使用秘密分享技術來分享數位影像是一個結合密碼學和影像處理的重要研究。自1979年Shamir提出(k, n)秘密分享(Secret Sharing; SS)機制，從此便開啟了門檻式秘密分享的廣泛研究。兩個重要領域(密碼與影像)的結合，在影像媒體引進了秘密分享的門檻特性，使得數位影像分享技術能有更多應用。在(k, n)祕密影像分享 (表示成(k, n)-SIS)，一個祕密影像會分享成n個子影像。任k個子影像會還原祕密影像，當小於k個子影像時，則無法獲得祕密影像的任何訊息。所謂的(2, 2)分享數位影像機制(Sharing Digital Image scheme; SDIS) (表示成(2, 2)-SDIS)引用了彩色調色盤處理影像，將秘密影像分成兩張子影像。
(2, 2)-SDIS只是(k, n)-SIS的另一種型態，其中k=n=2。由於使用了植基於調色盤的影像，能大幅降低子影像大小更方便於在網路上儲存傳輸。在此之前，研究調色盤影像的秘密影像分享技術均為(2, 2)的機制。在本篇論文中，我們擴展(2, 2)-SDIS至(n, n)-SDIS，其中n>=3(n, n)-SDIS。我們的(n, n)-SDIS方法是植基於(2, 2)-SDIS基礎上設計的。將原本在(2, 2)-SDIS中使用5黑4白(簡寫成5B4W)處理顏色索引0~255的區塊，改為6B3W區塊。藉由區塊中黑點數量的提升，我們的(n, n)-SDIS主要貢獻是有更高子影像的解析度，也就是有較高對比的黑白子影像，以及彩度較高的彩色子影像。我們的方法也能延伸成分享百萬色的秘密影像。另外，我們也提出了一個更進一步改善彩色子影像解析度的方法。
當區塊改為6B3W時，n個區塊的互斥或結果如何能成功地表示0~255的顏色索引，本論文提出一個基於之前(2, 2)-SDIS的解決方法。文章中，我們也理論證明了(n, n)-SDIS符合安全性，抵抗(n1)的共謀攻擊，以及比之前的(2, 2)-SDIS有較佳的性能。

Sharing digital image by secret sharing technology is an important research area combining cryptography and image processing. In 1979, Shamir published a landmark paper that a (k, n)-Secret Sharing (SS) hiding a secret data by using a (k1)-degree polynomial. A combination of secret sharing and image processing has the threshold property of sharing image may achieve more applications. In a (k, n) secret image sharing (SIS), a secret image is shared into n shadow images. Any k shadow images may reveal the secret image, while less than k shadow images have no information about the secret image. The so-called (2, 2) sharing digital image scheme (SDIS) (referred to as (2, 2)-SDIS) adopts a color palette to share a digital color secret image into two shadow images.
A (2, 2)-SDIS is just another type of (k, n)-SIS where k=n=2. When using col-or-palette image, we can substantially decrease the storage size of shadows and more effectively transmit via Internet. Up to date, the researches about SDIS is only (2, 2) scheme. In this thesis, we extend (2, 2)-SDIS to (n, n)-SDIS, where n>=3. Our (n, n)-SDIS scheme is designed based on (2, 2)-SDIS. However, we use 5 black 4 white (5B4W) block to 6B3W block, to represent the color index of 0~255. Via increasing the number of black pixels, our (n, n)-SDIS scheme has higher resolution of shadow images, i.e., the black and white pixels in block demonstrate the higher contrast of binary meaningful shadow images, as well as the higher resolution of colorful meaningful shadow images. Our scheme can also be applied on sharing true-color secret image. In addition, we also propose a scheme to improve the resolution of shadow images further.
When changing 5B4W block to 6B3W block, how to obtain the color index 0~255 from XOR-ing two blocks, we design a novel approach based on previous (2, 2) schemes. In this thesis, we also theoretically prove that our (n, n)-SDIS is secure, resistant to (n-1)-colluder attack theoretically, and has better performance than previous (2, 2)-SDIS schemes.

第一章 Introduction 1
1.1 Background 1
1.2 Contribution of the Thesis 2
1.3 Organization of the Thesis 3
第二章 Previous Works 5
第三章 The Proposed (n,n)-SDIS 9
3.1 Design Concept 10
3.2 Sharing and Recovering Algorithms 11
3.3 Extension of (n, n)-SDIS to Share True Color Secret Image 18
3.4 Enhance Visual Quality of Color Meaningful Shadow 21
第四章 The (n-1)-Colluder Attack 23
第五章 Experiment and Comparison 27
5.1 Experimental Results 27
5.2 Discussion and Comparison 34
5.2.1 Enhancing RP 34
5.2.2 Comparison 38
第六章 Conclusion and Future Work 41

[CHAN08] C.C. Chang, Y.P. Hsieh, and C.H. Lin,“Sharing secrets in stego images with authentication,”Pattern Recognition, Vol. 41, pp. 3130-3137, 2008.
[CHEN14] C.C. Chen and W.J. Wu,“A secure Boolean-based multi-secret image sharing scheme, ”Journal of Systems and Software, Vol. 92, pp. 107-114, 2014.
[CHEN17] Y. Chen,“Fully incrementing visual Cryptography from a succinct non-monotonic structure,”IEEE Transactions on Information Forensics and Security, Vol. 12, pp. 1082-1091, 2017.
[JIA18] X. Jia, D. Wang, D. Nie, and C. Zhang, “Collaborative visual cryptog-raphy schemes,”IEEE Transactions on Circuits and Systems for Video Technology, Vol. 28, pp. 1056-1070, 2018.
[LIN10] Y.Y. Lin and R.Z. Wang,“Scalable secret image sharing with smaller shadow images,”IEEE Signal Processing Letters, Vol. 17, pp. 316-319, 2010.
[LIU17] Y.X. Liu and C.N. Yang,“Scalable secret image sharing scheme with es-sential shadows,”Signal Processing: Image Communication, Vol. 58, pp. 49-55, 2017.
[LI18] P. Li, Z. Liu, and C.N. Yang,“A construction method of (t, k, n)-essential secret image sharing scheme,”Signal Processing: Image Communication, Vol. 65, pp. 210-220, 2018
[LIU18] X. Liu, C.N. Yang, S.Y. Wu, and Y.S. Chou,“Progressive (k, n) secret image sharing schemes based on Boolean operations and covering codes,” Signal Processing: Image Communication, Vol. 66, pp. 77-86, 2018.
[NAOR95] M. Naor and A. Shamir,“Visual cryptography,”Advances in Cryptolo-gy-EUROCRYPT’ 94, LNCS 950, pp.1-12, 1995.
[SHAM79] A. Shamir,“How to share a secret,”Communications of the Association for Computing Machinery, Vol. 22, pp. 612-613, 1979.
[SHYU13] S.J. Shyu and H.W. Jiang,“General constructions for threshold multi-ple-secret visual cryptography Schemes,”IEEE Transactions on Infor-mation Forensics Security, Vol. 8, pp. 733-743, 2013.
[THIE02] C.C. Thien and J.C. Lin,“Secret image sharing,”Computer & Graphics, Vol. 26, pp. 765-770, 2002.
[WU11] C.C. Wu, S.J. Kao, and M.S. Hwang,“A high quality image sharing with steganography and adaptive authentication scheme,”Journal of Systems and Software, Vol. 84, pp. 2196-2207, 2011.
[WEI15] S.C. Wei, Y.C. Hou, and Y.C. Lu,“A technique for sharing a digital im-age,”Computers Standards & Interfaces, Vol. 40, pp. 53-61, 2015.
[WU18] X. Wu, C.N. Yang, Y.T. Zhuang, and S.C. Hsu,“Improving recovered image quality in secret Image sharing by simple modular arithmetic,” Signal Processing: Image Communication, Vol. 66, pp. 42-49, 2018.
[YANG16] C.N. Yang, C.H. Chen, and S.R. Cai,“Enhanced Boolean-based multi secret image sharing scheme,”Journal of Systems and software, Vol. 116, pp. 22-34, 2016.
[YANG17] C.N. Yang, C.H. Wu, Z.X. Yeh, D. Wang, and C. Kim,“A new sharing digital image scheme with clearer shadow images,”Computer Standards & Interfaces, Vol. 51, pp. 118-131, 2017.
[YANG17] C.N. Yang, C.C. Wu, and Y.C. Lin“k out of n region-based progressive visual cryptography,”IEEE Transactions on Circuits and Systems for Video Technology, Vol.29, pp.252-262, 2019.
[YANG18] C.N. Yang, F.H. Wu, and S.L. Peng,“Enhancing multi-factor cheating prevention in visual cryptography based minimum (k, n)-connected graph,”Journal of Visual Communication and Image Representation, Vol. 55, pp. 660-676, 2018.
[ZHOU18] Z. Zhou, C.N. Yang, and Y. Cao, and X. Sun,“Secret image sharing based on encrypted pixels,”IEEE Access, Vol. 6, pp. 15021-15025, 2018.