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

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 ⊕ AB A⊕BA⊕B 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 ⊕ A⊕BA⊕B 236 a 1.1 =136= b 1.1 =100= a 1.1 ⊕ b 1.1 =236=

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 = , , b 1,1 = , , a 1,1 ⊕ b 1,1 = , , 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.

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

Example Ｓ 1 =77 ： B 1 =89 ： Ｓ 2 =20 ： G 假設 n=4 S1S1 B1B1 100 B2B2 B3B ⊕⊕ ⊕ 77 B 2 =89 ： B 3 =200 ： Ｓ 3 =133 ： B 3 = 200 ： G=100 ： Ｓ 4 =184 ： S2S2 20 S3S3 133 S4S4 184 心得報告

Example Ｓ 1 =77 ： B 1 =89 ： Ｓ 2 =20 ： G 假設 n=4 S1S1 B1B1 100 B2B2 B3B ⊕⊕ ⊕ 77 B 2 =89 ： B 3 =200 ： Ｓ 3 =133 ： B 3 = 200 ： G=100 ： Ｓ 4 =184 ： S2S2 20 S3S3 133 S4S4 184 B1 ⊕ B2 B2 ⊕ B3B3 ⊕ G B1

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 S4S ⊕⊕ ⊕ G 100 Ｓ 1 =77 ： Ｓ 2 =20 ： B 2 =89 ： Ｓ 3 =133 ： B 3 =200 ： Ｓ 4 =184 ： G=100 ： B1 ⊕ B2 B2 ⊕ B3B3 ⊕ G B1

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:

G2G2 G1G1 G0G0 S0S B 1 =G 1 ⊕ S 0 G 1 :85= S 0 :125= B 1 :40= B 2 =G 2 ⊕ S 0 G 2 :110= S 0 :125= B 2: 19= B1B1 40 B2B2 19 ⊕⊕ Example n=3

G2G2 G1G1 G0G0 S0S B 1 =G 1 ⊕ S 0 G 1 :85= S 0 :125= B 1 :40= B 2 =G 2 ⊕ S 0 G 2 :110= S 0 :125= B 2: 19= B1B1 40 B2B2 19 ⊕⊕ Example n=3 G2⊕S0G2⊕S0 G1 ⊕S0G1 ⊕S0

Example n=3 S 3 =G 0 ⊕ B 2 G 0 :132= B 2 :19= S 3 :151= S 2 =B 1 ⊕ B 2 B 1 :40= B 2 :19= S 2 :59= S0S0 125 S1S1 40 S2S2 59 S3S3 151 B2B2 B1B G0G0 132 ⊕ ⊕

Example n=3 S 3 =G 0 ⊕ B 2 G 0 :132= B 2 :19= S 3 :151= S 2 =B 1 ⊕ B 2 B 1 :40= B 2 :19= S 2 :59= S0S0 125 S1S1 40 S2S2 59 S3S3 151 B2B2 B1B G0G0 132 ⊕ ⊕ G1 ⊕S0G1 ⊕S0 G2⊕S0G2⊕S0 B1⊕B2B1⊕B2 G0⊕B2G0⊕B2 B1B1

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:

Example S1S1 S3S3 40 S2S ⊕ ⊕ G0G0 132 S 1 :40= S 2 :59= B 2 :19= S 3 :151= G 0 :132= B1B1 40 S1S1 B2B2 19 B1B1 40 S2S2 59 ⊕ S 2 :59= B 1 :40= B 2 :19= B1⊕B2B1⊕B2 G0⊕B2G0⊕B2 B1B1 B1⊕B2B1⊕B2

Example S0S0 125 B1B1 40 B2B2 19 G2G2 110 G1G1 85 ⊕ ⊕ G0G0 132 S 1 :40= S 0 :125= G 1 :85= B 2 :19= S 0 :125= G 2 :110= G1 ⊕S0G1 ⊕S0 G2⊕S0G2⊕S0

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 S0S G0G0 132 G2G2 110 G1G1 85 ⊕ ⊕ ⊕ S 1 :40= S 2 :59= B 2 :19= S 3 :151= G 0 :132= S 1 :40= S 0 :125= G 1 :85=

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 S0S 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

Functionality comparison between the related image sharing and the proposed scheme

Comparison between the related image hiding and the proposed scheme.

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

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.

⊕ ⊕ = = B1B1 B2B2 S 0 is random martix

B1B1 B2B2 ⊕ ⊕

⊕⊕ ⊕ ⊕⊕