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A Study of ( k, n )-threshold Secret Image Sharing Schemes in Visual Cryptography without Expansion. Presenter ： Ying-Yu Chen Authors: Ying-Yu Chen, Justie Su-Tzu Juan. Department of Computer Science and Information Engineering National Chi Nan University Puli, Nantou Hsien, Taiwan .
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A Study of (k,n)-threshold Secret Image Sharing Schemes in Visual Cryptography without Expansion Presenter： Ying-Yu Chen Authors: Ying-Yu Chen, Justie Su-Tzu Juan Department of Computer Science and Information Engineering National Chi Nan University Puli, Nantou Hsien, Taiwan
Outline • Introduction • Preliminary • The (k, n)-threshold Secret Sharing Scheme • Experimental Results • Conclusion
Introduction – Visual Cryptography • Visual cryptography (VC) encryption share decryption
Introduction – (k, n)-threshold Secret Sharing • (k, n) = (2, 3) decryption encryption
Introduction – Progressive Visual Secret Sharing • Progressive visual secret sharing (PVSS)
Introduction – Naor and Shamir (1995) • Theyconstruct a (k, n)-threshold secret sharing scheme in VC with expansion. • : The relative difference in weight between white pixel and black pixel of stacking k shares.If contrast is larger, it represents the image is clearer to visible.
Introduction – Naor and Shamir (1995) n × n n × 4 n VC scheme : 1 0 0 0 • C0 : white pixel; C1 : black pixel • (2, 4) • C0 = C1 = R1 0 1 0 0 R2 0 0 1 0 R3 0 0 0 1 OR R4 R1and R2 R1,R2and R3 R1, R2, R3 and R4
Introduction – Fang et al. (2008) • Theyconstruct a (k, n)-threshold secret sharing scheme in VC without expansion. • They use the “Hilbert-curve” method.
Preliminary • Definition 1. • An nm 0-1 matrix M(n, j) is called totally symmetric if each column has the same weight, say j, and m equals to Cj , where the weight of a column vector means the sum of each entry in this column vector. • M(4, 2) = n m = C2 = 6 4
Preliminary • Definition 2. • Given an n m1 matrix A and an n m2 matrix B, we define • 1. [A||B] be an n (m1 + m2) matrix that obtained by concatenating A and B; • [aA||bB]be an n (am1+ bm2) matrix that be obtained by concatenating A for a times and B for b times. • A＝ , B＝ ,[2A||B] ＝ B 2A
Preliminary • Definition 3. • Light transmission rate = #white pixel #all pixel = 1 (#black pixel#all pixel).
The (k, n)-threshold Secret Sharing Scheme • It must follow the two conditions : • (C0, t) = (C1, t) for 1 t k. • (C0, t) (C1, t) for t k.
Algorithm • Input : A binary secret S with size w h and the value of nand k. • Output : n shares R1, R2, …, Rn, each with size w h. • if (k mod 2 == 1) • C0 = • C1 = • else • C0 = • C1 =
Algorithm • for (1 i h; 1 j w) • x = random(1…m) • for (1 t n) • if ( S(i, j) == 0 ) • Rt(i, j) = C0(t, x) ; • else • Rt(i, j) = C1(t, x) ; R1 R2 C0 = … Rn m
Proof • Theorem 1. • In the proposed scheme, if we stack at least k shares, the secret can be revealed; and if we stack the number of share less than k, the secret cannot be revealed. • Proof • (C0, t) = (C1, t) for 1 t k. • (C0, t) (C1, t) for t k.
Experimental Results • Example: (4, 5) • C0:[M(5, 2) || 3 M(5, 0) ||2 M(5, 5)] • C1: [2 M(5, 1)||M(5, 4)]
Experimental Results • (4, 5)
Experimental Results • (4, 6)
Experimental Results • (5, 6)
Conclusion • There is no expansion in our scheme. • With larger contrast we proposed, the stacked image is clearer. • [1] M. Naor and A. Shamir, “Visual cryptography,” 1995. • [2] W.-P. Fang, S.-J. Lin, and J.-C. Li, “Visual cryptography (VC) with non-expanded shadow images: a Hilbert-curve approach,” 2008.
