An Image Secret Sharing Method Based on Shamir Secret Sharing

This paper presents an image secret sharing method based on Shamir secret sharing method. We use the matrix projection to construct secret sharing scheme. A secret image to be divided as n image shares such that: i) Any k image shares (k ≤n) can be used to reconstruct the secret image in lossless manner and ii) Any (k-1) or fewer image shares cannot get sufficient information too reveal the secret image. It is an effective, reliable and secure method to prevent the secret image from being lost, stolen and corrupted. In comparison with other image secret sharing method this approach’s advantages are its strong protection of the secret image and its ability for real time processing.


Introduction
The effective and secure protection for important message is a primary concern in commercial and military applications.
Numerous techniques, such as image hiding and watermarking, were developed to increase the security of the secret. The secret image sharing approaches are useful for protecting sensitive information [1]. The main idea of secret sharing is to transform an image into n shadow images that are transmitted and stored separately. The original image can be reconstructed only if the shadow images that participated in the revealing process from a qualified set [2]. The (k; n)-threshold image sharing schemes were developed to avoid the single point failure. Hence the encoded content is corrupted during transmission. In these schemes, the original image can be revealed if k or more of these n shadow images are obtained. Moreover, the users who with complete knowledge of k 1 shares cannot obtain the original image. Blakley [3] & Shamir [4] independently proposed original concepts of secret sharing in 1979. In these (k; n)-threshold schemes encode the input data D into n shares, which are then distributed among k recipients. D can be reconstructed by anyone who obtains a predefined number k, where 1 k n, of the images.
A better image secret sharing approach was presented by Thien & Lin [1]. They used Shamir's secret sharing scheme to share a secret image with some cryptographic computation. The method significantly reduces the size of the secret image and the secret image can be reconstructed with good quality. Ramp secret sharing schemes are another types of secret sharing schemes [8][9][10][11]. In ramp schemes, a secret can be shared among a group of participants in such way that only sets of at least k participants can reconstruct the secret and k1 participants cannot [12]. The rest of this paper is organized as follows. Section II reviews the Review of shamir's secret sharing scheme Shamir [4] developed the idea of a (k, n)-threshold based secret sharing technique (k ≤n). The technique allows a polynomial function of order (k -1) constructed as, single shareholder knows the secret value s 0 [7]. Actually, no groups of (k 1) or fewer secret shares can discover the secret s 0 . That is when k or more secret shares are available, then we may set at least k linear equations y i = f(x i ) for the unknown s i 's. The unique solution to these equations shows that the secret value s 0 can be easily obtained by using Lagrange interpolation [4].

Proposed Method
In this section, we examine the application of some secret sharing schemes. We have worked a new approach to construct secret sharing schemes based on field extensions in [13]. In this paper, we generalise the results of [13].

Proposed scheme
Consider the matrix I is an image with height of h and wideness of w. The height corresponds to row number of matrix and the wideness corresponds to column number. Let the secret space be Mq for a pixel, where This set consists of the elements of the matrix I. Let the secret be the image I and the threshold structure be (k; n). In this case, it can be constructed a secret sharing scheme as follows.
The matrix P(x) is generated which consisting of height of h and by using I.
The a ij entry corresponds to i th row and j th column of matrix I.
It is clear that the degree of polynomial pij is (k -1). The columns of the matrix I are divided into pieces that has length of (k -1 It is determined an ID number for each participant. The i v GF q i n = ≤ ≤ by Algorithm 1 [13]. Then the matrix This polynomial matrix is written as the matrix Yi by using  (2)) f x x x x x GF x = + + + + ∈ to construct GF (256). It can be constructed a (3; 5)-threshold schemes by using the following matrix I.
The matrix P(x) can be constructed as follows. The leading coefficient is randomly selected and the other coefficients are chosen from matrix I.
The coefficient of polynomial in the matrix P(x) is moved to GF(256). Therefore, it is obtained the elements of matrix T(x) = [tij(x)]; (t(x) 2 (GF(q))[x]).
( ) 2    The pieces of participants are as follows. These elements correspond to the following matrices in M256.
At least 3 participants can recover the image by combining their shares by using Lagrange Interpolation in [13]. It is seen that the original secret image in Figure (1a) and the secret pieces are seen (1b-1d). Reconstructed image is seen in Figure (1f).

Advantages
It is known that a file in the computer environment can be

Conclusion
We proposed an image secret sharing method based on Shamir secret sharing. We have two techniques. i) Secret sharing scheme using matrix projection and ii) Shamir's secret sharing scheme. A secret image can be successfully reconstructed from any k image shares but cannot be revealed from any (k-1) or fewer image shares.
The size of image shares is smaller than the size of the secret image.
Our scheme is defined over GF(256). So it is a lossless scheme. This is another advantage of our scheme. So the proposed scheme stands well, in terms of security.