Two-Decoding-Option Image Sharing Method - Visual Cryptography and Secret Image Sharing - page 457

p = 251 is chosen such that g(x) is constrained between 0 and 250 and suitable
to represent the conventional 8-bit grayscale or color images. Notice that the
possible values of an 8-bit gray pixel are from 0 to 255, so the grayscale values
(> 250) need to be modified to 250 and will cause distortion. Obviously, we
can use the Galois Field GF(2 8 ) rather than the ordinary arithmetic (mod
251) to achieve a lossless scheme. Thien and Lin further reduced shadows with
size 1=k times that of the secret image [13] by embedding the secret data in all
coecients of g(x). The formal encoding of Thien and Lin's scheme is briey
described below.
A secret image is rst divided into non-overlapping k -pixel blocks,
and every j -th (0 j 1) block includes the secret pixel values
p jk ;p jk+1 ;:::;p jk+k1 . The (k 1)-degree polynomial S j (x) represents a
shadow pixel associated with the j -th block on shadows.

S j (x) = (p jk + p jk+1 x + p jk+2 x 2 + ::: + p jk+k1 x k1 )inGF(2 8 );

(17.3)

where x is often an image identication and 0 j 1. The value of S j (x)
is generated using the original pixel values p jk ;p jk+1 ;:::;p jk+k1 included in
the j -th block. In this chapter, the Galois Field GF(2 8 ) was chosen to achieve
a lossless secret image. Because k pixels are processed each time, the size of
the shadow image is 1=k of the secret image. By reversing the encoding, the
polynomial in (17.3) can be reconstructed from k shadow pixels; hence, the
blocks can be recovered and finally the secret image is reconstructed.

17.2.2 VCS
The rst VCS was Naor{Shamir's (k;n)-VCS to encrypt a halftone secret im-
age into noise-like shadows. The authors used the whiteness (the number of
white subpixels in a m-subpixel block) to distinguish the black color from the
white color, i.e., "mh"B"h"W (respectively "m l"B"l"W) represents a
white (respectively black) color, where h > l. A black-and-white (k;n)-VCS
can be designed using two base n m matrices B 1 and B 0 with elements
"1" and "0" denoting black and white subpixels. When sharing a black (re-
spectively white) secret pixel, the dealer randomly chooses one row of the
matrix in the set C 1 (respectively C 0 ), which includes all matrices obtained
by permuting the columns in B 1 (respectively B 0 ) to a relative shadow. Let
OR (B i jr), i = 0; 1, denote the "OR"-ed vector of any r rows in B i , and H()
be the Hamming weight of a vector. The base matrices of the (k;n)-VCS
should satisfy the following conditions:
(V-1). H (OR (B 1 jr)) (ml) and H (OR (B 0 jr)) (mh)) for r = k,
where 0 l < h m.
(V-2). H (OR (B 1 jr)) = H (OR (B 0 jr)) for r (k 1).
The first condition is often referred to as the contrast condition, and the
secret image can be recognized due to their different contrasts of black and
white colors. The second condition is the security condition that assures the
(k;n)-VCS of perfect secrecy.