在 (k, n)-視覺式秘密分享技術( Visual Cryptography Scheme; VCS )，藉由將秘密影像加密得到n張子影像，任意的k張子影像，在不用電腦幫忙的情況下直接疊合子影像解密取得，可由人眼直接解密，若少於k張子影像則無法解回秘密影像。由於VCS視覺解密的獨特性，使得它在應用層面上也相當廣泛，現今已有越來越多的VCS應用，例如：數位影像索引、數位浮水印、安全顯示器、嵌入私人信息和數位影像智慧財產權保護機制等。
最近，Wang提出了一個創新的(2, n)累進式的視覺密碼機制(region incrementing visual cryptographic scheme; RIVCS)，可以逐步重建祕密在具有多個安全階層的單一影像。在RIVCS，一個秘密影像可以根據安全階層區分為多個區域，任意t張子影像疊合，2<=t<=n，可以解回第(t－1)階區域。無論如何，Wang的機制有顏色不正確的問題，被重建影像的顏色可能會相反（即黑白互換）。如果機密的資訊隱藏在顏色中，顏色不正確的問題會讓人誤解祕密。此外，Wang的機制只適用於(2, n)-RIVCS。在本篇論文，我們提出可以適用於k 和 n的一般性(k, n)-RIVCS，能夠在所有區域解回正確的顏色。我們也提出修改後的(k, n)-RIVCS可以適用於k 和 n，跟 Wang的機制比較加強了解密後影像的清晰度並減少了子影像的擴張。最後，理論證明我們所提出的(k, n)-RIVCS符合安全性和清晰度的條件。
我們的機制能夠在所有區域解回正確的顏色。若跟Wang 的機制一樣，可容忍重建影像的顏色互換，我們也設計了修改後的(k, n)-RIVCS。跟 Wang 的機制比較，這種修改後的(k, n)-RIVCS 減少了子影像的擴張，並且解密後的影像有較佳清晰度。實驗結果展現了我們的(k, n)-RIVCS 確實可得回正確顏色和有較小的子影像擴張。

In a (k, n) visual cryptography scheme (VCS), a secret image is encoded into n shadow images that distributed to n participants. Any k participants can reveal the secret image by stacking their shadow images to visually decode the secret image without the help of hardware or computation, but (k1) or fewer participants will not gain any information. The beautiful property of VCS is the ease of decoding, the stacking-to-see property, and it can be adopted in many applications such as digital image indexing, watermarking, securing display, embedding private information.
Recently, Wang introduced a novel (2, n) region incrementing visual cryptographic scheme, which can gradually reconstruct secrets in a single image with multiple security levels. In RIVCS, a secret image is subdivided into multiple regions in such way that any t shadow images, where 2tn, can be used to reveal the (t1)-th region. However, Wang’s scheme suffers from the incorrect-color problem, which the colors of reconstructed images may be reversed (i.e., the black and white are reversed). If the color of text is also the secret information, the incorrect-color problem will compromise the secret. Additionally, Wang's scheme is only suitable for the 2-out-of-n case, i.e., (k, n)-RIVCS where k=2. In this thesis, we propose the general (k, n)-RIVCS revealing the correct colors for all regions, where k and n can be any integers. We also propose the modified (k, n)-RIVCS to enhance the contrast and reduce the size of shadow image compared with Wang’s scheme, where k and n can be any integers. Finally, we theoretically prove that the proposed (k, n)-RIVCS satisfies the security and contrast conditions.

Chapter 1 Introduction 1
1.1 Background 1
1.2 Motivation 3
1.3 Contribution of the Thesis 3
1.4 Organization of the Thesis 3
Chapter 2 Preliminaries 5
2.1 (k, n)-VCS 5
2.2 Wang’s (2, n)-RIVCS 9
Chapter 3 The Proposed RIVCS 13
3.1 The (k, n)-RIVCS with Correct Color for All Regions 14
3.2 The Modified (k, n)-RIVCS 19
Chapter 4 Experimental Results and Comparison 27
4.1 Experimental Results 27
4.2 Discussion and Comparison 35
Chapter 5 Conclusion and Future Work 39
References 41

[NAOR94] M. Naor and A. Shamir, “Visual Cryptography,” Advances in Cryptology- EUROCRYPT’94, pp. 1-12, 1994.
[ITO99] R. ITO, H. Kuwakado, and H. Tanaka, “Image size invariant visual cryptography,” IEICE Trans. on Fund. of Elect. Comm. and Comp. Sci., vol. E82-A, pp. 2172-2177, 1999.
[YANG04] C.N. Yang, “New visual secret sharing schemes using probabilistic method,” Pattern Recognition Letters, vol. 25, pp. 481-494, 2004.
[CIMA06] S. Cimato, R. De Prisco and A. De Santis, “Probabilistic Visual Cryptography Schemes,” The Computer Journal, vol. 49, no. 1, pp. 97-107, 2006.
[WANG11] D. Wang F. Yi and X. Li, “Probabilistic visual secret sharing schemes for grey-scale images and color images,” Information Sciences, vol. 181, pp. 2189-2208, 2011.
[WU05] H.C. Wu and C.C. Chang, “Sharing visual multi-secrets using circle shares,” Computer Standards & Interfaces, vol. 28, pp.123-135, 2005.
[SHYU07] S.J. Shyu, S.Y. Huang, Y.K. Lee, R.Z Wang and K. Chen, “Sharing multiple secrets in visual cryptography,” Pattern Recognition, vol. 40, pp. 3633-3651, 2007.
[FENG08] J.B. Feng, H.C. Wu, C.S, Tsai, Y.F. Chang, and Y.P. Chu, “Visual Secret sharing for multiple secrets, Pattern Recognition,” vol. 41, pp. 3572-3581, 2008.
[FANG10] L.G. Fang, Y.M. Li, and B. Yu, “Multi-secret visual cryptography based on reversed images,” Information and Computing, vol. 4, pp. 195–198, 2010.
[LEE11] K.H. Lee and P.L. Chiu, “A high contrast and capacity efficient visual cryptography scheme for the encryption of multiple secret images,” Optics Communications, vol. 284, pp.2730–2741, 2011.
[YANG10A] C.N. Yang and T.H. Chung, “A general multi-secret visual cryptography scheme,” Optics Communications, vol. 283, pp. 4949-4962, 2010.
[TSAI07] D.S. Tsai, T.H. Chen and G. Horng, “A cheating prevention scheme for binary visual cryptography with homogeneous secret images,” Pattern Recognition, vol. 40, pp. 2356-2366, 2007.
[YANG09A] C.N. Yang A.G. Peng and T.S. Chen “MTVSS: (M)isalignment (T)olerant (V)isual (S)ecret (S)haring on Resolving Alignment Difficulty,” Signal Processing, vol. 89, pp. 1602-1624, 2009.
[LIU09] F. Liu, C.K. Wu, and X.J. Lin, “The alignment problem of visual cryptography schemes,” Designs, Codes and Cryptography, vol. 50, pp. 215-227, 2009.
[CIMA05] S. Cimato, A. DeSantis, A.L. Ferrara, and B. Masucci, “Ideal contrast visual cryptography schemes withreversing,” Information Processing Letters, vol. 93, pp. 199-206, 2005.
[YANG08A] C.N. Yang, C.C. Wang, and T.S. Chen, “Visual cryptography schemes with reversing,” The Computer Journal, vol. 51, pp. 710-722, 2008.
[VERH97] E.R. Verheul and H.C.A. Van Tilborg, “Constructions and properties of k out of n visual secret sharing scheme,” Designs, Codes and Cryptography, vol. 1, pp. 179-196, 1997.
[SHYU06] S.J. Shyu, “Efficient visual secret sharing scheme for color images,” Pattern Recognition, vol. 39, pp. 866-880, 2006.
[YANG08B] C.N. Yang and T.S. Chen, “Colored visual cryptography scheme based on additive color mixing,” Pattern Recognition, vol. 41, pp. 3114-3129, 2008.
[THIE02] C.C. Thien and J.C. Lin, “Secret image sharing,” Computer & Graphics, vol. 26, pp. 765-770, 2002.
[THIE03] C.C. Thien and J.C. Lin, “An image-sharing method with user-friendly shadow images,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 13, pp. 1161-1169, 2003.
[LIN04] C.C. Lin and W.H. Tsai, “Secret image sharing with steganography and authentication,” Journal of Systems & Software, vol. 73, p. 405-414, 2004.
[YANG07] C.N. Yang, T.S. Chen, K.H. Yu, and C.C. Wang, “Improvements of image sharing with steganography and authentication,” Journal of Systems & Software, vol. 80, pp. 1070-1076, 2007.
[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.
[YANG09B] C.N. Yang and C.B. Ciou, “A Comment on sharing secrets in stego images with authentication,” Pattern Recognition, vol. 42, pp. 1615-1619, 2009.
[ESLA11] Z. Eslami, S.H. Razzaghi, and J. Zarepour Ahmadabadi, “Secret image sharing with authentication-chaining and dynamic embedding,” Journal of Systems & Software, vol. 84, pp. 803–809, 2011.
[WANG07] R.Z. Wang and S.J. Shyu, “Scalable secret image sharing. Signal Processing,” Image Communication, vol. 22, pp. 363-373, 2007.
[YANG10B] C.N. Yang and Sin-Ming Huang, “Constructions and properties of k out of n scalable secret image sharing,” Optics Communications, vol. 283, pp. 1750-1762, 2010.
[YANG11] C.N. Yang and Y.Y. Chu, “A general (k, n) scalable secret image sharing scheme with the smooth scalability,” Journal of Systems & Software, vol. 84, pp. 1726-1733, 2011.
[NAOR97] Naor, M. and B. Pinkas, “Visual Authentication and Identification,’’ Advances in Cryptology-CRYPTO’ 97, Vol. LNCS 1294, pp. 322–336, 1997.
[MCCU05] J.M. McCune, A. Perrig, and M.K. Reiter, “Seeing-Is-Believing: using camera phones for human-verifiable authentication,” IEEE Symposium on Security and Privacy, pp. 110–124, 2005.
[YANG08C] C.N. Yang and T.S. Chen, “Security analysis on authentication of images using recursive visual cryptography,” Cryptologia, Vol. 32, pp. 131–136, 2008.
[BORC07] B. Borchert, “Segment-based visual cryptography,” Technical Report WSI-2007-04, University Tubingen (Germany), Wilhelm-Schickard-Institutfur Informatik, 2007.
[CIMA11] S. Cimato and C.N. Yang, Visual cryptography and secret image sharing, CRC Press, Taylor & Francis, 2011.
[SURE10] B. Surekha, G. Swamy, and K.S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Computer Applications, vol. 1, pp. 77-81, 2010.
[MONO10] T. Monoth, B. Anto P, “Tamperproof transmission of fingerprints using visual cryptography schemes,” Procedia Computer Science, vol. 2, pp. 143-148, 2010.
[WEIR10] J. Weir and W. Yan, “Resolution variant visual cryptography for street view of Google maps,” ISCAS, Paris, France, May 30–June 2, pp. 1695-1698, 2010.
[YANG06] C.N. Yang, T.S. Chen, and M.H. Ching, “Embed additional private information into two-dimensional barcodes by the visual secret sharing scheme,” Integrated Computer-Aided Engineering, vol. 13, pp. 189-199, 2006.
[WANG09] R.Z. Wang, “Region incrementing visual cryptography,” IEEE Signal Processing Letters, vol. 16, pp. 659-662, 2009.
[SHAM79] A. Shamir, “How to share a secret,” Communications of the Association for Computing Machinery, vol. 22, pp. 612-613, 1979.
[BLAK85] G. R. Blakley and C. Meadows, “Security of ramp schemes,” Crypto’84, vol. LNCS 196, pp. 242-2695, 1985.
[KURO93] K. Kurosawa, K. Okada, K. Sakano, W. Ogata, and T. Tsujii, “Nonperfect secret sharing schemes and matroids,” Eurocrypt’93, vol. LNCS 765, pp. 126-141, 1993.
[OGAT98] W. Ogata and K. Kurosawa, “Some basic properties of general nonperfect secret sharing schemes,” Journal of Universal Computer Science, vol. 4, pp. 690-704, 1998.
[LIN03] C.C. Lin and W.H. Tsai, “Visual cryptography for gray-level images by dithering techniques,” Pattern Recognition Letters, vol. 24, pp. 349-358, 2003.
[DROS96] S. Droste, “New results on visual cryptography,” in CRYPTO’96, vol. LNCS 1109, pp. 401-415, 1996.