Visual Cryptography Given By: Moni Naor Adi Shamir Presented By: Anil Vishnoi (2005H103017)

Contents: ► Introduction ► Terminology ► The Model ► Efficient Solution for Small K and n ► K out of K scheme ► K out of n Scheme ► Conclusion ► Reference

Introduction: ► Cryptography: Plain TextEncryptionCipher Text Plain Text DecryptionChannel

Visual Cryptography: Plaintext (in form of image) Encryption (creating shares) Channel (Fax, ) Decryption (Human Visual System)

Example: ► Secret Image Share1 Stacking the share reveals the secret Share2

Encoding of Pixels: Original Pixel Share1Share2overlaid Note: White is actually transparent

Computer Representation of pixels ► Visual Cryptography scheme represented in computer using n x m Basis matrices Original Pixel Original Pixelshare1 s1= s0= share2 overlaid Image

(2,2) Model 1. Construct two 2x2 basis matrices as: s0= 10s1= Using the permutated basis matrices, each pixel from the secret image will be encoded into two sub pixels on each participant's share. A black pixel on the secret image will be encoded on the ith participant's share as the ith row of matrix S1, where a 1 represents a black sub pixel and a 0 represents a white sub pixel. Similarly, a white pixel on the secret image will be encoded on the ith participant's share as the ith row of matrix S0.

Cont….. 3. Before encoding each pixel from the secret image onto each share, randomly permute the columns of the basis matrices S0 and S1 3.1This VCS (Visual Cryptography Scheme) divides each pixel in the secret image into m=2 sub pixels. ► 3.2It has a contrast of α(m)·m=1 and a relative contrast of α(m)=1/2.

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Conclusion:

Terminology: ► Pixel—Picture element ► Grey Level: The brightness value assigned to a pixel; values range from black, through gray, to white. ► Hamming Weight (H(V)): The number of non-zero symbols in a symbol sequence V- Vector of 1 and 1 of any length ► A qualified set of participants is a subset of Ρ whose shares visually reveal the 'secret' image when stacked together. ► A forbidden set of participants is a subset of Ρ whose shares reveal absolutely no information about the 'secret' image when stacked together.

Visual Cryptography (cont..) ► Visual Cryptography is a secret-sharing method that encrypts a secret image into several shares but requires neither computer nor calculations to decrypt the secret image. Instead, the secret image is reconstructed visually: simply by overlaying the encrypted shares the secret image becomes clearly visible ► A Visual Cryptography Scheme (VCS) on a set Ρ of n participants is a method of encoding a 'secret' image into n shares such that original image is obtained only by stacking specific combinations of the shares onto each other.

Terminology (cont…) ► The relative contrast (also called relative difference) of a VCS is the ratio of the maximum number of black sub pixels in a reconstructed (secret) white pixel to the minimum number of black sub pixels in a reconstructed (secret) black pixel. So, the lower the relative contrast in a scheme, the better. Note: the smallest relative contrast attainable in a VCS is 1/2, which is only achieved in a (2,2)-threshold VCS ► The contrast of a VCS is the difference between the minimum number of black sub pixels in a reconstructed (secret) black pixel and the maximum number of black sub pixels in a reconstructed (secret) white pixel.

The Model A solution to the k out of n visual secret sharing scheme consists of two collections of n x m Boolean (Basis) matrices S0 and S1. To share a white pixel, the dealer randomly chooses one of the matrices in S0, and to share a black pixel, the dealer randomly chooses one of the matrices in S1. The chosen matrix defines the color of the m sub pixels in each one of the n transparencies for a original pixel. The solution is considered valid if the following three conditions are met: 1. For any S in S0, the ``or'' V of any k of the n rows satisfies H(V ) <= d-α.m 2. For any S in S1, the ``or'' V of any k of the n rows satisfies H(V ) – d. n-Total Participant k-Qualified Participant

The Model (cont…) 3. For any subset {i1 ; i 2 ; : : : i q} of {1; 2; : : : n} with q < k, the two collections of q x m matrices D t for t ε {0,1} obtained by restricting each n x m matrix in C t (where t = 0; 1) to rows i1 ; i2 ; ::; iq are indistinguishable in the sense that they contain the same matrices with the same frequencies. ► Condition 3 implies that by inspecting fewer than k shares, even an infinitely powerful cryptanalyst cannot gain any advantage in deciding whether the shared pixel was white or black.

Advantage of Visual Cryptography ► Simple to implement ► Encryption don’t required any NP-Hard problem dependency ► Decryption algorithm not required (Use a human Visual System). So a person unknown to cryptography can decrypt the message. ► We can send cipher text through FAX or ► Infinite Computation Power can’t predict the message.

Basis Matrices ► Basis matrices are binary n x m used to encrypt each pixel in the secret image, where n is the number of participants in the scheme and m is the pixel expansion. The following algorithm is used to implement a VCS using basis matrices: If the n x m basis matrices S1 (used to encrypt black pixels) and S0 (used to encrypt white pixels) for any VCS are given, the secret image SI is encrypted as follows:

Basic Matrices (Cont..) for each pixel p in SI: { if (p is black) Let R = a random permutation of the columns of S1 else Let R = a random permutation of the columns of S0 for each participant i (1 <= i <= n): { The position on participant i’s share that corresponds to p is expanded into m pixels where each of these pixels j (1 <= j <= m) is black if Ri,j = 1 and white if Ri,j = 0. }}