3-Out-of-n Cheating Prevention Visual Cryptographic Schemes
Ching-Nung Yang
1
, Stelvio Cimato
2
, Jihi-Han Wu
1
and Song-Ruei Cai
1
1
Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, Taiwan
2
Dipartimento di Informatica, Università degli studi di Milano, Crema, Italy
Keywords: Visual Cryptography, Cheating, Deterministic Cheating, Cheating Prevention.
Abstract: In literature, (2, n) cheating prevention visual cryptographic schemes (CPVCSs) have been proposed,
dealing with the case of dishonest participants, called cheaters, who can collude together to force honest
participants to reconstruct a wrong secret. While (2, n)-CPVCSs resistant to deterministic cheating have
been presented, the problem of defining (k, n)-CPVCS for any k has not been solved. In this paper, we
discuss (3, n)-CPVCS, and propose three (3, n)-CPVCSs with different cheating prevention capabilities. To
show the effectiveness of the presented (3, n)-CPVCS, some experimental results are discussed as well.
1 INTRODUCTION
The cryptographic technique for the visual sharing
of secret images, denoted Visual Cryptography (VC)
or Visual Secret Sharing (VSS) was firstly proposed
in (Naor and Shamir, 1994). A VC scheme (VCS) is
usually implemented as a threshold (k, n) scheme,
where a secret image is decomposed into n shadow
images (called “shadows”) which are then
distributed to the n participants. Any set of k
participants is enabled to reconstruct the secret
image by simply stacking together the shadows they
own, while (k1) or fewer participants cannot obtain
any secret information. In a (k, n)-VCS, each pixel
of the secret image is “expanded” into m subpixels
in each shadow, where the value m is called the pixel
expansion. Following Naor and Shamir’s work, most
studies dealt with the pixel expansion of VCS
(Cimato et al., 2006; Ito et al., 1999; Kuwakado and
Tanaka, 2007; Wang et al., 2011; Yan et al., 2015;
Yang, 2004).
A (k, n)-VCS usually consider honest
participants, who can provide correct shadows
during the reconstruction phase. However, cheating
behaviour occurs in VCS, when some dishonest
participants, called cheaters, collude together to
forge shadows and force honest participants to
reconstruct a wrong secret. Several methods (Horng
et al., 2006; Hu and Tzeng, 2007; De Prisco and De
Santis, 2009; Hu and Tzeng, 2007; Liu et al, 2011;
Tsai et al., 2007) have been proposed to face the
cheating problem. In (Horng et al., 2006), the
problem of (n1)-colluder cheating has been defined
and a (2, n) cheating prevention VCS (CPVCS) has
been proposed by using (2, n+l)-VCS instead of (2,
n)-VCS. Horng’s (2, n)-CPVCS makes (n1)
collusive cheaters harder to predict the structure of
the honest participant’s shadow, and is immune to
deterministic cheating. However, Horng’s (2, n)-
CPVCS only prevents deterministic cheating for the
black secret pixel. De Prisco and De Santis in (De
Prisco and De Santis, 2009) extend Horng et al.’s
work and propose a new (2, n)-CPVCS, which does
not allow deterministic cheating for both black and
white colors. A (k, n)-CPVCS for any k still remains
unsolved.
In this paper, we study the (n1)-colluder
cheating problem in (k, n)-CPVCS for k=3. Our (3,
n)-CPVC
S can prevent deterministic cheating for
both black color and white color. The rest of the
paper is organized as follows. Section 2 introduces
(k,n)-VCS and reviews the previous (2, n)-CPVCSs.
In Section 3, we propose three (3, n)-CPVCSs with
different cheating prevention capabilities. Examples
in Section 4 are given to demonstrate the
effectiveness of our scheme. Conclusions are drawn
in Section 5.
400
Yang, C-N., Cimato, S., Wu, J-H. and Cai, S-R.
3-Out-of-n Cheating Prevention Visual Cryptographic Schemes.
DOI: 10.5220/0005740504000406
In Proceedings of the 2nd International Conference on Information Systems Security and Privacy (ICISSP 2016), pages 400-406
ISBN: 978-989-758-167-0
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
2 RELATED WORKS
2.1 (K, n)-VCS
In a black-and-white VCS, each pixel is subdivided
into m subpixels in each of n shadows. A (k, n)-VCS
uses h
1
black subpixels and (m-h
1
) white subpixels
(denoted as h
1
b(m-h
1
)w) to represent black and
white secret pixels, respectively, where 0≤h
0
<h
1
≤m.
The values of h
1
and h
0
are the blackness of black
color and white color. Let X be a set of involved
participants, and w(v) be the Hamming weight of v.
Suppose that M is an nxm matrix. The notation
(M|X) defines a |X|xm matrix, which selects the rows
of the corresponding participants in X from M. With
add(M|X) we denote the OR-ed vector of all rows in
(M|X), and with D(M|X) we denote a set including
all distribution matrices obtained by permuting all
the columns in (M|X). The formal definition of a
VCS is then given as follows:
Definition 1. A (k, n)-VCS is given by (nxm)
black and white base matrices
1
B
and
0
B
satisfying
the following two conditions.
(i) (Contrast condition): Given any qualified
set X, where |X|=k, we have that
11
(|)wB X h
(respectively,
00
(|)
wB X h
), where
0h
0
<h
1
m.
(ii) (Security condition): Given any forbidden
set X, where |X|<k, we have
1
(|)
D
BX
and
0
(|)
D
BX
.
The collection C
1
(respectively, C
0
) is obtained
by permuting the columns of the corresponding
matrix B
1
(respectively, B
0
) in all possible ways.
When sharing a black (respectively, white) secret
pixel, the dealer randomly selects one matrix from
C
1
(respectively, C
0
) and chooses each row of the
matrix to a relative shadow.
To easily describe base matrices of VCS, some
notations are introduced. The notation
,ni
represents a matrix composed of all n-bit columns
with Hamming weight i. Obviously,
,ni
is a matrix,
e.g.,
3,0
=
0
0
0
,
3,1
=
100
010
001
,
3,2
=
011
101
110
,and
3,3
=
1
1
1
. Let
,
ni
l
denote the concatenation of l
matrices, i.e.,
,, ,
(||||)
l
ni ni ni
l , where || is the
concatenation operation. Also, let
,
,
||
,
||…||
,
, e.g.,
,
,
|
|
,
||
,
.
2.2 (2, n)-CPVCS
In (Horng et al., 2006), authors introduced the
problem of (n
1)-colluder cheating in (2, n)-VCS,
where (n
1) cheaters collude together to force the
honest participant to reconstruct a wrong secret.
Consider as an example, the case of (2, 3)-VCS: If
two cheaters (say participants P
1
and P
2
) collude
together, they have S
1
and S
2
and can exactly know
the structure of S
3
. Then, they can provide fake
shadows
1
ˆ
S
and
2
ˆ
S
, and let P
3
obtain a wrong secret
image
ˆ
S
(i.e.,
13
ˆˆ
SS S
or
23
ˆˆ
SSS
). Horng et
al. proposed a (2, n)-CPVCS to make it harder for
(n
1) cheaters to predict the structure of the other
shadow. They adopted (2, n+l)-VCS, where l≥1,
instead of (2, n)-VCS. Afterwards, the dealer
randomly takes only n out of (n+l) shadows and
delivers them to n participants. Horng et al.’s
approach consists in adding l all-0 columns into the
base matrices of (2, n)-VCS (see Construction 5.5 in
(De Prisco and De Santis, 2009)). By the notation,
the base matrices of Horng et al.’s (2, n)-CPVCS are
0, ,0
|| ( 1)
nn n
Bnl
and
1,1,0
||
nn
Bl
.
,
||2
,
100
100
100
(1-1)
,
100
010
001
(1-2)
For n=3, base matrices of Naor and Shamir’s (2, 3)-
VCS and Horng et al.’s (2, 3)-CPVCS are shown in
Eq. (1) and Eq. (2), respectively.
,
||3
,
1000
1000
1000
(2-1)
,
||3
,
1000
0100
0010
(2-2)
Suppose that blocks B and W are black and
white m-subpixel blocks in shadows. Let
12
CC
P
be
the probability that cheaters can change the C
1
color
block to C
2
color block, where C
1
and C
2
{B, W}.
Obviously, from Eq. (1-1), cheaters exactly know
the structure of B and W, so that they can apply
deterministic cheating on Naor and Shamir’s (2, 3)-
3-Out-of-n Cheating Prevention Visual Cryptographic Schemes
401
(a)
13
SSS
(b)
13
ˆˆ
SSS
(c)
13
ˆˆ
SSS
(d)
13
ˆˆ
SSS
Figure 1: The 2-colluder cheating: (a) no cheating on (2, 3)-VCS (b) cheating on (2, 3)-VCS (c) cheating on (2, 3)-CPVCS
(d) cheating on (2, 3)-CPVCS by a left-right arrow faked image.
VCS (i.e.,
1
WB WB BW BW
PPPP
). For the
white secret pixel, from the matrix
0
B
in Eq. (1-2),
two cheaters P
1
and P
2
can exactly know the
structure of W in S
3
. Thus, they can change the white
block W to be any color block, i.e.,
1
WB
P
and
1
WW
P
. However, for the black pixel, from the
matrix
1
B
in Eq. (1-2), cheaters can only identify
the location of “1” of B in the shadow S
3
with 50%
probability. Therefore, cheaters only have
0.5
BW
P
to modify 2B2W (denotes 2 black
subpixels and 2 white subpixels in a block) to
1B3W. In general, for (2, n)-CPVCS, (n
1) cheaters
have
1/( 1)
BW
Pl
.
It is evident that cheaters have the probability
1
BB
P
since they just do not change their
subpixels in shadow. The following example shows
that 2-colluder cheating can perform deterministic
cheating on (2, 3)-VCS, but is not completely
effective for 100% in Horng et al.’s (2, 3)-CPVCS.
Example 2 Apply 2-colluder cheating on Naor
and Shamir’s (2, 3)-VCS and Horng et al.’s (2, 3)-
CPVCS.
Consider the case of 2-colluder cheating, where
the two cheaters P
1
and P
2
want to fool the honest
participant P
3
to get a wrong secret. Suppose that the
secret image is a left arrow
. Fig. 2(a) shows the
stacked result of (2, 3)-VCS (say S
1
+S
3
) without
cheating, where a left arrow
is correctly
recovered. Suppose that two cheaters produce a
forged shadow
1
ˆ
S
, and intentionally tamper the
secret image from a left arrow
to a right arrow
. Cheating results
13
ˆˆ
SSS
(a right arrow) on
(2, 3)-VCS and (2, 3)-CPVCS are shown in Figs.
2(b) and (c). As shown in Fig. 2(b), cheaters can
perform deterministic cheating successfully.
However, in Fig. 2(c), although cheaters can fake a
right arrowhead they cannot remove the left
arrowhead completely (note: cheaters can only
correctly identify the location of “1” of B in the
shadow S
3
with 50% probability). If we adopt a left-
right arrow
as the faked secret image, where
the black areas include all black areas of the left
arrow, then the cheating is still successful for (2, 3)-
CPVCS (see Fig. 2(d)).
In (De Prisco and De Santis, 2009), authors
proposed a new (2, n)-CPVCS with base matrices
0, , ,0
|| ||
nall nn n
Bn
and
1, ,1,0
|| ||
nall n n
B
,which can prevent deterministic cheating for both
black and white colors. However, this scheme has a
large pixel expansion m=(2
n
+n+1). For n=3, De
Prisco and De Santis’s (2, 3)-CPVCS has the base
matrices with m=12, reported in Eqs. (3).
,
,
,
010011011000
001010111000
000101111000
(3-1)
,
||
,
||
,
010011011000
001010110100
000101110010
(3-2)
The cheating prevention approach proposed by De
Prisco and De Santis’s (2, 3)-CPVCS is briefly
described below. Two cheaters P
1
and P
2
have
different ways of performing a cheating attack. In
the following we show the most effective cheating
attack with the maximum
WB
P
and
B
W
P
. By
modifying
1
1
1
and
0
0
0
in B
0
to
0
1
1
and
1
0
0
,
respectively, the white block 7B5W is changed to
the black block 8B4W in
13
ˆ
SS
. As shown in Eq.
(3), from S
1
and S
2
, cheaters have the probability 2/3
(see Eq. (4-1)) and 4/5 (see Eq. (4-2)) to correctly
select the column
1
1
1
and
0
0
0
, respectively, where
ICISSP 2016 - 2nd International Conference on Information Systems Security and Privacy
402
boldface denotes wrong choices. Finally, the
probability
WB
P
is 2/34/5=8/15.
0
0
0
1
0
0
0
1
0
0
0
1
1
0
1
0
1
1
1
1
1
1
1
1
000
000
000
(4-1)
0
0
0
10
01
00
11011
10111
01111
0
0
0
0
0
0
0
0
0
(4-2)
By the same argument, the most effective
cheating way of changing B to W is modifying
1
0
0
and
0
0
1
in B
1
to
0
0
0
and
1
0
1
, respectively, so that
the black. Block 8B4W is changed to the white
block 7B5W in
13
ˆ
SS
.
As shown in Eq. (5), from S
1
and S
2
, cheaters have
the probability 2/3 (see Eq. (5-1)) and 2/4 (see Eq.
(5-2)) to correctly select the column
1
0
0
and
0
0
1
,
respectively, where boldface denotes the wrong
choices. Finally, the probability
BW
P
is
2/32/4=1/3.
0
0
0
1
0
0
001
101
010
01
11
11
1
0
0
000
100
010
(5-1)
10
01
00
0
0
1
110110
101101
011100
0
0
1
(5-2)
Readers can found a detailed analysis of the
successful probabilities to change the color in (De
Prisco and De Santis, 2010). However, the examples
above are two cheating ways having maximum
WB
P
and
BW
P
.
Although Horng’s (2, 3)-CPVCS does not have
the cheating prevention capability for white color,
Horng’s (2, 3)-CPVCS with m=12 (adding 9 all-0
columns) has
1/9
BW
P
, that is much lesser than
1/3
BW
P
of De Prisco and De Santis’s (2, 3)-
CPVCS.
3 THE PROPOSED (3, n)-CPVCS
Horng’s (2, n)-CPVCS adopts a very simple
approach by adding l all-0 columns, but its cheating
prevention capability is only effective for the black
secret pixel. And, this will be a problem if the secret
image allows meaningful forging by only changing
W to B. De Prisco and De Santis’s (2, 3)-CPVCS
solves the weakness of Horng’s (2, n)-CPVCS by
using a larger pixel expansion, so that cheaters
cannot figure out the shadow of honest participant.
In this paper, we use a well-known Naor and
Shamir’s (3, n) to construct the proposed (3, n)-
CPVCS. Our scheme has the same cheating
prevention capability like De Prisco and De Santis’s
(2, n)-CPVCS that is effective for both black and
white colors (i.e.,
1
BW
P
and
1
WB
P
), and we
only adopt the simple approach (adding all-0
columns and all-1 like Horng’s (2, n)-CPVCS. Naor
and Shamir’s (3, n)-VCS has base matrices
0,1 ,0
|| ( 2)
nn n
Bn
and
10 ,1 ,
|| ( 2)
nnn
BB n
For n=4, Naor and Shamir’s (3, 4)-VCS has the base
matrices reported as Eq. (6).
,
||2
,
011100
101100
110100
111000
(6-1)
,
||2
,
100011
010011
001011
000111
(6-2)
Construction 1. By adding l all-0 columns,
where l≥2, into Naor and Shamir’s (3, n)-VCS with
base matrices B
0
and B
1
, we have a (3, n)-CPVCS
with white and black base matrices
00,0
(|| )
n
BBl
and
11 ,0
(|| )
n
BBl
Theorem 1. Under the (n1)-colluder cheating,
the proposed (3, n)-CPVCS from Construction 1 has
the probabilities
2/( 1)
BW
Pl
, and
1
BB W B WW
PP P
Proof. Suppose that (n
1)
cheaters (say participants P
1
, P
2
, …, and P
n
1
)
collude together to force the honest participant (P
n
)
to reconstruct a wrong secret. From Construction 1,
the base matrices of (3, n)-CPVCS are given in Eq.
(7).The black and white blocks B and W in the
reconstructed image are (n+1)B(n+l −3)W and
nB(n+l−2)W, respectively. For the white secret pixel
(see
0
B
in Eq. (6)), (n−1) cheaters exactly know the
locations of “1” and “0” in the other shadow. Thus,
cheaters can change the white block W to any color
block, i.e.,
1
WB WW
PP
. For the black secret
pixel, we obviously have probability
1
BB
P
since
cheaters just do not change their subpixels in
shadows.
3-Out-of-n Cheating Prevention Visual Cryptographic Schemes
403
2
0 0 ,0 , 1 ,0 ,0 , 1 ,0
01 11 0 0
10 11 0 0
|| ( || ( 2) ) || || ( 2)
11 01 0 0
11 10 0 0
n
nl
nnn n nnn n
BBl n l nl
(7-1)
2
11 ,0 ,1 , ,0
10 00 1 1 0 0
01 00 1 1 0 0
|| ( || ( 2) ) || .
00 10 1 0 0 0
00 01 1 1 0 0
n
nl
nn nnn
BBl n l
(7-2)
Cheaters do not exactly know all the locations of “1”
and “0”, but they do know (n2) locations of “1”
and (n1) locations of “0” (see
1
B
in Eq. (7)). By
this observation, cheaters can produce two forged
shadows (say
1
ˆ
S
and
2
ˆ
S
), from the matrix in Eq.
(8), where the underlined “0” and “1” denote the
known locations to cheaters (the locations of (n-2)
“1” and (n-1) “0”). Every one row and every two
rows of the matrix in Eq. (7) are indistinguishable in
the sense that they contain the same matrices with
the same frequency.
2
11
1
2
ˆ
0011101100
ˆ
0011011100.
0011111000
n
nl
n
S
S
S
(8)
Thus, the honest participant does not know that the
shadows
1
ˆ
S
and
2
ˆ
S
are fake. In the last (l+1)
columns, cheaters do not know the exact location of
the single “1” in S
n
, and they can only achieve the
probabilistic cheating. In the last (l+1) columns,
there are
1
2
l
C
possible cheating combinations, and
1
l
C
of them have nB(n+l−2)W. Cheaters can
successfully modify the black block
(n+1)B(n−3+l)W to the white block nB(n−2+l)W
with the probability
B
W
P
=
1
12
/2/(1)
ll
CC l
.
This probability is
B
W
P
=1 for l=1. So,
Construction 1 should have the value l2.
Construction 2. By adding l all-1 columns,
where l≥2, into Naor and Shamir’s (3, n)-VCS with
base matrices B
0
and B
1
, we have a (3, n)-CPVCS
with white and black base matrices
00,
(|| )
nn
BBl
and
11 ,
(|| )
nn
BBl
.
Theorem 2. Under the (n1)-colluder cheating,
the proposed (k, n)-CPVCS from Construction 2 has
the probabilities
2/( 1)
WB
Pl
, and
1
WW BB BW
PPP
.
Proof. Naor and Shamir’s (3, n)-VCS has base
matrix
10
BB
. By the same argument in the proof
of Theorem 1, we can prove that cheaters can
modify the white block (n+l)B(n−2)W to the black
block (n+l+1)B(n−3)W with the probability
2/( 1)
WB
Pl
, and
1
WW BB BW
PPP
.
Construction 3. By adding l
1
all-0 columns and l
2
all-1 columns, where l
1
≥2 and l
1
≥2, into Naor and
Shamir’s (3, n)-VCS with base matrices B
0
and B
1
,
we have a (3, n)-CPVCS with white and black base
2
1
2
1
1
1
2
ˆ
0 01 110110 0
ˆ
0011011100
0011111000
nl
l
n
n
S
S
S
(9-1)
2
1
2
1
1
1
2
ˆ
1 10 001001 1
ˆ
1 10 010001 1 .
1 10 000011 1
nl
l
n
n
S
S
S
(9-2)
matrices
001,02,
(|| || )
nnn
BBl l
and
111,02,
(|| || )
nnn
BBl l
matrices.
Theorem 3. Under the (n1)-colluder cheating,
the proposed (3, n)-CPVCS from Construction 3 has
the probabilities
1
2/( 1)
BW
Pl
,
2
2/( 1)
WB
Pl
,
and
1
BB WW
PP
.
Proof. From the matrix in Eq. (9-1), by the same
argument in the proof of Theorem 1, cheaters can
modify the black block (n+l
2
+1)B(n−3+l
1
)W to the
white block (n+l
2
)B(n−2+l
1
)W with
1
2/( 1)
BW
Pl
, and
1
BB
P
. Also, from the
matrix in Eq. (9-2), we can prove that cheaters can
modify the white block (n+l
2
)B(n−2+l
1
)W to the
ICISSP 2016 - 2nd International Conference on Information Systems Security and Privacy
404
black block (n+l
2
+1)B(n−3+l
1
)W with
2
2/( 1)
WB
Pl
, and
1
WW
P
.
4 EXAMPLES
Construction 1 has the cheating prevention
capability against the modifications of the black
color to the white color, while Construction 2 has
the cheating prevention capability against the
modifications of the white color to the black color.
Construction 3 has both prevention capabilities of
Construction 1 and Construction 2. Two examples
are given to test the effectiveness of Constructions 1,
2, and 3. Both examples adopt a left arrow
as
the secret image. To test different cheating
prevention capabilities, we use three fake secret
images, a right arrow
, a left-right arrow
,
and a rectangle without arrow
. In the right
arrow
, the black (respectively, white) areas do
not contain all black (respectively, white) areas in
the secret image (the left arrow). The black areas in
the left-right arrow
contain all black areas in
the secret image, and the white areas in the rectangle
without arrow
contain all white areas in the
secret image.
Example 2. Construct the (3, 4)-CPVCSs
Construction 1 and Construction 2, respectively.
To achieve the invariant aspect ratio, we add 3
(2) all-0 columns into Naor and Shamir’s (3, 4)-
CPVCS to form the (3, 4)-CPVCS with m=9 by
Construction 1.
Base matrices are
04,3 4,0
|| 5B
and
14,1 4,4 4,0
|| 2 || 3B
. The three colluders (say
participants P
1
, P
2
, and P
3
) get their shared pixels
from the first three rows in base matrices. Cheaters
can produce two forged shadows
1
ˆ
S
and
2
ˆ
S
to
maliciously modify the black block (5B4W) to the
white block (4B5W) by using the matrix
1
2
4
ˆ
000101100
ˆ
000011100
000111000
S
S
S
, where the underlined
positions denote the known locations to cheaters. In
the last four columns, cheaters do not know the
exact location of the single “1” in S
4
, and thus they
can only achieve the probabilistic cheating. There
are six possible cheating combinations
1100
1100
1000
,
0011
0011
1000
,
1001
1001
1000
,
0110
0110
1000
,
1010
1010
1000
, and
0101
0101
1000
.
Cheaters can change the black color (5B3W) to the
white color (4B4W) with 3/6=1/2 probability. Fig. 2
shows the cheating results by applying 3-colluder
cheating on this scheme. The stacked results
124
ˆˆ ˆ
SS S S
using the right arrow
and the
left-right arrow
as secret image, respectively,
are shown in Figs. 2(a) and (b). Because
Construction 1 only provides the cheating
prevention capability against the modification from
B to W, it has similar result to Horng et al.’s (2, n)-
CPVCS. As shown in Fig. 2(a), we have a lighter
residual of the left arrowhead. . However, Fig. 2(b)
shows that cheaters can completely fool the honest
participant to get the wrong secret, that is a left-right
arrow
(since the black areas in
contain all
black areas in
). Analyses of (3, 4)-CPVCS with
04,3 4,0 4,4
|| 2 || 3B
and
14,1 4,4
|| 5B
from
Construction 2, are the same as the above. Figs. 2 (c)
and (d) show the cheating results for using the right
(a) (b) (c) (d)
Figure 2: The 3-colluder cheating on (3, 4)-CPVCS by using the secret image: (a, b) Construction 1 (c, d) Construction 2.
3-Out-of-n Cheating Prevention Visual Cryptographic Schemes
405
(a) (b)
Figure 3: The 3-colluder cheating on the (3, 4)-CPVCS by using the secret image: (a) Construction 3 using the left-right
arrow (b) Construction 3 using the rectangle without arrow.
arrow and the rectangle without arrow as secret
images, respectively. As shown in Fig. 2(c), some
parts of the right arrowhead still remain. Fig. 2(d)
shows that cheaters can fool the honest participant to
get the wrong secret
, where the white areas
contain all the white areas in
.
Example 3. Construct a (3, 4)-CPVCS by
Construction 3.
To achieve the minimum pixel expansion, we
add 2 (l
1
=2) all-0 columns and 2 (l
2
=2) all-1 columns
into Naor and Shamir’s (3, 4)-CPVCS to form the
(3, 4)-CPVCS with m=10. Base matrices are
04,3 4,0 4,4
|| 4 || 2B
and
14,1 4,4 4,0
|| 4 || 2B
.
Cheaters can fake two forged shadows
1
ˆ
S
and
2
ˆ
S
to
maliciously modify the black block (7B3W) to the
white block (6B4W) with the probability
BW
P
=2/3
by using the matrix
1
2
4
ˆ
ˆ
S
S
S
0001110110
0001101110
0001111100
. Also,
this scheme can produce two forged shadows
1
ˆ
S
and
2
ˆ
S
to modify the white color (6B4W) to the black
color (7B3W) with probability
WB
P
=2/3 by using
the matrix
1
2
4
ˆ
1110001001
ˆ
1110010001
1110000011
S
S
S
. We use
and
as secret images, so that the cheating attack in
Construction 1 and Construction 2 is successful. Fig.
3 shows that Construction 3 can detect both
cheatings.
ACKNOWLEDGEMENT
This work was supported in part by Ministry of
Science and Technology, Taiwan, under Grant 104-
2918-I-259-001 and 104-2221-E-259-013.
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