1. A general (k, n) scalable secret image sharing scheme with the smooth scalability Ching-Nung Yang, Yu-Ying Chu The Journal of Systems and Software 84 (2011) 1726– 1733 2011/09/22

2. Outline Scalable secret image sharing (SSIS) This paper proposed smooth (k, n)-SSIS scheme for improving Yang et al.’s (k, n)-SSIS scheme (2010). Yang, C.N., Huang, S.-M., 2010. Constructions and properties of k out of n scalable secret image sharing. Optics Communications 283, 1750–1762.

3. Scalable secret image sharing (SSIS) The secret is gradually revealed in decoding phase by the increment of participants. It provides the scalability and flexibility that different versions of the secret image can be reconstructed by different combinations of participants.

4. Example: Yang et al.’s (3, 5)-SSIS scheme (2010) Secret image P={p 1,p 2,p 3,p 4,p 5 } n=5, k = 3 |P|=3 |P|=4 |P|=5 decoding phase

5. Problem Yang et al.’s (3, 5)-SSIS scheme (2010, Approach 1) : |P|=3 |P|=4 |P|=5 1/10 4/10 10/10 Yang et al.’s (k, n)-SSIS scheme did not have the smooth scalability.

6. |P|=3 |P|=4 |P|=5 3/10 6/10 10/10 Problem (con.) Yang et al.’s (3, 5)-SSIS scheme (2010, Approach 2) : Yang et al.’s (k, n)-SSIS scheme did not have the smooth scalability.

7. Smooth scalability (1) [Security condition]: I(R t ) = 0 for t ≤ (k−1). (2) [Scalability condition]: I(R t ) < I(R t+1 ) for k ≤ t ≤ n, and I(R n ) = 1. (2) [Smooth scalability condition]: I(R t ) = t/n for k≤ t ≤n. R t is a reconstructed image from any t shadows and I(·)is the percentage of the information amount of a reconstructed image to a secret image.

8. This paper (2011) Smooth (3, 5)-SSIS scheme: |P|=3 |P|=4 |P|=5 3/5 4/5 5/5 6/10 8/1010/10

9. Smooth (k, n)-SSIS scheme secret image Divided … O1O1 O sub-image O2O2 OnOn … sub-shadow shadow S2S2 SnSn S1S1 … … … OkOk O k+1 … Secret image Sharing Merge (k, n) (k+1, n) (n, n) … Distribute share … …

10. Combination of (t, n)-SIS schemes, k≤t≤n, for a smooth (k, n)-SSIS scheme.

11. Conclusion Different purposed: – Yang et al.’s (k, n)-SSIS scheme (2010) for smaller shadow size, and this paper for smooth scalability. SchemeShadow size (compare with secret image) Smooth scalability Yang et al.’s (2010) Approach 1 1/nNo Yang et al.’s (2010) Approach 2 1/kNo This paper (1+  n i=k+1 1/i)/n Between (1/k) and (1/n) Yes Wang et al.’s (2010) (2n  k)/n 2 Between (1/k) and (1/n) Yes

12. Yang et al.’s scheme (2010) Approach 1 secret image Divided O1O1 O sub-image O2O2 Om1Om1 … sub-shadow S2S2 … … … Secret image Sharing (k, k) … Distribute share … …

13. Distribute shares Yang’s (k, n) scalable secret image sharing (Approach 1) Divided secret S into m = disjoint sub-image S i. Each sub-image S i : (k, k)-threshold scheme Share size : 13 s 11 s 21 s 31  s 12 s 22  s 41 s 13  s 32 s 42 …  s 23 s 33 s 43 keys and locks table S 1 S 2 S 3 … S m p1p1 p2p2 pnpn 1 1 1 0 1 1 0 1 … 0 1 1 1 … m locks, Each lock has k keys  Each lock construct k shares

14. Approach 1 ex_1:(k, n)=(3,4) p1p1 p2p2 p3p3 p4p4 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 s 11 s 21 s 31  s 12 s 22  s 41 s 13  s 32 s 42  s 23 s 33 s 43 keys and locks table: Locks Sub-secret S 1 S 2 S 3 S 4 Share size = 14

15. Approach 1 ex_1:(k, n)=(3,4) p1p1 p2p2 p3p3 p4p4 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 s 11 s 21 s 31  s 12 s 22  s 41 s 13  s 32 s 42  s 23 s 33 s 43 keys and locks table: Locks Sub-secret S 1 S 2 S 3 S 4 p 1,p 2,p 3 can reconstruct Sub-Secret S 1 p 1,p 2, p 3, p 4 can reconstruct Sub-Secret S 1,…, S 6 15

16. Distribute shares Yang’s (k, n) scalable secret image sharing (Approach 2) Divided secret S into m = disjoint sub-image S i. Each sub-image S i : (k, k)-threshold scheme Share size : 16 s 11 s 21 s 31 s mk s 12 s 2k s 2k s m1 s 1k s 22 s 3k s m2 … s 1k s 23 s 33 s m3 keys and locks table S 1 S 2 S 3 … S m p1p1 p2p2 pnpn 1 1 1 0 1 0 0 1 0 1 … 0 1 1 1 … m locks, Each lock has (k  1) keys  Each lock construct k shares p3p3

17. Approach 2 ex_1:(k, n)=(3,4) p1p1 p2p2 p3p3 p4p4 keys and locks table: Locks Sub-secret S 1 S 2 S 3 S 4 S 5 S 6 Share size = 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 s 12 s 22 s 32 s 41 s 51 s 61 s 13 s 21 s 31 s 42 s 52 s 61 s 11 s 23 s 31 s 43 s 51 s 62 s 11 s 21 s 33 s 41 s 52 s 63 17

18. Approach 2 ex_1:(k, n)=(3,4) p1p1 p2p2 p3p3 p4p4 keys and locks table: Locks Sub-secret S 1 S 2 S 3 S 4 S 5 S 6 p 1,p 2,p 3 can reconstruct Sub-Secret S 1 & S 2 & S 4 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 s 12 s 22 s 32 s 41 s 51 s 61 s 13 s 21 s 31 s 42 s 52 s 61 s 11 s 23 s 31 s 43 s 51 s 62 s 11 s 21 s 33 s 41 s 52 s 63 Approach 2 reconstruct more information than Approach 1. 18