project complete without scaling

7/30/2019 Project Complete Without Scaling

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Certificate

Certified that this project titled A New Visual Cryptography Scheme For Color

Images With No Pixel Expansion is the bonafide work of Mr. Kapil Kumar

Chawala (Enroll. No. 321516) and Mr. Vikas Kumar (Enroll. No. 321522), who

carried out the work under my supervision, for the partial fulfillment of the

requirements for the award of the degree of Master of Computer Applications.

Certified further that to the best of my knowledge and beliefs, the work reported

here in does not form part of any other thesis or dissertation on the basis of which

a degree or an award was conferred on an earlier occasion on this or any other

candidate.

Head of the Department Project Guide

Dr. P. K. Mishra Dr. Achintya Singhal


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Acknowledgement

We take immense pleasure in thanking Dr. Achintya Singhal for having

permitted us to carry out this project work and feel great delight in expressing our

deep sense of gratitude to him under whose able guidance this work was

undertook. It was an honor and pleasure to work under him. Words are inadequate

in offering our thanks to him for his useful suggestions, encouragement and taking

good care to see that, we think and act in right direction, which helped us in

completing the project work successfully, in time.

We express our heartfelt gratitude to Dr. P. K. Mishra, Head of the

Department, for providing us the essential resources required to undertake this

project.

Our special thanks goes to our faculty members Mr. Shailendra Singh andMr. Sateyendra Kumar for their incredible and endless support.

And finally, yet importantly, we would like to express heartfelt thanks to

our beloved parents and rest of the family for their blessing and every individual

who was associated with the project.

Kapil Kumar Chawala Vikas Kumar


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Abstract

In 1994, Naor-Shamir [2] introduced the notion of Visual Cryptography

Scheme (VCS), which is the secret sharing of digitized images. A k-out-of-n VCS

splits an image into n secret shares which are indistinguishable from random noise.

These shares are then printed on transparencies. From any k 1 or less shares, no

information about the original image (other than the size of it) will be revealed.

The image can only be recovered by superimposing k or more shares. This

recovery process does not involve any computation. It makes use of the human

vision system to perform the pixel-wise OR logical operation on the

superimposed pixels of the shares. When the pixels are small enough and packed

in high density, the human vision system will average out the colors of

surrounding pixels and produce a smoothed mental image in a humans mind.

Early VCS schemes mainly focused on black-and-white and gray-scale

images. One of the most potentially useful types of visual cryptography scheme iscolor visual cryptography. Due to the nature of a color image, this again helps to

reduce the risk of alerting someone to the fact that information is hidden within it.

It allows high quality sharing of these color images. Color images are highly

popular and have a wider range of uses when compared to other image types.

Some significant work has been done on color images. However, an issue

that is common to most of the previous work is pixel expansion, which means that

the size of each secret share is several times larger than that of the original image.

Two important parameters which govern the quality of reconstructed images are m

(pixel expansion rate which represents the loss in resolution from the original

image to the shares) and (the relative difference in weight between the

superimposed shares that come from one color level (e.g. black) and another color

level (e.g. white)). For image integrity, a good VCS should make the value of m

close to one (i.e. no pixel expansion) and

as large as possible.It is uncommon if there is a scheme which satisfies all the following three


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commonly desired properties:

1. Supporting images of arbitrary number of colors;2. No pixel expansion;3. Supporting k-out-of-n threshold setting.

We answer this question affirmatively by proposing a new probabilistic k-

out-of-n threshold visual cryptography scheme which satisfies all these properties

for color images based on the past studies in black-and-white visual cryptography,

gray scale images, the halftone technology, and the additive color mixing model.

In particular, our scheme utilizes a probabilistic technique for achieving no pixel

expansion. In this project, we have given the new algorithm of proposed schemefor color images and considered the various constructions based on this algorithm

and experimental results related to the new scheme.


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Table of Contents

Certificate . i

Acknowledgement ii

Abstract .... iii

Table of Contents . v

List of Figures ...... vii

1. Introduction 11.1 Cryptography ... 21.2 Secret sharing ... 31.3 Visual Cryptography 41.4 Summary .. 5

2. Related work 72.1 VCS Schemes for Black-and-White Images . 8

2.1.1 Naor-Shamir Black-and-White VCS ... 82.1.1.1Preliminary Notation 92.1.1.2Two-out-of-two scheme 102.1.1.3Three-out-of-three scheme ........ 112.1.1.4A general k-out-of-kscheme . 142.1.1.5A general k-out-of-n scheme . 15

2.1.2 Other Black-and-White VCS Schemes 162.2 VCS Schemes for Gray-scale Images 16

2.2.1 Some Gray-Scale VCS Schemes . 162.2.2 Chen et al. Gray-Scale VCS with No Pixel Expansion ... 18

2.3 VCS Schemes for Color Images ... 192.3.1 Hou Colored VCS Schemes 192.3.2 Yang-Chen Colored VCS 25


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2.3.3 Other Colored VCS Schemes .. 272.3.4 Summary . 27

3. A newk-out-of-n colored VSS scheme 283.1 Basic Principle of Color 293.2 The Proposed Probabilistic VSS Scheme for Color Images . 313.3 Algorithm-For our proposed (k, n) ProbVSS scheme ... 393.4 Constructions of our proposed (k, n) ProbVSS scheme 40

3.4.1 A proposed 2-out-of-n ProbVSS scheme . 413.4.2 A proposed 3-out-of-n ProbVSS scheme . 493.4.3 A proposed k-out-of-kProbVSS scheme . 513.4.4 A proposed k-out-of-n ProbVSS scheme . 55

3.5 Summary ... 584. Comparison .. 605. Application ... 64

5.1 Electronic-Balloting System . 655.2 Encrypting Financial documents ... 665.3 Watermarking 665.4 Moire patterns 675.5 Person Authentication 685.6 Summary 69

6.

Conclusion and Future work .. 70

Bibliography .. 73

Appendix . 77


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List of Figures

Figure 2.1: Construction of 2-out-of-2 scheme ... 10

Figure 2.2: Secret image, two share and superimposed image (2, 2)....... 11

Figure 2.3: Three share and superimposed image (3, 3)... 12

Figure 2.4: Five share and superimposed image (3, 5)..... 13

Figure 2.5: Lena gray-scale and halftone image .. 17

Figure 2.6: Two share and their superimposed image (2, 2) ... 18

Figure 2.7: Scheme 1 of Hou colored VCS scheme .... 21

Figure 2.8 Four share and superimposed image (4, 4)..... 21

Figure 2.9: Scheme 2 of Hou colored VCS scheme 23

Figure 2.10: Two share and superimposed image (2, 2) ...... 23

Figure 2.11: Color composition and decomposition ... 24

Figure 2.12: Two share and superimposed image (2, 2)... 25

Figure 3.1: ACM and SCM color model .. 30

Figure 3.2: Continues tone and Half tone 35

Figure 3.3: Lena gray-scale and halftone image .. 36

Figure 3.4: Colored image with RGB component and their halftone .. 36

Figure 3.5: Original image with two shares and superimposed image (2, 2) ...... 42

Figure 3.6: Three share and superimposed image (2, 3) .. 45

Figure 3.7: Four share and superimposed image (2, 4) ... 47

Figure 3.8: Three share and superimposed image (3, 3) ...... 50

Figure 3.9: Four share and superimposed image (4, 4) .... 53

Figure 3.10: Four share and superimposed image (3, 4) .. 58

Figure 4.1: Comparison of VCS scheme . 63

Figure 5.1: Moire Patterns ... 68


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Chapter 1

INTRODUCTION


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1.IntroductionAs technology progresses and as more and more personal data is digitized,

there is even more of an emphasis required on data security today than there has

ever been. Protecting this data in a safe and secure way which does not restrict the

access of an authorized authority is an immensely difficult and very interesting

research problem. Many attempts have been made to solve this problem within the

cryptographic community. In this chapter, we will describe about the

cryptography, secret sharing and one of its branches visual cryptography.

1.1CryptographyCryptography probably began in or around 2000 B.C. in Egypt. On the

tombs of deceased kings, hieroglyphics, which are intentionally cryptic, were used

to tell their life stories and achievements. Cryptography was once used to make the

text more royal and important. Classic cryptography means only secret writing

(converting a meaningful message into an incomprehensible one). There are two

techniques most commonly used in the classic cryptography, i.e. transposition and

substitution. Transposition means that the order of the plaintext is rearranged

according to a specified rule (e.g. cityuofhk becomes uckoyfiht in a simple

rearrangement scheme). Substitution means that the original letter or groups of

letters are replaced by other letter or group of letters (e.g. cityuofhk becomes

djuzvpgil by replacing each letter with the one following it in alphabetical

order). Classic ciphers were used historically in transporting secret messages, e.g.

Caesar cipher (a type of substitution cipher), which was used by Julius Caesar to

communicate with his generals. However, almost all types of classic ciphers are

vulnerable to statistical analysis since they always reveal statistical information

about the original message. Now people still use these classic ciphers, but mainly

as puzzles.

The modern field of cryptography has been expanded to more topics such as


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secret sharing schemes, authentication codes, identification schemes and key

distribution. There are two classes of encryption algorithms for modern

cryptography, i.e., symmetric-key cryptography and public-key cryptography. Insymmetric-key cryptography, the sender and the receiver share the same key to

encrypt or decrypt a message. This algorithm leads to the difficulty in key

management since the number of keys grows quickly with the increase of the

communication groups. Public-key cryptography can solve the problem of key

management by providing a public key and a private key. The two keys are

mathematically related, but without the secret information, no one can get one key

from the other. Typically, the public key is freely distributed and used to encrypt a

message, while the private key is kept secret and used to decrypt a message.

1.2Secret sharingA secret is something which is kept from the knowledge of any but the

initiated or privileged. Secret sharing defines a method by which a secret can be

distributed between the groups of participants, whereby each participant is

allocated a piece of the secret. This piece of the secret is known as a share. The

secret can only be reconstructed when a sufficient number of shares are combined

together. While these shares are separate, no information about the secret can be

accessed. That is, the shares are completely useless while they are separated.

Within a secret sharing scheme, the secret is divided into a number of

shares and distributed among n persons. When any k or more of these persons

(where kn) bring their shares together, the secret can be recovered. However, if

k 1 persons attempt to reconstruct the secret, they will fail. Due to this threshold

scheme, we typically refer to such a secret sharing system as a (k, n)-threshold

scheme or k-out-of-n secret sharing.

In 1979, Adi Shamir published an article titled How to share a secret [1].

In this article, the following example was used to describe a typical secret sharing

problem:

Eleven scientists are working on a secret project. They wish to lockup the


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documents in a cabinet so that the cabinet can be opened if and only if six or more

of the scientists are present. What is the smallest number of locks needed? What is

the smallest number of keys to the locks each scientist must carry?.The minimal solution uses 462 locks and 252 keys per scientist.

In the paper [1], Shamir generalized the above problem and formulated the

definition of (k, n)-threshold scheme. The definition can be explained as follows:

Let D be the secret to be shared among n parties. A k-out-of-n threshold

scheme is a way to divide D into n piecesD1, . . . , Dn that satisfies the following

conditions:

Knowledge of any kor moreDi pieces makesD easily computable, Knowledge of any k-1 or fewer Di pieces leaves D completely

undetermined (in the sense that all its possible values are equally likely).

1.3Visual CryptographyWe present one of these data security methods known as visual

cryptography (VC). Specially, visual cryptography allows us to effectively and

efficiently share secrets between the numbers of trusted parties. As with many

cryptographic schemes, trust is the most difficult part.

Visual cryptography is a new type of cryptographic scheme that focuses on

solving the problem of secret sharing. Visual cryptography uses the idea of hiding

secrets within images. These images are encoded into multiple shares and later

decoded without any computation. This decoding is as simple as superimposing

transparencies, which allows the secret to be recovered. Visual cryptography is a

desirable scheme as it embodies both the idea of perfect secrecy (using a one time

pad) and a very simple mechanism for decrypting/decoding the secret without the

computer participation. The interesting feature about visual cryptography is that it

is perfectly secure. There is a simple analogy from one time padding to visual

cryptography. Considering the currently popular cryptography schemes, which are

usually conditionally secure, this is the second critical advantage of visual


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cryptography.

Based on the idea of threshold secret sharing, in 1994, Naor-Shamir [2]

introduced Visual Cryptography, which is the secret sharing of digitized images. Itsolves the problem of encrypting images in a secure way so that the decryption

process can be done by a person purely using his/her visual system without any

computation. A k-out-of-n Visual Cryptography Scheme (VCS) splits an image

into n secret shares which are then printed on transparencies. These shares when

separated will reveal no information about the original image (other than the size

of it). The image can only be recovered by superimposing kor more shares. This

recovery process does not involve any computation. It makes use of the human

vision system to perform the pixel-wise OR logical operation on the superimposed

pixels of the shares. When the pixels are small enough and packed in high density,

the human vision system will average out the colors of surrounding pixels and

produce a smoothed mental image in a humans mind.

Early VCSs [2-7] mainly focused on black-and-white or gray-scale secret

images. Additionally, pixel expansion is common to most of the previous work.This means that the size of each secret share is several times larger than that of the

original image and the resulting superimposed image is also several times larger

than the original secret image.

In this project, we proposed a new VCS scheme for color images. In our

constructions of color visual cryptography scheme (CVCS), we utilize a

probabilistic technique for achieving no pixel expansion. Our CVCS scheme

supporting the three desirable properties which are summarized as follows:

1. Supporting images of arbitrary number of colors;2. No pixel expansion;3. Supporting k-out-of-n threshold setting.1.4Summary

In this chapter, first we have described about cryptography, secret sharing

and then we have described the visual cryptography which is the branch of


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cryptography and the technique of secret sharing. On visual cryptography, we have

proposed a new probabilistic k-out-of-n scheme based on color images.

The rest of the project report is organized as follows. In chapter 2, we willreview the literature and related results in visual cryptography scheme. In chapter

3, we will propose new k-out-of-n colored VCS. In chapter 4, we will compare our

scheme with the others schemes. In chapter 5, we will give the brief introduction

about the applications of visual cryptography. In chapter 6, we will give

conclusion and future work of our scheme.


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Chapter 2

RELATED WORK


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2.Related WorkIn this chapter, we briefly review the literature on black-and-white VCS

schemes, gray-scale VCS and colored VCS schemes. By reviewing these schemes,

we find several commonly desirable properties which should be supported by VCS

schemes.

2.1Visual cryptography scheme for black and white imagesNaor-Shamir [2] first introduced the concept of VCS and proposed a

general k-out-of-n threshold VCS for black-and-white images. Their scheme acts

as the building block of other VCS schemes. In this section, we will give basic

model of 2-out-of-2, 3-out-of-n, k-out-of-k and k-out-of-n threshold VCS, some

results obtained by implementation of their basic model and also summarize the

features of several other black-and-white VCS schemes proposed after Naor-

Shamirs and show their merits and demerits.

2.1.1 Naor-Shamir Black-and-White VCSIn [2], Naor-Shamir introduced visual cryptography in which written

material (printed text, handwritten notes, etc.) can be encrypted in a secure way so

that the decryption process only needs the human visual system without any

computation.

In visual secret sharing (VSS) schemes, problem message considered as a

collection of black-and-white pixels and each pixel is handled separately. Each

original pixel appears in n modified shares, one for each transparency. Each share

is a collection of m black and white sub-pixels. The resulting structure can be

described as n m Boolean matrix, S= [sij], where sij= 1 iff thejth

sub-pixel in the

ith

transparency is black. When transparencies i1, i2 . . . ir are stacked together, a

combined share whose black sub-pixels are represented by the Boolean OR of

rows i1, i2 . . . ir are in S. The gray level of the combined share is proportional to

the Hamming weight H(V) of the OR-ed m-vector V and visual system


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interpreted it as black ifH(V) dand as white ifH(V) < d m for some fixed

threshold 1 dm and relative difference > 0. The formal definition ofk-out-

of-n visual secret sharing scheme is described as follows:Definition 1: A solution to the k-out-of-n visual secret sharing scheme

consists of two collections ofn m Boolean matrices C0 and C1. To share a white

pixel, the associate randomly chooses one of the matrices in C0 and to share a

black pixel, the associate randomly chooses one of the matrices in C1. The chosen

matrix defines the color of the m sub-pixels in each one of the n transparencies.

The solution is considered valid if the following three conditions are met:

For Contrast:

1. For any Sin C0, the OR Vof any kof the n rows satisfiesH(V) d m.2. For any Sin C1, the OR Vof any kof the n rows satisfiesH(V) d.

For Security:

3. For any subset {i1, i2 . . . iq} of {1, 2,...., n} with q < k, the two collections ofq m matricesDt for t

{0, 1} obtained by restricting each n m matrix in

Ct (where t {0, 1}) to rows i1, i2 . . . iq are indistinguishable in the sensethat they contain the same matrices with the same frequencies.

2.1.1.1 Preliminary Notationn, Group Sizek, Thresholdm, the number of pixels in a share. This parameter represents the loss in

resolution from the original image to the recovered one.

, the relative difference in the weight between the combined shares thatcome from a white pixel and a black pixel in the original image. This

parameter represents the loss in contrast.

, the size of the collection ofC0 and C1. C0, Collection ofn m Boolean matrices for shares of white pixel. C1, Collection ofn m Boolean matrices for shares of black pixel.


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V, OR-ed krowsH(V), Hamming weight (H) of a string is the number of symbols that are

different from the zero symbol.

d, number in [1, m]r, Size of collections C0 and C1.

2.1.1.2 Two-out-of-two schemeThe original problem of visual cryptography is the special case of a 2-out-

of-2 visual secret sharing problem. The 2-out-of-2 VSS problem is solved by the

following collection of 2 2 matrices:

C0= {all the matrices obtained by permuting the columns of0 01 1 0 10 1 0 11 0}C1= {all the matrices obtained by permuting the columns of1 10 0 1 01 0 1 00 1}

The decryption process through human vision system is described in the

following Figure 2.1:

Figure 2.1 Construction of 2-out-of-2 scheme

When using the black and white image in Figure 2.2(a) as secret image, the

experimental results are shown in the following figures. The two shadow images

are shown in the Figure 2.2(b, c) and their reconstructed image using OR operationis shown in the Figure 2.2(d):


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(a).Original Image (b) Share 1

(c) Share 2 (d) 1+2 using OR

Figure 2.2

2.1.1.3 Three-out-of-three schemeThe 3-out-of-3 VSS problem is solved by the following collection of 3 4

matrices:

C0 = {all the matrices obtained by permuting the columns of0 0 1 10 1 0 10 1 1 0}C1 = {all the matrices obtained by permuting the columns of1 1 0 01 0 1 0

1 0 0 1}

Each matrix in either C0 or C1 contains one horizontal share, one vertical

share and one diagonal share. Each share contains a random selection of two black

sub-pixels, and any pair of shares from one of the matrices. By stacking all the

three transparencies from C0 and C1 we can observe C0 is only 34 black whereas a

C1 is completely black. When using the black and white image of Figure 2.2(a) as

a secret image, the experimental results are shown in the following figures. The

three shadow images are shown in the Figure 2.3(a, b, c) and their reconstructed

image using OR operation is shown in the Figure 2.3(d, e). Stacking any k-1 i.e. 2


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shadow images does not reveal any information as shown in Figure 2.3(d).

(a) Share 1 (b) Share 2 (c) Share 3

(d) 1+2 using OR (e) 1+2+3 using OR

Figure 2.3

The above 3-out-of-3 VSS scheme can be generalized to 3-out-of-n for anarbitrary n 3. LetB be the blackn (n2) matrix which contains only 1's and let

Ibe an identity n n matrix which contains 1's on the diagonal and 0's elsewhere,

then BI (concatenation ofB and I) denote n (2n2) Based upon the following

properties we can design the matrix for 3(k)-out-of-5(n) scheme.

Here, black matrix B = n (n2) = 5 (52) = 5 3; an identity n n

matrix which contains 1's on the diagonal and 0's elsewhere,I= n n = 5 5. Two

collections of n m Boolean matrices C0 and C1 are obtained by permuting the

columns ofc(BI) andBI, m = 2n 2 = 2(5) 2 = 8. i.e., 5 8 Boolean matrices.

Share 1

Share 2

c(BI) = Share 3

Share 4

Share 5

0 0 0 0 1 1 1 1

0 0 0 1 0 1 1 1

0 0 0 1 1 0 1 1

0 0 0 1 1 1 0 1

0 0 0 1 1 1 1 0


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Hamming weight ofc(BI) for white is:H(V) = 5

Share 1

Share 2

BI= Share 3

Share 4

Share 5

Hamming weight ofBIfor black is:H(V) = 8

C0 = {all the matrices obtained by permuting the columns ofc (BI)}

C1 = {all the matrices obtained by permuting the columns ofBI}

If the columns are not permuted then there is a possibility to reveal the

secret information in any single share and therefore the process fails. When using

Figure 2.2 (a) as a secret image, the experimental results shown in the following

figure. The five shadow images are shown in the Figure 2.4 (a, b, c, d & e) and

their reconstructed image using OR operation is shown in the Figure 2.4(h, i).

Stacking any k-1 i.e. 2 shadow images does not reveal any information as shown in

Figure 2.4(f, g).

(a) Share 1 (b) Share 2

(c) Share 3 (d) Share 4

1 1 1 1 0 0 0 0

1 1 1 0 1 0 0 0

1 1 1 0 0 1 0 0

1 1 1 0 0 0 1 0

1 1 1 0 0 0 0 1


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(e) Share 5 (f) 1+2 using OR

(g) 3+4 using OR (h)1+2+3 using OR

(i) 2+3+4 using OR

Figure 2.4

In above example each pixel is divided into 8 sub-pixels which is not a

perfect square which distorts the aspect ratio of the image. So, to avoid distorting

the image, the dummy sub-pixels are added to keep the aspect ratio unchanged. In

the 3-out-of-5 scheme any single share contains 4 black and 4 white pixels. To

make it a complete square array without distorting their aspect ratio, we need to

add one more pixel; it should be either black or white.

2.1.1.4 A Generalk-out-of-k schemeFor all k there exists a general construction of k-out-of-k visual secret

sharing scheme, the pixel expansion must use at least 2k-1

pixels, and the relative

contrast should be 1/(2k-1

). There is a need to construct two collections of k 2n-1

Boolean matrices i.e., S0 and S1.


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1. S0 handles the white pixels.2. S1 handles the black pixels.

All 2k-1

column have an even number of1s in S0 and odd number of1's in

S1 and no two krows are same in both S0 & S1. C0 and C1 contain all permutations

of columns in S0 and S1.

2.1.1.5 A Generalk-out-of-n SchemeA general k-out-of-n scheme is designed from k-out-of-kscheme. Let Cbe

k-out-of-k visual secret sharing scheme with parameters m, r, . The scheme C

consists of two collections ofk m Boolean matrices and C0 = T0

1, T0

2, . . . . , T0

rand C1 = T

11, T

12, . . . . , T

1r.His a collection ofl functions hH, :1

{1 k}. Let B be the subset of {1 n} of size k and q is probability that

randomly chosen function yields q different values onB, 1 qk.We construct from CandHbe a k-out-of-n scheme C with parameters

m = m.l k, r = rl

1. The ground set is V= UH2. Each 1 trl is indexed by a vector (t1, t2, , tl) where 1 tr.3. The matrix Sbtfor t= (t1, t2, , tl) where b {0,1} is defined as

Sbt[i, (j, h)] = T

btj [h(i), j]

Contrast: Contrast should be k:

1. krows is, Sbtmapped to q < kdifferent values by h.2. Hamming weight of OR ofq rows isf(q)3. Difference is m white and black pixels occur when h is one and happens

at k.

4. White: !"! $%&5. Black: ' ( ) $%!"! *Security: Security properties of the k-out-of-kscheme imply the security ofk-

out-of-n scheme because we are using (k, k) scheme to create (k, n) scheme. The


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expected hamming weight of OR ofq rows, q < kis irrespective of white or

black pixel.

The above constructions are provably secure and the pixel expansion rate isgetting larger when the values of kand n grow. Moreover, the scheme does not

support images of arbitrary number of colors.

2.1.2 Other Black-and-White VCS SchemesSince the introduction of VCS, there have been many other schemes proposed

[3-4]. In 2003, by removing the same column which appears in both basic

matrices, Blundo et al. [3] proposed VCS schemes with optimal contrast in (n 1)-out-of-n (where n > 3) and 3-out-of-n schemes. They also conjectured that the k-

out-of-n scheme, where k=4 or 5, had an optimal contrast. In [4], Yang proposed

another one which achieved no pixel expansion by using the frequency of white

pixels to show the contrast of the recovered image. Yangs scheme can be easily

implemented on the black-and-white VCS with pixel expansion. We will describe

the Yangs approach in section 3.2 (a) and also distinguish his approach from

conventional approach. All the above schemes only support black-and-white

images.

2.2VCS Schemes for Gray-scale ImagesDue to the limitations on applicability of the black-and-white VCS

schemes, in 1997, Verheul-Tilborg [5] proposed the k-out-of-n VCS for gray-scale

images. In this section, we give a review on their scheme and also figures several

problems about it.

2.2.1 Some Gray-Scale VCS SchemesIn 1997, Verheul-Tilborg [5] introduced a general method for k-out-of-n

VCS for gray-scale images. For an image of c gray levels, cn q matrices are

constructed to represent the secret sharing process for each gray level. The

elements of these matrices are in {0, 1, , c 1}. And they use i (i

{0, 1, ,

c1}) to represent the gray level. The superimposed color is i if all the


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corresponding sub-pixels of all shares are i, otherwise the superimposed color is

black. To encrypt a pixel with the gray level i (where 0 ic 1), they choose

the matrix which represents the gray level i. The n rows of the matrix correspondto the n shares and the q columns correspond to the gray levels of the q sub-pixels

of each share. In this scheme, the pixel expansion rate has to be at least c(k1)

. The

scheme was restricted to small k and n as large values resulted in exponential

image size and high computation cost.

Lin and Tsai [6] proposed another VCS scheme for gray-scale images by

applying dithering techniques in 2003. The gray-scale image is first converted into

a binary image with the same size by dithering and then Naor-Shamirs VCS for

black-and-white images is employed. This scheme improved the pixel expansion

of Verheul-Tilborgs VCS so that the pixel expansion rate is the same as Naor-

and-Shamirs black-and-white VCS. Furthermore, only two collections of matrices

are needed for any number of gray levels. In dithering technique, we can use

various halftone techniques such as Floyd and Jarvis halftoning method to

implement the algorithm of Lin and Tsai for gray-scale image. Halftone

technology is discussed briefly in section 3.2 (b).

When using the Figure 2.5 (a) as a secret image, the experimental results

shown in the following figures. Halftone image of Figure 2.5 (a) is shown in the

Figure 2.5 (b). The two shadow images are shown in the Figure 2.6 (a, b) and their

reconstructed image using OR operation is shown in the Figure 2.6 (c):

(a) Original image (b) Halftone image

Figure 2.5


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(a) Share 1 (b) Share 2 (c) 1+2 using OR

Figure 2.6

2.2.2 Chen et al. Gray-Scale VCS with No Pixel ExpansionIn 2007, Chen et al. [7] extended the results of [4] to gray-scale images and

proposed a gray-scale VCS. Chen et al.s VCS maps a block in a secret image to

one corresponding equal-sized block in each share image without image size

expansion. For a secret block, this method generates the corresponding share

blocks containing multiple levels rather than two levels based on the density of

black pixels on the block. It uses two types of techniques, namely, histogram

width-equalization and histogram depth-equalization.

Chen et al.s VCS scheme can reach the goal of no pixel expansion.

However, their scheme does not support colored images and only supports k-out-

of-kthreshold setting.

Besides, before secret sharing, it needs to do preprocessing on the original

images (block averaging). The details of the algorithm are as follows:

In the k-out-of-k scheme where each secret block contains s pixels, the

algorithm BASIS MATRIX (k, s, and l) is designed to randomly generate a

Boolean Basis Matrix for a secret block. Here, s is the block size and l (0 l s/2)

is the gray-scale intensity level.

Algorithm: BASIS MATRIX(k, s, l)

1. Randomly select s/2 + l positionsp1,p2 ,. . . ,ps/2+l from the s positions.2. For i = 1 to k-1, randomly select s/2 positions fromp1,p2 ,. . . ,ps/2+l; LetRi

be a variable consisting of s bits and set the bits located at the s/2 selected


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positions in Ri as 1-bits and others as 0-bits.

3. R =R1 orRi or . . . orRk1;4. Let p1, p2 ,. . . , ph be the positions of all the 0-bits which are located at

positionsp1,p2 ,. . . ,ps/2+l inR.

5. Set the bits located at p1, p2 ,. . . , ph ofRk as 1-bits; Set the s/2-h bits byrandomly selecting from the bits located at p1,p2 ,. . . ,ps/2+l fromRk, except

the positions p1, p2 ,. . . , ph , as 1-bits, and set the other bits as 0-bits.

Regard each member ofR1,R2, . . . ,Rkas one row of the basic matrix.

This algorithm adopts block-to-block mapping as a key measure in itsimplementation. It makes significant progress in the aspect of pixel expansion

since it is the first scheme which achieves no pixel expansion. However, the

authors did not provide a general solution for the k-out-of-n scheme. And there

was no solution for colored images in their scheme as well. Besides, it needs to

preprocess the original image before doing secret sharing (block averaging). This

process reduces the quality of reconstructed image.

2.3VCS Schemes for Color ImagesThe requirement of encrypting natural image makes researchers focus on

the VCS schemes for color images. In [8], Hou firstly proposed three methods for

encrypting color images. In this section, we describe Hous scheme in detail and

review Yang-Chen colored VCS [9] for its no pixel expansion property. Moreover,

we discuss some other colored VCS schemes proposed recently to show the

development of colored VCS.

2.3.1 Hou Colored VCS SchemesFor color VCS schemes, [8-14], Hous schemes [8] are believed to be the

first set of color VCS schemes. In his paper, he proposed three methods for gray-

scale and color images based on the previous studies in black-and-white visual

cryptography, halftone technology, and color decomposition method. His methods

have the backward compatibility with the previous results in black-and-white


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visual cryptography and can be easily applied to gray-scale and color images.

Subtractive model is used in all the methods.

Hous Method 1: The first method produces four shares, namely black mask,C(Cyan) share,M(Magenta) share and Y(Yellow) share. By superimposing these

shares, it shows the best reconstructed quality among Hous three methods.

Algorithm for the first methods is given below:

Algorithm of color visual cryptographymethod1

1. Transform the color image into three halftone images: C,M, and Y.2. For each pixel P

ijwith color components (C

ij, M

ijor Y

ij) of the composed

image P, do the following:

a) Select a black mask with a size of 22, and assign a black pixelrandomly to two of these four positions and leave the rest positions

blank (transparent or white). This step will make the black mask a

half black-and-white block.

b) After selecting a mask, determine the positions of the cyan pixels inthe block of the corresponding sharing images. This is done

according to the positions of the black pixels in the mask and the

value ofCij .

If Cij = 1 (the cyan component will be revealed), fill the

positions corresponding to the positions of the white pixels in the

mask with a cyan pixel and leave the rest positions blank.

IfCij = 0 (the cyan component will be hidden), fill the colors

in the opposite way. i.e., fill the positions corresponding to the

positions of the black pixels in the mask with a cyan pixel and leave

the rest positions blank.

Finally, add the block to the corresponding position of Share 1.

c) In accord with (b), determine the positions of magenta pixels of theblock in Share 2 with the value ofMij and those in Share 3 with the

value ofYij.


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3. Repeat Step 2 until every pixel of the composed image is decomposed,hence obtaining four transparencies (cyan, magenta, yellow and black) of

visual cryptography to share the secret image.4. After stacking the four sharing images, the secret image can be decrypted

by human eyes.

Figure 2.6 summarizes this method by giving an example in which a random

block of the black mask is chosen and shown in the first column. The second

column shows the eight possible combinations of the original (dithered) C,M, and

Ypixel values. The following three columns show the encoding of the blocks in

the corresponding shares of C, M, and Y. The last column illustrates the

superimposed image ofC,M, and Yshares with the black mask.

Figure 2.7: Scheme 1 of Hou Colored VCS Schemes

When using the Figure 2.8 (a) as a secret image, the experimental results for

Hous first method are shown in the following figures. Four shadow images are

shown in Figure 2.8 (b, c, d, e) and the reconstructed image using OR operation is

shown in Figure 2.8 (f)

(a) Original image (b) Cyan share (c) Magenta share


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(d) Yellow share (e) Black mask (f) Stacked image using OR

Figure 2.8Hous Method 2: The second method expands every pixel of a halftone image

into a 22 block on two sharing images and fills the block with cyan, magenta,yellow and transparent, respectively. Using these four colors, two stacked images

can generate various colors through different permutations. Algorithm for this

method is given below:

Algorithm of color visual cryptographymethod2

1. Transform the color image into three halftone images: C,M, and Y.2. For each pixel Pij of the composed image, do the following:

a) Expand a 22 block in Share 1 and fill the block with cyan,magenta, yellow, and transparent randomly.

b) Generate a 22 block in Share 2 according to the permutation of thefour colors of the block in Share 1 and the values ofCij,Mij, Yij , and

determine the color distribution of the corresponding block in Share

2 as illustrated in Figure 2.9. Take a pixel with Pij = (1, 1, 0) for

example. When deciding the clockwise permutation of the block in

Share 1 to be cyan magenta white yellow, swap the

positions of cyan and magenta and form the permutation of the

corresponding block in Share 2 as magenta cyan white

yellow. The stacked result will be a blue-like block, just as the pixel

(1, 1, 0) should be on the secret image.


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3. Repeat Step 2 until every pixel of the composed image is decomposed,hence obtaining two visual cryptography transparencies to share the secret

image.4. After stacking the two sharing images, the secret image can be decrypted by

human eyes.

Figure 2.9: Scheme 2 of Hou Colored VCS Schemes


Hous second method are shown in the following figures. The two shadow images

are shown in Figure 2.10 (a, b) and the reconstructed image using OR operation is

shown in Figure 2.10 (c):


Figure 2.10

Hous Method 3: This method needs two sharing images and does not

sacrifice too much contrast for color visual cryptography. It transforms a color


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secret image into three halftone images C, M, and Yand generates six temporary

sharing images C1, C2,M1,M2, Y1, and Y2. Each of these sharing images will have

two white pixels and two color pixels in every 22 block. The method thencombines C1,M1, and Y1 to form a colored halftone Share 1 and C2,M2, Y2 to form

Share 2. Algorithm for this method is given below:

Algorithm ofvisualcryptographymethod 3

1. Transform the color image into three halftone images: C,M, and Y.2. For each pixel Pij of the composed image, do the following:

a) According to the traditional method of black-and-white visualcryptography, expand Cij, Mij and Yij into six 22 blocks, C1ij, C2ij,

M1ij,M2ij and Y1ij, Y2ij.

b) Combine the blocks C1ij; M1ij and Y1ij and fill the combined blockcorresponding to Pij in Share 1.

c) Combine the blocks C2ij; M2ij and Y2ij and fill the combined blockcorresponding to Pij in Share 2.

3. Repeat Step 2 until every pixel of the composed image is decomposed,hence obtaining two visual cryptography transparencies to share the secret

image.

4. After stacking the two sharing images, the secret image can be decrypted byhuman eyes.

Figure 2.11: Color pixel decomposition and reconstruction


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Hous third method are shown in the following figures. The two shadow images

are shown in Figure 2.12 (a, b) and the reconstructed image using OR operation isshown in Figure 2.12 (c).


Figure 2.12

Hous three methods for encrypting colored images are considered to be the

first colored schemes. Since Hou did not provide any security analysis on these

methods, we can not guarantee the security of them. The first method has been

proved to be insecure recently by Leung et al. [10]. The second one is secure,

whereas the quality of reconstructed image is not satisfying. The third one is

considered to be secure with better quality of reconstructed image.

However, the three methods have several common drawbacks. First, the pixel

expansion rates of these methods are all 4 which make the reconstructed image

four times larger than the original one. Second, they have limitation on the color

levels of original images, so dithering is needed before secret sharing. Third, no

general k-out-of-n solution was proposed, since the first method is for 4-out-of-4

and the last two are for 2-out-of-2. Finally, no security analysis for these methods

was provided.

2.3.2 Yang-Chen Colored VCSIn all the previous methods, a secret pixel is represented by several color

sub-pixels and the number of these sub-pixels is referred to as the pixel expansion.


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Generally, they require a larger pixel expansion to produce more colors. In [9],

Yang and Chen use additive color mixing in a probabilistic way so that the pixel

expansion is fixed on 3 regardless of the number of colors in the reconstructedimages. They use the appearance frequencies ofR, G and B to simulate a secret

color. The drawback of the scheme is that to improve the color contrast of the

reconstructed images, the only way is to modify the original images. The details of

the scheme are discussed in the following.

A secret pixel is divided into three colored sub-pixels (R, G, B) where the

first R-colored sub-pixel has the appearance probability piR , i.e., this sub-pixel

may be R color with probability piR and NULL color with probability (1-piR).

Accordingly, the other two sub-pixels (G-color and B-color) have the appearance

probabilitiespj

G andpkB, respectively.

Step 1: Calculate LR, LG and LB which represent the number of colors on the

three primary colors of the original image respectively.

Step 2: Construct the primary color sets: CiR (R-colored sets), C

jG (G-colored

sets) and Ck

B (B-colored sets). Suppose the basic matrices of black-and-white

scheme are n l Boolean matrices, then the primary color sets are n l LX

matrices (XR, G,B).CR

(255r) / (LR

-1)= CW+ +CW+ CB+ +CB1R;0N

LR

-1-r r

Where, r [0,LR 1] andLR is the number of colors on the primary color red. andCW and CB are the basic matrices of black-and-white scheme. The constructions ofC

jG (G-colored sets) and C

kB are similar with C

iB .

Step 3: Scan each pixel of the original image and apply the corresponding

matrices in Step 2 by randomly choosing a column of each of the three Basis

Matrices. Consider this column as an n-bit vector. For the first bit, assign the

corresponding pixel black color (i.e. 0 color intensity) if the bit is 1, otherwise we

assign it X(X {R, G, B}) color (i.e. 255 color intensity). Continue this process


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until the corresponding color in each of the n shares is assigned a color value.

There are 3 sub-pixels for each pixel in the original image so the pixel expansion is

3.It is quite a new idea for its using probabilistic method to make sure the pixel

expansion maintains 3. And it doesnt limit the number of colors in the original

image. Besides, they provided a general k-out-of-n scheme. It can deal with

colored image and is provably secure. The quality of the reconstructed image is

good when the size of the original image is large. However, it still needs

preprocessing, since the only way of improving the color contrast is to modify the

number of colors in the original image.

2.3.3 Other Colored VCS SchemesLukac-Plataniotis [11] proposed a colored VCS scheme in 2005, which only

supports 2-out-of-2 threshold setting and has pixel expansion. Shyu [12] proposed

a colored VCS scheme in 2006. Shyus scheme applies any k-out-of-n black-and-

white VCS on decomposed images of the original secret image so that it supports

the general k-out-of-n threshold setting. The scheme has less pixel expansion

compared with previous ones while it has relatively good quality of superimposed

image. Hou and Tus colored VCS [13] supports k-out-of-n threshold setting with

no pixel expansion. Dithering is required for preprocessing the original image

before secret sharing and the number of colors supported is fixed on 8. Cimato et

al.s scheme [14], at the cost of large pixel expansion, solves the problem that

superimposing many pixels of the same color results in a dark version of the color.

2.4SummaryIn this chapter, we have reviewed the literature of visual cryptography

schemes of black and white images, gray-scale images, color images and various

experimental results related to it. These schemes give the strong foundation for our

proposed scheme for color images which we will consider in the next chapter.


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Chapter 3

A NEWK-OUT-OF-NCOLORED VSS SCHEME


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3.A Newk-out-of-n Colored Visual Secret Sharing SchemeIn this chapter, we propose new scheme to construct the visual secret shares.

Our scheme uses a different approach, the probabilistic method. The major

difference between schemes is that our scheme uses pixel operation and the

conventional scheme uses sub-pixel operation.

First, we give the basic principle of color, notations used in the context and

then we give the proposed probabilistic VSS (ProbVSS) scheme for colored

images, their various constructions, theorems and their proofs.

3.1Basic Principle of ColorThe additive (ACM) and subtractive models (SCM) in the Figure 3.1 (a, b)

are commonly used to describe the constitutions of colors [15, 16]. In the additive

system, the primaries are red, green and blue (RGB), with desired colors being

obtained by mixing different RGB components. By controlling the intensity of red

(green or blue) component, we can modulate the amount of red (green or blue) in

the compound light. The more the mixed colored-lights, the more is the brightness

of the light. When mixing all red, green and blue components with equal intensity,

white color will result. The computer monitor is a good example of the additive

color model.

In the subtractive color model, color is represented by applying the

combinations of colored-lights reflected from the surface of an object (becausemost objects do not radiate by themselves). Take an apple under the natural light

for example. The surface of the apple absorbs green and blue part of the natural

light and reflects the red light to human eyes, so it becomes a red apple. By mixing

cyan (C) with magenta (M) and yellow (Y) pigments, we can produce a wide range

of colors. The more the pigment we add, the lower is the intensity of the light, and

thus the darker is the light. So, it is called the subtractive color model. C,Mand Y


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are the three primitive colors of pigment, which cannot be composed from other

colors. The color printer is a typical application of the subtractive model.

In computer systems, Application Interfaces (APIs) provided by most imageprocessing software as well as the Windows operating system are based on the

RGB model. This is mainly because they use monitors as the primary output

media. Monitors themselves generate color images by sending out RGB light into

humans retina. In true color systems, R, G, andB are each represented by 8 bits,

and therefore each single color ofR, G, and B can represent 0255 variations of

scale, resulting in 16.77 million possible colors. When using (R, G,B) to describe

a color pixel (0, 0, 0) represents full black and (255, 255, 255) represents full

white.

(a) Additive model (ACM) (b) Subtractive model (SCM)

Figure 3.1

The previous CVSSs can be categorized into the following three cases:

(Cases A, B and C). The principle means of color mixing for CVSSs has been

either to use SCM (Cases A and B) or a combination of SCM and ACM (Case

C).These three color mixing methods are briefly described as follows:

(Case A) The stacked sub pixel in a reconstructed image may be the color

of the sub pixel or a black color. Verheul-Tilborg's scheme [5], Yang-Laih's

scheme [17]and Cimato et al.'s scheme [14]used the color of the sub-pixel except

for a black color to directly show the color of a secret pixel. These schemes

covered the unexpected colors by the black color.


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(Case B) The remaining color in a reconstructed image for Koga-

Yamamoto's scheme [18] is not only the color of the sub pixel but also their

mixture. In these schemes, the authors believed that the unexpected mixed colorswill be ignored by human vision.

(Case C) This is similar to Case B using SCM except for that the simulated

color in the reconstructed image is the optical mixture of colors according to ACM

using the technique of juxtaposition color mixture. Hou's scheme [8], Ishihara-

Koga's scheme [19]and Shyu's scheme [12]belong to this type.

All three types require SCM mixing for stacking sub pixels in different

shadows. Cases A and B use only SCM, while Case C first obtains the color of the

sub pixel using SCM and then decodes the secret color by the juxtaposition color

mixture with the help of HVS.

In the proposed probabilistic color visual secret sharing scheme, we have

used the ACM directly. The secret color is encrypted by using ACM in a

probabilistic way. The operation for the ACM color mixing model is formally

defined as follows:

Definition 1: For the ACM color mixing model, each mixed color M is

produced by a light mixing functionL () shown as follows:

M=L (Rl, Gm, Bn), (3.1)

WhereRl, Gm, Bn areR, G andB primary colors with the luminance level l, m and

n belongs to[0, 255], respectively. There are totally (LRLGLB)colors that can

be produced, whereLR,LG andLB are luminance levels ofR, G andB.

3.2The Proposed Probabilistic VSS Scheme for Color ImagesWe have used the probabilistic strategy in our proposed scheme. A

probabilistic VCS uses different frequencies of whiteness in black and white areas

to distinguish the color. Finally, the secret image can be reconstructed while the

edge is blurred. The probabilistic VCS have no pixel expansion, i.e. the shadow

size is the same as the secret image.


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In our proposed VSS scheme for color images, we have extended the

Yangs [4] approach for black and white images and Y.C. Hous approach for

color images. Although Yang had extended their work with Chen and given theconstructions for color images, but they did not achieved the no pixel expansion

and they talked about the color levels mixing. Our VSS scheme is supporting the

three desirable properties which are summarized as follows:

1) Supporting images of arbitrary number of colors;2) No pixel expansion;3) Supporting k-out-of-n threshold setting.

Our proposed ProbVSS scheme consists of the following steps:

a) Construction of matrices.b) Creation of halftone images.c) Creating the shares.

a) Construction of matrices :For constructing the two set of matrices, we use Yangs probabilistic

method with no expansible shadow size. Instead of expanding the pixel into m sub

pixels, he only uses one pixel to represent one pixel.

Conventional VSS Probabilistic VSS

x0By0W White pixel xByW A black or white pixel

x1By1W Black pixel

Where,

x0+y0= x1+y1=m x+y=1, sox=0,y=1 ORx=1,y=0

i.e. m=1

So, in conventional VSS for different values ofxi, yi and m will cause

different contrast and shadow extension but in probabilistic VSS there will no

pixel expansion. As a replacement for using n m Boolean matrix, here we define

n 1 matrix, S= [si]

Where, si=1, if the pixel in the ith

shadow is black pixel,


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si=0, if the pixel in the ith

shadow is white pixel.

When shadows i1, i2 . . . ir are stacked, we can represent it by OR-ed

operation of rows i1, i2 . . . ir in S. The black or white level of this combined pixelL(V) is determined by the OR-ed operation of this r-tuple column vector V,

i.e. L (V) = si1 + si2 + ..+ sir,

Where + denotes OR-ed operation.

This method is to use the frequency of white pixels in the black and white

areas of the recovered image for interpreting black and white pixels by human

visual system. Define p0 (resp.p1) as the appearance probability of white pixel in

the white (resp. black) area of the recovered image. For the fixed threshold

probabilityp0pTH 1 and relative contrast > 0, ifp0pTHandp1pTH , the

frequency of white pixels in the white area of the recovered image will be higher

than that in the black area. So, the human visual system can distinguish with high

probability between black and white areas.

Next, we use Definition 2 to show the formal required conditions of this

probabilistic VSS scheme. Here the term probabilistic to point out that our

visual system distinguishes the contrast of the recovered image based on the

difference of the frequency of white color in black and white areas.

Definition 2: A (k, n) ProbVSS scheme can be shown as two sets, white set C0

and black set C1, consisting of n and n , n 1 matrices, respectively. When

sharing a white (resp. black) pixel, the dealer first randomly chooses one n 1

column matrix in C0

(resp. C1), and then randomly selects one row of this column

matrix to a relative shadow. The chosen matrix defines the color level of pixel in

every one of the n shadows. A ProbVSS Scheme is considered valid if the

following conditions are met:

1. For these n (resp. n) matrices in the set C0(resp. C1), the OR-ed valueof any k-tuple column vector VisL (V).These values of all matrices form a

set (resp.).


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2. The two sets and satisfy that p0pTH and p1pTH-, wherep0 andp1 arethe appearance probabilities of the 0 (white color) in the set and ,

respectively.3. For any subset {i1, i2,., iq} of {1, 2, , n} with q < k, thep0 andp1 are the

same.

The first two conditions are called contrast and the third is called security.

From the above definition, the matrices in C0 and C1 are n 1 matrices, so the

pixel expansion is one, howeverB0 andB1. In the conventional VSS scheme are n

m matrices, and thus the pixel expansion is m.

For the description of the construction, we first define the notation i,j to

represent the set of all n 1 column matrices with the Hamming weight i of every

column vector, andj denotes the matrices belonging to Cjwherej{0,1}.For example n=3,2, 0 are three 3 1 column matrices shown as:

2, 0 = ,011

100

110

.and2, 0 belongs to C0.

Example: For (2, 2) ProbVSS i.e. n=2,

C0 = {0, 0,2, 0}

=/00 11And

C1 = {1, 1}

= /01 10Thus, for any (k, n) VSS scheme first we will construct the collection of

matrices C0 and C1 according to the above discussion. In section 3.3 we will give

the proposed algorithm for any k-out-of-n VSS scheme and then in section 3.4 we

will illustrate the various constructions using the above approach in our proposed

ProbVSS scheme.


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In the next section, we describe how we transformed the image into three

halftone images and then we will describe how to create shares for any k-out-of-n

ProbVSS scheme. The approach for creating shares is based on the Hous methodbased on color images but we have modified his approach according to our need

for our proposed scheme.

b) Creation of halftone images:A. The halftone technologyAccording to their physical characteristics, different media uses different

ways to represent the color levels of images. The computer screen uses the electric

current to control the lightness of the pixels. The diversity of the lightness

generates different color levels. The general printer, such as dot matrix printers,

laser printers, and jet printers, can only control a single pixel to be printed (black

pixel) or not to be printed (white pixel), instead of displaying the gray level or the

color tone of an image directly. As such, the way to represent the gray level or

color level of images is to use the density of printed dots. For example, the printed

dots in the bright part of an image are sparse, and those in the dark part are dense

as in Figure 3.2 (a, b). The method that uses the density of the net dots to simulate

the gray level or color level is called Halftone and transforms an image with

gray level or color level into a binary image before processing.

Figure 3.2: (a) Continuous tone and (b) Halftone

Take the color level image in Figure 3.3 (a). For example, every pixel of the

transformed halftone image in Figure 3.3 (b) has only two possible color levels


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(black or white). Beca

when viewing a dot, te

levels through the denactually has only two co

Figure 3.3: (a) Ori

B. For our propIn our proposed

image i.e. Red, Green a

each component of th

halftone images corresp

There are variou

halftone image. In our

image into halftone im

following figure we sh

component images whic

se human eyes cannot identify too tiny

d to cover its nearby dots, we can simul

ity of printed dots, even though the trlorsblack and white.

ginal image (b) Halftone i

sed scheme

scheme, we first obtained the three c

d Blue components of the image, then

image into halftone image i.e. we o

nding to the each component of the ima

s methods for transforming the image i

roposed scheme we use two methods fo

age which are Floyd method and Jarvi

w the process of converting an image i

h are obtained by Floyd method:

(a) Original image

36

printed dots and,

te different color

ansformed image

mage

mponents of the

e transformed the

tained the three

e.

to corresponding

transforming the

s method. In the

to three halftone


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(b) Red component (c) Green component (d) Blue component

(e) Red halftone image (f) Green halftone image (g) Blue halftone image

Figure 3.4

c) Creating the sharesIn our proposed scheme for any k-out-of-n VSS scheme after obtaining the

three halftone images, we create the shares. For this, we create 3n temporary

sharing images and each of these sharing images will contain the corresponding

pixels according to the corresponding matrix of the corresponding component.

Then, we combined the each component of the temporary sharing images to create

the n shares. In this scheme any kshares reveal the secret image but any k- 1 share

can not reveal the secret image.

For example: For (2, 2) ProbVSS scheme, we create the 6 temporary sharing

images which areR1,R2, G1, G2,B1 andB2. After creating these sharing images we

combine theR1, G1 andB1 to create the first share and R2, G2 andB2 to create the

second share.


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In the following paragraph we describe how to assign color pixels to the

individual temporary shares and how to combine those temporary shares to form

sharing images.Let the three halftone images areRij, Gij andBij. For each component of the

image if the pixel value in the halftone image is 1 it means that the presence of

the primary color or if the pixel value in the halftone image is 0 it means that the

absence of the primary color. For any k-out-of-n ProbVSS scheme after creating

the two collection of matrices C0 and C1 which consist ofn 1 column matrices, if

the pixel value in the halftone image 0 then we randomly select any one of the

matrix from the collection of matrices C0 or if the pixel value in the halftone image

is 1 then we randomly select one of the matrices from the collection of matrices

C1. After choosing the matrix we assign the corresponding color pixel to the

corresponding share. If the pixel value in the chosen matrix is 1, then we assign

the corresponding color pixel value to the corresponding share or if the pixel value

in the chosen matrix is 0, then we assign the zero value to the corresponding

share. We repeat this process for all the pixels of the every component of the

image. After creating the temporary sharing images, we combined the temporary

sharing images as component of the sharing images as described above. In this

way we create the kshares and any kshares reveal the secret image but any k-1

shares can not reveal the secret image.

Since, each chosen matrices consist of the n pixels, we assign only one pixel to

the shares of the corresponding pixels, so there is no pixel expansion which is the

most desirable property in the visual secret sharing scheme. Moreover, we

randomly choose the matrices from the collection of matrices for each pixel of the

each component image, so no one can determine any information from individual

sharing images. Thus, our proposed (k, n) ProbVSS scheme is perfectly secure and

valid one.

In the next section, we will consider the formal algorithm of our proposed

(k, n) ProbVSS scheme, various constructions based on the proposed algorithm,


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security analysis of the constructions and we also prove that every construction of

our scheme is valid one.

3.3Algorithm-For our proposed (k,n) ProbVSS scheme1. Transform the color image P into three halftone images:R, G andB2. Construct the two collections of matrices C0 and C1.3. For each pixel Pij with color components(Rij, Gij and Bij) of the

composed image P, do the following:

A. Determine the pixel value of theRij If Rij = 0, we randomly choose one matrix from the

collection of matrices C0.

If Rij = 1, we randomly choose one matrix from thecollection of matrices C1.

B. Determine the pixel of the chosen matrix:If the pixel is 1 (Red component will be revealed),

Assign the corresponding color pixel to the corresponding Red

temporary sharing image.

If the pixel is 0 (Red component will be hidden),

Assign the corresponding color pixel to another corresponding

Red temporary sharing image.

C. Repeat step B n times according to the (k, n) ProbVSS scheme.4. Repeat the step 3 for each Green and Blue component i.e. until every pixel

of the composed image Pij is considered.

5. After obtaining the n temporary sharing images for the each component ofthe image (Red, Green, and Blue) i.e. 3n temporary sharing images, we

combine the components of the temporary sharing images to form the

sharing images as described in above section.

6.

After stacking the any kshares, the secret image can be decrypted by humaneyes.


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In the next section, we consider the various constructions of our proposed (k, n)

ProbVSS scheme and their proofs.

3.4Constructions of our proposed (k,n) ProbVSS schemeThe proposed (k, n) ProbVSSadopts the probabilistic additive color mixing

defined in section 3.1. For each component pixel is represented by one pixel only

i.e. each color pixel value of the Red component has appearance probabilities Pi

and zero color value with probability (1 Pi). The other two components (G-color

and B-color) have the appearance probabilities Pj and Pk.

When stacking any kor more shadows, the color appearance probabilities of(R, G, andB) will be P

iR, P

jGand P

kBin such a way that one can decode the color

via HVS. However, when stacking less than k shadows, the color appearance

probabilities are all the same.

We first give a definition of our scheme, the contrast and security

conditions for our (k, n) ProbVSS scheme, and then describe various constructions

of the scheme.

Definition 3: A (k,n) ProbVSS of color images can be represented as two

collection of matrices C0 and C1, including n 1 column Boolean matrices where

the elements 1 and 0 represent the primary color and zero color value in the

shadows. In this first we create 3n temporary sharing images and then we

combined it into n sharing images. If the value in the halftone image is 0, we

randomly choose one of the matrices from the set C0 or if the value in the halftone

image is 1, we randomly choose one of the matrices from the set C1. For sharinga secret colorRi, Gjor Bk, If the value in the chosen matrix is 1 then we put the

primary color value or if the value in the chosen matrix is 0 then we put the zero

color value to the corresponding mth

temporary sharing image according to the

element of the mth

row of the chosen column matrix. A ProbVSS scheme is

considered valid if the following conditions are met:


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Contrast condition: Apply ''OR'' operation for any k or more rows for these

column matrices in these sets. The appearance probabilities PiR, P

jGand P

kB for the

OR-ed results are proportional to the luminance levelsRi, GjandBk.Security condition: When stacking less than kshadows, the color appearance

probabilities PiR, P

jGand P

kBare all the same.

In this scheme, when we apply XOR instead of OR the image is revealed

better because the contrast is not sacrificed too much in this case.

3.4.1A proposed 2-out-of-n ProbVSS scheme for color images

Proposed (2, 2) ProbVSS: Let C0= {0, 0, 2, 0} and C1 = {1, 1} are the

two collection of matrices consisting of 2 1 matrices for a (2, 2) ProbVSS

scheme. In this we create 3n=3 2 i.e. 6 temporary sharing imagesR1,R2, G1, G2,

B1 andB2 using the above matrices and then we combined R1, G1 andB1 to form

the 1st

share and combined theR2, G2,B2 to form the 2nd

share.

Theorem 1: The proposed (2, 2) ProbVSS scheme is for non expansible shadow

size color images.

Proof: The two sets are:

C0 = {0, 0,2, 0} =/00 11And

C1 = {1, 1} =

/01

10

For each halftone image (Red, Green and Blue) of the secret image, when

the pixel value is 0 we randomly choose one matrix from C0 or when the pixel

value is 1 we randomly choose one matrix from C1.

For Red color the possible set is:

CR0= / 22(1-R, 0-Z) and CR1= /2 2(1-R, 0-Z)Using the same approach the possible sets for Green and Blue is:


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CG0= / 33(1-G, 0-Z) and CG1= /3 3(1-G, 0-Z),CB0=

/ 44(1-B, 0-Z) and CB1=

/4 4(1-B, 0-Z)

The chosen matrix is only one for each component from the above set and

the chosen matrix is selected randomly for each component. From the chosen

matrix, since each matrix is 2 1, so for each component we assign first row pixel

to the first temporary shadow and second row pixel to the second temporary

shadow of the corresponding components. Thus, non expansible shadow size for

(2, 2) ProbVSS scheme.

From the above sets we observe that when stacking two shadows R+R=R,

R+Z=R, G+G=G, G+Z=G, B+B=B, B+Z=B and Z+Z=Z. The color appearance

probabilities of primary colors are: P0R=P

0G=P

0B=0.5 and P

1R=P

1G=P

1B=1

respectively. So, the appearance probabilities of every primary color for each

shadow are all 0.5.

When using the Figure 3 5 (a) as the secret image, the experimental results

are shown in the following figures. Two shadow images are shown in the Figure

3.5 (b, c) and the reconstructed image using the OR operation is shown in Figure

3.5 (d) and using the XOR operation is shown in the Figure 3 5 (e).

(a) Original image


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(b) Share 1 (c) Share 2

(d) 1+2 using OR (e) 1+2 using XOR

Figure 3.5

Proposed (2, 3) ProbVSS: Let C0 = {0,0,3,0} and C1 = {3,1,2,1} are the

two collection of matrices consisting of 3 1 matrices for a (2, 3) ProbVSS

scheme. In this we create 3n=3 3 i.e. 9 temporary sharing images R1,R2,R3, G1,

G2, G3 and B1, B2 and B3 using the above matrices and then we combined R1, G1

andB1 to form the 1st

share, combined theR2, G2,B2 to form the 2nd

share and the

R3, G

3,B

3to form the 3

rdshare.


size color images.


C0 = {0,0,3,0} = ,000 111.

And


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C1 = {3,1,2,1} = ,00

1

01

0

10

0

11

0

10

1

01

1

.For each halftone image (Red, Green and Blue) of the secret image, when




CR0= ,

5

22

2

.(1-R, 0-Z) and CR1= ,

2

2

2

22

2Z

R

2

2

.(1-R, 0-Z)Using the same approach the possible sets for Green and Blue is

CG0= ,5 333.(1-G, 0-Z) and CG1= ,

3 3

3 33

3ZG 33.(1-G, 0-Z)

CB0= ,5 444.(1-B, 0-Z) and CB1= ,

4 4

4 44

4ZB 44.(1-B, 0-Z)


the chosen matrix is selected randomly for every component. From the chosen


to the first temporary shadow, second row pixel to the second temporary shadow

and third row pixel to the third temporary shadow of the corresponding

components. Thus, non expansible shadow size for (2, 3) ProbVSS scheme.

Using the above approach, we can conclude that the appearance

probabilities of every primary color for each shadow are all 0.33. When using the

same Figure 3 5 (a) as the secret image, the experimental results are shown in the

following figures. Three shadow images are shown in Figure 3 6 (a, b, c) and the

reconstructed images using the OR operation is shown in Figure 3.6 (d, e, f) and

using the XOR operation is shown in Figure 3 6 (g, h, i).


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(a) Share1 (b) Share2 (c) Share3

(d) 1+2 using OR (e) 2+3 using OR (f) 1+3 using OR

(g) 1+2 using XOR (h) 2+3 using XOR (i) 1+3 using XOR

Figure 3.6

Proposed (2, 4) ProbVSS: Let C0 = {0,0,4,0} and C1 = {2,1} are the two

collection of matrices consisting of 4 1 matrices for a (2, 4) ProbVSS scheme. In

this we create 3n=3 4 i.e. 12 temporary sharing imagesR1,R2,R3,R4,G1, G2, G3,

G4,B1,B2,B3 andB4 using the above matrices and then we combinedR1, G1 andB1

to form the 1

st

share, combined theR2, G2,B2 to form the 2

nd


rdshare and combined theR4, G4,B4 to form the 4

thshare.


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size color images.


C0 = {0,0,4,0} = ;0000< ;1111


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probabilities of every primary color for each shadow are all 0.33. When using the

Figure 3 5 (a) as the secret image, the experimental results are shown in thefollowing figures. Four shadow images are shown in the Figure 3 7 (a, b, c, d) and

the reconstructed images using the OR operation is shown in the Figure 3 7 (e, f,

g, h, i, j) and using the XOR operation is shown in the Figure 3 7 (k, l, m, n, o,

p).


(d) Share4 (e) 1+2 using OR (f) 2+3 using OR

(g) 3+4 using OR (h) 1+3 using OR (i) 1+4 using OR


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(j) 2+4 using OR (k) 1+2 using XOR (l) 2+3 using XOR

(m) 3+4 using XOR (n) 1+3 using XOR

(o) 1+4 using XOR (p) 2+4 using XOR

Figure 3.7

Proposed (2, n) ProbVSS: Let C0 and C1 are the two collection of matrices

consisting of n 1 matrices for a (2, n) ProbVSS scheme. In this we create 3n

temporary sharing images R1, R2,., Rn, G1, G2,.,Gn, B1, B2,., Bn using the

above matrices and then we combined R1, G1 and B1 to form the 1st

share,

combined theR2, G2,B2 to form the 2

nd

share and combined theRn, Gn,Bn to formthe n

thshare. C0 and C1 are defined as follows:


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C0 = {0,0,n,0} and

C1 = {n/2,1, for even n}or

C1 = {a,1,a+1,1 , where a is least integer function ofn/2, for odd n}

We follow the same procedure to construct the proposed (k, k) ProbVSS as

illustrated above.

3.4.2 A proposed 3-out-of-n ProbVSS scheme for color imagesProposed (3, 3) ProbVSS: Let C0 = {0,0,2,0} and C1 = {1,1,3,1} are the

two collection of matrices 3 1 matrices for a (3, 3) ProbVSS scheme. In this we

create 3n=3 3 i.e. 9 temporary sharing images R1,R2,R3, G1, G2, G3 andB1,B2

andB3 using the above matrices and then we combined R1, G1 andB1 to form the

1st

share, combined the R2, G2,B2 to form the 2nd

share and combined the R3, G3,

B3 to form the 3rd

share.


size color images.


C0 = {0,0,2,0} = ,000 110 101 011.and

C1 = {1,1,3,1} = ,001 010

100 111.


the pixel value is 0, we randomly choose one matrix from C0 or when the pixel



CR0=

,

22

22

22.

(1-R, 0-Z) and CR1=

,2

2

2

222.

(1-R, 0-Z)


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Using the same approach the possible sets for Green and Blue is:

CG0=

,

33

33

33.

(1-G, 0-Z) and CG1=

,3

3

3

333.

(1-G, 0-Z)

CB0= , 44

44 44.(1-B, 0-Z) and CB1= ,

4 4

4 444.(1-B, 0-Z)


the chosen matrix is selected randomly for every component. From the chosen


to the first temporary shadow, second row pixel to the second temporary shadowand third row pixel to the third temporary shadow of the corresponding

components. Thus, non expansible shadow size for (3, 3) ProbVSS scheme.


probabilities of every primary color for each shadow are all 0.25 .When using the

same Figure 3.5 (a) as the secret image, the experimental results are shown in the

following figures. Three shadow images are shown in Figure 3.8 (a, b, c) and the

reconstructed images using the OR operation is shown in Figure 3.8 (j) and

using the XOR operation is shown in the Figure 3.8 (k). It is also shown in the

Figure 3.8 (d, e, f) and in the Figure 3.8 (g, h, i) that stacking k-1 i.e. 2 shadows

can not reveal any secure information.



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(d) 1+2 using OR (e) 2+3 using OR (f) 1+3 using OR

(g) 1+2 using XOR (h) 2+3 using XOR (i) 1+3 using XOR

(j) 1+2+3 using OR (k) 1+2+3 using XOR

Figure 3.8

Proposed (3, n) ProbVSS: we can construct any proposed 3-out-of-n

ProbVSS scheme in the same way as the above illustrated approach.

3.4.3 A proposedk-out-of-k ProbVSS scheme for color imagesProposed (4, 4) ProbVSS: Let C0 = {0,0, 2,0,4,0} and C1 = {1,1, 3,1}

are the two collection of matrices consisting of 4 1 matrices for a (4, 4) ProbVSS


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scheme. In this we create 3n=3 4 i.e. 12 temporary sharing imagesR1,R2,R3,R4,

G1, G2, G3,G4,B1,B2, B3 andB4 using the above matrices and then we combined

R1, G1 andB1 to form the 1

st


nd

share,combined theR3, G3,B3 to form the 3

rdshare and combined theR4, G4,B4 to form

the 4th

share.


size color images.


C0 = {0,0,2,0,4,0} = ;0000< ;0011< ;1100< ;1010< ;0101< ;1001< ;0110< ;1111


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CG1 = ;

3< ;

3< ;3

< ;3

< ;33

3< ;33

3< ;3

33< ;3

33


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(d) Share4 (e) 1+2 (f) 2+3

(g) 3+4 (h) 1+3 (i) 1+4

(j) 2+4 (k) 1+2+3 (l) 2+3+4

(m) 1+2+4 (n) 1+3+4 (o) 1+2+3+4using XOR

Figure 3.9


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Proposed (k,k) ProbVSS: Let C0 and C1 are the two collection of matrices

consisting of k 1 matrices for a (k, k) ProbVSS scheme. In this we create 3k

temporary sharing images R1, R2,., Rk, G1, G2,.,Gk, B1, B2,., Bk using theabove matrices and then we combined R1, G1 and B1 to form the 1

stshare,

combined theR2, G2,B2 to form the 2nd

share and combined theRk, Gk,Bk to form

the kth

share. C0 and C1 are defined as follows:

C0 = { i,0, where i is even and 0ik}

C1 = { i,1, whre i is odd and 0ik}.

We follow the same procedure to construct the proposed (k, k) ProbVSS as

illustrated above.

3.4.4 A proposedk-out-of-n ProbVSS scheme for color imagesProposed (3, 4) ProbVSS: Let C0 and C1 are the two collection of matrices

consisting of 4 1 matrices for a (3, 4) ProbVSS scheme. In this we create 3n=3

4 i.e. 12 temporary sharing imagesR1,R2,R3,R4,G1, G2, G3,G4,B1,B2,B3 andB4

using the above matrices and then we combinedR1, G1 andB1 to form the 1st

share,

combined theR2, G2,B2 to form the 2nd

share, combined theR3, G3,B3 to form the

3rd

share and combined theR4, G4,B4 to form the 4th

share.


size color images.


C0 = ;0000< ;0000< ;1110< ;1101< ;1011< ;0111


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value is 1 we randomly choose one matrix from C1.For Red color the possible set is:

CR0 = ;< ;< ;

222< ;222< ;

222< ;222


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probabilities of every primary color for each shadow are all 0.167 .When using the

Figure 3.5 (a) as the secret image, the experimental results are shown in thefollowing figures. Four shadow images are shown in the Figure 3.10 (a, b, c, d)

and the reconstructed images using the OR operation is shown in Figure 3.10 (k,

l, m, n) and using the XOR operation is shown in Figure 3.10 (o, p, q, r).


(d) Share4 (e) 1+2 (f) 2+3

(g) 3+4 (h) 1+3 (i) 1+4


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(j) 2+4 (k) 1+2+3 using OR (l) 2+3+4 using OR

(m) 1+2+4 using OR (n) 1+3+4 using OR (o) 1+2+3 using XOR

(p) 2+3+4 using XOR (q) 1+2+4 using XOR (r) 1+3+4 using XOR

Figure 3.10

3.5SummaryIn this chapter, we have propsed a k-ot-of-n ProbVSS scheme for color

images. Our scheme supports any type of color images. We elaborated the three

steps, namely, construction of matrices, creatinge the halftone images and creating

the shares. We have given the algorithm of our propsed scheme and various


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constructions of our proposed scheme. We have also defined the valid proposed k-

out-of-n ProbVSS scheme and proved that our scheme is valid one. Our scheme

supported the three desirable properties which are:1. Supporting images of arbitrary number of colors,2. No pixel expansion,3. Supporting k-out-of-n threshold setting.In the next chapter, we will compare our scheme with other novel schemes

and proved that our scheme is fully secure and valid one.


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Chapter 4

COMPARISION


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4.ComparisonVarious parameters are recommended by researchers to evaluate the

performance of visual cryptography schemes. In this chapter, we have considered

some of those aspects to compare our VCS with other VCS schemes namely Naor

Shamir [2], Verheul-Tilborg [5], Chang-Tsai [6], Y.C. Hou [8], Yang [4], Chan

[20], Hou and Tu [13], Shyu [12], Chen et al. [7] and Yang-chen [9] VCS

schemes. We have compared our scheme with these schemes in mainly four

aspects which are the most desirable properties of any secret sharing scheme.

The four aspects of comparison are also the objectives of any visual

cryptography scheme, which are as follows-

1. Dealing with colored images.2. No Pixel expansion.3. Supporting general (k, n) threshold setting.4. Supporting arbitrary no of color levels.The summary of t

project complete without scaling

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