# American Institute of Mathematical Sciences

## An algorithm for reversible information hiding of encrypted medical images in homomorphic encrypted domain

 School of Management, Guangdong University of Technology, Guangzhou 510520, China

* Corresponding author: Xueyan Wu

Received  June 2017 Revised  November 2017 Published  November 2018

At present, in reversible information hiding algorithm of image, the difference expansion idea is used. After the carrier image encryption, in encrypted image, information bits are embedded in the low value, resulting in the fact that in the image embedded with watermarking information, a part of the boundary pixel value has flipped. After being extracted, the carrier image cannot be recovered completely that is not only a large quantity of calculation, and the image quality has also been some damage. A algorithm for reversible information hiding of encrypted the medical image in the homomorphic encryption domain is proposed. Combining wavelet and fast fuzzy algorithm, the image edge is extracted from the high frequency and low frequency parts of medical image, and the medical image is reconstructed in the compressed boundary part. Combined with the thought of block compressed sensing and block edge pixels, the reconstructed medical image is divided to multiple non-overlapping blocks. The pixel in the right lower edge of the block is made homomorphic encryption operation, the remaining pixels are made compressed sensing operation, and the two parts are combined to a ciphertext to be sent to the owner of the channel According to the information hiding key the secret information is embedded into the ciphertext by the channel owners. The receiver can extract the information and restore the original image based on the encryption key and the information hiding key. The experimental results show that the proposed algorithm has high embedding capacity, the image quality after recovery is high, and the computational complexity is low.

Citation: Xueyan Wu. An algorithm for reversible information hiding of encrypted medical images in homomorphic encrypted domain. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019099
##### References:
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##### References:
 [1] M. A. Akhaee, S. M. E. Sahraeian, B. Sankur and F. Marvasti, Information hiding with maximum likelihood detector for correlated signals, IEEE Transactions on Multimedia, 36 (2015), 144-155. doi: 10.1016/j.dsp.2014.09.003. [2] M. Aziz, M. H. Tayarani-N and M. Afsar, A cycling chaos-based cryptic-free algorithm for image steganography, Nonlinear Dynamics, 80 (2015), 1271-1290. [3] B. Bank, J. Heintz, G. Matera, J. L. Montana, L. M. Pardo and A. R. Paredes, Quiz games as a model for information hiding, Journal of Complexity, 34 (2016), 1-29. doi: 10.1016/j.jco.2015.11.005. [4] A. Basu, K. Nandy, A. Banerjee, S. Giri, S. Sarkar and S. K. Sarkar, On the implementation of ip protection using biometrics based information hiding and firewall, International Journal of Electronics, 103 (2016), 177-194. [5] H. Chen, J. Ni, W. Hong and T. S. Chen, Reversible data hiding with contrast enhancement using adaptive histogram shifting and pixel value ordering, Information Sciences, C (2017), 250-265. [6] K. Dong, H. J. Kim, S. C. Yong, H. J. Sang and B. H. Chung, Reversible binary image watermarking method using overlapping pattern substitution, Etri Journal, 37 (2015), 990-1000. [7] W. F., W. L. and W. T. T, Hidden optimization algorithm simulation analysis of large capacity images, Computer Simulation, 3 (2016), 409-412. [8] N. Jiang, N. Zhao and L. Wang, Lsb based quantum image steganography algorithm, International Journal of Theoretical Physics, 55 (2016), 3722-3736. doi: 10.1007/s10773-016-3001-3. [9] H. J. L. and Z. X. H., Adjustable reversible data hiding algorithm with large embedded capacity based on image gradient prediction, Journal of Jilin University (Engineering and Technology Edition), 2074-2079. [10] M. Li, D. Xiao, A. Kulsoom and Y. Zhang, Improved reversible data hiding for encrypted images using full embedding strategy, Electronics Letters, 51 (2015), 690-691. [11] T. Luo, G. Jiang, M. Yu, H. Xu and F. Shao, Inter-view local texture analysis based stereo image reversible data hiding, Digital Signal Processing, 48 (2016), 116-129. doi: 10.1016/j.dsp.2015.09.007. [12] K. Martin and K. Shankar, How often should you clean your room?, Discrete Mathematics and Theoretical Computer Science, 17 (2015), 413-441. [13] Z. Qian, H. Zhou, W. Zhang and X. Zhang, Comments on Steganography Using Reversible Texture Synthesis, 4, IEEE Transactions on Image Processing, 2017. [14] S. S. H., Ubiquitous data gathering algorithm based on maximization of information gain in wireless sensor networks, in Journal of China Academy of Electronics and Information Technology, 4 (2017), 371-377. [15] L. B. Si and H. Y. Qiao, Evaluation of technological innovation efficiency in equipment manufacturing industry based on input orientation-panel data analysis based on data envelopment model, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 1381-1386. [16] H. Unno, R. Yamkum, C. Bunporn and K. Uehira, A new displaying technology for information hiding using temporally brightness modulated pattern, IEEE Transactions on Industry Applications, 53 (2017), 596-601. [17] C. Y. Wang, L. Li, H. R. Li, C. L. Qin and H. Y. Zhao, Research on the digital image encryption algorithm based on double chaos, Bulletin of Science and Technology, 12 (2016), 169-173. [18] Y. J. Wang Dong—Chen, Knowledge management of web financial reporting in human-computer interactive perspective., Eurasia Journal of Mathematics Science & Technology Education, 13 (2017), 3349-3373. [19] L. Yong, S. Tuo and J. Shi, Sparse solution of some special optimization problems, Journal of Interdisciplinary Mathematics, 20 (2017), 595-602. [20] X. Y. Zhang and J. Q. Wang, Consensus-based framework to mcgdm under multi-granular uncertain linguistic environment, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 33 (2017), 1263-1274. [21] H. Zheng, Y. Yang, D. Xiao and J. He, Rdh in bcs images based on block edge pixel separation, Electronics Letters, 53 (2016), 18-20.
coefficient of medical images after block compression perception processing
grouping rules of medical image
standard test medical images
average computing time for LE in medical images
the number of inlay pixels for eight test images
 LE AI BA BN BO HI PE LA b/($\uparrow$) 0 3 514 285 26 0 2 1
 LE AI BA BN BO HI PE LA b/($\uparrow$) 0 3 514 285 26 0 2 1
maximum embedding rate and maximum pure embedding rate of eight test images
 LE AI BA BN BO HI PE LA Maximum embedding rate/bpp 0.4855 0.4855 0.4802 0.4812 0.4855 0.4855 0.4932 0.4932 Maximum embedding rate/bpp 0..4854 0.4852 0.4525 0.4758 0.4825 0.4821 0.4921 0.4924
 LE AI BA BN BO HI PE LA Maximum embedding rate/bpp 0.4855 0.4855 0.4802 0.4812 0.4855 0.4855 0.4932 0.4932 Maximum embedding rate/bpp 0..4854 0.4852 0.4525 0.4758 0.4825 0.4821 0.4921 0.4924
Comparison of the results of different algorithms performed on LE
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 1.425 38.1 54.25 14.25 36.58 45.8 ZF 0.074 38.1 68.47 2.58 36.58 51.41 LO 0.185 38.27 61.58 3.68 36.74 48.52 The proposed algorithm 0 26.85-42.1 $\uparrow$ 0 22.14-49.55 $\uparrow$
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 1.425 38.1 54.25 14.25 36.58 45.8 ZF 0.074 38.1 68.47 2.58 36.58 51.41 LO 0.185 38.27 61.58 3.68 36.74 48.52 The proposed algorithm 0 26.85-42.1 $\uparrow$ 0 22.14-49.55 $\uparrow$
Comparison of the results of different algorithms performed on AI
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 4.362 37.58 50.25 16.25 37.58 42.58 ZF 0.124 37.58 65.25 4.584 36.58 48.47 LO 0.685 36.78 48.25 5.147 36.14 48.62 The proposed algorithm 0 26.85-41.8 $\uparrow$ 0 22.14-52.41 $\uparrow$
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 4.362 37.58 50.25 16.25 37.58 42.58 ZF 0.124 37.58 65.25 4.584 36.58 48.47 LO 0.685 36.78 48.25 5.147 36.14 48.62 The proposed algorithm 0 26.85-41.8 $\uparrow$ 0 22.14-52.41 $\uparrow$
Comparison of the results of different algorithms performed on BA
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 33.25 36.25 38.58 25.25 36.25 39.58 ZF 20.54 36.25 40.25 9.25 36.25 43.25 LO 22.14 36.84 41.05 12.54 36.87 40.14 The proposed algorithm 0 22.14-32.14 $\uparrow$ 0 21.15-50.15 $\uparrow$
 Algorithm Embedding rate = 0.0145 Embedding rate = 0.0524 Error rate/% Direct decryption image restore image Error/% rate/% Direct decryption image restore image KF 33.25 36.25 38.58 25.25 36.25 39.58 ZF 20.54 36.25 40.25 9.25 36.25 43.25 LO 22.14 36.84 41.05 12.54 36.87 40.14 The proposed algorithm 0 22.14-32.14 $\uparrow$ 0 21.15-50.15 $\uparrow$
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