1. 國立暨南國際大學 National Chi Nan University 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

2. Outline Introduction Preliminary The (k, n)-threshold Secret Sharing Scheme Experimental Results Conclusion 2

3. Introduction – Visual Cryptography Visual cryptography (VC) 3 encryption decryption share

4. Introduction – (k, n)-threshold Secret Sharing (k, n) = (2, 3) 4 decryption encryption

5. Introduction – Progressive Visual Secret Sharing 5 Progressive visual secret sharing (PVSS)

6. Introduction – Naor and Shamir (1995) 6 They construct 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.

7. Introduction – Naor and Shamir (1995) 7 C 0 : white pixel; C 1 : black pixel (2, 4) C 0 = C 1 = OR R 1 and R 2 R 1, R 2 and R 3 R 1, R 2, R 3 and R 4 R1R1 R2R2 R3R3 R4R4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 VC scheme : n × nn × 4 n

8. Introduction – Fang et al. (2008) 8 They construct a (k, n)-threshold secret sharing scheme in VC without expansion. They use the “Hilbert-curve” method.

9. Preliminary 9 Definition 1. An n  m 0-1 matrix M(n, j) is called totally symmetric if each column has the same weight, say j, and m equals to C j, where the weight of a column vector means the sum of each entry in this column vector. M(4, 2) = m = C 2 = 6 4 n

10. Preliminary 10 Definition 2. Given an n  m 1 matrix A and an n  m 2 matrix B, we define 1. [A||B] be an n  (m 1 + m 2 ) matrix that obtained by concatenating A and B; 2.[a  A||b  B] be an n  (a  m 1 + b  m 2 ) matrix that be obtained by concatenating A for a times and B for b times. A ＝, B ＝, [2A||B] ＝ B 2A

11. Preliminary 11 Definition 3. Light transmission rate  = #white pixel  #all pixel = 1  (#black pixel  #all pixel).

12. The (k, n)-threshold Secret Sharing Scheme 12 It must follow the two conditions :  (C 0, t) =  (C 1, t) for 1  t  k.  (C 0, t)   (C 1, t) for t  k.

13. Algorithm 13 Input : A binary secret S with size w  h and the value of n and k. Output : n shares R 1, R 2, …, R n, each with size w  h. 1.if (k mod 2 == 1) C 0 = C 1 = else C 0 = C 1 =

14. Algorithm 14 2.for (1  i  h; 1  j  w) x = random(1…m) for (1  t  n) if ( S(i, j) == 0 ) R t (i, j) = C 0 (t, x) ; else R t (i, j) = C 1 (t, x) ; C 0 = m R1R1 R2R2 RnRn …

15. Proof 15 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  (C 0, t) =  (C 1, t) for 1  t  k.  (C 0, t)   (C 1, t) for t  k.

16. Experimental Results 16 Example: (4, 5) C 0 : [M(5, 2) || 3  M(5, 0) || 2  M(5, 5)] C 1 : [2  M(5, 1) || M(5, 4)]

17. Experimental Results 17 (4, 5)

18. Experimental Results 18 (4, 6)

19. Experimental Results (5, 6) 19

20. Conclusion 20 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. NS scheme[1]FLL scheme[2]Our scheme contrast  in (4, 5)  1/4261 1/15 contrast  in (4, 6)  1/4261 1/24 contrast  in (5, 6)  1/12820 1/30 contrast  in (6, 8)  1/152200 1/128

21. Thanks for your listening 21