Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 48

Theorem 2 In the SC model, the pixel expansion of a c-color (n;n)-threshold
scheme, for any c;n 2, is lower bounded by

( c 2 n1 1; if n is even
c 2 n1 c + 1; if n is odd.

m

Note that the above lower bound implies that the schemes of [7, 11] have
optimal pixel expansion. In [3] an alternative construction of c-color (n;n)-
threshold schemes with optimal pixel expansion is provided. The construction
is the following:

Construction 2 Fix any color i; base matrix Ci i consists of the following
columns:
1. for r = 0; 1;:::;dn=2e 1 include the 2r

columns, having 2r
entries equal to and the remaining ones of color i;
2. for any color j 6= i, for r = 0; 1;:::;d n 2 e1 include the n

2r1

columns having 2r1 entries equal to and the remaining ones of
color j;

Below is an example for c = 3 and n = 4. For such a scheme m = 23 and
= 1=23.

2 3

1 2 2 23 3 31 1123
1 2 22 3 3311123
1 22 2 33 3 11123
12 2 23 3 3 11 123

4 5

C 1 =

2 3

2 1 1 13 3 32 2213
2 1 11 3 3322213
2 11 1 33 3 22213
21 1 13 3 3 22 213

4 5

C 2 =

2 3

3 1 1 12 2 23 3312
3 1 11 2 2233312
3 11 1 22 2 33312
31 1 12 2 2 33 312

4 5

C 3 =

Other results of [3] are

A characterization of maximal contrast (k;n)-thresholds schemes. The char-
acterization describes the schemes with a linear programming problem.

A construction of c-color (2;n)-threshold schemes with improved pixel ex-
pansion with respect to [10, 11].