Visual Cryptography and Contrast Bounds - Visual Cryptography and Secret Image Sharing - page 243

T f1;:::;ng), the number h f1;:::;ng is minimal if for all S(f1;:::;ng
inequality (8.30) is satisfied with equality.
To this end, suppose

X X

h T < l T

STf1;:::;ng
jSjjTj mod 2

STf1;:::;ng
jSj6jTj mod 2

for some S(f1;:::;ng. But the contrast levels

( h T

for T S

h T =

h T 1 otherwise

and

( l T

for T S

l T =

l T 1 otherwise

satisfy (8.30), since

jfT j S T f1;:::;ng;jTjjSj mod 2gj =
jfT j S T f1;:::;ng;jTj6jSj mod 2gj :

This proves that if inequality (8.30) is not satisfied with equality we can
find smaller values for the parameters lT T and h T , which also satisfy (8.30).
Thus, in an optimal scheme (i.e., a scheme with the smallest possible values for
l T and h T ) inequality (8.30) is satised with equality for each T(f1;:::;ng.
Next we claim that

X X

T 0 2 jT 0 j1 T 0 2 jT 0 j1jTj h T = (8.32)

;6=T 0 f1;:::;ng T ( T 0 f1;:::;ng

for ;6= S f1;:::;ng satisfy (8.30) with equality.
To prove this we have to show that

T 0 2 jT 0 j1jTj

X X X

T 0 2 jT 0 j1 =

STf1;:::;ng
jSjjTj mod 2

;6=T 0 f1;:::;ng T ( T 0 f1;:::;ng

!

T 0 2 jT 0 j1jTj

X X X

T 0 2 jT 0 j1

T

STf1;:::;ng
jSj6jTj mod 2

;6=T 0 f1;:::;ng T ( T 0 f1;:::;ng