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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
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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].
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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
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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].
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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,
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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.
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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.
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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
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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:
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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).
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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.
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Table 2.2 The pixel pattern for 2-out-of-2 VCS with 4-subpixel layout
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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
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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
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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.
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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.
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(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
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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
Table 2.4 The details of the pixels in SI2 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 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
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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:
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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
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(a) (b)
(c) (d)
(e) (f)
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(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
Table 2.5 The details of the pixels in SI1 for the 2-out-of-3 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
Share2 + Share 3 250 100 19195 5805 25000
Table 2.6 The details of the pixels in SI2 for the 2-out-of-3 VCS
Image No. of Columns
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
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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
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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:
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)
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(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)
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(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
Table 2.7 The details of the pixels in SI1 for the 3-out-of-3 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+
Share3 250 100 20673 4327 25000
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Table 2.8 The details of the pixels in SI2 for the 3-out-of-3 VCS
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 +
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 :
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Department of Information Technology, Kannur University 44
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
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Department of Information Technology, Kannur University 45
The basis matrices S0 and S1 are met the three conditions in the
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)
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Department of Information Technology, Kannur University 46
(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
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Department of Information Technology, Kannur University 47
(a) (b)
(c) (d)
(e) (f)
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(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
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Table 2.9 The details of the pixels in SI1 for the 3-out-of-6 VCS
Image No. of Columns
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
Table 2.10 The details of the pixels in SI2 for the 3-out-of-6 VCS
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.