1. Efficient multi-secret image sharing based on Boolean operations Signal Processing Tzung-Her Chen, Chang-Sian Wu

2. 110 110 001 100 011 100 Definition 1. Image matrix exclusive-OR operation Case 1: Binary image matrix A ⊕ B=[a i,j,b i,j ], where i,j=0,1,…p-1 and ⊕ is bit-wise exclusive-OR operation ⊕ 1 1 1 1 1 00 0 0 AB A⊕BA⊕B 110 101 011 000 Case 2: For 256-gray-level image, each pixel is represented with eight bits. 256-gray-level image matrix A ⊕ B=[a i,j,k b i,j,k ], where i,j=0,1…..,p-1, k=0,1,….,7. AB 136100 ⊕ A⊕BA⊕B 236 a 1.1 =136=10001000 b 1.1 =100=01100100 a 1.1 ⊕ b 1.1 =236=11101100 100 011 100 110 110 001

3. Case 3: For 24-bit color image,assume the additive model is adoptedfor example. Each color pixel is represented with three primary colors, red(R), green(G), and blue(B),in which each is represented with eight bits.24-bit color image matrix A ⊕ B=[a i,j,kr,kg,kb ⊕ b i,j,kr,kg,kb ], where i,j=0,1,…,p-1, kr=0,1,…,7, kg=0,1,….,7, kb=0,1,…,7. AB (56,100,73)(110,54,196) ⊕ A⊕BA⊕B (86,82,141) a 1,1 =00111000,01100100,11000100 b 1,1 =01101110,00110110,01001001 a 1,1 ⊕ b 1,1 =01010110,01010010,10001101 Thus, it satisfies (1)A ⊕ B=B ⊕ A; (2)A ⊕ A=0; (3)A ⊕ C is random ; and (4)A ⊕ C=D implies C ⊕ D=A.

4. Definition 2. Image matrix chain exclusive-OR operation A i, i=1,….,k, are image matrices with the same dimension p×p. The exclusive-OR operation to all A i is defined as A i = A 1 ⊕ A 2 ⊕ A 3 …, A k. The encoding process involves two steps: Step 1: Generate n-1 random matrices B 1, B 2, ….., and B n-1. Note that if a well-defined random number generator, such as a ‘‘linear feedback shift register’’, the generated random matrices B 1, B 2, ….., and B n-1 will be distinct Step 2: Compute the share images S i (i=1,…..,n) by the following operations : G is secret the image

5. Example Ｓ 1 =77 ： 01001101 B 1 =89 ： 01011001 Ｓ 2 =20 ： 00010100 G 假設 n=4 S1S1 B1B1 100 B2B2 B3B3 7789220 ⊕⊕ ⊕ 77 B 2 =89 ： 01011001 B 3 =200 ： 11011100 Ｓ 3 =133 ： 10000101 B 3 = 200 ： 11011100 G=100 ： 01100100 Ｓ 4 =184 ： 10111000 S2S2 20 S3S3 133 S4S4 184 心得報告

6. Example Ｓ 1 =77 ： 01001101 B 1 =89 ： 01011001 Ｓ 2 =20 ： 00010100 G 假設 n=4 S1S1 B1B1 100 B2B2 B3B3 7789220 ⊕⊕ ⊕ 77 B 2 =89 ： 01011001 B 3 =200 ： 11011100 Ｓ 3 =133 ： 10000101 B 3 = 200 ： 11011100 G=100 ： 01100100 Ｓ 4 =184 ： 10111000 S2S2 20 S3S3 133 S4S4 184 B1 ⊕ B2 B2 ⊕ B3B3 ⊕ G B1

7. Definition 2. Image matrix chain exclusive-OR operation In the decoding phase, n share images are used to reveal the secret image by exclusive-OR operation as follows: S1S1 S2S2 S3S3 S4S4 7720133184 ⊕⊕ ⊕ G 100 Ｓ 1 =77 ： 01001101 Ｓ 2 =20 ： 00010100 B 2 =89 ： 01011001 Ｓ 3 =133 ： 10000101 B 3 =200 ： 11011100 Ｓ 4 =184 ： 10111000 G=100 ： 01100100 B1 ⊕ B2 B2 ⊕ B3B3 ⊕ G B1

8. The proposed secret sharing scheme for multiple secret images n secret images G i, i=0,1,….. n-1, are encoded into n+1 share images S m, m=0,1,2,…..n. The encoding phase consists of three steps: Step 1:Generate a random integer matrix as the first share image S 0 with the same size of secret images.The random values in S 0 are in the range between 0 and 255. Step 2:Generaten1 random matrices B i by the following operations: Step 3: Compute the other share images S i by the following operations:

9. G2G2 G1G1 G0G0 S0S0 125 13285110 B 1 =G 1 ⊕ S 0 G 1 :85=01010101 S 0 :125=01111101 B 1 :40=00101000 B 2 =G 2 ⊕ S 0 G 2 :110=01101110 S 0 :125=01111101 B 2: 19=00010011 B1B1 40 B2B2 19 ⊕⊕ Example n=3

10. G2G2 G1G1 G0G0 S0S0 125 13285110 B 1 =G 1 ⊕ S 0 G 1 :85=01010101 S 0 :125=01111101 B 1 :40=00101000 B 2 =G 2 ⊕ S 0 G 2 :110=01101110 S 0 :125=01111101 B 2: 19=00010011 B1B1 40 B2B2 19 ⊕⊕ Example n=3 G2⊕S0G2⊕S0 G1 ⊕S0G1 ⊕S0

11. Example n=3 S 3 =G 0 ⊕ B 2 G 0 :132=10000100 B 2 :19=00010011 S 3 :151=10010111 S 2 =B 1 ⊕ B 2 B 1 :40=00101000 B 2 :19=00010011 S 2 :59=00111011 S0S0 125 S1S1 40 S2S2 59 S3S3 151 B2B2 B1B1 40 19 G0G0 132 ⊕ ⊕

12. Example n=3 S 3 =G 0 ⊕ B 2 G 0 :132=10000100 B 2 :19=00010011 S 3 :151=10010111 S 2 =B 1 ⊕ B 2 B 1 :40=00101000 B 2 :19=00010011 S 2 :59=00111011 S0S0 125 S1S1 40 S2S2 59 S3S3 151 B2B2 B1B1 40 19 G0G0 132 ⊕ ⊕ G1 ⊕S0G1 ⊕S0 G2⊕S0G2⊕S0 B1⊕B2B1⊕B2 G0⊕B2G0⊕B2 B1B1

13. The proposed decoding scheme for multiple secret images The decoding phase consists of three steps: Step 1: All n+1 share images collected together are used to reconstruct the first secret image as follows Step 2: Generaten1 random matrices Bi, which are obtained as follows: Step 3: Reconstruct the other(n-1) secret images Gk by the following operations:

14. Example S1S1 S3S3 40 S2S2 59 151 ⊕ ⊕ G0G0 132 S 1 :40=00101000 S 2 :59=00111011 B 2 :19=00010011 S 3 :151=10010111 G 0 :132=10000100 B1B1 40 S1S1 B2B2 19 B1B1 40 S2S2 59 ⊕ S 2 :59=00111011 B 1 :40=00101000 B 2 :19=00010011 B1⊕B2B1⊕B2 G0⊕B2G0⊕B2 B1B1 B1⊕B2B1⊕B2

15. Example S0S0 125 B1B1 40 B2B2 19 G2G2 110 G1G1 85 ⊕ ⊕ G0G0 132 S 1 :40=00101000 S 0 :125=01111101 G 1 :85=01010101 B 2 :19=00010011 S 0 :125=01111101 G 2 :110=01101110 G1 ⊕S0G1 ⊕S0 G2⊕S0G2⊕S0

16. Theorem Assume that n (n>1) distinct secret images Gk with high entropy, k=0,1,…, n-1, are encoded into n+1 share images Sm, m=0,1,2,…., n. The secret images can be reconstructed correctly by the following formula: S1S1 S3S3 S2S2 S0S0 125 4059151 G0G0 132 G2G2 110 G1G1 85 ⊕ ⊕ ⊕ S 1 :40=00101000 S 2 :59=00111011 B 2 :19=00010011 S 3 :151=10010111 G 0 :132=10000100 S 1 :40=00101000 S 0 :125=01111101 G 1 :85=01010101

17. Theorem Assume that n (n>1) distinct secret images Gk with high entropy, k=0,1,…, n-1, are encoded into n+1 share images Sm, m=0,1,2,…., n. The secret images can be reconstructed correctly by the following formula: S1S1 S3S3 S2S2 S0S0 125 4059151 G0G0 132 G2G2 110 G1G1 85 ⊕ ⊕ ⊕ S 1 =G 1 ⊕ S 0 S 2 =G 1 ⊕ S 0 ⊕ G 2 ⊕ S 0 S 3 =G 2 ⊕ S 0 ⊕ G 0 G 0 =G 1 ⊕ S 0 ⊕ G 1 ⊕ S 0 ⊕ G 2 ⊕ S 0 ⊕ G 2 ⊕ S 0 ⊕ G 0 G 1 =S0 ⊕ G 1 ⊕ S 0 G 2 =S0 ⊕ G 1 ⊕ S 0 ⊕ G 1 ⊕ S 0 ⊕ G 2 ⊕ S 0

18. Functionality comparison between the related image sharing and the proposed scheme

19. Comparison between the related image hiding and the proposed scheme.

20. Computational complexity comparison between the related works and the proposed scheme in the decoding phase

21. Conclusions Compared with tradition VSS-based image sharing schemes, the proposed scheme benefits valuable merits including lossless secret reconstruction, no pixel expansion, generalization of image format, no not-easy- to-align problem, and no codebook required. Compared with the schemes combining VSS and image hiding technique, the proposed scheme has the two main advantages: high sharing capacity and computational efficiency of multi-secret sharing.

22. ⊕ ⊕ = = B1B1 B2B2 S 0 is random martix

23. B1B1 B2B2 ⊕ ⊕

24. ⊕⊕ ⊕ ⊕⊕