A New Approach for Visual Cryptography

1. A New Approach for Visual Cryptography Wen-Guey Tzeng and Chi-Ming Hu Designs, codes and cryptography, 27, 207-227,2002 Reporter:李惠龍

2. Outline • Introduction • The model • Naor and Sharmir definition • Improved definition

3. Introduction • Definition • Visual Cryptography is a way encrypt visual information (ie. pictures, texts) so humans can perform the decryption without the help of computers, unlike almost any other cryptographic techniques. • Visual Cryptography was first introduced by Naor and Shamir at EUROCRYPT ’94.

4. Introduction • Encryption • Break image into n shares or shadow images.

5. Introduction • Decryption • n shares are required to decrypt the image. • Print out the pictures on transparencies. • Stack the images on top of each other.

6. Introduction • Suppose a k out of n threshold scheme if used, then k shadow images must be stacked together to reveal the original image. • Shadow images are combined using the OR operator. • No cryptographic computation is required.

7. Introduction • Example • Secure image “IC” is divided into 4 shares, which is denoted by • Qualified sets are all subsets of containing at least one of the three sets {1,2},{2,3},{3,4} • (2,4)-threshold VCS

8. Introduction

9. The model (Naor and Sharmir) • (k, n)-threshold VCS scheme • Definition 2.0.1 Hamming weight: The number of non-zero symbols in a symbol sequence. H(0110)=2 • Definition 2.0.2 OR-ed k-vector: Given a j x k matrix, it’s the k-vector where each tuple consists of the result of performing boolean OR operation on its corresponding j x 1 column vector

10. The model (Naor and Sharmir) • Definition 2.0.3 VCS scheme is a 6 tuple (n, m, S, V, α, d) n: shares, one for each transparency m: each share is a collection of m black and white sub-pixels (pixel expansion) S: n x m boolean matrix V: the grey level of the combined share If H(V) ≥ d black, if H(V) < d-αm white α: contrast, α>0 d: threshold, 1≤d≤m

11. The model (Naor and Sharmir) • Definition 2.0.4 VCS schemes where a subset is qualified iff its cardinality is k are called (k, n)-threshold visual cryptography schemes. • Two collections of n x m boolean matrices ζ0 andζ1 • To construct a white pixel, randomly choose one of the matrices in ζ0, and to share a black pixel, randomly choose a matrices in ζ1. • The chosen matrix will define the color of the m sub-pixels in each one of the n transparencies.

12. The model (Naor and Sharmir) • For any matrix S in ζ0, the “or” operation on any k of the n rows satisfies H(V) ≤ d-αm. • For any matrix S in ζ1, the “or” operation on any k of the n rows satisfies H(V) ≥ d. • For any subset of {1,2,…,n} with q<k, the two collection of q x m matrices Bt obtained by restricting each n x m matrix in ζt (where t ={0,1}) to rows are indistinguishable in the sense that they contains exactly the same matrices with the same frequencies.

13. The model (Naor and Sharmir) • 1 and 2 defines the contrast of a VCS. • 3 states the security property of (k, n)-threshold VCS. • Example • (3, 3)-threshold VCS • Each pixel is divided into 4 sup-pixel (m=4)

14. The model (Naor and Sharmir) • A black pixel: all 4 black sub-pixels • A white pixel: 3 black sub-pixels and 1 white sub-pixel • Verify the security property: 3 white sub-pixels and 1 black pixel ?

15. Improved definition for VCS

16. Improved definition for VCS • m is called pixel expansion • α(m) is called contrast, which should be as large as possible. • the set of thresholds.

17. Improved definition for VCS

18. Improved definition for VCS • Example for definition 2.2 • Definition 2.1: VCS1 • Definition 2.2: VCS2

19. Improved definition for VCS

20. Properties of VCS2 • VCS2 is a generalization of VCS1, any VCS1 is a VCS2.

21. Properties of VCS2 • If basis matrices S0 and S1 have a common column, we can delete it from S0 and S1 to reduce pixel expansion.

22. Properties of VCS2 • Exchange the roles of S0 and S1 in a VCS2.

23. Properties of VCS2 • Add a participant such that Q is augmented.

24. Properties of VCS2 • Complete access structure: (P, Q, F) complete if F=2P-Q, which denoted by (P, Q) is short.

25. Properties of VCS2

26. Properties of VCS2 • Construct a VCS2: add an additional participant x to Γ such that some set containing x are forbidden.

27. Properties of VCS2 • Concatenate the basis matrices of two VCS2 if their access structures satisfy some conditions.

28. Properties of VCS2

29. Properties of VCS2

30. Properties of VCS2

31. To be continued