A New Approach for Visual Cryptography Wen-Guey Tzeng and Chi-Ming Hu Designs, codes and cryptography, 27, ,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 Hamming weight: The number of non-zero symbols in a symbol sequence. H(0110)=2 Definition 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 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 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 ζ 0 and ζ 1 To construct a white pixel, randomly choose one of the matrices in ζ 0, and to share a black pixel, randomly choose a matrices in ζ 1. 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) 1. For any matrix S in ζ 0, the “or” operation on any k of the n rows satisfies H(V) ≤ d-αm. 2. For any matrix S in ζ 1, the “or” operation on any k of the n rows satisfies H(V) ≥ d. 3. For any subset of {1,2,…,n} with q<k, the two collection of q x m matrices B t obtained by restricting each n x m matrix in ζ t (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 S 0 and S 1 have a common column, we can delete it from S 0 and S 1 to reduce pixel expansion.

22 Properties of VCS2 Exchange the roles of S 0 and S 1 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=2 P -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