图像。随后进行解密操作以恢复原始图像。在所提出的方法中，隐藏位在一些典型的内容保留操作下仍然可以正确提取，例如JPEG/JPEG2000压缩和加性高斯噪声。与以往的纯文本域可逆水印方法相比，该方法在图像质量和鲁棒性方面具有令人满意的性能。实验结果表明了所提出方法的有效性。例如 JPEG/JPEG2000 压缩和加性高斯噪声。与以往的纯文本域可逆水印方法相比，该方法在图像质量和鲁棒性方面具有令人满意的性能。实验结果表明了所提出方法的有效性。例如 JPEG/JPEG2000 压缩和加性高斯噪声。与以往的纯文本域可逆水印方法相比，该方法在图像质量和鲁棒性方面具有令人满意的性能。实验结果表明了所提出方法的有效性。