Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
Paper Number: TCSVT 7078
1
Abstract—A (k, n) Visual Cryptographic Scheme (VCS)
encodes a secret image into n shadow images (printed on
transparencies) distributed among n participants. When any k
participants superimpose their transparencies on an overhead
projector (OR operation), the secret image can be visually
revealed by human visual system without computation. However,
the monotone property of OR operation degrades the visual
quality of reconstructed image for OR-based VCS (OVCS).
Accordingly, XOR-based VCS (XVCS), which uses XOR
operation for decoding, was proposed to enhance the contrast. In
this paper, we investigate the relation between OVCS and XVCS.
Our main contribution is to theoretically prove that the basis
matrices of (k, n)-OVCS can be used in (k, n)-XVCS. Meantime,
the contrast is enhanced 2
(k1)
times.
Index Terms—Secret sharing, Image secret sharing, Visual
cryptography, Visual secret sharing.
I. INTRODUCTION
Visual Cryptographic Scheme (VCS) is a category of secret
image sharing schemes. In a threshold (k, n)-VCS, where k n,
a secret image is encrypted into n shadow images (shadows) by
expanding a secret pixel into m subpixels. The value of m is
referred to as the pixel expansion. There is no difference
between the pixel and the subpixel except that the “pixel”
denotes the secret pixel located in the secret image, and the
“subpixel” is the pixel located in shadows. Their sizes are equal
and thus the shadow size is expanded m times compared with
the size of secret image. Any k or more shadows can be stacked
for reconstructing the secret image by the human visual system
(HVS) without computation, while k1 or fewer shadows
cannot recover the secret image. The first VCS encrypted the
black/white secret image into shadows [1]. Because the visual
quality of a reconstructed image is degraded by a large pixel
expansion, most studies try to reduce the pixel expansion [2-8].
Some even have a non-expansible shadow size (i.e., m=1), and
they are known as the probabilistic VCSs [5-8]. VCS can also
be applied on gray or colored secret images. A trivial solution is
Copyright (c) 2013 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending an email to pubs-permissions@ieee.org.
This work was supported in part by the project under Grant NSC
101-2221-E-259-028, and was also supported in part by the National Natural
Science Foundation of China (Grant No. 61170032).
C.N. Yang is with Department of CSIE, National Dong Hwa University,
Taiwan, (corresponding author to provide phone: +886-3-8634025; fax:
+886-3-8634010; e-mail: cnyang@mail.ndhu.edu.tw).
D.S. Wang is with Department of Computer Science and Technology,
Tsinghua University, China.
to convert the gray or color secret image into a binary image by
the halftoning technique and then processes it by the
black-and-white VCS [9, 10]. Other approaches for sharing the
gray and color images were chronologically proposed [11-16].
VCS has poor visual quality of a reconstructed image.
Another polynomial-based secret image sharing scheme can
recover a distortion-less secret image, but its decoding needs
Lagrange interpolation [17-19]. The authors combined VCS
and polynomial-based secret image sharing to design and
develop a two-in-one VCS with different decoding options
[20-22]. In this two-in-one VCS, the first phase is stacking to
see a vague reconstructed image like VCS, and the second
phase is to perfectly reconstruct the secret image by Lagrange
interpolation.
The stacking operation in VCS is OR operation, and thus the
conventional VCS is also referred to as OR-based VCS
(OVCS). To enhance the visual quality, some XOR-based
VCSs (XVCSs) allowing participants to perform XOR
operations were accordingly proposed [23-29]. In this paper,
we are not to design a new XVCS. Our main contribution is
theoretically to prove the basis matrices in (k, n)-OVCS also
satisfy the contrast and security conditions of (k, n)-XVCS.
Meantime, the contrast is 2
(k1)
times enhanced by XOR
operation. This new observation implies that VCS
simultaneously performs two roles- OVCS and XVCS-
providing more decoding choices when using VCS in
applications. The rest of this paper is organized as follows.
Section II describes (k, n)-OVCS and (k, n)-XVCS. The main
result that shows the basis matrices in OVCS satisfying the
contrast and security conditions in XVCS is given in Section
III. In Section IV, some experiments are shown. Section V is
the conclusion.
II. P
REVIOUS OR-BASED AND XOR-BASED VISUAL
CRYPTOGRAPHIC SCHEMES
A. OR-based Visual Cryptographic Scheme
Naor and Samir [1] used the whiteness to distinguish the
black color from the white color, i.e., “mh”B“h”W
(respectively “ml”B“l”W), where 0l<hm, represents a
white (respectively black) secret pixel. The values of l and h are
the black whiteness and the white whiteness (the number of
white subpixels in a m-subpixel block), respectively. A (k,
n)-OVCS can be designed using two n×m Boolean matrices B
1
and B
0
with elements “1” and “0” denoting black and white
subpixels. When sharing a black (respectively white) secret
pixel, the dealer chooses one row of the matrix in the black set
C
1
(respectively the white set C
0
) which includes all matrices
Property Analysis of XOR Based Visual
Cryptography
Ching-Nung Yang, Senior Member, IEEE, and Daoshun Wang
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