Polynomial-Based Image Sharing - Visual Cryptography and Secret Image Sharing - page 397

were created so that they would be detected by those malicious shareholders
in the target secret image (t;n)-threshold system, thus acting as a deterrent.
In 2008, Wang et al. [10] took advantage of the CRC (Cyclic Redundant
Code) in conjunction with the hash function to make authentication-based
steganography more robust, in order to prevent the attacks of fake share of-
ferings. In addition to making it more robust, both the capacity and the image
quality remain comparable to the scheme suggested by the scheme in [12]. In
this chapter, we introduce the polynomial-based image sharing scheme by
reviewing some preliminaries and related works and depicting Wang et al.'s
approach. Moreover, some improvements of Wang et al.'s approach are pro-
posed in this chapter.
The presentation of this chapter is organized as follows. In Section 15.2,
the concept of the polynomial-based sharing scheme is introduced. In Section
15.3, we introduce some preliminaries and related works about the polynomial-
based image sharing scheme. Wang et al.'s scheme and its improvements are
presented in Section 15.4. Observations and experiments of benchmark images
are presented and discussed in Section 15.5. Finally, conclusions are offered in
Section 15.6.

15.2 Polynomial-Based Sharing Scheme
15.2.1 Shamir's Secret Sharing Scheme
A (t;n)-threshold scheme is a method of sharing a secret among n participants
such that any subset consisting of t participants can reconstruct the secret,
but no subset of smaller size can reconstruct the secret. The first method
was provided in 1979 by Shamir [7] and is known as Shamir's secret sharing
scheme. In the initial state, a dealer chooses a large prime number p, and makes
p public. The following calculation is performed on Z p : A secret, s 2 Z p exists,
so the dealer randomly generates a (t-1)th degree polynomial, in Z q :

q(x) x + q 1 x + q 2 x 2 + + q t1 x t1 (modp)

where q(0) = s is the secret. The dealer distributes the shadow, yi i q(x i )(mod
p) to node P i , where x i is the ID number for P i .
Suppose that t participants get together to reconstruct the polynomial.
Also, assume that their pairs are (x 1 ;y 1 ), (x 2 ;y 2 ), , (x t ;y t ). The polyno-
mial q(x) can be reconstructed by solving the following equation:
2 3 2 3 2 3

1 x 1 x t1 s

q 1
.
q t1

y 1
y 2
.
y t

1
1 x 2 x t1

4 5 4 5 4 5

2
. . . . . .
1 x t x t1

(mod p)

t