XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 165

the result that the Hamming weight of the stacking (XOR operation) of the
rows of B 0 is less than that of B 1 . Hence, the contrast condition is satisfied.
Second, for the security condition, we need to prove that the submatrices
of any less than k rows of B 0 and B 1 have the same columns, and only in such
a case, all the column permutations of the two submatrices will generate the
same collection, that is, the security condition is satisfied. Denote B 0 (resp.
B 1 ) as the submatrix generated by restricting to arbitrary t rows of B 0 (resp.
B 1 ), where t < k. Denote B 0 (resp. B 1 ) as the submatrix generated by con-
catenating B 0 (resp. B 1 ) and arbitrary kt rows chosen from the remaining
rows of B 0 (resp. B 1 ) (other than the rows in B 0 and B 1 ). As discussed above,
we have B 0 = I 0 [R;B 1 = I 1 [R, where I 0 (resp. I 1 ) is the matrix that con-
tains all the even (resp. odd) columns of length k. Note that I 0 and I 1 are the
basis matrices of a (k;k)-VC scheme proposed in References [16, 3]. We have
the result that the submatrices generated by restricting to any t rows of B 0
and B 1 have the same columns. Hence, the submatrices generated by restrict-
ing to any t rows of B 0 and B 1 have the same columns, that is, the security
condition is satisfied.
Besides the above schemes, Liu et al. [15] proposed a step construction to
construct XOR-based visual cryptography for general access structure by ap-
plying (2; 2)-VCS recursively, where a participant may receive multiple share
images. Readers can refer to [15] for more detail. An approach to construct
extended XOR-based visual cryptography was proposed in Reference [14].
6.4 Visual Cryptography Scheme with Reversing
First we introduce the following notations that will be used throughout the
section. Let AjjB denote the concatenation of two matrices A and B of the
same number of rows. Let jXj be the number of elements in set X. The symbol
""denotes an XOR operation. Let GRAY (P) be the gray level of a pixel P
and dened as GRAY (P) = jblack pixelj=m.
6.4.1 (k,n) VC Scheme Using Cyclic-Shift Operation
Let the shadow image s = [s ijk ], and the element s ijk is the secret pixel s ij in
a (W H)-pixel secret image replaced by m subpixels (s ij1 ;s ij2 ; ;s ijm ),
where i 2 [1;W], j 2 [1;H], and k 2 [1;m]. The cyclic-shift operation is
([s ijk ]) = [(s ijk )], where () is a 1-bit cyclical right shift function, i.e.,
(s ij1 ;s ij2 ; ;s ijm ) = (s ijm ;s ij1 ; ;s ijm1 )
A matrix operation () cyclically shifts right one subpixel in every m
subpixels (for a secret pixel) in the shadow image.
When the dierence of whiteness "hl" is odd, we can design an ideal