Cryptography Reference
In-Depth Information
image is the same as the original one. The scheme can also be used to share
other secret messages among the participants.
4.2 Related Works
In Naor and Shamirs (t, n)-VSS scheme, the secret image is divided into n
shadows for the different participants. Each pixel in the image is then ex-
panded into sub-pixels for each shadow according to the encryption function
and two codebooks, C
0
and C
1
. Each codebook includes a number of pattern
sets, and is used to generate the shadows for the participants. For a white
pixel, the VSS scheme randomly chooses one pattern set from C
0
. While for
a black pixel, the scheme randomly chooses one pattern set from C
1
.
Fig. 4.1 gives a (2, 2)-VSS example that is encrypted by using Naor and
Shamirs VSS scheme. In Fig. 4.1, a secret image I is divided into two secret
shadows, S
1
and S
2
. Assume that the codebooks used in this example are C
0
and C
1
as shown in Fig. 4.2. Each codebook consists of six pattern sets. Each
pattern set includes two patterns used to generate the shadows for S
1
and S
2
.
The VSS scheme splits each pixel into two patterns, which are chosen from
the codebooks C
0
and C
1
, and then pastes the patterns in the shadows. For
example, the i-th pixel of I in Fig. 4.1 is a black pixel. Hence, the scheme
randomly chooses the fourth pattern set from C
1
and respectively pastes the
patterns to S
1
and S
2
. After encryption, the secret image is divided into two
binary shadows, each of which is 22 times larger than the secret image.
When we need to decrypt the secret image, the VSS scheme stacks a certain
number of shadows to reveal the image. Following the same procedure, the
reconstructed image I
′
is shown in Fig. 4.1.
Let us consider a real image with size 12864 as the secret image. Fig. 4.3
demonstrates the (2, 2)-VSS scheme, where Fig. 4.3(a) is the input image and
Fig. 4.3(b) and Fig. 4.3(c) are two generated shadows. Fig. 4.3(d) is the output
which is four times larger than the original input.
In the decryption process, the VSS scheme stacks the shadows by using
the decryption function f
d NS
to reconstruct the secret image. The function
f
d NS
maps a certain number of blocks in each shadow into a binary pixel.
This function is defined as follows:
1, if S
i
=1∪S
i
=1,
0, otherwise,
f
d NS
(S
i
,S
i
)=
(4.1)
where S
i
and S
i
are the i-th bit of S
1
and S
2
, respectively. In Eq. (4.1), if
both values of S
i
and S
i
are 1, it means that the color of the bit is black.
And then, the color of the reconstructed bit is black.
The Naor and Shamirs scheme is simple and intuitive. However, the visual
quality of the reconstructed image using the scheme is poor. In addition, the
scheme cannot provide a perfect reconstruction of the pixel intensity aspect.
