Polynomial-Based Image Sharing - Visual Cryptography and Secret Image Sharing - page 399

a polynomial of q 0 (x) is constructed as follows:

g(x)=(249 (x4)
(x1)

+ 147 (x2)

(x4) )mod251
=(249 (2) 1 (x 4) + 147 (2) 1 (x 2))mod251
= (200x + 100)mod251

Following up the result, the first two pixels of the target secret image, S 0
= 100 and S 1 = 200 are obtained from the coecients of q 0 (x) = 200x + 100
mod251.

15.3 Preliminaries and Related Works
15.3.1 Thien-Lin Scheme
The (t;n)-threshold scheme in visual cryptography was proposed by Naor
and Shamir [6], where the target secret (image) could only be recognized by
the human eye if there were exactly t or more than t share-images stacked.
However, the share-images generated could only reveal one target secret. When
there are more than 2 target secrets, the numbers of share-images required,
increases. As a result, the occupied share-images, held by participants, also
increase. Inspired by these observations, Thien and Lin (Thien-Lin scheme)
[8] proposed a specific method to accommodate a higher number of share-
images, with the normal storage requirements, using LaGrange polynomial
construction. In this method, there are a total of n share-images, with each
of them being only

1

t times the target secret image. This is also done in the
(t;n)-threshold scheme.

15.3.2 Lin-Tsai Scheme
In [4], Lin and Tsai (Lin-Tsai scheme) proposed a scheme of sharing im-
age, where the requirements of capacity and detecting ability are further
considered when compared to the study of [8]. Similarly, a polynomial of
q(x)=(a 0 + a 1 x + a 2 x 2 + + a i1 x i1 ) of degree t-1 is applied upon the
concept of the (t;n)-threshold scheme. The q(x) is then devised for exact re-
construction when t pairs of (xi, i , q(x i )) are offered among n share holders,
where x i is one of the pixels of a pixel-value and q(xi) i ) is the share held in
the share holder. Accordingly, the pixels of the target secret image can be
obtained from the coecients of the new one of q 0 (x) via the (t;n)-threshold
polynomial construction. The procedures in [4] are briefly given as follows: