傳統的(k, n)秘密影像分享(Secret Image Sharing; SIS)方案是一種常用的密碼技術，通常用於安全地分享祕密訊息。在這種方案中，一個秘密影像會被分成n個子影像、並且每分送給參與者。只有k個或更多參與者合作時，才能回復原始的秘密影像。反之，若少於k個子影像無法解回秘密影像。儘管(k, n)-SIS已廣泛應用在很多場合，但是一般而言所有參與者在解密過程都只具有同等的重要性、並沒有差異性。
在實際應用情境，某一些參與者確實會比其他人重要，例如公司董事會成員。這些重要參與者確實應該擁有一些特權，以(k, n)-SIS而言可以是降低門檻值讓這些特權參與者可以使用更小的門檻來解回秘密影像。因此，本論文的研究主題是需要設計一種具有特權集合 (SIS with privileged set; SISPS)的方案。具特權集合的(k, n)-SISPS機制的成員有一般參與者、和一組特權參與者。任何k個參與者(含特權參與者)可回復秘密，而kp (<k) 個特權參與者也能解回秘密。本論文題目是"具有特權集合的秘密影像分享及其應用"，所以除了設計(k, n)-SISPS另外一個重點就是如何使用特權集合做不同的應用。我們提出了五種不同的應用情境: (i)使用特權集合的當作隱藏集合、(ii)使用特權集合當作驗證集合以驗證解回的秘密是否正確、(iii)使用特權集合驗證解回的秘密但特權集合無法得知秘密、(iv) 使用特權集合完成漸進式解回秘密影像(註:特權參與者可以使用更小的門檻值恢復秘密圖像但解回秘密並不完整、而k個參與者可以完整解回秘密影像)、(v)使用特權集合於兩個SISPS當作共同的特權集合(註:兩個SISPS的共同特權集合可以解出相同的秘密影像、但是兩個SISPS的任何k個參與者能各自解回不同的秘密影像)。上述這些不同的應用情境都是植基於(k, n)-SISPS的特權集合特性。
總而言之，具有特權集合的(k, n)-SISPS方案是一種新型的SIS方案，可以有更多樣性的實際應用。理論分析證明了(k, n)-SISPS符合一般的門檻特性k值、及特權的門檻特性kp值。另外，實驗結果也展現了五種不同應用的有效性。

The (k, n) secret image sharing (SIS) scheme is commonly used for securely sharing secret image. For (k, n)-SIS, a secret image is subdivided into n shadow images (referred to as shadows) and delivered to all participants. Any k or more participants can collaborate together to recover secret image; however, the less than k shadows cannot be used for recovery. The (k, n)-SIS is already widely used in many applications. In general, all participants have the same privilege and their shadows are with the same importance.
In real application scenarios, some participants may be more important than others, such as a CEO of a company. These important participants should be provided with privilege, on which they may recover secrete image with a smaller threshold. In this thesis, we study a SIS with a privileged set (SISPS). There are two types of involved participants of the (k, n)-SISPS: one is common participant and the other is privileged participant. Any k participants (including common and privileged participants) may recover secret, and any kp (< k) participants can also recover secret image. The thesis is entitled “Secret Image Sharing with Privileged Set and It’s Applications”. Except designing a (k, n)-SISPS, our major contribution is to use privileged set for various applications. We provide five application scenarios: (i) using the privileged set as hidden set, (ii) using privileged set as authentication set, (iii) using privileged set for verification and meantime the secret can only be recovered from any k involved participants, but cannot be obtained from kp privileged participants, (iv) using privileged set for progressive recovery (note: using privileged set may recover secret with low resolution but can recover the secret image when using k shadows), (v) using privileged set as a common privileged set in two (k, n)-SISPSs (note: the common privileged set may recover the same secrete, and any k participants in two SISPSs can recover the respective secret images).
In summarize, a (k, n)-SISPS is a new type of SIS and could be applied to diversities of various applications. Theoretical derivation shows that the (k, n)-SISPS satisfies two thresholds: the common threshold “k” and the privileged threshold “kp”. In addition, experimental results also demonstrate the effectiveness of the above five application scenarios.
Keywords: Secret sharing, Secret image sharing, Privileged set, Polynomial interpolation, Authentication, Progressive recovery

Chapter 1 Introduction 1
1.1 Background 1
1.2 The Organization of The Thesis 3
Chapter 2 Previous Works 5
2.1 Polynomial-based secret image sharing scheme 5
2.2 Secret image sharing scheme with essential participants 6
2.3 Threshold changeable secret image sharing scheme 7
Chapter 3 The Proposed (k, n)-SISPS 9
Chapter 4 How to use SISPS 21
4.1 Using the Privileged Set as Hidden set 21
4.2 Using Privileged Set as Authentication Set 22
4.3 Reconstruction from k Shadows and Using Privileged Set as Authentication Set 23
4.4 Using SISPS in Progressive Mode 25
4.5 Using Privileged Set as Common Set in Two (k, n)-SISPSs 27
Chapter 5 Experiment and Comparison 29
5.1 Experimental Results 29
5.2 Comparison and discussion 35
Chapter 6 Conclusion and Future Work 41
References 43

[SHAM79] A. Shamir, How to share a secret, Communications of the Association for Computing Machinery 22 (11) (1979) 612-613.
[THIE02] C.C. Thien, J.C. Lin, Secret image sharing, Computers & Graphics 26 (5) (2002) 765-770.
[LIN04] C.C. Lin and W.H. Tsai, Secret image sharing with steganography and authentication, Journal of Systems and Software 73(3) (2004) 405-414.
[WANG07] R.Z. Wang and S.J. Shyu, Scalable secret image sharing, Signal Processing-Image Communication 22 (4) (2007) 363-373.
[CHAN08] C.C. Chang, Y.P. Hsieh and C.H. Lin, Sharing secrets in stego images with authentication, Pattern Recognition 41(10) (2008) 3130-3137.
[SHYU09] S.J. Shyu, C.C. Chuang, Y.R. Chen and A.F. Lai, Weighted Threshold Secret Image Sharing, Proceedings of Advances in Image and Video Technology, LNCS 5414 (2009) 988-998.
[YANG10] C.N. Yang and S.M. Huang, Constructions and properties of k out of n scalable secret image sharing, Optics Communications 283(9) (2010) 1750-1762.
[YANG11] C.N. Yang and Y.Y. Chu, A general (k, n) scalable secret image sharing scheme with the smooth scalability, Journal of Systems and Software 84(10) (2011) 1726-1733.
[LI12] L. Li, A.A. El-Latif, X. Yan, S. Wang, X. Niu, A Lossless Secret Image Sharing Scheme Based on Steganography, Proceedings of the 2012 Second International Conference on Instrumentation & Measurement, Computer, Communication and Control (2012) 1247-1250.
[YANG12] C.N. Yang, J.F. Ouyang and L. Harn, Steganography and authentication in image sharing without parity bits, Optics Communications 285(7) (2012) 1725-1735.
[LI13] P. Li, C.N. Yang, C.C. Wu, Q. Kong, Y. Ma, Essential secret image sharing scheme with different importance of shadows, Journal of Visual Communication and Image Representation 24 (7) (2013) 1106-1114.
[LEE14] J.S. Lee, C.C. Chang and H.Y. Tsai, A User-Friendly and Authenticatable Secret Image Sharing Scheme, Journal of Internet Technology 15 (3) (2014) 433-439.
[LIU14] Y.X. Liu, C.N. Yang and P.H. Yeh, Reducing shadow size in smooth scalable secret image sharing, Security and Communication Networks 7(12) (2014) 2237-2244
[LI15] M.C. Li, S. Ma and C. Guo, A novel weighted threshold secret image sharing scheme, Security and Communication Networks 8(17) (2015) 3083-3093.
[LI16] P. Li, C.N. Yang and Z.L. Zhou, Essential secret image sharing scheme with the same size of shadows, Digital Signal Processing 50 (2016) 51-60.
[YUAN16] L.F. Yuan, M. Li, C. Guo, W. Hu, X. Luo, Secret Image Sharing Scheme with Threshold Changeable Capability, Mathematical Problems in Engineering 2016 (2016) 1-11.
[ZARE16] J. Zarepour-Ahmadabadi, M.S. Ahmadabadi and A. Latif, An adaptive secret image sharing with a new bitwise steganographic property, Information Sciences 369 (2016) 467-480.
[LEE17] J.S. Lee and Y.R. Chen, Selective scalable secret image sharing with verification, Multimedia Tools and Applications 76 (1) (2017) 1-11.
[LIU17] Y.X. Liu and C.N. Yang, Scalable secret image sharing scheme with essential shadows, Signal Processing-Image Communication 58 (2017) 49-55.
[CHEN18] Y.C. Chen, J.S. Lee, and H.C. Su, Selective scalable secret image sharing with adaptive pixel-embedding technique, Multimedia Tools and Applications 77 (20) (2018) 27107-27121.
[YAN19] X.H. Yan, Y.L. Lu, and L.T. Liu, A general progressive secret image sharing construction method, Signal Processing-Image Communication 71 (2019) 66-75.
[LI18a] P. Li, Z.Q. Liu and C.N. Yang, A construction method of (t, k, n)-essential secret image sharing scheme, Signal Processing-Image Communication 65 (2018) 210-220.
[LI18b] P. Li, Z. Liu, An Improved Essential Secret Image Sharing Scheme with Smaller Shadow Size, International Journal of Digital Crime & Forensics 10 (3) (2018) 78-94.
[DESO18] R.L. de Souza, M. Vigil, R. Custodio, F. Caullery, L. Moura, D. Panario, Secret Sharing Schemes with Hidden Sets, Procedding of 2018 IEEE Symposium on Computers and Communications (2018) 718-723.
[GUO19] Y.Z. Guo, Z. Ma, and M. Zhao, Polynomial Based Progressive Secret Image Sharing Scheme With Smaller Shadow Size, IEEE Access 7 (2019) 73782-73789.
[HE19] Q. He, S. Yu, H. Xu et al., A Weighted Threshold Secret Sharing Scheme for Remote Sensing Images Based on Chinese Remainder Theorem, CMC-Computers Materials & Continua 58 (2) (2019) 349-361.
[WU19a] X.T. Wu and C.N. Yang, Partial reversible AMBTC-based secret image sharing with steganography, Digital Signal Processing 93 (2019) 22-33.
[WU19b] X.T. Wu and C.N. Yang, Invertible secret image sharing with steganography and authentication for AMBTC compressed images, Signal Processing-Image Communication 78 (2019) 437-447.
[LIU19] Y.X. Liu, C.N. Yang, C.M. Wu, Q. Sun, W. Bi, Threshold changeable secret image sharing scheme based on interpolation polynomial, Multimedia Tools and Applications 78 (13) (2019) 18653-18667.
[LIU20] Y.X. Liu, C.N. Yang, Q.D. Sun, (k, n) scalable secret image sharing with multiple decoding options, Journal of Intelligent & Fuzzy Systems 38 (1) (2020) 219-228.
[AZZA20] A.A. Azza and S.G. Lian, Multi-secret image sharing based on elementary cellular automata with steganography, Multimedia Tools and Applications 79 (29-30) (2020) 21241-21264.
[SARD20] M.K. Sardar and A. Adhikari, Essential secret image sharing scheme with small and equal sized shadows, Signal Processing-Image Communication 87 (2020) 115923.
[XION20] L.Z. Xiong, X.W. Zhong and C.N. Yang, DWT-SISA: a secure and effective discrete wavelet transform-based secret image sharing with authentication, Signal Processing 173 (2020) 107571.
[WU20] X.T. Wu, C.N. Yang and Y.Y. Yang, Sharing and hiding a secret image in color palette images with authentication, Multimedia Tools and Applications 79 (35-36) (2020) 25657-25677.
[WANG21] Y. Wang, J. Chen, Q. Gong, X. Yan and Y. Sun, Weighted Polynomial-Based Secret Image Sharing Scheme with Lossless Recovery, Security and Communication Networks 2021 (2021) 5597592.
[QIN21] S. Qin, Z. Tan, F. Zhou, J. Xu, Z. Zhang, A Verifiable Steganography-Based Secret Image Sharing Scheme in 5G Networks, Security and Communication Networks 2021 (2021) 6629726.
[XIONG21] L. Xiong, X. Zhong, C.N. Yang, X. Han, Transform Domain-Based Invertible and Lossless Secret Image Sharing With Authentication, IEEE Transactions on Information Forensics and Security 16 (2021) 2912-2925.
[GAO21] K. Gao, J.H. Horng and C.C. Chang, An Authenticatable (2,3) Secret Sharing Scheme Using Meaningful Share Images Based on Hybrid Fractal Matrix, IEEE Access 9 (2021) 50112-50125.
[BHAT21] K. Bhat, U.K. Reddy, R. Kumar, D. Mahto, A novel scheme for lossless authenticated multiple secret images sharing using polynomials and extended visual cryptography, IET Information Security 15(1) (2021)13-22.
[YAN21] X.H. Yan, Y. Lu, C.N. Yang, X. Zhang, A Common Method of Share Authentication in Image Secret Sharing, IEEE Transactions on Circuits and Systems for Video Technology 31(7) (2021) 2896-2908.
[WU21] G. Wu, M. Wang, Q. Wang, Y. Yao, L. Yuan, G. Miao, A Novel Threshold Changeable Secret Image Sharing Scheme, Symmetry-Basel 13 (2021) 286.
[YADA22] M. Yadav and R. Singh, Essential secret image sharing approach with same size of meaningful shares, Multimedia Tools and Applications 81 (2022): 22677-22694.