Secret Sharing Secret Sharing Scheme Visually On Scheme Visually On Virtual World (Visual Virtual World (Visual

Cryptography)Cryptography)

Presented By:Presented By:

MADHULIKA SINGHMADHULIKA SINGH

Contents:Contents:

► IntroductionIntroduction►TerminologyTerminology►The ModelThe Model►Efficient Solution for Small K and nEfficient Solution for Small K and n►K out of K schemeK out of K scheme►K out of n SchemeK out of n Scheme►ConclusionConclusion►ReferenceReference

Introduction:Introduction:

► Cryptography:Cryptography:

Plain TextPlain Text EncryptionEncryption Cipher Cipher TextText

Plain Text DecryptionPlain Text Decryption ChannelChannel

Visual Cryptography:Visual Cryptography:

Plaintext (in form of image)Plaintext (in form of image)

Encryption (creating shares)Encryption (creating shares)

Channel (Fax, Email)Channel (Fax, Email)

Decryption (Human Visual System)Decryption (Human Visual System)

Example:Example:

► Secret ImageSecret Image

Share1Share1

Stacking the Stacking the shareshare

reveals the secretreveals the secret

Share2Share2

Encoding of Pixels:Encoding of Pixels:

Original PixelOriginal Pixel

Share1Share1

Share2Share2

overlaidoverlaid

Note: White is actually transparentNote: White is actually transparent

Computer Representation of pixelsComputer Representation of pixels► Visual Cryptography scheme represented in Visual Cryptography scheme represented in

computer using n x m Basis matricescomputer using n x m Basis matrices

Original PixelOriginal Pixel

share1share1

s1=s1= s0= s0= share2share2

overlaid Imageoverlaid Image

(2,2) Model(2,2) Model

1. 1. Construct two 2Construct two 2xx2 basis matrices as: 2 basis matrices as:

s0= s0= 11 00 s1= 1s1= 1 0 000 11 1 1 0 0

2.Using the permutated basis matrices, each pixel from the 2.Using the permutated basis matrices, each pixel from the

secret image will be encoded into two sub pixels on each secret image will be encoded into two sub pixels on each participant's share. A black pixel on the secret image will participant's share. A black pixel on the secret image will be encoded on the be encoded on the ithith participant's share as the participant's share as the ithith row of row of matrix matrix S1S1, where a 1 represents a black sub pixel and a 0 , where a 1 represents a black sub pixel and a 0 represents a white sub pixel. Similarly, a white pixel on represents a white sub pixel. Similarly, a white pixel on the secret image will be encoded on the the secret image will be encoded on the ithith participant's participant's share as the share as the ithith row of matrix row of matrix S0S0. .

Cont…..Cont…..

3. Before encoding each pixel from the secret image onto 3. Before encoding each pixel from the secret image onto each share, randomly permute the columns of the basis each share, randomly permute the columns of the basis matrices matrices S0S0 and and S1S1

3.13.1 This VCS (Visual Cryptography Scheme) divides This VCS (Visual Cryptography Scheme) divides each each pixel in the secret image into pixel in the secret image into mm=2 sub pixels. =2 sub pixels.

► 3.23.2 It has a contrast of It has a contrast of α(m)·mα(m)·m=1 and a relative contrast =1 and a relative contrast of of α(m)α(m)=1/2. =1/2.

Queries?Queries?

Conclusion:Conclusion:

Terminology:Terminology:► Pixel—Picture elementPixel—Picture element

► Grey Level: The brightness value assigned to a pixel; Grey Level: The brightness value assigned to a pixel; values range from black, through gray, to white.values range from black, through gray, to white.

► Hamming Weight (H(V)): The number of non-zero Hamming Weight (H(V)): The number of non-zero symbols in a symbol sequencesymbols in a symbol sequenceV- Vector of 1 and 1 of any length V- Vector of 1 and 1 of any length

► A qualified set of participants is a subset of A qualified set of participants is a subset of ΡΡ whose shares whose shares visually reveal the 'secret' image when stacked together. visually reveal the 'secret' image when stacked together.

► A forbidden set of participants is a subset of A forbidden set of participants is a subset of ΡΡ whose shares whose shares reveal absolutely no information about the 'secret' image reveal absolutely no information about the 'secret' image

when stacked together.when stacked together.

Visual Cryptography (cont..)Visual Cryptography (cont..)

► Visual Cryptography is a secret-sharing method that Visual Cryptography is a secret-sharing method that encrypts a secret image into several shares but requires encrypts a secret image into several shares but requires neither computer nor calculations to decrypt the secret neither computer nor calculations to decrypt the secret image. Instead, the secret image is reconstructed visually: image. Instead, the secret image is reconstructed visually: simply by overlaying the encrypted shares the secret image simply by overlaying the encrypted shares the secret image becomes clearly visible becomes clearly visible

► A Visual Cryptography Scheme (VCS) on a set A Visual Cryptography Scheme (VCS) on a set ΡΡ of of nn participants is a method of encoding a 'secret' image into participants is a method of encoding a 'secret' image into nn shares such that original image is obtained only by shares such that original image is obtained only by stacking specific combinations of the shares onto each stacking specific combinations of the shares onto each other. other.

Terminology (cont…)Terminology (cont…)

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

► The contrast of a VCS is the difference between the The contrast of a VCS is the difference between the minimum number of black sub pixels in a reconstructed minimum number of black sub pixels in a reconstructed (secret) black pixel and the maximum number of black sub (secret) black pixel and the maximum number of black sub pixels in a reconstructed (secret) white pixel. pixels in a reconstructed (secret) white pixel.

The ModelThe Model

A solution to the k out of n visual secret sharing A solution to the k out of n visual secret sharing scheme consists of two collections of n x m Boolean scheme consists of two collections of n x m Boolean (Basis) matrices S0 and S1. To share a white pixel, the (Basis) matrices S0 and S1. To share a white pixel, the dealer randomly chooses one of the matrices in S0 , dealer randomly chooses one of the matrices in S0 , and to share a black pixel, the dealer randomly and to share a black pixel, the dealer randomly chooses one of the matrices in S1. The chosen matrix chooses one of the matrices in S1. The chosen matrix defines the color of the m sub pixels in each one of the defines the color of the m sub pixels in each one of the n transparencies for a original pixel. The solution is n transparencies for a original pixel. The solution is considered valid if the following three conditions are considered valid if the following three conditions are met: met:

1. For any S in S0 , the ``or'' V of any k of the n rows 1. For any S in S0 , the ``or'' V of any k of the n rows satisfies H(V ) <= d-satisfies H(V ) <= d-αα.m.m

2. 2. For any S in S1 , the ``or'' V of any k of the n rows For any S in S1 , the ``or'' V of any k of the n rows satisfies H(V ) – d. satisfies H(V ) – d.

n-Total Participantn-Total Participant

k-Qualified Participantk-Qualified Participant

The Model (cont…)The Model (cont…)

3. For any subset 3. For any subset {i{i1 ; i 2 ; : : : i q1 ; i 2 ; : : : i q} } of of {{1; 2; : : : n1; 2; : : : n}} with with q < k, the two collections of q x m matrices Dq < k, the two collections of q x m matrices Dtt for t for t εε {0,1} {0,1} obtained by restricting each n x m matrix in Cobtained by restricting each n x m matrix in Ctt (where t = 0; 1) to rows i1 ; i2 ; ::; iq are (where t = 0; 1) to rows i1 ; i2 ; ::; iq are indistinguishable in the sense that they contain the indistinguishable in the sense that they contain the same matrices with the same frequencies. same matrices with the same frequencies.

► Condition 3 implies that by inspecting fewer than k Condition 3 implies that by inspecting fewer than k shares, even an infinitely powerful cryptanalyst shares, even an infinitely powerful cryptanalyst cannot gain any advantage in deciding whether the cannot gain any advantage in deciding whether the shared pixel was white or shared pixel was white or

black. black.

Advantage of Visual CryptographyAdvantage of Visual Cryptography

► Simple to implementSimple to implement► Encryption don’t required any NP-Hard problem Encryption don’t required any NP-Hard problem

dependencydependency► Decryption algorithm not required (Use a human Decryption algorithm not required (Use a human

Visual System). So a person unknown to Visual System). So a person unknown to cryptography can decrypt the message. cryptography can decrypt the message.

► We can send cipher text through FAX or E-MAILWe can send cipher text through FAX or E-MAIL► Infinite Computation Power can’t predict the Infinite Computation Power can’t predict the

message.message.

Basis MatricesBasis Matrices

► Basis matrices are binary Basis matrices are binary nn x x mm used to encrypt each used to encrypt each pixel in the secret image, where pixel in the secret image, where nn is the number of is the number of participants in the scheme and participants in the scheme and mm is the pixel is the pixel expansion. The following algorithm is used to expansion. The following algorithm is used to

implement a VCS using basis matrices:implement a VCS using basis matrices: If the If the nn x x mm basis matrices basis matrices SS1 (used to encrypt black 1 (used to encrypt black pixels) and pixels) and SS0 (used to encrypt white pixels) for any 0 (used to encrypt white pixels) for any VCS are given, the secret image VCS are given, the secret image SISI is encrypted as is encrypted as follows: follows:

Basic Matrices (Cont..)Basic Matrices (Cont..)

for each pixel for each pixel pp in in SISI: : { { if (if (pp is black) is black)

Let Let RR = a random permutation of the columns = a random permutation of the columns of of SS1 1

else else Let Let RR = a random permutation of the columns = a random permutation of the columns

of of SS0 0 for each participant for each participant ii (1 <= (1 <= ii <= <= nn): ): { {

The position on participant i’s share that The position on participant i’s share that corresponds to corresponds to pp is expanded into is expanded into mm pixels where pixels where each of these pixels each of these pixels jj (1 <= (1 <= jj <= <= mm) is black if ) is black if Ri,jRi,j = = 1 and white if 1 and white if Ri,jRi,j = 0. = 0.

} } } }