XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 168

6.4.2 A Scheme for General Access Structure
Let P = f1; 2; ;ng be a set of participants, 2 P represents the set of all
subsets of P. Let qual 2 P and forb 2 P , where qual \ forb = . The
members of qual (resp. forb ) is called qualified sets (resp. forbidden sets).
The = ( qual ; forb ) is called the access structure. Define 0 = fA 2 qual :
A 0 2 qual for all A 0 Ag be all the minimal qualied sets [1].
Suppose 0 = f Q 1 ; ; Q t g, by employing the optimal (k;k)-scheme
[16], the basis matrices L 0 and L 1 are constructed as follows:
Let k p = j Q p j, and Q p = fp 1 ; ;p k p g, for 1 p t. We will construct
a n2 k p 1 matrix E p , i 2f0; 1g according to the following steps: the p i row of
E p is the i-th row of the basis matrix B 0 of the (k p ;k p )-scheme. The elements
of other rows of E p are all 1's.
Then L 0 = E 1 . The construction of E p is similar to E p except we replace
the p i row of E p from the basis matrix B 1 of the (k p ;k p )-scheme instead of
B 0 . Then L 1 = E 1 kkE t .
Lemma 10 [10] The L 0 and L 1 are a pair of basis matrices of a perfect black
VC scheme for 0 such that the expansion rate is m = 2 jQ 1 1j ++ 2 jQ t 1j
and GRAY (white) = 1 1=m.
We now construct an n 2 k p 1 matrix F p , which has the property that
the elements in pi i row of F p are all 0's and the other rows of F p are all 1's,
here 1 p t. Then an auxiliary basis matrix A 0 = F 1 jjF i .