Multi-secret Sharing Visual Cryptography Scheme Based on XOR Algorithm
Yanyan Han1, Shuai Ji2, Wencai He3 and Yao Zhang2
(1. Department of Scientific Research, Beijing electronic science & technology institute, Beijing
100070, China)
(2. Communication Engineering Department, Xidian University, Xi'an 710071, China)
(3. Communication Engineering Department, Beijing electronic science & technology institute,
Beijing 100070, China)
Abstract — XOR algorithm can be used
to improve the recovery effects of visual
cryptography scheme. In this paper, we
present a multi-secret visual cryptography
scheme by using the matrices of the (k,
k)-VCS. We can reveal multiple secret
images perfectly by the XOR algorithm and
shifting operation on the shares. The visual
qualities of recovered images are ideal
without any pixel expansion.
Keywords — Visual cryptography,
Multiple secret images, XOR algorithm, No
pixel expansion.
I.
Introduction
As a new secret sharing technique, visual
cryptography scheme (VCS) was introduced
by Naor and Shamir in Euro-Crypt conference
in 1994 [1]. In the (k, n)-VCS, a secret image
is encoded into n shares, which consist of
some random binary patterns. Then stacking
any k or more than k shares can reveal the
secret image by human visual system without
any computation. However, if the number of
shares is less than k, any information about the
secret image cannot be caught.
In recent years, there are a lot of
achievements about VCS [2-6]. However, the
schemes above can only extract a secret image.
Nowadays, a visual cryptography with extra
ability of reveal more data was proposed by
Fang [7]. However, the pixel expansion is
serious and the contrast is low. After, Huang
[8] proposed a non-expanded VCS based on
block encoding [9], which encodes a
black-and-white image into the same size of
share images as the original secret image. But
it only can be applied on the (2, 2)-VCS.
In this paper, we propose a multi-secret (k,
k)-VCS for gray images based on XOR
algorithm. Through halftone technology and
encryption algorithm, we generate k shares.
Simply stacking the k shares can reveal a
secret image; later, fix one of the shares and
move the next share half down, we can extract
another two small images. We can see that the
number of the recovered image is 2(k-1)+1.
Compared
with
the
original
visual
cryptography scheme, the scheme is
increasing the number of secret images. Each
participant only needs to keep a share and they
can recover lots of images. The scheme
reduces the difficulty of manage the shares. It
is simple with good security and the recovery
secret image is characterized by the good
contrast without pixel expansion.
II. Related Work
In this paper, the proposed scheme adopts
the (k, k)-VCS which is based on XOR
algorithm [10]. Structuring matrix set 𝐶0 , 𝐶1,
which express white pixels and black pixels
fundamental matrix set in the secret image.
And structuring white and black pixels
fundamental matrix set 𝑆0 , 𝑆1 in the sharing
images, which select from matrix set 𝐶0, 𝐶1
respectively
and
randomly.
In
the
reconstruction of the secret image, a white
pixel or a black pixel of the original secret
image is expressed as a subpixel. The 𝐶0 , 𝐶1
for (2, 2)-VCS and (3, 3)-VCS are as shown in
Eq. (1) and Eq. (2):
𝐶0 =
𝐶0
1 0
1 0 , 𝐶1 =


1 0
0 1 


(1)
1 1 0 0
0 0 1 1 
 0 1 0 (2)


=
,𝐶 = 1


0 1 0 1 
1 0 0 1
0 1 1 0
1
III. The Design of Scheme
In this paper, we propose a novel visual
cryptography scheme without any pixel
expansion for gray images. The proposed
scheme can hide a big secret image and 2(k-1)
small secret images which are a quarter of the
big one. In this paper, let k=3. Firstly, we use
halftone technology to process all of the five
secret gray images respectively and transform
them into binary images. Halftone is a method
to transform an image with gray level into a
binary image. In this paper, we use the
Floyd-Steimberg error diffusion [11] to get the
halftone images. Then we use XOR algorithm
to generate three shares. The shares have the
following three properties: (a) stacking the
three shares will obtain the big secret images;
(b) stacking any two of them will obtain
nothing; (c) if we fix one of the shares and
then move the next share half down, we will
obtain another two secret images. The detail
method is given as following.
1. Process of Encryption
The input is a 512×512 gray image P1
and four 256×256 gray images P2 , P3 , P4 , P5.
Firstly, we process the gray images P1 , P2 , P3 ,
P4 and P5 into binary images. Then, the
encryption of the secret image is as shown in
the following steps.
(1) For each coordinate (i, j) in the area
where {(i, j)|256≤i≤511, 0≤j≤511} do the
following:
(1a) Randomly assign “1” and “0” to
S1 (𝑖, 𝑗) and S2 (𝑖, 𝑗);
(1b) If secret image P1 (𝑖, 𝑗) is a white
pixel, then S3 (𝑖, 𝑗) = S1 (𝑖, 𝑗) ⊕ S2 (𝑖, 𝑗) ⊕ 0;
otherwise, S3 (𝑖, 𝑗) = S1 (𝑖, 𝑗) ⊕ S2 (𝑖, 𝑗) ⊕ 1.
(“⊕” denotes XOR algorithm).
(2) For each coordinate (i, j) in the area
where {(i, j)|0≤i≤255, 0≤j≤255} do the
following:
(2a) If P2 (𝑖, 𝑗) is a white pixel, then copy
the S1 (𝑖 +256, j) to S2 (𝑖, 𝑗); otherwise, copy
the complement of S1 (𝑖 +256, j) to S2 (𝑖, 𝑗);
(2b) If P3 (𝑖, 𝑗) is a white pixel, then copy
the S2 (𝑖 +256, j) to S3 (𝑖, 𝑗); otherwise, copy
the complement of S2 (𝑖 +256, j) to S3 (𝑖, 𝑗);
(2c) If P1 (𝑖, 𝑗) is a white pixel, then
S1 (𝑖, 𝑗) = S2 (𝑖, 𝑗) ⊕ S3 (𝑖, 𝑗) ⊕ 0; otherwise,
S1 (𝑖, 𝑗) = S2 (𝑖, 𝑗) ⊕ S3 (𝑖, 𝑗) ⊕1.
(3) For the remaining area (where {(i,
j)|0≤i≤255, 256≤j≤511}) do the following:
(3a) If P4 (𝑖, 𝑗) is a white pixel, then copy
the S1 (𝑖 +256, j) to S2 (𝑖, 𝑗); otherwise, copy
the complement of S1 (𝑖 +256, j) to S2 (𝑖, 𝑗);
(3b) If P5 (𝑖, 𝑗) is a white pixel, then copy
the S2 (𝑖 +256, j) to S3 (𝑖, 𝑗); otherwise, copy
the complement of S2 (𝑖 +256, j) to S3 (𝑖, 𝑗);
(3c) If P1 (𝑖, 𝑗) is a white pixel, then
S1 (𝑖, 𝑗) = S2 (𝑖, 𝑗) ⊕ S3 (𝑖, 𝑗) ⊕ 0; otherwise,
S1 (𝑖, 𝑗) = S2 (𝑖, 𝑗) ⊕ S3 (𝑖, 𝑗) ⊕1.
Share images S1 ， S2 and S3 are
obtained.
2. Process of Recovery
The recovery of the secret image is as
shown in the following steps:
(1) Through the XOR algorithm, the secret
image P1 is recovered after stacking the three
shares.
(2) Fix share S1 and move share S2 half
down, it can reveal the image P2 and P4 . Fix
share S2 and move share S3 half down, it
can reveal the image P3 and P5 .
IV. Simulation Results
In Fig. 1, it shows the five secret images.
In Fig. 2, it shows the three shares 𝑆1 , 𝑆2 and
𝑆3. Fig. 3(a) is the result of stacking 𝑆1 , 𝑆2
and 𝑆3; Fig. 3(b) is the result of stacking 𝑆1
with shifted 𝑆2 ; Fig. 3(c) is the result of
stacking 𝑆2 with shifted 𝑆3.
(c) Share S3
Fig. 2 The Share Images
(a) Image P1
(a) 𝑆1 ⊕ 𝑆2 ⊕ 𝑆3
(b) Image P2
(c) Image P3
(d) Image P4
(e) Image P5
Fig. 1 Original Secret Images
(b) 𝑆1 ⊕ "𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑆2 "
(a) Share S1
(c) 𝑆2 ⊕ "𝑠ℎ𝑖𝑓𝑡𝑒𝑑 𝑆3 "
Fig. 3 The Result of Stacking
V.
(b) Share S2
Performance Analysis
In this paper, a visual cryptography with
multiple secret images was proposed. In (k,
k)-VCS, stacking all of the k shares, we can
obtain a secret image which is the same size of
the shares. Fix the i-th share and move the
(i+1)-th share half down, we can extract
another two small images which are 1/4 of the
shares (i=1, 2, …, k-1). In order to improve
the quality of recovery images and reduce the
pixel expansion, we use XOR algorithm.
1. Capability Analysis
According to the simulation results, there
is no pixel expansion and the contrast is ideal.
It also can reveal more images after shifting.
An extended visual cryptography scheme for
extract more images was proposed in previous
studies. In Fang’s scheme [7], when we fix the
first share and shift the other share for certain
units, we can obtain the extra data. But the
pixel expansion is serious and it only deals
with binary image. In Huang’s scheme [8],
although it solves the problem of pixel
expansion, but the contrast is low. Compared
with [7, 8], our scheme has a great
improvement on the recovery effect. The
shares and the recovery images are as the
same size as the original images and there is
no pixel expansion. The contrast of the
recovered image is ideal. Therefore, this
scheme has significant improvement.
2. Security Analysis
To share a white (black) pixel, we
randomly choose one of the matrices in 𝐶0
(𝐶1). The chosen matrix defines the color of
the subpixels in each one of the sharing image.
If the scheme satisfies the following three
conditions, we consider that the scheme has
security.
(1) For any S in 𝐶0, the “XOR” 𝑉 of k
rows satisfies 𝐻(𝑉) ≤ 𝑑 − 𝛼𝑚.
(2) For any S in 𝐶1, the “XOR” 𝑉 of k
rows satisfies 𝐻(𝑉) ≥ 𝑑.
(3) For any subset {𝑖1, 𝑖2,…, 𝑖𝑞 } of {1, 2, …,
the same matrices with the same frequencies.
The first two conditions imply that by
stacking of k shares, we can gain the secret
image. And the third condition implies that by
stacking of fewer than k shares, even an
infinitely powerful cryptanalyst cannot gain
any advantage in deciding whether the shared
pixel was white or black. This means that the
(k, k)-VCS which is proposed in this paper is
safety.
3. The Extension of This Scheme
In this paper, we reveal the big secret
image after stacking the k shares. If we fix one
of the shares and move down the next share
for 1/2, we will obtain another two secret
images. The number of the secret image which
we recover is 2(k-1)+1. Therefore, we can
extend the scheme. For example, we adopt the
similar encryption method and fix one of the
shares and move down the next share for
(1-1/u), we will obtain another u secret images.
The number of the secret image which we
recover is u(k-1)+1. (u is the number of small
images we revealed after stacking two shares).
We can conclude that the quantity of the secret
images are no less than 2(k-1)+1.
k} with 𝑞 < 𝑘, the two collections of q×m
matrices 𝐷𝑡 for t ∈ {0, 1} obtained by
restricting each k × m matrix in 𝐶𝑡 (where
t = 0, 1) to rows
𝑖1, 𝑖2,…, 𝑖𝑞
are
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VI. Conclusions
In this paper, we discuss a multi-secret
VCS for gray images which are based on
XOR algorithm. This scheme can be applied
on any k. We generate k sharing images which
have a random distribution of black and white
pixels, and we also can recover lots of secret
images. The visual qualities of recovered
images are ideal without any pixel expansion.
But the disadvantage of this scheme is that it
also has a problem of aligning the two shares.
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