Reversible Data Hiding in Encrypted Images Based on Polynomial Secret Sharing
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摘要: 针对密文域可逆信息隐藏在多用户场景下算法嵌入率低、载体图像容灾性能较弱等问题，该文提出一种基于多项式秘密共享的图像密文域可逆信息隐藏方案。通过将图像分割成多幅影子图像并存储在不同的用户端，可以增强图像的容灾性，为了实现额外信息在图像重构前后提取的可分离性，该方案包括两种嵌入算法：算法1在图像分割的过程中，将额外信息嵌入多项式的冗余系数中得到含有额外信息的影子图像，该算法支持在图像重构之后提取额外信息；算法2针对图像分割后的任一影子图像，利用秘密共享的加法同态特性实施嵌入，该算法支持直接从影子图像中提取额外信息。实验在不同门限方案和影子图像压缩率的条件下进行测试，当压缩率为50%时，(3, 4)门限方案的嵌入率达4.18 bpp(bit per pixel)，(3, 5)门限方案的嵌入率达3.78 bpp。结果表明，两种嵌入算法分别支持从影子图像与重构图像中提取额外信息，实现了方案的可分离性；与现有方案相比，所提算法嵌入率较高、计算复杂度较低，具有较强的实用性。Abstract: In order to solve the problems of low embedding rate of reversible data hiding in encrypted domain with multi-users and weak disaster tolerance of cover image, a reversible data hiding in encrypted images based on polynomial secret sharing scheme is proposed. The cover image is divided into multiple shadow images and stored to multiple users, so that the image disaster tolerance can be enhanced. Separability is achieved by combining two embedding algorithms, where Algorithm 1 embeds additional data into the redundancy coefficients of the polynomial during the process of image sharing to generate shadow images containing additional data, which can extract data after image reconstruction; Algorithm 2, for any shadow image, embeds additional data by using the additive homomorphism of secret sharing, which supports extracting additional data directly from the shadow image. Experiments are performed in different threshold and shadow image compression rates. When the compression rate is 50%, the embedding rate reaches 4.18 bit per pixel (bpp) for the (3, 4) threshold scheme and 3.78 bpp for the (3, 5) threshold scheme. The results show that the algorithms support extracting additional data from the shadow images and the reconstructed images respectively, which achieves the separability of the scheme. The proposed scheme has a higher embedding rate and lower computational complexity than the existing schemes, and is more practical.
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表 1 相关变量说明
变量 变量说明 变量 变量说明 ${\boldsymbol{I}}$ 原始图像 $ {\boldsymbol{K}}_{\text{s}}^{[i]},i = 1,2, \cdots ,n $ 用户共享密钥 ${ {\boldsymbol{I} }_{\rm{e}}}$ 置乱图像 ${{\boldsymbol{K}}_{\text{a}}}$ 多项式嵌入时，额外信息隐藏密钥 ${ {\boldsymbol{m} }_{{a} } }$ 多项式嵌入的额外信息 ${\boldsymbol{K} }_{{b} }^{[i]},i = 1,2, \cdots ,n$ 同态嵌入时，额外信息隐藏密钥 ${ {\boldsymbol{m} }_{{b} } }$ 同态嵌入的额外信息 ${\boldsymbol{I}}_{{\text{ema}}}^{[i]},i = 1,2, \cdots ,n$ 包含额外信息的影子图像 ${{\boldsymbol{K}}_{\text{e}}}$ 图像置乱密钥 ${\boldsymbol{I}}_{{\text{emb}}}^{[i]},i = 1,2, \cdots ,n$ 包含额外信息和标记信息的影子图像 
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表 2 译码表
生成多项式 ${\boldsymbol{E}}$ 0000000 0000001 0000010 0000100 0001000 0010000 0100000 1000000 ${q_1}(x) = {x^3} + x + 1$ ${{\boldsymbol{S}}_1}$ 000 001 010 100 011 110 111 101 $ {q_2}(x) = {x^3} + {x^2} + 1 $ ${{\boldsymbol{S}}_2}$ 000 001 010 100 101 111 011 110
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表 3 实际最大嵌入率
最大嵌入率
(bpp)n = 4 n = 5 n = 6 n = 7 w = 2 w = 2 w = 3 w = 2 w = 3 w = 4 w = 2 w = 3 w = 4 w = 5 k = 3 4.1818 3.7818 \ 3.5152 \ \ 3.3247 \ \ \ k = 4 \ 5.3818 3.7818 4.8485 3.5152 \ 4.4675 3.3247 \ \ k = 5 \ \ \ 6.1818 4.8485 3.5152 5.6104 4.4675 3.3247 \ k = 6 \ \ \ \ \ \ 6.7532 5.6104 4.4675 3.3247
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表 4 同态解密前后重构图像质量
测试图像 同态解密前重构密文图像 同态解密前重构明文图像 同态解密后重构明文图像 PSNR(dB) 信息熵 PSNR(dB) 信息熵 PSNR(dB) 信息熵 Lena 10.166119 7.955290 10.870086 7.955290 $ + \infty $ 7.218498 Baboon 10.571863 7.951466 11.136816 7.951466 $ + \infty $ 7.139099 Boat 9.628571 7.928695 10.478454 7.928695 $ + \infty $ 7.046737 Goldhill 9.688918 7.967631 10.387593 7.967631 $ + \infty $ 7.472315
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表 5 信息嵌入前后含密影子图像安全性分析
测试图像 未嵌入信息的影子图像 多项式嵌入后的影子图像 同态嵌入后的影子图像 信息熵 NPCR(%) UACI(%) 信息熵 NPCR(%) UACI(%) 信息熵 Lena 7.998589 99.617195 33.423751 7.998618 99.611473 33.433949 7.998648 Baboon 7.998515 99.599838 33.412525 7.998594 99.601936 33.402735 7.998660 Boat 7.998591 99.606895 33.435428 7.998544 99.605942 33.412868 7.998607 Goldhill 7.998551 99.619293 33.476432 7.998505 99.608231 33.459180 7.998645
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表 6 不同算法特性对比
算法 算法框架 加密方式 时间复杂度 密文扩展 可分离性 信息隐藏者 文献[8] VRBE 流密码 $O(N)$ 1 否 单用户 文献[18] VRBE 多秘密共享 $O(N)$ 1 否 单用户 文献[25] VRBE Paillier $O({N^3})$ 128 是 单用户 文献[3] VRAE 流密码 $O(N)$ 1 否 单用户 文献[15] VRAE Paillier $O({N^3})$ 128 否 单用户 文献[17] VRAE 秘密共享 $O(N)$ n 否 单用户 文献[20] VRAE 秘密共享 $O(N)$ n 是 多用户 文献[13] VRIE LWE $O({N^2})$ 16 否 单用户 文献[14] VRIE 流密码 $O(N)$ 1 否 单用户 本文算法 VRIE 秘密共享 ${\mathbf{O(N)}}$ n/w 是 多用户
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