XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 153

[17]. Biham and Itzkovitz [2] investigated a VC system based on passive light
polarization. This is more flexible than the Naor and Shamir schemes but
cannot be modeled by an XOR [17]. Tuyls et al. [17, 18] gave a VC system
that uses the polarization of light. The operation of the VC system is mathe-
matically described by an XOR operation. In Section 6.3, we will show that a
(n;n)-VC scheme with optimal resolution and contrast exist, and that (2;n)-
VC schemes are equivalent to binary codes. Three explicit constructions for
general k out of n schemes are also introduced [17, 13].
Viet and Kurosawa [21] noticed the phenomenon that most copy machines
nowadays can change a black image into a white one and vice versa, then
first gave a VC scheme with reversing for binary image. In the VC scheme
with reversing, a dealer needs run the distribution phase of a VC scheme c
times (with c arbitrary constant), hence requiring each participant to held
c shadows. The almost ideal contrast of a recovered secret image is almost
obtained for a large number of runs c. Cimato et al. [7] presented two ele-
gant construction methods to improve the contrast and pixel expansion of
the scheme in Reference [21]. In order to reduce run times of two schemes
[21, 7], Yang et al. [25, 26] overcame the weakness of reversing only a based
perfect VC scheme and first introduced a nonperfect black VC scheme, this
approach uses a Boolean XOR operation for decoding. Reducing the stacking
and reversing operations and minimizing number held by each participant,
Hu and Tzeng [10] proposed an ideal contrast VC scheme with less reversing
and stacking operations in only two runs. The scheme needs to perform XOR
operations to decode the secret image. In section 6.4, we will introduce Yang
et al.'s scheme and Hu and Tzeng's [25, 26] scheme with ideal contrast.
The construction of the above schemes is all based on basis matrices, so
they may suffer pixel expansion and loss of contrast. Probabilistic VC schemes
are proposed in [11, 24, 6] with no pixel expansion. However, the recovered
secret image has low contrast. Wang et al. [23] proposed two secret sharing
schemes based on a Boolean operation and the recovering operation is XOR.
One scheme is (2;n) for the binary image, and the other is (n;n) for the
grayscale image. Both have no pixel expansion. Chao and Lin [5] improved
Wang et al.'s [23] scheme in order to obtain a (k;n) scheme, which is fast and
with a reasonable pixel expansion rate. In section 6.5, we introduce the (2;n),
(n;n), and (k;n) schemes.
6.2 Preliminaries
In a black and white (k;n)-visual cryptography (VC) scheme [16], the secret
image consists of a collection of black and white pixels and each pixel is
subdivided into a collection of m black and white subpixels in each of the n
shares. These subpixels are printed in close proximity to each other so that the