Visual Cryptography for Multiple Secrets - Visual Cryptography and Secret Image Sharing - page 82

a j = permute(s A ; j )

3.2.1

for (each secret image i, 1 i x) do q i = (p i ) j 3.2.2

3.2.3 = btod(rotate(q 1 q 2 :::q x ;k 1))

b j = permute(s B ; j ) 3.2.4 g

g

output(A, B) // A and B are composed by all a j 's and b j 's, respectively

4.

Note that a new permutation j is determined to permute the subpixels
in each pair of a j and b j for 1 k x to ensure the entire randomness that
the subpixels in a j and b j can provide. Further, a j and b j are encoded by
using the same permutation j so that the numbers of the white and black
subpixels in a j b j and s A b B are exactly the same for 1 k x.
The pixel expansion of Shyu et al.'s scheme is 2x when x secrets are shared.
In the case of x = 2, the pixel expansion is 2x = 4 which is the same as that of
Wu and Chen [12] as well as Wu and Chang [13]. The number of all possible
patterns in an extended block in S 1 [12] (see Figure 3.2(c)) or any sector
block in A [13] (see Figure 3.4(a)) is 4, which contains two white and two
black subpixels, while that in b j of the scheme is (4!)=(2! 2!) = 6. The
randomness of this scheme, in the case of x = 2, is surely better than that of
Wu and Chen as well as Wu and Chang.
It is seen from Algorithm 1 that we do not physically store any information
about Tables 3.5, 3.7, and 3.8 in memory. The elementary blocks s A 's and s B 's
are generated in the run time (Step 2 in Algorithm 1) according to formulae
(3.5) and (3.6). The encoding process is guaranteed by formulae (3.7) and
(3.8) (Step 3 in Algorithm 1).

3.4.2 Feng et al.'s Scheme
Regarding Feng et al.'s (2, 2, x) scheme [3], x secret images P 1 ;P 2 ;:::;P x are
encoded into two shares A and B. Each set of x corresponding pixels (in x
secret images) is encoded into two blocks, namely is A 2 A and s B 2 B, each
of which consists of x rows containing 3 pixels each. The pixel expansion is
thus 3x. The rotation relationship for revealing each of the x secrets is similar
to Shyu et al.'s scheme where the ith secret is revealed by A B (i1) for
1 i x. One special design in their scheme is that the encoded shares are
in the form of cylinders to avoid the distortion of the revealed secrets.
They chose four types of 3-pixel patterns, referred to as the effective block
B e = , ineffective block Bi= i = , white block Bi= w = and black block
B b = to construct s A and s B . It is evident that Bi i B w = B i B b =
, B e B w = , and B e B b = .
Figure 3.19 specifically illustrated the stacked results of these chosen pat-
terns in their scheme.