Image Secret Sharing - Visual Cryptography and Secret Image Sharing - page 381

14.3 Approaches for Image Sharing
14.3.1 Shamir's Secret Sharing Scheme
Shamir's secret sharing scheme is based on the Lagrange interpolation [2].
Given a set of points (xi, i ;y i )(i = 0; 1; 2;:::;k), the Lagrange interpolation
polynomial L k (x) can be constructed using:

k X k Y

xx i
x j x i

f k (x) =

y i (14.2)

i = 0 j=0;i6=j

Given a secret, it can be easily shared using this interpolation scheme. If
GF(q) denotes a Galois field (q > n), the following polynomial is constructed
by choosing proper coecients a 0 , a 1 , , a k from GF(q), which satisfy:

k X

f k (x) = s + a i x i

(14.3)

i=0

where s is the secret key. The coecients are randomly chosen over the
integers [0;q) and the details are provided in [2]. Suppose s i = f(ai),(i i ),(i =
0; 1; ;k), each S i is known as a share and they all can be delivered to
different persons.
Now we would like to reconstruct the original secret. Suppose k people
have provided their shares si, i , (i = 0; 1; ;k). The following Lagrange inter-
polation polynomial is utilized to reconstruct the original secret:

k X k Y

aa i
a j a i

P k (x) =

s i (14.4)

i=0 j=0;i6=j

where addition, subtraction, multiplication, and division are defined over
GF(q).

P k (a i ) = s i ; i = 0; 1; 2; ;k; s = P k (0);

(14.5)

Thus, we can obtain the original secret s [1][2].

14.3.2 Color Image Sharing Based on the Lagrange Interpo-
lation
When an image is treated as a secret, we can share the secret based on the
Lagrange interpolation. We consider the image to be a matrix; and Lagrange
interpolation is generalized for matrices.
We assume a grayscale image corresponds to a matrix A. Given ma-
trix A i = (a i w;h ) WH , where w and h are integer (w = 1; 2; ;W; h =