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32 Int'l Conf. Foundations of Computer Science | FCS'11 | α in (4, 5)-threshold threshold secret sharing scheme of NS scheme is approach to 1 / 4000. Because Fang et al. use NS scheme to design their scheme, α in FFL scheme is in the same as α in NS scheme scheme. In our scheme, α is equal to 3 / 15 − 2 / 15 = 1 / 15 in the (4, (a) (c) (e) (g) (i) Figure 1. Experimental result of (4, 5) 5)-threshold threshold secret sharing scheme. Also, for any (4, n)-threshold threshold secret sharing scheme in VC for n ≥ 6, our r proposed scheme has better performance perform than NS scheme [9] ] and FLL scheme [6] [ in the value of α. (a) Secret image S, (b) Share 1, (c) Share 2, (d) Share 3, (e) Share 4, (f) ) Share 5, (g) ) The result of stacking share 1 and share 2, (h)The )The result of stacking share 1, share2 and share 3, 3 (i) The result of stacking share 1, share2, share 3 and share 4, (j) The result of stacking all the shares. (b) (d) (f) (h) (j)
Int'l Conf. Foundations of Computer Science | FCS'11 | 33 Besides, there is no expansion in our scheme which is smaller than the NS scheme [9] and we also generate shares by randomly without using Hilbert-curve while [6] need. Overall, our proposed scheme reveals the secret image when stacking at least four shares and the secret image will be clearer if stacking more and more shares together. That is, our scheme is a (4, n) threshold PVSS scheme. The advantages of our proposed scheme are that it has no pixel expanded, and has the larger α that the stacked image will be clearer. The future work is to generate the proposed scheme to be a general k out of n secret sharing scheme in VC. ACKNOWLEDGMENT This work was partially supported by National Science Council of Taiwan, ROC under Grant No. NSC99-2628-E-2-60-011-. REFERENCES [1] D. Hilbert, “Ü eber die stetige Abbildung einer Linie aufein Flächenstück,” Mathematische Annalen, vol. 38, pp. 459—460, 1891. [2] C. Blundo, A. De Santis and D. R Stinson. “On The Contrast in Visual cryptography schemes,” Journal Of Cryptology, vol. 12, pp. 261—289, 1999. [3] Young-Chang Hou, Zen-Yu Quan, “Progressive Visual Cryptography with Unexpanded Shares,” IEEE Transactions on Circuits and Systems for Video Technology, accepted 2011. [4] P. A. Eisen, D.R. Stinson, “Threshold visual cryptography schemes with specified whiteness levels of reconstructed pixels,” Des. Codes Cryptogr. 25, pp. 15—61, 2002. [5] W. P. Fang, J. C. Lin, “Progressive viewing and sharing of sensitive images,” Pattern Recognition and Image Analysis, Vol.16, no. 4, pp. 638—642, 2006 [6] Wen-Pinn Fang, Sen-Jen Lin, Ja-Chen Lin, “Visual cryptography (VC) with non-expanded shadow images: a hilbert-curve approach,” Proceeding on IEEE International Conference on Intelligence and Security Informatics (ISI2008). Grand Formosa Regent Hotel, Taipei, Taiwan, pp. 271—272, 2008. [7] N. Linial and N. Nisan, “Aprroximate inclusion-exlusion,” Combinatorica 10, pp. 349—365, 1990. [8] C. C. Lin and W. H. Tsai, “Secret multimedia information sharing with data hiding capability by simple logic operations,” Pattern Recognition and Image Analysis, vol. 14(4), pp. 594—600, 2004. [9] M. Naor and A. Shamir, “Visual cryptography,” Eurocrypt’94, Lecture Notes in Computer Science, Springer-Verlag, Berlin, vol. 950, pp. 1—12, 1995. [10] T. Hofmeister, M. Krause and H. U. Simon, “Contrast-Optimal k out of n Secret Sharing Schemes in Visual Cryptography,” Theory of Computer Science, vol. 240, pp. 471—485, 2000. [11] S.-J. Shyu, “Image encryption by multiple random grids,” Pattern Recognition, vol. 42, no. 7, pp. 1582—1596, 2009. [12] C. N. Yang, “New Visual secret sharing schemes using probabilistic method,” Pattern Recognition Letters, vol. 25(4), pp.481—495, 2004.
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