An Introduction to Visual Cryptography

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Nov 21, 2013 (3 years and 7 months ago)

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5

An Introduction to Visual Cryptography


Soo
-
Chang Pei

(
貝蘇章
)

,
Io
-
Kuong Tam

(
譚耀光
), and
Ru
-
Fang Torng

(
童如芳
)


Department of Electrical Engineering, National Taiwan University,

Taipei, Taiwan, R. O. C.

Email: pei@cc.ee.ntu.edu.tw, Fax: 886
-
2
-
23671909



Abstrac
t

In this paper we give a tutorial introduction to visual secret sharing scheme (VSSS).
VSSS is a secret sharing scheme that uses human visual system to decrypt secret image without
performing any cryptographic computation. A
k
-
out
-
of
-
n

VSSS,
)
,
(
n
k

VSSS, is to encrypt a
secret image into
n

shadow images called shares. To decrypt the secret image, we simply
xerox
k

shares onto transparencies, and then stacking them. We can “visually” decrypt the secret
image, but a
ny
)
1
(

k

shares gain no information about it.


I.

Introduction

In this section, we review the idea of traditional secret sharing scheme that was invented by
Shamir [1] and Blakle
y

[2] independently. Here is an example to illustrate the idea
. Assume
that a bank has a vault that must be opened by a secret key. The bank employs three senior
tellers, but
the bank

do
es

not want to trust any of them individually. Hence, they would like to
design a system such that any two of the three senior te
llers can open the vault together. This
problem can be viewed as a
)
3
,
2
(

secret sharing scheme.

In general, a
)
,
(
n
k

secret sharing scheme is a method to share a secret
K

among
n

participants such that the following conditions hold:



A
ny
k

participants together can compute
K
.



A
ny
t

participants,
k
t

, gain no information about
K
.

Here is an example of a
)
2
,
2
(

secret sharing scheme. Assume that the secret
K

is a
binary sequence of length
m
, i.e.
)
,
,
,
(
2
1
m
k
k
k
K


. The two shares,
1
s

and
2
s

can be
constructed as follow. The first share is chosen to be a random binary sequence of length m, say
)
,
,
,
(
1
12
11
1
m
s
s
s
s


. Then, we can compute the second share by doing “exclusive
-
or” on
K

and
1
s
.

i
i
i
s
k
s
1
2



,
m
i
,
,
1




(1)

For example, assume that
2

m
,
)
1
,
0
(

k
. Then the two shares can be
constructed
as
follow:

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6

)
0
,
0
(
1

s

, then
)
1
,
0
(
1
2



K
s
s
.

)
1
,
0
(
1

s

,then
)
0
,
0
(
1
2



K
s
s
.

)
0
,
1
(
1

s

, then
)
1
,
1
(
1
2



K
s
s
.

)
1
,
1
(
1

s

, then
)
0
,
1
(
1
2



K
s
s
.

However, looking only at one share, say
1
s
, any fou
r values of
K

are possible. In other
words, it gains no information about
K

if another share
2
s

is unknown.


II.

Visual Secret Sharing Scheme

Naor and Shamir [3] proposed a visual secret sharing

scheme (VSSS) that uses human
visual system to decrypt the secret image without performing any cryptographic computation.
The difference between a VSSS and a traditional secret sharing scheme is in how the secret is
decrypt
ed
. Usually, the traditional s
ecret sharing scheme requires computation over a finite field.
In a VSSS, however, the computation is
simply
performed by the human visual system of the
users.

It is important to realize that the construction of a secure VSSS is difficult. Suppose that a

particular pixel
P

on a share
i
s

is black. Whenever a set of shares (including
i
s
) is stacked
together, the result must be black. It means that in the secret image, the pixel
P

must be black.
In other words, we gain
“some”

information about the secret image be examining one of the
shares, and the security condition does not allow this. Naor and Shamir [3] proposed a VSSS
that solved this problem by splitting each orig
inal pixel into
m

subpixels. In this section, we will
introduce this idea and explain how to decrypt
“visually”
.

In general, a VSSS assumes that the secret is a collection of black and white pixels, or a
binary image, and each pixel is encrypt
ed

separatel
y. Each original pixel encrypts into
n

shares
,
and e
ach share is a collection of
m

black and white subpixels, which are printed near to each other
such that human visual system averages their individual black/white contribution. The VSSS
can be described

by an
m
n


Boolean matrix
M

where
1
]
,
[

j
i
M

iff the
j
-
th
subpixel in
the
i
-
th

shares is black
, and
0
]
,
[

j
i
M

iff the
j
-
th
subpixel in the
i
-
th

shares is
white
.

To decrypt the secret ima
ge, we simply xerox
t

shares onto transparencies, and then
stacking them together with perfect alignment. We can see a stacked version share
V

whose
black subpixels are represented by the Boolean
“or”

of row
t
s
s
s
,
,
,
2
1


in
M
.

t
s
s
s
V





2
1


(2)

The gray level of this stacked share
V

is proportional to the Hamming weight
)
(
V
H

of
V
. This gray level is interpreted by the visual system of the users as black if
d
V
H

)
(

and
as white if
m
d
V
H



)
(

for some fixed threshold
m
d


1

and relative difference
0


.

Here is a
)
2
,
2
(

example to illustrate the idea. A
)
2
,
2
(

VSSS can be described by the
following
2
2


Boolean matrices.

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0
1
0
1
0
M
,







1
0
0
1
1
M
.

In this example, a particul
ar pixel
P

in the secret image is split into two subpixels, i.e.
2

m
, in each of the two shares. If the given pixel
P

is white, we use
0
M

to encrypt the
pixel by setting
the first row to
1
s

and setting the second row to
2
s
,
)
0
,
1
(
1

s
,
)
0
,
1
(
2

s
.
The Hamming

weight of the stacked version share
V

is
1
)
(

V
H
, where
)
0
,
1
(
2
1



s
s
V
. If the given pixel
P

is black, we use
1
M

to encrypt the pixel, and the
Hamming weight is
2
)
(

V
H
, where
)
1
,
1
(
2
1



s
s
V
. In this example, the fixed
threshold
1

d
, and the relative difference
5
.
0


. By stacking
1
s

and
2
s

together, a
pixel
P

is interpreted by the visual system of the users as white if the Hamming weigh
t
1
)
(

V
H

and as black if
2
)
(

V
H
.

By permuting the columns of
0
M

and
1
M
, we obtain two collections of
2
2


Boolean
matrices.







































0
1
1
0
,
1
0
0
1
,
1
0
1
0
,
0
1
0
1
1
0
C
C


To share a w
hite pixel, we randomly choose one of the matrices in
0
C
, and to share a



pixel
M






0
1
0
1






1
0
1
0






1
0
0
1






0
1
1
0
)
(
V
H
2
1
1
2
1
s
2
s
2
1
s
s
V



Figure 1

Encrypting algorithm of a
)
2
,
2
(

VSSS

black pixel, we randomly choose one of the matrices in
1
C
. Figure 1 illustrates the scheme by

specifying the algorithm for encrypting one pixel.

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Note that permuting the column of
0
M

and
1
M

does not change the Hamming weight
of the matrix. However, this p
rocedure is required in order to satisfy the security condition.

In the discussion above, we introduce the algorithm for encrypting one pixel. This
algorithm is to be applied for every pixel in the secret image to construct the two shares. Figure
2 is an

experiment example of a
)
2
,
2
(

V
SSS.



(a) the secret image


(b) share
1
s



(c) share
2
s



(d) decrypted image
2
1
s
s
V




Figure 2

Experiment example of a
)
2
,
2
(

VSSS

We can extend this algorithm to a
)
,
(
n
k

VSSS as below:



Design
0
M

and
1
M
.

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Construct
0
C

and
1
C
.



To share a white pixel, we randomly choose one of
the matrices in
0
C
, and to share a
black pixel, we randomly choose one of the matrices in
1
C
.

The scheme is valid if the following three conditions are satisfied:



For any
M

in
0
C
, the
“or” stacked version share

V

of any
k

of the
n

rows satisfies
m
d
V
H



)
(
.



For any
M

in
1
C
, the
“or” stacked version share

V

of any
k

of the
n

r
ows satisfies
d
V
H

)
(
.



For
k
t

, the
“or” stacked version share
V

of any
t

of the
n

rows is a function of
t
,
i.e.
)
(
)
(
t
f
V
H

, regardless of whether the matrix were
taken from
0
C

or
1
C
. In
other words, it gains no information about the secret image by examining less than
k

shares.

In this stage, we already introduce the VSSS idea of Naor and Shamir. The

problem is,
however, how to design
0
M

and
1
M
.

In the next section, we introduce the design method of
a general
)
,
(
k
k

VSSS, i.e. the design method of
0
M

and
1
M
. A more general
)
,
(
n
k

VSSS can be extend fr
o
m a
)
,
(
k
k

solution.


III.

A General

)
,
(
k
k

V
SSS

We now introduce the construction method of a
)
,
(
k
k

VSSS that minimizes
m
. In this
method, an original pixel is split into
1
2


k
m

subpixels, and the relative difference
1
2
/
1


k

.

Consider a set
}
,
,
,
{
2
1
k
e
e
e
W



with
k

elements and let
}
,
,
,
{
2
1
m




,
in

which
1
2


k
m
, be a list of all the subsets of
W

with even cardinality and let
}
,
,
,
{
2
1
m





be a list of all the subsets of
W

with odd cardinality. Each list defines the
following
m
k


matrices
0
M

and
1
M
: For
k
j


1

and
m
i


1
, let
1
]
,
[
0

j
i
M

iff

i
e
j


and
1
]
,
[
1

j
i
M

iff
j
i
e


.

Here is an example to construct a (3,3) VSSS.
For
3

k
,
}
,
,
{
3
2
1
e
e
e
W


and the set
of all subsets of
W

is


)
,
,
(
),
,
(
),
,
(
),
,
(
),
(
),
(
),
(
),
(
3
2
1
3
2
3
1
2
1
3
2
1
e
e
e
e
e
e
e
e
e
e
e
e

, where
)
(


denotes empty set
. By t
he definition of
i


and
i

, we have





)
,
(
),
,
(
),
,
(
),
(
,
,
,
3
2
3
1
2
1
4
3
2
1
e
e
e
e
e
e











)
,
,
(
),
(
),
(
),
(
,
,
,
3
2
1
3
2
1
4
3
2
1
e
e
e
e
e
e






Then, we can construct
0
M

and
1
M

according to
i


and
i

.

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10























1
1
0
0
1
0
1
0
1
0
0
1
,
1
1
0
0
1
0
1
0
0
1
1
0
1
0
M
M

In this section, we describe the construction method of a
)
,
(
k
k

V
SSS. The construction
method
of a general
)
,
(
n
k

VSSS can be extend from a
)
,
(
k
k

solution. We do not
introduce
the construct
ion

method of a
)
,
(
n
k

VSSS because it is too complicated for a tutorial paper here.
The reader
can
refer to [3] for the construction of a
)
,
(
n
k

VSSS.


IV.

General Access Structure

Scheme

Ateniese, Blu
ndo, Santis and Stinson [4] extend the VSSS to a general access structure
VSSS. Here is an example to illustrate the idea. Assume that a ba
n
k has a vault. In this time,
the bank employs three senior tellers and a manager. They would like to design a sy
stem such
that one of the three senior tellers together
with

the manager can open the vault. However, two
of the three senior tellers can not obtain the permission. This problem can be viewed as a
general access structure scheme.

In a

)
,
(
n
k

VSSS, the secret image is decrypted by stacking any

k

s
hares together. In a
general access structure VSSS, however, we can specify some qualified subsets of shares that can
decrypted the secret image, but other forbidden subsets

of shares have no information about the
it
.
For example, assume that there are four shares. Let


4
3
2
1
,
,
,
s
s
s
s



be the set of all shares,
and let

2

denotes the set of all subsets of

. Suppose that we

want to construct a VSSS such
that the qualified sets are all subsets of


containing at least one of the three sets



2
1
,
s
s
,


3
2
,
s
s

or


4
3
,
s
s
.
In order words,
1
s

and
2
s

together can decrypt the secret, so as


3
2
,
s
s

and


4
3
,
s
s
.
Hence, the family of qualified sets is



)
,
,
,
(
),
,
,
(
),
,
,
(
),
,
,
(
),
,
,
(
),
,
(
),
,
(
),
,
(
4
3
2
1
4
3
2
4
3
1
4
2
1
3
2
1
4
3
3
2
2
1
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
Qual



Let
all remaining sets of


be the
forbidden se
ts.



)
,
(
),
,
(
,
)
,
(
),
(
),
(
),
(
),
(
4
2
4
1
3
1
4
3
2
1
s
s
s
s
s
s
s
s
s
s
Forb



The pair
)
,
(
Forb
Qual



is called the access structure of the scheme.

Figure 3 is a
graphical exp
lanat
ion of the access structure. We are interested in constructing a VSSS where
the
i
-
th share
and

the
j
-
th share
toget
her can decrypt the secret image iff
i
,
j

are neighbor.


1
s
4
s
3
s
2
s


Figure 3

A general access structure example

We now introduce the construction method of a general access structure VSSS by us
ing
“cumulative arrays”. Let
M
Z

denote the collection of the maximal forbidden sets which has
t

elements.

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11




,
}
{
:
,
,
,
2
1
Qual
i
Forb
t
M
s
B
B
F
F
F
Z









for all

B
s
s
i
i



}
{
,
}
{

A cumulative array (
CA
) is a
t
n


Boolean matrix such that
1
]
,
[

j
i
CA

iff
j
i
T
s

.
Consider the example above,
)}
,
(
),
,
(
),
,
{(
4
2
4
1
3
1
s
s
s
s
s
s
Z
M

. Therefore,
3

t
, and the

cumulative array is the following:














0
0
1
1
1
0
0
1
1
1
0
0
CA

Let
0
M


and
1
M


be the Boolean matrices for a
)
,
(
t
t

VSSS. The Boolean matrices
0
M

and
1
M

for a
)
,
(
Forb
Qual



VSSS can be constructed as follo
ws. For any fixed
i
,
assume that there are
l

s
'
1

elements in the
i
-
th row of
CA
, and let
l
j
j
,
,
1


be the integers
j
such that
1
]
,
[

j
i
CA
. The

i
-
th row of
0
M

consists of the “or” of the row
l
j
j
,
,
1


of
0
M


and the
i
-
th row of
1
M

consists of the “or” of the row
l
j
j
,
,
1


of
1
M

. In the
above
example,
3

t
. The
)
3
,
3
(

VSSS solution is

.
1
1
0
0
1
0
1
0
1
0
0
1
,
1
1
0
0
1
0
1
0
0
1
1
0
1
0
























M
M

The Boolean matrices
0
M

and
1
M

can be constructed as:

.
1
0
0
1
1
1
1
0
1
0
1
1
1
1
0
0
,
0
1
1
0
1
1
1
0
1
1
1
0
1
1
0
0
1
0


























M
M

The second row of
0
M

is the “or” of rows 1 and 2 of
0
M

, that is,
)
1
,
1
,
0
,
0
(
)
1
,
0
,
1
,
0
(
)
1
,
1
,
1
,
0
(


, and the third row of
0
M

is the “or” of rows 2 and 3 of
0
M

.

Figure 4 is an experiment example of the above g
eneral structure VSSS. In this section,
we introduce a general access structure VSSS and described the construction method of this
scheme. It is more general than the VSSS proposed in [3] and we can specify the qualified set in
this method. It makes the

construction of VSSS become more flexible.


V.

VSSS for Color Images

In the VSSS introduced before, the secret image is a binary image. Koga and Yamamoto
[5] proposed a VSSS that can be applied to encrypt color images. The main idea of their scheme
is to e
xtend a basic VSSS into a finite lattice
-
based problem. In this section, we will describe
their method and provide some experiment examples to illustrate the idea.

Let
L

be an arbitrary finite lattice. For arbitrary elements
L
b
a

,
, the least upper
bound of
}
,
{
b
a

is denoted by
b
a

. Figure 5 shows the Hasse diagram of the binary lattice.
The lattice is denoted by
bin
L
, and it is the lattice that we concern in
the basic VSSS before. It
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12

has only two colors, i.e.
}
1
,
0
{

C
. The least upper bound of the two colors is defined as
follow:
0
0
0


,
1
1
0


, and
1
1
1


. Note that


m
eans the “or” operation of the
two colors, and it is the “

” operation that we defined in (2).

Figure 6 shows another finite lattice
col
L
. It has eight colors
,
,
,
,
0
{
C
C
M
Y


}
1
,
,
,
B
G
R
, white

(0), yellow (Y), magenta (M), cyan (C), red (R), green (G), blue (B) and black
(1). The mixture of arbitrary two colors means finding the least upper bound of the two colors.
For example,
M
M


0
,
R
Y
M


, and
1
1


R
.




(a) share
1
s







(b) share
2
s



(c) share
3
s







(d) share
4
s



(e)
2
1
s
s









(f)
3
2
s
s




(g)
4
3
s
s









(h)
3
1
s
s




(i)
4
2
s
s









(j)
4
1
s
s



Figure 4
Experiment example of a general access structure VSSS

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Figure 5
Hasse diagram of
bin
L



Figure 6
Hasse diagram of
col
L


Here is an example of
)
2
,
2
(
VSSS with three colors
}
,
,
,
{
C
R
M
Y

.






















1
0
0
1
,
1
0
1
0
,
1
0
1
0
Y
M
M
Y
M
Y
M
Y
M
M
M
Y
M
Y
M
R
M
Y

In this example, an original pixel is split into four subpixels,
4

m
, and each share
contains one yellow subpixel, one cyan subpixel, one white subpixel and one black subpixel. To
share a yellow pixel, we use
Y
M

to encrypt the two shares. To share a magenta pixel or a red
pixel, we use
M
M

or
R
M
. Figure 7 is an experiment example to share a secret image with
three colors.

Note that each stacked version
V

of
Y
M
,
M
M
, and
R
M

has two
s
'
1

elements.
This means that two out of four subpixels of
V

are recognized as a color in
C
, while the others
become black.

We now describe the construction method of a
)
,
(
k
k

VSSS for color images. We first
design
0
M

and
1
M

of a
)
,
(
k
k

VSSS by using method described in section


. For a
)
3
,
3
(

VSSS , we have

.
1
0
0
1
0
1
0
1
0
0
1
1
,
0
1
1
0
1
0
1
0
1
1
0
0
1
0






















M
M

Define
)
(
0
x
M

and
)
(
1
x
M

by replacing
s
'
0

in
0
M

and
1
M

by
s
x
'
.























1
1
1
1
1
1
)
(
,
1
1
1
1
1
1
)
(
1
0
x
x
x
x
x
x
x
M
x
x
x
x
x
x
x
M

Assume that
}
,
{
C
C
Y

, we can construct
Y
M

by concatenating
)
(
0
Y
M

with
)
(
1
C
M
, and
construct
C
M

by concatenating
)
(
0
C
M

with
)
(
1
Y
M
.

)
(
0
Y
M
M
Y













1
1
1
1
1
1
1
1
1
1
1
1
)
(
1
C
C
Y
Y
C
C
Y
Y
C
C
Y
Y
C
M

1
0
1
0
R
G
C
M
Y
B
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(a) the secret image


(b) share
1
s


(c) share
2
s


(d) decrypted image
2
1
s
s



Figure 7
Experiment example of color VSSS


)
(
0
C
M
M
C













1
1
1
1
1
1
1
1
1
1
1
1
)
(
1
Y
Y
C
C
Y
Y
C
C
Y
Y
C
C
Y
M

Increasing the number of colors is easy. For example, consider the construction of a
)
3
,
3
(
VSSS with
}
,
,
{
C
G
C
Y

.

)
(
0
Y
M
M
Y


)
(
1
C
M

)
(
1
G
M

)
(
0
C
M
M
C


)
(
1
G
M

)
(
1
Y
M

)
(
0
G
M
M
G


)
(
1
Y
M

)
(
1
C
M

In this section, we describe a VSSS for color images. Theoretically, this method is able to
encrypt color images with arbitrary number of colors.

The number of subpixels
m
, however,
will be much larger than a binary VSSS unless the number of colors is small. The reader is
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referred to [6~8] for other color VSSS methods.






(a) share
1
s


(b) share
2
s

(c)
2
1
3
s
s
s




Figure 8
Experiment example of concealing the share image
3
s

under two innocent images
1
s

and
2
s


VI.

Ext
ensions

There are many other extensions and improvements of the basic VSSS. In this section, we
will describe some useful ideas and some practical applications.




Contrast Improvement

For a
)
2
,
2
(

VSSS, a pixel is interpreted by the visual

system of the users as white if
1
)
(

V
H
, and as black if
2
)
(

V
H
. It means that there is
%
50

loss of contrast in the
decrypted image. For
)
,
(
k
k

VSSS, the contrast loss will be much mo
re serious as
k

become larger. Therefore, the construction method to improve the contrast is required. The
reader can refer to [9~10] for the contrast improved VSSS.




Concealing the Share Image under Innocent images

Since the shar
e image looks like a random noise. It makes everyone know that there are
some secret hidden under the share image. It might be useful to conceal the share image under
innocent images. Figure 8 is an experiment example of this idea. We can conceal the sh
are
image
3
s

under two innocent images
1
s

and
2
s
. By stacking
1
s

and
2
s

together, we
can decrypt the secret image
3
s
. The reader

is referred to [3] for more information.




Share Multiple Secret Images

Droste [11] considered the problem of sharing more than one secret image among a set of
shares. A construction method is proposed to obtain VSSS such that different subsets of shares
recover different secret images.



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VII.

Conclusions

In this tutorial paper, we provide an introduction to visual secret sharing scheme (VSSS).
VSSS is a low
-
cost, simple cryptographic technique that can be used by anyone without any
knowledge of cryptography.

We have described the construction method of a
)
,
(
k
k

VSSS,
and we also explain how to decrypt the secret image “visually”. Finally, we introduced the
general access structure VSSS and the extension to encrypt color images.


Reference:

[1]

A. Shamir, “How to share a secret.” Communications of the ACM 22 (1979), 612
-
613.

[2]

G. R. Blakley, “Safeguarding cryptographic keys.” In “Proceedings of the National Computer
Conference, 1979”, American Federation of Information Processing Societies Proceed
ings
48 (1979), 313
-
317

[3]

M. Naor and A. Shamir, “Visual Cryptography”, Advances in Cryptography
EUROCRYPT’94, Perugia, Italy, May 1994, 1
-
12

<http://www.wisdom.weizmann.ac.il/~naor/PAPERS/vis.gs>

[4]

G. Ateniese, C. Blundo, A. De Santis and D.R. Stinson, “Visua
l Cryptography for general
access structures” Information and Computation 129 (1996), 86
-
106

<www.eccc.uni
-
trier.de/eccc/>

[5]

H. Koga and H. Yamamoto, “Proposal of a Lattice
-
Based Visual Secret Sharing Scheme for
Color and Gray
-
Scale Images” IEICE Trans. Fund
amentals, Vol. E81
-
A, No. 6, June 1998,
1263
-
1269

[6]

E. R. Verheul, H. C. A. Van Tilborg, “Constructions and Properties of
k

out of
n

Visual
Secret Sharing Scheme” Designs, Codes and Cryptography, Vol. 11, No. 2, M
ay 1997,
179
-
196

[7]

V. Rijmen, B. Preneel, “Efficient Color Visual Encryption of ‘Shared Colores of Benetton’ ”
EUROCRYPT ’96 rumpsession talk.

<
http://www.east.kuleuven.ac.be/%7Eri
jmen/vc/euro96/tekst.html
>

[8]

D. Naccache, “Colorful Cryptography


a Purely Physical Secret Sharing Scheme Based in
Chromatic Filters” Coding and Information Integrity, French
-
Israeli Workshop, Dec 1994

[9]

M. Naor and A. Shamir, “Visual Cryptography

: Improving the Contrast via the Cover
Base”, Security in Communication Networks, Amalfi, Italy, Sep. 1996

<ftp://theory.lcs.mit.edu/pub/tcryptol/96
-
07.ps>

[10]

C. Blundo, A. De Santis, D.R. Stinson, “On the Constrast in Visual Cryptography Schemes”
Theory of
Cryptography Library, report 96
-
13

<frp://theory.lcs.mit.edu/pub/tcryptol/96
-
13.ps>

[11]


S. Droste, “New Results on Visual Cryptography” Advances in Cryptography CRYPTO’96,
Lecture Notes in Computer Science, No. 1109, Spinger
-
Verlag, 1996, 401
-
415

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