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Chapter 2

REVIEW OF VISUAL CRYPTOGRAPHY SCHEMES

2.1 Introduction

This chapter provides an overview of the developments and the

research works done by other researchers in visual cryptography. The

different perspectives on visual cryptography, such as access structure,

generation of shares and other aspects reported by different authors are

discussed in this chapter. The general construction of various visual

cryptography schemes such as 2-out-of-2, 2-out-of-n, n-out-of-n, k-out-of-

n and the consistent image size visual cryptography scheme are also

demonstrated with examples.

2.2 Developments in Visual Cryptography

In 1995, Naor and Shamir introduced a very interesting and simple

cryptographic method called visual cryptography to protect secrets [11].

Basically, visual cryptography has two important features. The first

feature is its perfect secrecy and the second is its decryption method

which requires neither complex decryption algorithms nor the aid of

computers. It uses only human visual system to identify the secret from

Chapter-2

Department of Information Technology, Kannur University 18

the stacked image of some authorized set of shares. Therefore, visual

cryptography is a very convenient way to protect secrets when computers

or other decryption devices are not available. The simple decryption

method is the reason that attracts many researchers to make further

detailed enquiries in this research area. Nowadays, many related methods

concerning the theory and the applications of visual cryptography are

proposed.

An extended visual cryptography scheme (EVCS) was proposed

by Ateniese et al. [12]. Extended visual cryptography schemes permits the

construction of visual secret sharing schemes within which the shares are

meaningful as opposed to having random noise on the shares. After the

sets of shares are superimposed, this meaningful information disappears

and the secret is recovered. This is the basis for the extended form of

visual cryptography [13-14].

The image size invariant visual cryptography was proposed by Ito

et al. [15]. The traditional visual cryptography schemes employ pixel

expansion. In pixel expansion, each share is m times the size of the secret

image. Thus, it can lead to the difficulty in carrying these shares and

consumption of more storage space. Ito’s scheme removes the need for

this pixel expansion. That is, the reconstructed image is identical to the

original image. There are also some other studies which focus on the

methods without pixel expansion [16-22].

Review of Visual Cryptography Schemes


In order to provide perfect secrecy and the maximum clarity of the

recovered secret images, most researchers use the concept of pixel

expansion, which was first introduced by Naor and Shamir to design their

visual cryptography schemes. That is, each pixel of the binary secret

image is encoded into m subpixels on each share, where m is called the

parameter of pixel expansion of the scheme. By analyzing any block of m

subpixels of the forbidden set of shares, one cannot distinguish which

color was used in the secret pixel. But when the shares of the qualified set

are stacked up, the block of any m subpixels corresponding to a black

secret pixel will provide more black subpixels than a block of subpixel

corresponding to a white secret pixel. Thus, the blocks corresponding to

black secret pixels will have more blackness and those blocks

corresponding to white secret pixels will have less blackness. This

property makes it possible for someone to distinguish black and white

blocks, and hence the secret image can reveal the stacked image by losing

some contrast of the original secret image.

In 1996, Naor and Shamir proposed an alternative VCS model for

improving the contrast in [23]. In 1999, Blundo et al. [24-26] analyzed

the contrast of the reconstructed image in k-out-of-n VCS schemes .

Blundo et al. gave a complete characterization of 2-out-of-n VCS schemes

having optimal contrast and minimum pixel expansion in terms of certain

balanced incomplete block designs. Blundo et al.’s research results are

Chapter-2


valuable for the researchers who are interested in the area of visual

cryptography. The other research works done by different authors are

found in [27-31].

Viet and Kurosawa [32] proposed a VCS with reversing, in which

the participants are also allowed to reverse their transparencies. But in this

scheme there is a loss of resolution, since the number of pixels in the

reconstructed image is greater than that in the original secret image. Other

studies related to VCS with reversing are found in [33-35].

The concept of recursive hiding of secrets in visual cryptography

was proposed by Gnanaguruparan and Kak [36]. This provides a method

of hiding secrets recursively in the shares of threshold schemes, which

permits an efficient utilization of data. In recursive hiding of secrets,

several additional messages can be hidden in one of the shares of the

original secret image [37-39]. By using recursive threshold visual

cryptography in network application, network load can be reduced.

Recently, visual cryptography schemes were also proposed to deal

with gray-level images [40]. The use of halftoning techniques make it

possible that the readymade schemes designed for binary secret images

can be directly applied to gray-level images [41]. The different gray-level

visual cryptography schemes are studied by researchers in [42-47].



Applying visual cryptography techniques to colour images is a

very important area of research because it allows the use of natural colour

images. Color images are also highly popular and have a wider range of

uses when compared to other image types.

In 1997, Verheul and Van Tilborg [48] proposed a colored visual

cryptography scheme. In 2000, Yang and Laih [49] proposed a different

construction mechanism for the colored visual cryptography scheme.

They argued that their method can be easily implemented and can get

much better block length than Verheul and Van Tilborg’s scheme.

A major common disadvantage of the above reviewed colored

VCS schemes is that the number of colors and the number of subpixels

determine the resolution of the revealed secret image. If many colors are

used, the subpixels require a large matrix to represent it. Also, the contrast

of the revealed secret image will go down drastically. Consequently, how

to correctly stack these shared transparencies and recognize the revealed

secret image are the major issues. The different color visual cryptography

schemes are studied in [50-63]. Almost all color visual cryptography

schemes proposed required few computations.

Recently, more and more applications of visual cryptography, such

as authentication, human identification [64], copyright protection,

Chapter-2


watermarking, mobile ticket validation, visual signature checking etc. are

introduced [65-74].

The print and scan application of VCS can be found in [75]. In this

application, scan the shares into a computer system and then digitally

superimpose their corresponding shares. This would make possible secure

verification of e-tickets or other documents.

The developments and the research works done by other

researchers in the different perspectives on visual cryptography, such as

access structure, generation of shares and other aspects reported by

different authors are discussed in [76-97].

2.3 Visual Cryptography Schemes

The visual cryptography schemes (VCS) describe the way in

which an image is encrypted and decrypted. There are different types of

visual cryptography schemes [98-100]. For example, there is the k-out-of-

n scheme that says n shares will be produced to encrypt an image, and k

shares must be stacked to decrypt the image. If the number of shares

stacked is less than k, the original image is not revealed. The other

schemes are 2-out-of-n and n-out-of-n VCS. The most of the constructions

of visual cryptography schemes are realized using two n × m matrices, S0

and S1, called basis matrices. The general method for the construction of

basis matrices is discussed below.



2.3.1 Basis Matrices

Any black-and-white visual cryptography scheme can be described

using two n x m Boolean matrices S0 and S1, called basis matrices, to

describe the subpixels in the shares [11]. The basis matrix S0 is used if the

pixel in the original image is white, and the basis matrix S1 is used if the

pixel in the original image is black. The use of the basis matrices S0 and

S1 can have small memory requirements (it keeps only the basis matrices

S0 and S1), and it is efficient (to choose a matrix in C0 or C1) because it

only generates a permutation of the columns of S0 or S1. Basically, the

two basis matrices S0 and S1 should satisfy the following definition.

Definition 2.1: A k-out-of-n visual cryptography scheme with parameters

1 ≤ d≤ m and α > 0 can be constructed from two n x m Boolean matrices

S0 and S1 if the following three conditions are met:

1. The OR m-vector V of any k of the n rows in S0 satisfies ωH(V) ≤ d

– α .m.

2. The OR m-vector V of any k of the n rows in S1 satisfies ωH (V) ≥ d.

3. For any set {r1, r2, . . . , rt}⊂ {1,2,. . .,n} with t < k, the t x m

matrices obtained by restricting S0 and S1 to rows r1, r2,…, rt, are

equal up to a column permutation.

Chapter-2


Where ωH (V) is the hamming weight (the number of ones) of the

m-vector V of any k of the n rows, m is the pixel expansion and α is the

relative difference.

The conditions (1) and (2) related to contrast in a reconstructed

image and condition (3) related to security.

Formula 2.1(Relative Difference):

Let ωH (S0) and ωH (S1) be the hamming weight corresponding to

the basis matrices S0 and S1. Then relative difference (α) is defined as:

α = (ωH (S1) – ωH (S0 ))/ m

Formula 2.2(Contrast):

Let α be the relative difference and m be the pixel expansion. The

formula to compute contrast in different VCS is:

β = α.m, β ≥ 1

The basic idea of visual cryptography can be best described by

considering a 2-out-of-2 VCS.

2.3.2 The Construction of 2-out-of-2 VCS

Let us consider a binary secret image S containing exactly m pixels.

The dealer creates two shares (binary images), S1 and S2, consisting of

exactly two pixels for each pixel in the secret image as shown in Table 2.1. If

the pixel in S is white, the dealer randomly chooses one row from the first



two rows of Table 2.1. Similarly, if the pixel in S is black, the dealer

randomly chooses one row from the last two rows of Table 2.1.

Table 2.1 Pixel pattern for 2-out-of-2 VCS with 2-subpixel

layout Original Pixel Pixel Value Share1 Share2 Share1+ Share2 0

0

1

1

To analyse the security of the 2-out-of-2 VCS, the dealer randomly

chooses one of the two pixel patterns (black or white) from the Table 2.1

for the shares S1 and S2. The pixel selection is random so that the shares

S1 and S2 consist of equal number of black and white pixels. Therefore,

by inspecting a single share, one cannot identify the secret pixel as black

or white. This method provides perfect security. The two participants can

recover the secret pixel by superimposing the two shared subpixels. If the

superimposition results in two black subpixels, the original pixel was

black; if the superimposition creates one black and one white subpixel, it

indicates that the original pixel was white.

In visual cryptography, the white pixel is represent by 0 and the

black pixel by 1. For the 2-out-of-2 VCS, the basis matrices, S0 and S1 are

designed as follows:

Chapter-2


S0= ⎥⎦

⎤⎢⎣

⎡0101

S1= ⎥⎦

⎤⎢⎣

⎡1001

The relative difference α and contrast β, for the above basis matrices

can be computed as:

α = ½

β = 1

There are two collections of matrices, C0 for encoding white pixels

and C1 for encoding black pixels. Let C0 and C1 be the following two

collections of matrices:

C0 = { π (S0) } C1 = { π (S1) }

Where π(S0) and π(S1) represents the collection of all matrices

obtained by permuting the columns of matrices S0 and S1 respectively.

That is,

C0 = { ⎥⎦

⎤⎢⎣

⎡0101

, ⎥⎦

⎤⎢⎣

⎡1010

} and C1 = { ⎥⎦

⎤⎢⎣

⎡1001

, ⎥⎦

⎤⎢⎣

⎡0110

}

To share a white pixel, the dealer randomly selects one of the

matrices in C0, and to share a black pixel, the dealer randomly selects one

of the matrices in C1. The first row of the chosen matrix is used for share

1(S1) and the second row for share 2 (S2).



The Figure 2.1 depicts example for 2-out-of-2 VCS with two

subpixels layouts.

(a)

(b)

( c)

(d)

Figure 2.1 The 2-out-of-2 VCS with 2-subpixel layout (a) SI,(b) S1, (c) S2, and (d) S1 + S2

The problem that arises with this scheme is that for every pixel

encoded from the original image into two subpixels and placed on each share

in a horizontal or vertical fashion (here horizontal), the shares have a size of s

x 2s if the secret image is of size s x s. Hence there is distortion.

2.3.3 The Construction of 2-out-of-2 VCS with Square Pixel Expansion

To avoid the horizontal or vertical distortion of the reconstructed

image, can be use 4-subpixel layout. In the 4-subpixel layout, the shares

have a size of 2s x 2s if the secret image is of the size s x s. Hence there is

no horizontal or vertical distortion in the reconstructed image. The only

change is that the image is m times larger than the original [101].

In order to explain the 2-out-of-2 VCS with 4-subpixel layout each

pixel is expanded into 2×2 subpixels as shown in Table 2.2.

Chapter-2


Table 2.2 The pixel pattern for 2-out-of-2 VCS with 4-subpixel layout



For the 2-out-of-2 VCS with 4-subpixels, the basis matrices S0 and S1 are

designed by using Table 2.2 as follows:

S0= ⎥⎦

⎤⎢⎣

⎡01010101

S1= ⎥⎦

⎤⎢⎣

⎡01101001

The relative difference α and contrast β for the above basis

matrices are computed as:

α = ½

β = 2

Let C0 and C1 be

C0 = { π

⎥⎦

⎤⎢⎣

⎡01010101

}

C1= { π ⎥⎦

⎤⎢⎣

⎡01101001

}

To analyze the security of the 2-out-of-2 with 4-subpixel layout

VCS, the dealer randomly chooses one of the pixel patterns (black or

white) from the Table 2.2 for the shares S1 and S2. The pixel selection is

random and the shares S1 and S2 consist of equal number of black and

white pixels. Therefore, by inspecting a single share, one cannot identify

whether the secret pixel is black or white. Thus, this method provides

perfect security. The two participants can recover the secret pixel by

Chapter-2


superimposing the two shared subpixels. If the superimposition results in

four black subpixels, the original pixel was black; if the superimposition

creates two black and two white subpixels, it indicates that the original

pixel was white.

The Figure 2.2 depicts 2-out-of-2 VCS with 4-subpixel layout

(a)

(b)

(c )

(d )

Figure 2.2 The 2-out-of-2 VCS with 4-subpixel layout (a) SI,

(b) S1, (c) S2, and (d) S1+S2

2.3.4 The Construction of VCS with Consistent Image Size

It is also possible to do VCS without enlargement of the

reconstructed image. In this method, the generation of shares with

consistent share size using random basis column pixel expansion



technique [15-22]. In this technique, the shares have a size of s x s if the

secret image is of the size s x s. That is, the reconstructed image is

identical to the original image.

In order to demonstrate the random basis column pixel expansion

technique, use 2-out-of-2 VCS with 4-subpixel layout. To implement this

method, the above two basis matrices S0 and S1 are used. The matrix S0 is

for encoding white pixels and S1 is for encoding black pixels. The

matrices S0 and S1 as:

S0 = ⎥⎦

⎤⎢⎣

⎡10101010

, S1 = ⎥⎦

⎤⎢⎣

⎡11000011

Generally, one of the columns is randomly chosen from S0 for

encoding white pixels and one of the columns from S1 is taken for

encoding black pixels. That is, the chosen column vector V = [vi]:

V = ⎥⎦

⎤⎢⎣

⎡21

vv

The ith element determines the color of the pixel in the ith share

image; that is, the element v1 is interpreted as a pixel in the first share

image, and element v2 is interpreted as a pixel in the second share image

and so on.

Chapter-2


For example, in 2-out-of-2 VCS, to share a black pixel, one of the

columns in S1 is chosen at random. Suppose the chosen column vector

from S1 is:

V = ⎥⎦

⎤⎢⎣

⎡10

In the column vector the 1st element determines the color of the

pixel in the 1st shared image; that is, the element 0 represents the white

pixel in the first share image, and element 1 represents the black pixel in

the second share image. Similarly, to share a white pixel, one of the

columns in S0 is chosen at random.

To implement this method in 2-out-of-2 VCS with 4-subpixel

layout, the above two basis matrices S0 and S1 are used. The matrices S0

and S1 are divided into different column matrices as:

S0 = { ⎥⎦

⎤⎢⎣

⎡00

, ⎥⎦

⎤⎢⎣

⎡11

, ⎥⎦

⎤⎢⎣

⎡00

, ⎥⎦

⎤⎢⎣

⎡11

}

S1 = { ⎥⎦

⎤⎢⎣

⎡01

, ⎥⎦

⎤⎢⎣

⎡01

, ⎥⎦

⎤⎢⎣

⎡10

, ⎥⎦

⎤⎢⎣

⎡10

}

The Figures 2.3 and 2.4 represents 2-out-of-2 VCS on two

different secret images using random basis column pixel expansion

technique.



(a) (b)

(c) (d)

Figure 2.3 The 2-out-of-2 VCS with consistent share size using the image SI1 (a) SI1, (b)S1, (c)S2 and (d) S1+S2

(a) (b)

(c) (d)

Figure 2.4 The 2-out-of-2 VCS with consistent share size using the image SI2 (a) SI2, (b) S1, (c) S2 and (d) S1+S2

Chapter-2


Table 2.3 The details of the pixels in SI1 for the 2-out-of-2 VCS

Image No. of Columns

No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 250 100 7510 17490 25000

Share1 + Share 2 250 100 16297 8703 25000


Image No. of

Columns No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 200 200 16665 23335 40000

Share1 + Share 2 200 200 28391 11609 40000

In this thesis, we are using random basis column pixel expansion

technique for all examples and experiments.

2.4 The Construction of 2-out-of-n VCS

In the 2-out-of-n scheme, n shares will be produced to encrypt an

image, and any two shares must be stacked to decrypt the image

[99][102]. The construction of a 2-out-of-n visual cryptography scheme

can be done with the following definition.

Definition 2.2: Let S0 and S1 be two n x m Boolean matrices for 2-out-of-

n visual cryptography scheme. Then S0 and S1 are said to be basis

matrices, if

1. S0 should have n – 1 columns of weight n and remaining columns

of weight zero.

2. S1 is such that



1ij

0 if i = jS

1 if i j⎧

=⎨ ≠⎩

For 2-out-of-n visual cryptography scheme m = n.

2.4.1 The Basis Matrices for 2-out-of-n VCS

The 2-out-of-n visual cryptography can be best described by

considering a 2-out-of-3 -VCS case. The basis matrix S0 designed as

S0 = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

011011011

Similarly the basis matrix S1 as:

S1 = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

011101110

The relative difference α and contrast β for the above basis

matrices are computed as:

α = 1/3

β = 1

The basis matrices S0 and S1 are met the three conditions in the

definition 2.1. There are two collections of matrices, C0 for encoding

white pixels and C1 for encoding black pixels. The matrices C0 and C1 can

be designed as:

Chapter-2


C0 = { π ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

011011011

}

C1 = { π

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

011101110

}

Figures 2.5 and 2.6 shows a 2-out-of-3 VCS obtained by using

the collections C0 and C1 defined above.

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 2.5 The 2-out-of-3 VCS using the image SI1 (a) SI1 (b) S1, (c) S2, (d) S3, (e) S1+S2, (e) S1+S3, and (f) S2+S3



(a) (b)

(c) (d)

(e) (f)

Chapter-2


(g)

Figure 2.6 The 2-out-of-3 VCS using the image SI2 (a) SI2(b) S1,

(c) S2, (d) S3, (e) S1+S2, (e) S1+S3, and (f) S2+S3


Image No. of

Columns No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 250 100 7510 17490 25000

Share2 + Share 3 250 100 19195 5805 25000



No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 200 200 16665 23335 40000

Share2 + Share 3 200 200 32239 7761 40000

2.5 The Construction of n-out-of-n VCS

In the n-out-of-n visual cryptography scheme, n shares will be produced

to encrypt an image, and n shares must be stacked to decrypt the image

[99][102]. If the number of shares stacked is less than n, the original image is not

revealed. Let G be the ground set given by



G = { g1, g2, …….gn } of n elements.

Let E = { π1, π 2, ………………. π 2n-1 } be the collection of all subsets

with even cardinality and let O = { σ1, σ2,………. σ2n-1 } be the collection of

all subsets with odd cardinality. Each of them is defined by a matrix which is

denoted by S0 and S1 with order n x 2n-1. For 1 ≤ i ≤ n and 1 ≤ j ≤ 2n-1, all the

matrices are obtained by permuting the columns of S0 = [ sij0] and S1 = [sij1] in

which

sij0 = 1⇔ gi π j ,

sij1 = 1⇔ gi σj

2.5.1 The Basis Matrices for n-out-of-n VCS

The n-out-of-n visual cryptography can be best described by

considering a 3-out-of-3 -VCS case. The basis matrix S0 is designed as :

S0=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

110001101010

Similarly the basis matrix S1 is :

S1= ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

110010101001

For the above basis matrices, the relative difference α and contrast

β are computed as:

α = 1/4

β = 1

Chapter-2



definition 2.1.There are two collections of matrices, C0 for encoding

white pixels and C1 for encoding black pixels. The matrices C0 and C1 can

be designed as:

C0 = { π

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

110001101010

}

C1 = { π

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

110010101001

}

Figure 2.7 and 2.8 shows a 3-out-of-3 VCS applied to two different

images:

(a) (b)

(c) (d)



(e) (f)

(g) (h)

Figure 2.7 The 3-out-of-3 VCS using the image SI1 (a) SI1 (b) S1, (c)

S2, (d) S3, (e) S1+S2, (f) S1+S3, (g) S2+S3, and (h) S1+S2+S3

(a) (b)

(c) (d)

Chapter-2


(e) (f)

(g) (h)

Figure 2.8 The 3-out-of-3 VCS using the image SI2 (a)SI2 (b) S1, (c)

S2, (d) S3, (e) S1+S2, (f) S1+S3, (g) S2+S3, and (h)

S1+S2+S3



No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 250 100 7510 17490 25000

Share1+ Share 2+

Share3 250 100 20673 4327 25000





No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 200 200 16665 23335 40000

Share1 + Share 2 +

Share3 200 200 34175 5825 40000

2.6 The Construction of k-out-of-n VCS

The general k-out-of-n visual cryptography schemes were introduced by Naor and Shamir in [11]. However, their constructions were quite inefficient, since a large number of subpixels were needed to encode a given pixel. A much more efficient construction mechanism was introduced by Ateniese et al.[13-14]. This construction significantly reduces the number of subpixels needed for encoding. Consider a starting n × l matrix SM (n, l, k) whose entries are elements of a ground set {a1, a2, …, ak }, with the property that, for any subset of k rows, there exists at least one column such that the entries in the k given rows of that column are all distinct, where l is the whiteness of a black pixel. If there exists a SM(n, l, k), then there exists a k-out-of- n VCS with pixel expansion m = l*2k-1. The n×m basis matrices S0 and S1 are constructed by respectively replacing the symbols a1, a2, …, ak with the 1st, …, kth rows of the corresponding basis matrices of the k-out-of-k VCS.

2.6.1 The Basis Matrices for k-out-of-n VCS

The k-out-of-n visual cryptography can be illustrated by a 3-out-of-

6-VCS case. The starting matrix SM designed as :

Chapter-2


SM=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

123213132312231321

aaaaaaaaaaaaaaaaaa

Substituting the rows of the basis matrix S0 of 3-out-of-3 VCS for

a1, a2, a3 in SM to obtain the basis matrix S0 in 3-out-of-6 VCS. That is:

S0 =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

010100110110001101010110010101100011011001010011001101100101011000110101

Similarly the basis matrix S1 is obtained by replacing a1, a2, a3 of

SM by the rows of the basis matrix S1 of 3-out-of-3 VCS.

S1 =

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

100110101100101010011100100111001010110010011010101011001001110010101001

For the above basis matrices, the relative difference α and contrast β are computed as:

α = 1/12

β = 1




definition 2.1. The matrices C0 and C1 can be designed as:

C0 = { π

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

010100110110001101010110010101100011011001010011001101100101011000110101

}

C1 = {π

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

100110101100101010011100100111001010110010011010101011001001110010101001

}

The Figures 2.9 and 2.10 depict 3-out-of-6 VCS using above

matrices C0 and C1 applied to images SI1 and SI2:

(a) (b)

(c) (d)

Chapter-2


(e) (f)

(g) (h)

(i) (j)

(k)

Figure 2.9 The 3-out-of-6 VCS using the image SI1 (a) SI1 (b) S1,(c) S2, (d) S3, (e) S4, (f) S5, (g) S6, (h) S3+S4, (i) S1+S2+S3, (j) S1+S2+S4, (k) S2+S3+S4



(a) (b)

(c) (d)

(e) (f)

Chapter-2


(g) (h)

(i) (j)

(k)

Figure 2.10 The 3-out-of-6 VCS using the image SI2 (a) SI2 (b) S1,(c) S2, (d) S3, (e) S4, (f) S5, (g)S6,(h) S3 +S4, (i) S1+S2+S3, (j) S1+S2+S4, (k) S2+S3+S4





No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 100 250 7510 17490 25000

Share1 + Share2 +

Share3 100 250 17559 7441 25000


Image No. of

Columns No. of Rows

No. of Black Pixels

No. of White Pixels

Total Pixels

Secret image 200 200 16665 23335 40000

Share1 + Share2 +

Share3 200 200 29030 10970 40000

2.7 Conclusion

The developments and proposals by different authors in visual

cryptography schemes are reviewed here. The different perspectives on

visual cryptography such as types of access structures, types of shares,

and color models of secret images etc. are discussed in this chapter. The

construction of basis matrices for 2-out-of-n, n-out-of-n and k-out-of-n

VCS is demonstrated with examples. The consistent image size visual

cryptography scheme has also been explained with example.

REVIEW OF VISUAL CRYPTOGRAPHY SCHEMESshodhganga.inflibnet.ac.in/bitstream/10603/6102/12/12_chapter...

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