Visual Cryptography and Random Grids - Visual Cryptography and Secret Image Sharing - page 184

Lemma 1 If R 1 and R 2 are two independent random grids with T (R 1 ) =

T (R 2 ) = 1=2;

(1) T (R 1 R 2 ) = T (R 2 R 1 ) = 1=4;

(2) T (R 1 R 1 ) = 1=2:

R to be the inverse random grid of R if and only if

We dene

r 0 = r

for each r 0 in R where r is the corresponding pixel of r 0 in R and r denotes
the inverse of r. It is easy to see that r r = 1 and R R = 1 (1 denotes a
grid in which all pixels are opaque), that is

t (r r) = 0 andT(R R) = 0

(7.6)

P rob (r 0 = 0) =P rob (r = 0) =P rob (r =

For each pixel r 0 in R, since

1

1) =

2 , we obtain

1
2

1
2

t (r) =

( R) =

and T (7.7)

In general, the relationship of the average light transmissions of R 2 R is
given in Lemma 2.

Lemma 2 If R is a random grid with T (R) = , T (R) = 1 .

P rob (r = 0) =

Proof From T (R) = , we know t ha t for any pixel r 2 R

P rob (r = 1) = 1 . Let r 0 2 R be the corresponding pixel of r. Thus,
P rob (r 0 = 0 ) =P rob (r = 1) = 1 andP rob (r 0 = 1) =P rob (r = 0) = .
We have

and

T (R) = 1 .

Consider two independent random grids X and Y . There is another im-
portant property called the principle of combination : if we cut a section, say
A, from X and replace it with section B (with the same size as A) from Y ,
the result, denoted as Z = (X nA) \B, is another random grid, i.e.,

1
2

T (Z) = T (X) = T (Y ) = (7.8)

We shall expose how to use the aforemnetioned characteristics of random
grids to accomplish visual secret sharing in the next section.