Block-DCT Based Secret Image Sharing over GF(2 8 )

Transcription

1 178 Int'l Conf. Security and Management SAM'15 Block-DCT Based Secret Image Sharing over GF(2 8 ) Rosemary Koikara 1,Mausumi Goswami 1,Pyung-Han Kim 2, Gil-Je Lee 2,Kee-Young Yoo 2 1 Computer Science and Engineering, Christ University, Faculty of Engineering Bangalore , Karnataka, India 2 School of Computer Science and Engineering, Kyungpook National University 80 Daehakro, Bukgu, Daegu , Republic of Korea Abstract In this paper, we are concerned with securing secret information in the form of a secret image in such a way that the secret image is shared among the participants and no one share gives information about the secret hidden. Secret image sharing is a method of sharing secret message among multiple cover images, making it difficult to trace the message. We concentrate on performing secret image sharing in the frequency domain. Here, secret image sharing is done on grayscale images using Block-DCT (Discrete Cosine Transform). This method has the advantage of DCT-based data hiding schemes, i.e., since the secret data is embedded into DCT coefficients of the cover images. In the proposed scheme we improve Koikara et al. s scheme by using operations over the GF(2 8 ) to share the secret image and also to reconstruct it back. In our method the security of the secret information is maintained and the quality of the stego image and secret image is improved. Keywords: Secret Image Sharing, Block-DCT, Image Security. 1 Introduction Security threats have always been a concern when it comes to transferring information over the Internet. Over the past 3 decades several approaches have been proposed to protect secret messages being passed in the network [1]. Cryptographic algorithms and protocols have been developed to secure data. Most of these algorithms involve transformation of information into unrecognizable data. Some examples of these transformation algorithms are RSA [2], DES [3], etc. and are called encryption algorithms. Encryption algorithms generally encrypt data with a key which a receiver will not be able to transform the data back into recognizable form without it. Though cryptographic keys can be used for protecting data we need efficient key management schemes to protect the key [4]. The issue with key management scheme is that most of them keep the key in one location. Hence, the key may become inaccessible due to some misfortune. Therefore, threshold schemes are needed due to the mentioned reason. In 1979, Shamir [4] and Blakley [5] introduced the idea of secret sharing. Shamir proposed a (k, n)-threshold scheme which divides the secret data into n independent shares such that the secret data can be reconstructed using k or more shares. His scheme is based on polynomial arithmetic operation and Lagrange s interpolation. The goal was to take k points, and it is guaranteed that a unique polynomial with those points would exist such that. Blakley s scheme was based on hyperplane intersections instead of polynomial interpolation. The proposed scheme uses Shamir s (k, n)- threshold scheme as it is more efficient than the one developed by Blakley [4]. There have been many methods proposed for secret image sharing based on Shamir s scheme. In 2002, Thien and Lin [6] first proposed the use of Shamir s (k, n)- threshold sharing scheme for sharing images. In this scheme the size of the generated shares is only 1/k of original image. But, the disadvantage is that there is a loss of information as the grayscale values were limited to a range of {0, 250}. So, a process was included to remove this loss, but this in turn increased the shadow size. Later in 2006, Bai [7] proposed a method similar to Thien and Lin s method which uses a combination of matrix projection and Shamir s method. Though the share size in this case was significantly less than the size of the secret image, there was still an increase in the size of the shares. In 2010, Alharthi and Atrey [8] proposed an improvement over Thien and Lin by reducing the computing time. Though this scheme reduces the time complexity and the security, it still has Thien and Lin s disadvantage of the share size being larger. In 2015, Koikara et al. [9] proposed a scheme that uses Shamir s scheme [4] by sharing images in cover images that have been transformed to the frequency domain using block-dct. In this paper, we improve Koikara et al. s scheme [9] using a Galois field (GF) polynomial arithmetic operation over GF(2 8 ). The proposed scheme is based on Shamir s (k, n)-threshold scheme. Shamir s secret sharing scheme has been used for images by researchers because of the utility and high security it provides. But we carry out the sharing and reconstruction procedures in the frequency domain using block-dct based transformation to embed the secret data because performing the data hiding in the frequency domain increases the security of the secret information. As a result, we can conclude that the quality of the stego images are superior to the previous work. This paper is organized as follows. Section 2 explains some related works in the field of secret image sharing. Section 3 describes the proposed scheme. The experimental results are shown in Section 4. Section 5 gives the conclusion of this paper.

2 Int'l Conf. Security and Management SAM' BACKGROUND There are different techniques that have been proposed for sharing a secret image over a set of cover images. Some techniques perform it in the spatial domain while others in the frequency domain. A simple way of embedding the shadow image into the respective cover image is the use of least significant bit (LSB) substitution. In certain cases the LSBs are randomly visited and in certain cases the pixel values are incremented or decremented. In this section, we briefly review the related works and introduce the basic concepts. 2.1 Discrete Cosine Transform Discrete Cosine Transform transforms an image into the frequency domain and removes the sine component of the image. For 2-dimensional DCT we first divide the image into blocks of size 8 8 pixels each. Then we perform the 2-D DCT as given in Eq. (1) on each of these blocks., Here, F(u, v) and f(x, y) represent a DCT coefficient at coordinate (u, v) and pixel value at (x, y) respectively. Eq. (2) gives the mathematical expression of inverse DCT., An 8 8 block is used to apply DCT because of basically the following major reasons: 1) Many experiments were performed for various block sizes and 8 8 gave the best results. 2) It is more complex to perform DCT on matrices of sizes greater than ) Matrices of size less than 8 8 do not retain enough information to continue along the pipeline. 2.2 Galois Field We need to perform modular arithmetic for the sharing of shares instead of real arithmetic so that we have a field in which interpolation is possible. Hence, we use the modular operation with a prime number. We can also use Galois Field arithmetic operation for this. From [10] we come to know that the order of a finite field must be a power of a prime p n, where is a positive integer. The finite field of order p n is generally written as (1) (2) GF(p n ) and it stands for Galois field. A characteristic 2 finite field with 256 elements is called Galois Field GF(2 8 ). In GF(2 8 ), we use an irreducible polynomial as Eq. (3) to compute the modular operation. (3) 2.3 Shamir s (k, n)-threshold Secret Sharing Adi Shamir [4] introduced a (k, n)-threshold secret sharing scheme and devised a scheme to divide a secret data into shares,,,, in such a way that: 1) Any k or more pieces makes easily computable. 2) Any or fewer pieces leaves D completely undetermined. Suppose, (, ), (, ),, (, ) be points in the 2-dimensional plane such that all the s are distinct and data is a number; and we need to divide it into shares. We can do it using Eq. (4). (4) where, = and,,, are randomly chosen integers from a uniform distribution over the integers in [0, ). Now we need a minimum of of these shares for the secret to be extracted. The coefficients of can be found out using interpolation, and will be the secret,. 2.4 Thien and Lin s Secret Image Sharing Thien and Lin [6] proposed a secret image sharing scheme based on Shamir s (k, n)-threshold sharing scheme. Unlike Shamir s scheme, a random coefficient is not used in the polynomial equation for creating the secret shares. The coefficients are pixels from the secret image. Since grayscale values of pixels have the range {0, 255}, the prime number is taken as 251, as it is the greatest prime between 0 and 255. The polynomial for each share can be represented by Eq. (5). (5) Where,,,, are r pixels from the secret image that have not been shared yet. This is a form of multi-secret sharing. The size of each shadow image is 1/k of the secret image. For the reconstruction of the secret image they use k of the n shares in Lagrange s interpolation as given in Eq. (6). (6) Where the participant s share value. Their scheme is a lossy method because the grayscale value is limited to the range {0, 250}. So, a process can be used to

3 180 Int'l Conf. Security and Management SAM'15 handle grayscale values larger than 250. But this increases the size of the shadow image. 2.5 Review of Koikara et al. s scheme In 2015, Koikara et al. [9] proposed a novel secret sharing algorithm in the frequency domain. It is based on the (, )-threshold scheme and has two phases: embedding and reconstruction phases. Embedding of the secret is done in the frequency domain by using Block-DCT. The polynomial equation used in this scheme is: In Eq. (7) and (8), is the section of secret image for which the polynomial is generated. Each section, has pixels. The coefficients to are pixel s intensity values from the secret image. The value of is changed according to the share. Here, 256 is used so as to prevent loss of information. When we use = 251 in Eq. (5) a loss in information is there for pixels greater than intensity value 251. But as 256 is not a prime number, both the remainder value and the quotient value has to be embedded. A parity check is performed to reduce the round-off error that occurs. But this parity check does not eliminate the round-off error, it only reduces it. The round-off error occurs during the processing of the cover images. Their scheme used the floor and modular operation on integers. Hence it has a few disadvantages. Since both the floor values and the modular values are stored in the share there is an increase in the size of each share. Also there will be additional complexity during the secret sharing and the secret restoration phases. 3 PROPOSED SCHEME In this section, we propose an improved scheme for secret sharing in the frequency domain using modular (8) (7) operation over GF(2 8 ). This has various advantages. There is a significant decrease in the share size of each shadow image. Since there are fewer number of operations there will also be a reduction in the complexity of the algorithm. In our scheme we use a (k, n) threshold scheme similar to Shamir s. The polynomial equation can be mathematically expressed by Eq. (9) with irreducible polynomial m(x) as Eq. (3). In Eq. (9) are k pixels from the secret image that are yet to be shared. The value of x determines the share. Hence, we use a multi-secret sharing scheme as in Thein and Lin s scheme in which the polynomial equation has more than one secret. All the coefficients in the polynomial equation represent a secret pixel. Using Eq. (9) we will obtain n shares that will be used to embed into the cover image. As with the other threshold schemes even our scheme does not require all the n shares for extraction of the secret image. A minimum of k shares are required for reconstructing the secret image. The k shares have to be valid shares. Once the secret information is extracted from each of the shares, the Lagrange s interpolation as given by Eq. (10) is used on the secret image. In Eq. (10), indicates the bitxor operation and implies the multiplicative inverse of a. Fig. 1(a) shows the basic block diagram of the secret sharing module and Fig. 1(b) shows the basic block diagram of the secret restoration module. In the following subsections we describe the algorithms used in our proposed scheme. Let, be nm N cover images and S be a P Q secret image such that we have n participants. Let,,, be nm N stego images. Also, let k be the threshold and S' be the extracted secret image. (9) (10) Fig. 1 Schematic Diagram: (a) Secret Sharing Module and (b) Secret Reconstruction Module

4 Int'l Conf. Security and Management SAM' Fig. 2 An example of a secret sharing phase for a 8 8 block of one cover image 3.1 Sharing Algorithm In this subsection, the sharing algorithm is described. Input: n cover images,,, of size and a secret image SI of size. Output: n stego images,,, of size. Step 1: Divide each cover image,,, into non-overlapping blocks of size 8 8 pixels and apply 2-D DCT to each of these blocks. This is done by using the formula given in Eq. (1) in each one of the 8 8 blocks. Step 2: Calculate the minimum value in each cover image and add it to each pixel of the cover images. This is done to avoid numbers less than 0. Now, round off-the pixels. Step 3: Take k number of not yet shared pixels from SI and use the polynomial equation given in Eq. (9) to find n shares. Step 4: Embed the n shares found in Step 2 into the 3 rd LSBs of corresponding DCT transformed,,,. Step 5: Add the parity bit information to reduce the error due to round-off. When we perform DCT on a block of pixels we get floating point numbers which are truncated. Hence, there is a loss of information. Parity check reduces this loss of information to some extent. Let be the pixel to which parity information must be added. Now we use the following substeps: 5.1: Calculate the even parity of and embed it into the 1 st LSB position. 5.2: Calculate the odd parity of x and embed it into the 2 nd LSB position. Step 6: Repeat Step 3, Step 4 and Step 5 until all pixels of SI are embedded into,,,. Step 7: Obtain n stego images,,, by performing inverse DCT on the new,,,. IDCT is also done by dividing the images into non-overlapping blocks of size 8 8 pixels. The equation for IDCT is given in Eq. (2). Fig. 2 shows the example of secret sharing for a 8 8 block for cover image 1. Suppose, for a (4, 5) threshold scheme, k = 4 and n = 5. If in Step 3 the 4 not yet shared pixels are =156, =159, =158, =155. Then, the polynomial equation is. The 5 shares are calculated as follows, From Fig. 2 after Step 2 we have the 1 st pixel of the cover as 552. Now we need to insert the 1 st bit of f(1) to the 3 rd LSB of 552. After embedding it remains the same. For the parity check in Step 5, x = (552) 10 = ( ) 2. Then x can be changed as follows, = ( ) 2 = (555) Reconstruction Algorithm In this subsection, the data extraction algorithm for reconstructing the secret image is given. Input: k stego images,,,, of size Output: Extracted secret image. Step 1: Divide the k stego images,,,, into non-overlapping blocks of size 8 8 pixels and apply 2-D DCT on each of these blocks. This is done by using the formula given in Eq. (1). Step 2: Calculate minimum value from the stego image and add it to all the pixels of the stego image and round-off the pixels.

5 182 Int'l Conf. Security and Management SAM'15 Fig. 3 An example of a secret reconstruction phase for an 8 8 block of one stego image. Step 3: Take eight pixels from each of,,, and perform parity check for each of the 8 pixels. Let be one pixel, then parity is performed the following sub-steps: 3.1 Calculate the even parity of x and embed it in x s 1 st LSB 3.2 Calculate the odd parity of x and embed it in x s 2 nd LSB. 3.3 The computed x is called x'. 3.4 If x = x' then it is correct else increment or decrement the x and go to Substep 3.1. Step 4: Extract the 3rd LSB of each of the parity checked pixels to get k shares of the k pixels of the secret image. Step 5: Use above k pixels in Lagrange s interpolation to retrieve k pixels of the secret image. The equation of Lagrange s interpolation is given in Eq. (10). Step 6: Repeat Step 3, Step 4 and Step 5 until all the pixels of the secret image are processed. Fig. 3 shows the reconstruction phase for 8 8 pixels block of stego image 1. Suppose, after Step 2 we have the first pixel of the block as 554. So, for Step 3 we have the first pixel of block as 554. So, for Step 3 we have if x = (554) 10 = ( ) 2, then we have x' = ( ) 2 = (555) 10. Hence, we get back the secret bit from the 3rd LSB. 4 EXPERIMENTAL RESULTS In this section some of the results of experiments carried out are given in order to evaluate our scheme. For the evaluation we measured the quality of the stego images when compared to the respective cover images. Fig. 4 Cover Image (a) Set I (b) Set II (c) Set III and (d) Secret Image SI

6 Int'l Conf. Security and Management SAM' Fig. 5 Reconstructed Secret Image for set I, set II and set III using (a) Koikara et al. s and (b) Proposed. This comparison was carried out using the peak-signal-tonoise rate/ratio which we further refer to as PSNR. The PSNR is defined as follows, (11) The mean square error (MSE) of an image of size M N pixels is defined as follows, (12) where f(x, y) is the pixel intensity value at (x, y) of the cover image, i.e., the original pixel value of g(x, y) is the pixel intensity value at (x, y) of the stego image. The PSNR value is expressed in db. The higher the PSNR value, the higher is the quality of the stego image when compared to the cover image. But, a low value of PSNR indicates higher distortion. When the PSNR value is less than 30dB the image undergoes a lot and losses significant information. We evaluate the scheme by using 3 sets of images. The three sets of cover images used are shown in Fig. 3. The results for (4, 5) threshold scheme is evaluated in this section. The resulting secret image for Koikara et al. s scheme and our proposed scheme is given in Fig. 5(a) and Fig. 5(b) respectively. We compared the results of Koikara et al. s scheme in the frequency domain with the new one. In Table I, we compare the PSNR values of the stego images. The PSNR value of stego images using both Koikara et al. s [9] and the proposed scheme is given. By observing Table I we notice that there is an increase in the PSNR values of the stego images. This increase implies that there is an improvement in the quality of the stego images. The PSNR values of all stego images are more than 35dB hence they are acceptable. We also have Table II which gives us the comparison of PSNR values of the extracted secret image. From Table II, we notice that there is a significant increase in the quality of the secret images when compared to Koikara et al. s scheme. This is due to the fact that in Koikara et al. s field there was a greater margin for error since both the floor as well as mod values had to be embedded. But in the proposed scheme just one value i.e., the mod value is stored. To make sure that the amount of data embedded in both the schemes are same we embed random bits in the pixels of the stego images into which secret has not been embedded. This is done because the secret shares are not big enough to be embedded into the entire cover images. As the data hiding is done in the frequency domain the choice of the cover image used plays a huge role in this scheme. Cover Image Set TABLE I PSNR COMPARISON OF STEGO IMAGE FOR (4,5) -THRESHOLD SCHEME Stego Image -1 Stego Image -2 Stego Image -3 Stego Image -4 Stego Image -5 Previous Proposed Previous Proposed Previous Proposed Previous Proposed Previous Proposed Set I Set II Set III

7 184 Int'l Conf. Security and Management SAM'15 TABLE II PSNR COMPARISON OF EXTRACTED SECRET IMAGE PSNR (db) Extracted Secret Image Secret Image Set Previous Proposed (I) (II) (III) CONCLUSION We have proposed the (t, n)-threshold secret sharing in the frequency domain using 2D-DCT and performing operations over GF(2 8 ). Koikara et al. s scheme was done using integer arithmetic. The main aim of the proposed scheme is to improve the quality of the stego image when compared to Koikara et al. s scheme as well as to improve the extracted secret image. From Table II we can conclude that the PSNR value of the extracted secret image has increased by roughly 6dB to 8dB. As we observe in Table III the embedding capacity of the stego images have been maintained. All the operations were done using GF(2 8 ) to prevent any loss of information that may take place. When we perform secret sharing in the frequency domain the security of the information hidden increases as the embedding is done into the DCT coefficients of the cover image and not directly into the pixels of the cover image. While using this particular scheme we must be vary about the type of cover image used. The future works in secret sharing in the frequency domain would be to make it more generalized, such that a set of any cover images may be used for sharing the secret. 6 ACKNOWLEDGEMENTS This research work was supported by an academic student exchange program between Christ University, India and Kyungpook National University, Republic of Korea signed on 21 st February 2013 and the IT R&D program of MSIP/IITP. [ , Self-Organized Software platform (SoSp) for Welfare Devices]]. 7 REFERENCES [1] Imai, Hideki, et al. "Cryptography with information theoretic security." Information Theory Workshop, Proceedings of the 2002 IEEE. IEEE, [2] R. L. Rivest, A. Shamir and L. Adelman. 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