Reversible Data Hiding in Encrypted Image Using Multiple Data-Hiders Sharing Algorithm

Reversible Data Hiding in Encrypted Image (RDHEI) is a technology for embedding secret information in an encrypted image. It allows the extraction of secret information and lossless decryption and the reconstruction of the original image. This paper proposes an RDHEI technique based on Shamir’s Secret Sharing technique and multi-project construction technique. Our approach is to let the image owner hide the pixel values in the coefficients of the polynomial by grouping the pixels and constructing a polynomial. Then, we substitute the secret key into the polynomial through Shamir’s Secret Sharing technology. It enables the Galois Field calculation to generate the shared pixels. Finally, we divide the shared pixels into 8 bits and allocate them to the pixels of the shared image. Thus, the embedded space is vacated, and the generated shared image is hidden in the secret message. The experimental results demonstrate that our approach has a multi-hider mechanism and each shared image has a fixed embedding rate, which does not decrease as more images are shared. Additionally, the embedding rate is improved compared with the previous approach.


Introduction
Multimedia security technology is used to prevent unauthorized users from copying, sharing, and modifying media content. To prevent this problem, encryption and information hiding are often used to protect media content. As far as information hiding technology is concerned, traditional information hiding technology will destroy the content of the cover image. However, in some exceptional cases, such as military, medical, and legal document images, the slight distortion of the image is entirely unacceptable. Therefore, whether these images can be completely restored is very important. Reversible data hiding scheme (RDH) can correspond with the requirement of being lossless. RDH methods applied the methodology of changing context to hide the secret data in cover media. After data extracting, the changing context will be fully recovered to the cover media. On the other hand, RDHEI (Reversible Data Hiding in Encrypted Images) technology can combine encryption technology with RDH technology, which can not only hide secret information in the image, but can also encrypt the image to protect the image content.
RDH techniques can be broadly classified into three types: (1) Difference Expansion [1][2][3]: Difference expansion is performed by expanding the difference between adjacent pixels, and then secret information is embedded into the difference. Since the difference will be expanded after the secret information is embedded, this technique will inevitably produce a larger distortion; (2) Histogram Shift [4][5][6]: through Histogram Shift, the histogram is shifted by the original image or the histogram of the predicted discrepancy, and the empty position after the shift is used to embed the secret information; (3) Lossless Compression [7,8]: The secret information is hidden in the extra space after compressing the original image. Since lossless compression may lead to significant degradation of visual quality, it has received less attention. By and large, the RDH approach is based on the above with some other added strategies. For example, in 2019, Zhao et al. proposed a RDHEI approach [22], which expands the original single information hider into multiple information hiders. The total embedding volume of Chen et al.'s method is fixed, so the embedding rate decreases as the number of shared images increases.
In this study, we propose a new RDHEI technique where we hide the pixel values into the polynomial coefficients by dividing multiple pixels into a group. First, the image owner confidentially shares the image, generating a shared image and leaving space in each shared image for information embedding. Then, the information recipient can embed a confidential message after receiving the shared image. Finally, the recipient can decrypt the image and remove the hidden information. The recipient needs to obtain at least a Threshold before the original image can be recovered. Our approach enables multiple information hiders and has a more suitable fixed embedding rate. Furthermore, it solves the problem that the embedding rate becomes smaller as the number of shared images becomes larger and successfully improves the embedding rate compared with the previous method.
The remaining sections of this study are organized as follows. Section 2 reviews Chen's existing work [22]. Section 3 describes our novel secret sharing images with multiple data hiders method. This is followed by its experimental results along with the analysis and the existing works comparisons in Section 4. Finally, the conclusion is drawn in Section 5.

Preliminary
This section introduces Chen et al.'s approach, which develops RDHEI techniques through Shamir's Secret Sharing method. Chen et al.'s method has the role of multiple information hiders, but their total embedding is fixed, so the embedding rate decreases as the number of shared images increases. Chen et al.'s approach is introduced in the following. Figure 1 shows the flowchart of encryption, embedding, extraction, and decryption by Chen et al. First, the content-owner encrypts the image and uses the key ke to encrypt the original image into n. Each encrypted image is the same size as the original image. Then, each encrypted image is assigned to a different information hider. Each information hider, Data-hider t, hides the confidential information in the encrypted image t through the key kh t . Finally, the 1 ≤ t ≤ n receiver simply collects any k encrypted images with embedded information and related keys to extract the embedded information and decrypt it to obtain the original image.
embedding rate decreases as the number of shared images increases.
In this study, we propose a new RDHEI technique where we hide the pixel v into the polynomial coefficients by dividing multiple pixels into a group. First, the i owner confidentially shares the image, generating a shared image and leaving spa each shared image for information embedding. Then, the information recipient can e a confidential message after receiving the shared image. Finally, the recipient can de the image and remove the hidden information. The recipient needs to obtain at le Threshold before the original image can be recovered. Our approach enables multip formation hiders and has a more suitable fixed embedding rate. Furthermore, it solv problem that the embedding rate becomes smaller as the number of shared image comes larger and successfully improves the embedding rate compared with the pre method.
The remaining sections of this study are organized as follows. Section 2 rev Chen's existing work [22]. Section 3 describes our novel secret sharing images with tiple data hiders method. This is followed by its experimental results along with the ysis and the existing works comparisons in Section 4. Finally, the conclusion is draw Section 5.

Preliminary
This section introduces Chen et al.'s approach, which develops RDHEI techn through Shamir's Secret Sharing method. Chen et al.'s method has the role of mu information hiders, but their total embedding is fixed, so the embedding rate decrea the number of shared images increases. Chen et al.'s approach is introduced in the fo ing. Figure 1 shows the flowchart of encryption, embedding, extraction, and decry by Chen et al. First, the content-owner encrypts the image and uses the key ke to en the original image into n. Each encrypted image is the same size as the original im Then, each encrypted image is assigned to a different information hider. Each inform hider, Data-hider t, hides the confidential information in the encrypted image t thr the key kht. Finally, the 1 ≤ ≤ receiver simply collects any k encrypted images embedded information and related keys to extract the embedded information and de it to obtain the original image.  The content-owner performs the image encryption by first dividing image I into parts A and B. Part A is the first 10 pixels of the image, and the remaining part is part B. Figure 2 Entropy 2023, 25, 209 4 of 16 is the representation. Part A is used to store the parameters of the information hiding and the parameters of the histogram displacement process. First, the following bits are extracted from part A: all bits of the first two pixels and 3 LSBs of the last 8 pixels. Next, all the extracted bits are defined as AP. Then, scan all pixels in the part B, recording the location of the pixel values ≥ 250 with the location map, modify these pixel values to 249, and create a histogram of part B. From the histogram, we found the most suitable embedding point for embedding the location map and AP, which was defined as PP. In the histogram, all the values between o PP + 1 and 249 are shifted one place to the right so that the location map and AP can be using PP for 0 and for 1, as shown in Figure 3. Finally, the PP value is embedded into the 3-LSB part of the last 8 pixels of the A part. After the above pre-processing, image I is modified to image I .
The content-owner performs the image encryption by first dividing imag A and B. Part A is the first 10 pixels of the image, and the remaining part is pa 2 is the representation. Part A is used to store the parameters of the informa and the parameters of the histogram displacement process. First, the follow extracted from part A: all bits of the first two pixels and 3 LSBs of the last 8 p all the extracted bits are defined as AP. Then, scan all pixels in the part B, re location of the pixel values 250 with the location map, modify these pixel v and create a histogram of part B. From the histogram, we found the most suita ding point for embedding the location map and AP, which was defined as PP togram, all the values between o PP + 1 and 249 are shifted one place to the the location map and AP can be using PP for 0 and for 1, as shown in Figure 3 PP value is embedded into the 3-LSB part of the last 8 pixels of the A part. Aft pre-processing, image I is modified to image I′.  Using Equation (1), the B part of the image I′ is shared confidentially v Secret Sharing, where Ti,j (0) is a constant, ai,j (α) is an integer random number, a pixel value of position (i,j). The image owner uses key ke to generate n non-z integers xi,j (t) , = 1, 2, … , , to bring xi,j (t) into Equation (1) within x to obtain th tial sharing result Fi,j (xi,j (t) ), = 1, 2, … , . It then hides the parameters t and n two pixels of part A. From this we can obtain n encrypted images E (t) , = 1, 2 The content-owner performs the image encryption by first dividing image I into parts A and B. Part A is the first 10 pixels of the image, and the remaining part is part B. Figure  2 is the representation. Part A is used to store the parameters of the information hiding and the parameters of the histogram displacement process. First, the following bits are extracted from part A: all bits of the first two pixels and 3 LSBs of the last 8 pixels. Next, all the extracted bits are defined as AP. Then, scan all pixels in the part B, recording the location of the pixel values 250 with the location map, modify these pixel values to 249, and create a histogram of part B. From the histogram, we found the most suitable embedding point for embedding the location map and AP, which was defined as PP. In the histogram, all the values between o PP + 1 and 249 are shifted one place to the right so that the location map and AP can be using PP for 0 and for 1, as shown in Figure 3. Finally, the PP value is embedded into the 3-LSB part of the last 8 pixels of the A part. After the above pre-processing, image I is modified to image I′.  Using Equation (1), the B part of the image I′ is shared confidentially via Shamir's Secret Sharing, where Ti,j (0) is a constant, ai,j (α) is an integer random number, and I′i,j is the pixel value of position (i,j). The image owner uses key ke to generate n non-zero random integers xi,j (t) , = 1, 2, … , , to bring xi,j (t) into Equation (1) within x to obtain the confidential sharing result Fi,j (xi,j (t) ), = 1, 2, … , . It then hides the parameters t and n into the first two pixels of part A. From this we can obtain n encrypted images E (t) , = 1, 2, … , .
Data embedding is performed using the n information hiders separately. The t-th information hider obtains the shared encrypted image E (t) by scanning the first 2 pixels to obtain the t and n values. It divides the pixels into groups. Each group contains n pixels Using Equation (1), the B part of the image I is shared confidentially via Shamir's Secret Sharing, where T i,j (0) is a constant, a i,j (α) is an integer random number, and I i,j is the pixel value of position (i,j). The image owner uses key ke to generate n non-zero random integers x i,j (t) , t = 1, 2, . . . , n, to bring x i,j (t) into Equation (1) within x to obtain the confidential sharing result F i,j (x i,j (t) ), t = 1, 2, . . . , n. It then hides the parameters t and n into the first two pixels of part A. From this we can obtain n encrypted images E (t) , t = 1, 2, . . . , n.

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Data embedding is performed using the n information hiders separately. The t-th information hider obtains the shared encrypted image E (t) by scanning the first 2 pixels to obtain the t and n values. It divides the pixels into groups. Each group contains n pixels and replaces the l (1 ≤ l ≤ 7) LSB of the t-th pixel in each group with the secret information. As a result, the n encrypted images Em (t) , t = 1, 2, . . . , n with the secret information embedded can be obtained. The embedding rate of this method is W H−2 n ·l W H ≈ l n bpp (bit per pixel), where W and H are the width and height of the image, respectively.
In this part of information extraction and image decryption, it is assumed that the first k shared images are Em (t) , t = 1, 2, . . . , k and the key is kd. In the information extraction phase, for any shared image Em (t) , t = 1, 2, . . . , k. By sharing the first 2 pixels of the image, t and n can be obtained. Next, divide all the pixels in the B part of Em (t) into a group, and extract the l-LSB of the t-th pixel in each group to obtain the secret information of the shared image Em (t) . In image decryption, for the pixel Em (t) i,j , t = 1, 2, . . . , k at the coordinates (i,j), and the non-zero random integers x i,j (t) , and t = 1, 2, . . . , k generated by the key kd, equation can be obtained through the Lagrange polynomial. (1) Polynomial . . , k may not be able to obtain the correct F i,j (x) because l bits are hidden into l-LSB. Without a loss of generality, assuming that the first pixel to share an image is hidden in l-bit, i.e., Em (1) i,j is hidden in l-bits, then when the Em (1) i,j of l-LSB is modified to 0, 1, 2, ..., 2 l − 1, one of them must be Em (1) i,j before embedding l-LSB, which is called the correct Em (1) i,j . The correct Em (1) i,j can be identified by using where T i,j (0) is the constant used in the encryption step. After constructing F i,j (x), from Equation (1) pixel I i,j of the coordinates (i,j) can be obtained and therefore the image I can also be obtained. Finally, the image I is restored to the original image I using the information in part A.

Proposed Method
In this section, we propose a new approach to the interplay of confidential image sharing and information hiding. Our approach is also based on Shamir's Secret Sharing technique of confidential sharing. The traditional Shamir (k, n)-threshold secret sharing technique shares confidential information D into n shared information and the original confidential information D can be calculated by taking out k of the shared information. This method requires the construction of a k−1 polynomial, as follows: where a 0 = D, p is a prime, constant, or coefficient a i < p, i = 0, 1, . . . , k − 1. If this mechanism is applied to image sharing, for example, for a grayscale image with a gray scale of 256, to share a pixel value P, the above polynomial is usually used with a 0 = P and p = 251. Because the value range of pixel value P is 0~255, all pixel values of the image must be pre-processed so that the range of all pixel values becomes 0~250. We propose the finite field (GF(2 n )) strategy, modifying Shamir's polynomial formulation, as follows: where the constants or coefficients are a i ∈ GF 2 8 , i = 0, 1, . . . , k − 1 and both multiplication and addition are performed using the GF(2 n ) operator. Since the GF(2 n ) quadrature is closed, i.e., if a, b ∈ GF(2 n ), then a + b ∈ GF(2 n ), a − b ∈ GF(2 n ), a × b ∈ GF(2 n ), and a/b ∈ GF(2 n ), the polynomial can be directly applied to image sharing, and in the case of grayscale images with 256 (=2 8 ) gray levels, no pixel-specific pre-processing is required at all. Our method has three roles: Content-owner, Data-hiders, and Receivers, as shown in Figure 4, which shows the operation flow of our method in image encryption, data embedding, data extracting, and image decryption. We take Shamir's (k, n)-threshold as an example, where n means n shared images are generated and k means k shared images are obtained to decrypt the original image. First of all, the image owner can generate n shared images through Shamir's Secret Sharing technology and Galois field 256 for confidential sharing of the original image M. The key X 1 , . . . , X n are the keys required to generate C (1) , C (2) , . . . , C (n) , respectively. Next, the data-hider has a separate pair of n shared images to embed the confidential data S (n) embedded in n in a shared image to generate an embedded shared image C (1) , C (2) , . . . , C (n) . Lastly, n image recipients receive the embedded shared image, respectively C (1) , C (2) , . . . , C (n) . Any image recipient with an embedded key X 1 , . . . , X n can retrieve the confidential information embedded by the data-hider from the shared image. In addition, as long as any k of the embedded shared images can be decrypted, assuming that the k embedded shared images are then using the key X 1 , . . . , X k can decrypt the original image M and retrieve confidential data S. Linear operations on polynomials through a Galois domain, whose addition and multiplication are similar to normal addition and multiplication, except that the result of the operations are elements of the domain.
ropy 2023, 25, x FOR PEER REVIEW 6 o ( ) , ( ) , … , ( ) , respectively. Next, the data-hider has a separate pair of n shared ima to embed the confidential data S (n) embedded in n in a shared image to generate an e bedded shared image ′ ( ) , ′ ( ) , … , ′ ( ) . Lastly, n image recipients receive the embedd shared image, respectively ′ ( ) , ′ ( ) , … , ′ ( ) . Any image recipient with an embedd key , … , can retrieve the confidential information embedded by the data-hider fr the shared image. In addition, as long as any k of the embedded shared images can decrypted, assuming that the k embedded shared images are ′ ( ) , … , ′ ( ) then using key , … , can decrypt the original image M and retrieve confidential data S. Lin operations on polynomials through a Galois domain, whose addition and multiplicat are similar to normal addition and multiplication, except that the result of the operati are elements of the domain.

Image Encryption
For the image encryption part, we use Shamir's confidential sharing and pixel gro ing strategy. Figure 5 shows our confidential sharing process. We group the original p els, if for Shamir's sharing (k, n)-threshold, each group contains k pixels, in the shar process of group (Pi,j, …, Pi,j+k−1), we use these pixel values to build k − 1 times polynom then use X1, X2, …, Xn substituted into the polynomial to obtain the shared pixels y (1) i,j, y (n) i,j. Each shared pixel, y (t) i,j, can be split into k pixels to produce C (t) i,j, …, C (t) i,j+k−1. Since group (C (t) i,j, …, C (t) i,j+k−1) holds only y (t) i,j with 8 bits, so the group can be empty of 8 − 8( − 1) bits of space.

Image Encryption
For the image encryption part, we use Shamir's confidential sharing and pixel grouping strategy. Figure 5 shows our confidential sharing process. We group the original pixels, if for Shamir's sharing (k, n)-threshold, each group contains k pixels, in the sharing process of group (P i,j , . . . , P i,j+k−1 ), we use these pixel values to build k − 1 times polynomial, then use X 1 , X 2 , . . . , X n substituted into the polynomial to obtain the shared pixels y (1) i,j , . . . , y (n) i,j . Each shared pixel, y (t) i,j , can be split into k pixels to produce C (t) i,j , . . . , C (t) i,j+k−1 . Since the group (C (t) i,j , . . . , C (t) i,j+k−1 ) holds only y (t) i,j with 8 bits, so the group can be empty of 8k − 8 = 8(k − 1) bits of space.
ing strategy. Figure 5 shows our confidential sharing process. We group the original pixels, if for Shamir's sharing (k, n)-threshold, each group contains k pixels, in the sharing process of group (Pi,j, …, Pi,j+k−1), we use these pixel values to build k − 1 times polynomial, then use X1, X2, …, Xn substituted into the polynomial to obtain the shared pixels y (1) i,j, …, y (n) i,j. Each shared pixel, y (t) i,j, can be split into k pixels to produce C (t) i,j, …, C (t) i,j+k−1. Since the group (C (t) i,j, …, C (t) i,j+k−1) holds only y (t) i,j with 8 bits, so the group can be empty of 8 − 8 = 8( − 1) bits of space. Figure 5. Secret sharing process for pixel group (Pi,j, …, Pi,j+k−1). Figure 5. Secret sharing process for pixel group (P i,j , . . . , P i,j+k−1 ).
Here is our cryptographic algorithm. Our Algorithm 1 is illustrated with Shamir (k, n)-thresholds, so the polynomial is a (k − 1)-th polynomial.
Step 2: Substitute the Encryption key X t , t = 1, 2, . . . , n through 256 Galois field into the polynomial P i,j+k−1 x k−1 + P i,j+k−2 x k−2 + . . . + P i,j to obtain n encrypted pixel values Step 3: The encrypted pixel values y (t) i,j are divided into 8-bits, and each bit is put into the pixels of the shared image C (t) in order (C (t) i,j , . . . , C (t) i,j+k−1 ), and it is called the P part, if k ≤ 7, because k < 8 cannot put all the bits at once, so the remaining bits are put in order again and the remaining vacant part is called Part B.
Step 4: Repeat Step 2 and Step 3 until all set of pixels (P i,j , . . . , P i,j+k−1 ) have been processed.

Data Embedding
In the image embedding part, when the t th data-hider receives the shared image C (t) , it embeds each pixel of C (t) , and through the B part, it embeds the secret message S (t) into C (t) one after another. Figure 6 shows the embedding process of data-hider t for the pixel group (C (t) i,j , ..., C (t) i,j+k−1 ). Since group (C (t) i,j , ..., C (t) i,j+k−1 ) has a space of 8(k − 1) bits, part of the confidential data S (t) can be directly embedded in this space, resulting in the embedded shared group (C (t) i,j , ..., C (t) i,j+k−1 ). The embedding rate of our method is fixed, and the embedding rate does not decrease as the number of shared images increases. The following is our embedding Algorithm 2.

Algorithm 2: Data Embedding
Input: Sharing images C (t) , t = 1, 2, . . . , n, Data S (t) Output: Marked sharing images C (t) , t = 1, 2, . . . , n Step 1: For a pixel C (t) i,j of Sharing images C (t) , the message is taken from the secret message S (t) and embedded in the B part of pixel C (t) i,j .
Step 2: Repeat Step 1 until all the B parts of C (t) i,j are embedded in the secret message.
Step 3: The output of the embedded shared image C (t) .

Data Extracting and Image Decryption
In the part of data extraction and image decryption, the t th image recipient receives the shared image C (t) , and can extract the confidential information S (t) embedded by the t th data-hider. In addition, by collecting the shared images and decryption keys above the Threshold, a polynomial can be constructed for each set of pixels, through which the pixels of the original image and the secret information of the image owner can be extracted and the original image M can be recovered.
is our embedding Algorithm 2.
Algorithm 2: Data Embedding Input: Sharing images C (t) , = 1, 2, … , , Data S (t) Output: Marked sharing images C' (t) , = 1, 2, … , Step 1: For a pixel C (t) i,j of Sharing images C (t) , the message is taken from the secr sage S (t) and embedded in the B part of pixel C (t) i,j.
Step 2: Repeat Step 1 until all the B parts of C (t) i,j are embedded in the secret mes Step 3: The output of the embedded shared image C' (t) .   i,j , ..., C (t) i,j+k−1 ) has been embedded in a confidential message of 8 (k − 1) bits and can be taken out directly into a partial S (t) . After extraction, there is no need to reduce this group to (C (t) i,j , ..., C (t) i,j+k−1 ), because the group can obtain y (t) i,j whether it is reduced or not.

Data Extracting and Image Decryption
In the part of data extraction and image decryption, the tth image recipient the shared image ′ ( ) , and can extract the confidential information S (t) embedde tth data-hider. In addition, by collecting the shared images and decryption keys ab Threshold, a polynomial can be constructed for each set of pixels, through which els of the original image and the secret information of the image owner can be e and the original image M can be recovered. Figure 7 shows the process of extracting the cluster (C′ (t) i,j, ..., C′ (t) i,j+k−1) by Re The pixel group (C′ (t) i,j, ..., C′ (t) i,j+k−1) has been embedded in a confidential messag − 1) bits and can be taken out directly into a partial S (t) . After extraction, there is to reduce this group to (C (t) i,j, ..., C (t) i,j+k−1), because the group can obtain y (t) i,j whe reduced or not.    Figure 8 shows the response flow diagram of the pixel group (P i,j , . . . , P i,j+k−1 ). Assuming that the first k shared images are obtained from C (1) , C (2) , . . . , C (k) , the pixels of each shared image are also grouped, and each group contains k pixels. For the 7th shared image C (t) for which the pixel group (C (t) i,j , ..., C (t) i,j+k−1 ) can be taken out to share pixels y (t) i,j . Using y (t) i,j , t = 1, 2, ..., k and X t , t = 1, 2, ..., k and Lagrange polynomials, the originally constructed (k − 1)-th polynomial can be recovered, and therefore the pixel group (P i,j , . . . , P i,j+k−1 ) can be recovered.  Figure 8 shows the response flow diagram of the pixel group (Pi,j, …, Pi,j+k−1). Assuming that the first k shared images are obtained from ( ) , ( ) … , ( ) , the pixels of each shared image are also grouped, and each group contains k pixels. For the 7th shared image C (t) for which the pixel group (C (t) i,j, ..., C (t) i,j+k−1) can be taken out to share pixels y (t) i,j. Using ,j, t = 1, 2, ..., k and , t = 1, 2, ..., k and Lagrange polynomials, the originally constructed (k − 1)-th polynomial can be recovered, and therefore the pixel group (Pi,j, …, Pi,j+k−1) can be recovered.  The following is our Algorithm 3 for recovering images and extracting secret information. For the sake of illustration, we assume that we obtain the first k marked shared images C (1) , . . . , C (k) and use the key X 1 , . . . , X k to decrypt the original images M and extract the secret information S (1) , ..., S (k) .
where y (t) i,j , X t GF(256), t = 1, 2, . . . , k, and add, subtract, multiply, and divide are all in GF(256). Step 2. For sharing a set of pixels of the image i,j+k−1 ), extract the P part of the pixel set and merge it into the encrypted pixel values y (t) i,j , t = 1, 2, . . . , k.
Step 4. The coefficients of the polynomial P i,j+k−1 x k−1 + P i,j+k−2 x k−2 + . . . + P i,j are used to obtain the pixel sets (P i,j , . . . , P i,j+k−1 ) of the original images.
Step 6. Combine all pixel groups (P i,j , . . . , P i,j+k−1 ) into the original image M.

Decryption and Extracting Steps
We use 3 shared images to illustrate the steps of decryption and extracting. Take out the pixel sets of 3 shared image images (C i,j+k−1 ) and the decryption keys X 1 , X 2 , X 3 and take out each pixel set of B to obtain the secret information S. The P parts of each pixel group are merged individually to obtain the shared image pixels y (1) , y (2) , and y (3) . Substitute X 1 , X 2 , X 3 , and y (1) , y (2) , y (3) into Equation (4) The polynomial P i,j+k−1 x k−1 + P i,j+k−2 x k−2 + . . . + P i,j = 8x 2 + 4x + 2 is obtained to obtain the pixel group of the original image (P i,j , . . . , P i,j+k−1 ) = (2, 4, 8).

Experimental Results
In this section, we perform experiments and analysis. All the tested images are gray 425 level sized by 512 × 512. Figure 9 shows the experimental results of Image Boat. We use the three-out-of-four threshold secret sharing method to group every three original images and encrypt them into four shared images. Figure 9a is the original image, Figure 9b-e are different shared images, and Figure 9f is the image after decryption and information retrieval. From the image we can see that our method can fully recover. Similarly, we did the same for image Couple1 and the result is shown in Figure 10. Figure 11 shows the maximum embedding rate comparison among the proposed method and state-of-the-art methods. We use three-out-of-three, three-out-of-four, and three-out-of-five threshold secret sharing to make the comparison. Here is a three-out-offour to illustrate that, in our way, a shared image pixel can be embedded with 5 or 6 bits of secret information at an embedding rate of 5.  Figure 11, we can see that the embedding rate of my method is larger than other methods, and it will be more obvious as the number of shared images increases.
In this section, we perform experiments and analysis. All the tested images are gray 425 level sized by 512 × 512. Figure 9 shows the experimental results of Image Boat. We use the three-out-of-four threshold secret sharing method to group every three original images and encrypt them into four shared images. Figure 9a is the original image, Figure  9b-e are different shared images, and Figure 9f is the image after decryption and information retrieval. From the image we can see that our method can fully recover. Similarly, we did the same for image Couple1 and the result is shown in Figure 10.   Figure 11 shows the maximum embedding rate comparison among the proposed method and state-of-the-art methods. We use three-out-of-three, three-out-of-four, and three-out-of-five threshold secret sharing to make the comparison. Here is a three-out-offour to illustrate that, in our way, a shared image pixel can be embedded with 5 or 6 bits of secret information at an embedding rate of 5.  method and state-of-the-art methods. We use three-out-of-three, three-out-of-four, and three-out-of-five threshold secret sharing to make the comparison. Here is a three-out-offour to illustrate that, in our way, a shared image pixel can be embedded with 5 or 6 bits of secret information at an embedding rate of 5.3 bpp (Bit Per Pixel). In Chen et al.'s method, their embedding rate is ⌊ ⌋⋅ ≈ bpp, which is about 1.75 bpp since n = 4 and l = 7. From Figure 11, we can see that the embedding rate of my method is larger than other methods, and it will be more obvious as the number of shared images increases.  Table 1 shows the feature comparison among the proposed scheme and state-of-theart schemes, which shows the comparison of the features of our approach with other approaches.   Table 1 shows the feature comparison among the proposed scheme and state-of-the-art schemes, which shows the comparison of the features of our approach with other approaches.  Table 2 shows the comparisons of embedding capacity (bits) and embedding rate (bpp), here the experimental data is used in a three-out-of-four threshold secret sharing approach, in our method, a pixel of a shared image can embed 5 or 6 bits of secret information, so a shared image can embed 1,398,096 bits. The bpp (bit per pixel) for embedding rate is 1,398,096 (512×512) ∼ = 5.3. We share the image as four shared images, so you can embed 1,398,096 × 4 = 5,592,384 bits. Figure 12 shows the images used in Table 2.  Table 2 shows the comparisons of embedding capacity (bits) and embedding rate (bpp), here the experimental data is used in a three-out-of-four threshold secret sharing approach, in our method, a pixel of a shared image can embed 5 or 6 bits of secret information, so a shared image can embed 1,398,096 bits. The bpp (bit per pixel) for embedding rate is 1,398,096 (512 × 512) ≅ 5.3. We share the image as four shared images, so you can embed 1,398,096 × 4 = 5,592,384 bits. Figure 12 shows the images used in Table 2.  (h) Girl. Table 3 shows the comparison of embedding rata (bpp) with different k based on Shamir (k, n). In our approach, each original pixel Pi,j could be reconstructed by collecting k sharing pixels. It means that the k sharing pixels holds only Pi,j with 8 bits, so k sharing pixels has a space of 8 × ( − 1) bits, thus the embedding ratio is estimated by  Table 3 shows the comparison of embedding rata (bpp) with different k based on Shamir (k, n). In our approach, each original pixel P i,j could be reconstructed by collecting k sharing pixels. It means that the k sharing pixels holds only P i,j with 8 bits, so k sharing pixels has a space of 8 × (k − 1) bits, thus the embedding ratio is estimated by 8×(k−1) k .
Assume that k = 8 indicates that a pixel P i,j can be split into 8 pixels to produce shared pixels C (1) i,j , . . . , C (8) i,j by Shamir's confidential sharing. Therefore, the embedding ratio is If two-out-of-two threshold secret sharing is used for the experiment, the maximum number of entries in the multiplex is only one, which may make the sharing images appear contoured and therefore the encryption effect is not satisfactory. In this case, you just need to add a procedure to encrypt the shared image once more, and then the encrypted image will have the encryption effect. Of course, the decryption process should also add one additional step to the image decryption. We experiment with image Boat. Figure 13a is the original image, Figure 13b,c are the two-out-of-two threshold secret sharing different sharing images, and we can find some contours appear in Figure 13b,c. Figure 13d,e are the results of re-encryption of Figure 13b,c, respectively, which already have the effect of image encryption, while Figure 13d,e are the results of the re-encryption of Figure 13b,c. Figure 13f is the image after decryption and information retrieval.  If two-out-of-two threshold secret sharing is used for the experiment, the maximum number of entries in the multiplex is only one, which may make the sharing images appear contoured and therefore the encryption effect is not satisfactory. In this case, you just need to add a procedure to encrypt the shared image once more, and then the encrypted image will have the encryption effect. Of course, the decryption process should also add one additional step to the image decryption. We experiment with image Boat. Figure 13a is the original image, Figure 13b,c are the two-out-of-two threshold secret sharing different sharing images, and we can find some contours appear in Figure 13b,c. Figure 13d,e are the results of re-encryption of Figure 13b,c, respectively, which already have the effect of image encryption, while Figure 13d,e are the results of the re-encryption of Figure 13b,c. Figure 13f is the image after decryption and information retrieval.

Conclusions
This paper proposes an RDHEI technique based on the polynomial construction technique of Shamir's Secret Sharing technique, which divides k pixels into a group and uses k pixels to construct a polynomial. Then, our approach uses the polynomial and Galois Field calculation to generate n sharing pixels. Because of the finite field ( (2 )) strategy, our method can avoid the pixel-specific pre-processing. The 8 bits of each shared pixel are

Conclusions
This paper proposes an RDHEI technique based on the polynomial construction technique of Shamir's Secret Sharing technique, which divides k pixels into a group and uses k pixels to construct a polynomial. Then, our approach uses the polynomial and Galois Field calculation to generate n sharing pixels. Because of the finite field (GF(2 n )) strategy, our method can avoid the pixel-specific pre-processing. The 8 bits of each shared pixel are split into k small parts and placed in each of the k pixels of the shared image so that the constructed shared image has B part of the embedding space, and each shared image is encrypted. Information hiders can hide secret information in Part B of the shared image. Compared with other methods, our method has a higher embedding rate, and the embedding rate does not decrease due to more shared images. In the future work, we will consider how to apply the proposed model to other applications or to some specified multimedia such as video or audio.