
Citation: Osmar Alejandro Chanes-Cuevas, Adriana Perez-Soria, Iriczalli Cruz-Maya, Vincenzo Guarino, Marco Antonio Alvarez-Perez. Macro-, micro- and mesoporous materials for tissue engineering applications[J]. AIMS Materials Science, 2018, 5(6): 1124-1140. doi: 10.3934/matersci.2018.6.1124
[1] | Kiyotaka Obunai, Daisuke Mikami, Tadao Fukuta, Koichi Ozaki . Microstructure and mechanical properties of newly developed SiC-C/C composites under atmospheric conditions. AIMS Materials Science, 2018, 5(3): 494-507. doi: 10.3934/matersci.2018.3.494 |
[2] | Hendra Suherman, Yovial Mahyoedin, Afdal Zaky, Jarot Raharjo, Talitha Amalia Suherman, Irmayani Irmayani . Investigation of the mechanical properties of bio-composites based on loading kenaf fiber and molding process parameters. AIMS Materials Science, 2024, 11(6): 1165-1178. doi: 10.3934/matersci.2024057 |
[3] | Wisawat Keaswejjareansuk, Xiang Wang, Richard D. Sisson, Jianyu Liang . Electrospinning process control for fiber-structured poly(Bisphenol A-co-Epichlorohydrin) membrane. AIMS Materials Science, 2020, 7(2): 130-143. doi: 10.3934/matersci.2020.2.130 |
[4] | Alejandro Sandá, Rocío Ruiz, Miguel Ángel Mafé, Jon Ander Sarasua, Antonio González-Jiménez . Scrapping of PEKK-based thermoplastic composites retaining long fibers and their use for compression molded recycled parts. AIMS Materials Science, 2023, 10(5): 819-834. doi: 10.3934/matersci.2023044 |
[5] | Araya Abera Betelie, Yonas Tsegaye Megera, Daniel Telahun Redda, Antony Sinclair . Experimental investigation of fracture toughness for treated sisal epoxy composite. AIMS Materials Science, 2018, 5(1): 93-104. doi: 10.3934/matersci.2018.1.93 |
[6] | Iketut Suarsana, Igpagus Suryawan, NPG Suardana, Suprapta Winaya, Rudy Soenoko, Budiarsa Suyasa, Wijaya Sunu, Made Rasta . Flexural strength of hybrid composite resin epoxy reinforced stinging nettle fiber with silane chemical treatment. AIMS Materials Science, 2021, 8(2): 185-199. doi: 10.3934/matersci.2021013 |
[7] | Araya Abera Betelie, Anthony Nicholas Sinclair, Mark Kortschot, Yanxi Li, Daniel Tilahun Redda . Mechanical properties of sisal-epoxy composites as functions of fiber-to-epoxy ratio. AIMS Materials Science, 2019, 6(6): 985-996. doi: 10.3934/matersci.2019.6.985 |
[8] | Prashant Tripathi, Vivek Kumar Gupta, Anurag Dixit, Raghvendra Kumar Mishra, Satpal Sharma . Development and characterization of low cost jute, bagasse and glass fiber reinforced advanced hybrid epoxy composites. AIMS Materials Science, 2018, 5(2): 320-337. doi: 10.3934/matersci.2018.2.320 |
[9] | Ninis Hadi Haryanti, Suryajaya, Tetti Novalina Manik, Khaipanurani, Adik Bahanawan, Setiawan Khoirul Himmi . Characteristics of water chestnut (Eleocharis dulcis) long fiber reinforced composite modified by NaOH and hot water. AIMS Materials Science, 2024, 11(6): 1199-1219. doi: 10.3934/matersci.2024059 |
[10] | Kator Jeff Jomboh, Adele Dzikwi Garkida, Emmanuel Majiyebo Alemaka, Mohammed Kabir Yakubu, Vershima Cephas Alkali, Wilson Uzochukwu Eze, Nuhu Lawal . Properties and applications of natural, synthetic and hybrid fiber reinforced polymer composite: A review. AIMS Materials Science, 2024, 11(4): 774-801. doi: 10.3934/matersci.2024038 |
Since Shamir proposed (k,n)-threshold secret sharing technology [1] in 1979, this scheme has become the basis of other secret sharing schemes. In 1995, Naor and Shamir [2] extended secret sharing technology to image field and proposed the concept of visual cryptography: visual secret sharing (VSS). In 2002, Thien and Lin [3] proposed a secret image sharing (SIS) scheme based on (k,n)-threshold, and formally introduced Shamir's polynomial-based secret sharing method into the field of image encryption. Due to the unique advantages of image in information hiding and data processing, SIS has gradually developed into a special branch in the field of secret sharing.
SIS is a multi-channel secret information distribution technology [4]. In the process of SIS with (k,n)-threshold (2<k<n), the secret image is generated into n shadow images (also known as shadows or shares) by using method which ensures that insufficient shadow images can not recover the original secret image. When recovery is needed, k or more shadow images must participate in the process of effective recovery. Therefore, a SIS always consists of two phases: shadow images generation (secret image sharing) and secret image recovery. The measurement characteristics mainly include pixel expansion or compression, recovery loss or lossless, calculation efficiency high or low, and so on. Compared with the traditional image encryption and image hiding methods, the SIS method has the advantages of higher information security, lower computational complexity and loss tolerance. Therefore, the research on SIS has become a very popular direction in recent years.
At present, according to the different strategies of secret image sharing and recovery, SIS is mainly divided into two categories: polynomial-based scheme [5] and VSS [6,7].
(k,n)-threshold polynomial-based scheme depends on a k−1 degree polynomial in an integer field, in which k coefficients are embedded with a secret integer and k−1 random integers. n values are computed as shared values with n different non-zero independent variables. In the recovery phase, the coefficients of polynomial can be calculated by Lagrange interpolation using at least k shares, so as to recover the secret information. This kind of strategy is flexible in setting (k,n) threshold, and the visual quality of the recovered image is high, but it has higher computational complexity for recovery, O(klog2k).
The (k,n)-threshold VSS-based scheme can directly decrypt the secret information just by superposition or simple logic operation with enough shares. This kind of methods need low calculation for recovery, but may have some problems, such as pixel expansion, poor visual quality of recovered image and the limited (k,n) threshold. The random-grid VSS [8,9,10] scheme has been studied on these problems such as improving the contrast of restored image and no pixel expansion.
These two kinds of schemes are mainly studied from the mechanism of how to construct SIS. In application field, after obtaining shadow images from multiple channels, how to verify the authenticity and integrity of each shadow image is also a problem to be solved. A fake share is easily forged by the adversary, and has a destructive impact on the recovery of the secret image. Even more serious is that an adversary with a fake share can easily collect shares of other honest participants. As a result, the application of SIS scheme in data transmission is seriously restricted. In this paper, we mainly study the compressed SIS method with the ability of shadow image verification. Based on the current polynomial-based SIS scheme, we add VSS technique to achieve the ability of both secret image lossless recovery and shadow image high-efficiency verification.
The main structure of this paper is provided as follows. Section 2 introduces the related research status, while Section 3 analyzes the research basis. The design concept and implementation of this scheme are provided in Section 4. Section 5 discusses the experimental situation and comparative analysis, and finally the conclusion is drawn in Section 6.
In 1989, Tompa and woll [11] first proposed that there might be fake shadow images in the process of secret sharing, which would not only result in the failure of the reconstruction of the secret image, but also increase the risk of disclosure. In the field of SIS, the verification problem of shadow images has attracted more and more attention.
The traditional method to check the integrity of shadow image is to use hash function [12] and information hiding [13], calculate the hash value of the generated shadow image and hide this value by using information hiding technologies. Before the recovery of the secret image, each participant must compute the hash value of shares from others, and then compare the result with the hidden hash value to confirm whether the shadow image has been tampered or forged. Liu and Chang [14] proposed a VSS scheme with reversibility and verification by setting up turtle shell reference matrix. Liu et al. [15] studied the shadow image forgery problem by introducing the results of two polynomial of random value association, based on Thien and Lin's polynomial-based scheme. These schemes have more time cost on the verification phase rather than the sharing and recovery phases. Moreover, the share privately hold by each participant should be handed out for verification. Therefore, the private share is still disclosed, even though the verification fails. In addition, these scheme cannot achieve precise recovery.
Yan et al. [16] proposed an SIS with separate shadow verification ability, which is a combination of polynomial-based SIS and random-grid VSS. In their scheme, a binary secret image used for verification is shared by (2,2)-threshold random-grid VSS. Then, the highest bit of pixel of each grayscale shadow images is required to be the same as 1-bit pixel value of one of binary shares. Before recovery, the binary share can be extracted from the grayscale shares and then stacked with the other binary share privately hold by a trusted third party: if the original binary secret image is reconstructed, the grayscale share is valid for recovery, and vice versa. Since 1-bit verification information is fixedly embedded into the highest bit of the pixel of each grayscale shares, n shares have the identical highest bit plane. However, if information of the highest bit plane is leaked, the adversary can easily forge a authenticated grayscale share.
Inspired by Yan et al.'s scheme, we proposed a novel compressed SIS with shadow image verification capability, which is also a combination of polynomial-based SIS and random-grid VSS. In the proposed scheme, the size of each grayscale shadow image is reduced to half of the original secret image; and the verification information is hidden into two lowest bits of the grayscale pixel. As a result, the special design improves the efficiency and security. In addition, an improved version of polynomial-based SIS is utilized to achieve precisely recovery.
The specific idea of the proposed scheme is stated as follows. The framework of polynomial-based scheme is selected, which is built on the preciseness of mathematical calculation with the main characteristics of (k,n) threshold and lossless recovery; then a VSS scheme with simple stacking-to-see decryption is used for the verification of shadow images; finally, the above two schemes are fused with experimental verification to achieve the goal of this paper.
In a polynomial-based SIS scheme for (k,n) threshold, k−1 degree polynomial over a prime field P is usually used to share and recover secret information. The form of the polynomial is in formula 3.1.
f(x)=(a0+a1x+a2x2+...+ak−1xk−1)modP | (3.1) |
In Shamir's scheme [1], one secret pixel is assigned to a0 of the polynomial each time, and other coefficients of the polynomial are obtained by random values in [0,P−1]. Because the value of P is 251, but the maximum pixel value of grayscale image is 255, when the pixel value of image exceeds 251, this part of pixel cannot be recovered correctly and lossless recovery cannot be realized. On the basis of Shamir's scheme, in order to reduce the size of shadow image, Thien and Lin's scheme [3] embeds k secret pixel values into the coefficients a0 to ak−1, and the specific sharing and recovery methods are given in algorithms.
Thien and Lin's sharing algorithm |
Input: grayscale secret image S1, with size of (H,W), threshold (k,n) |
Output: n shadow images {S1Ci,1≤i≤n}, with size of (H,W/k) |
Algorithm: 1. Set the prime number P=251; 2. For s1(h,w)∈{S1(h,w)|1≤h≤H,1≤w≤W}, if s1(h,w)≥251, set s1(h,w)=250 3. Encrypt S1 with key to get S′1 4. Select k pixels in S′1 in order, denoted by b0,b1,...,bk−1 5. Construct k−1 degree polynomial: f(x)=(b0+b1x+...+bk−1xk−1)modP 6. Calculate s1c1=f(1),s1c2=f(2),...,s1cn=f(n) 7. Assign sc1,sc2,...,scn to S1C1(h,w),S1C2(h,w),...,S1Cn(h,w), 8. If there are still pixels in S′1, go to 4 9. Generate n shadow images S1C1,S1C2,...,S1Cn |
The original Thien and Lin's scheme [3] inherits the lossy recovery issue of Shamir's scheme. At the same time, because k secret pixels are embedded each time, although the size of the shadow image will be reduced to 1/k times of the original image, the randomness is missing. If no additional information encryption method is used, the more serious image information disclosure problem will be caused. Figure 1 shows an example of building (3, 4) threshold image sharing without additional encryption in Thien and Lin's scheme.
Random-grid VSS [17] scheme is a VSS scheme based on probability, mainly for binary images. Because this scheme has no pixel expansion and can achieve rapid recovery through logical "OR" and "XOR" operation, many improved methods based on this scheme have emerged in recent years [18,19].
Suppose that 1 represents black pixel and 0 represents white pixel. For the (2, 2) threshold, the sharing and recovery methods based on the random-grid VSS scheme are given in algorithms.
Thien and Lin's recovery algorithm |
Input: any k shadow images SCi1,SCi2,...,SCik,ik∈(1,n), with size of (H,W/k) |
Output: recovered grayscale secret image S11, with size of (H,W) |
Algorithm: 1. Set P = 251 2. Take out the pixels from the k shadow images in turn SCi1(h,j),SCi2(h,j),...,SCik(h,j)(1≤h≤H,1≤j≤W/k) 3. Set f(i1)=SCi1(h,j),f(i2)=SCi2(h,j),...,f(ik)=SCik(h,j). Lagrange interpolation is used to solve the following equations: f(i1)=(a0+a1i1+...+ak−1i1k−1)modPf(i2)=(a0+a1i2+...+ak−1i2k−1)modP⋯f(ik)=(a0+a1ik+...+ak−1ikk−1)modP 4. Arrange the calculated coefficients a0,a1,...,ak−1 in turn 5. If the shadow image pixels are not analyzed, go to 2 6. Generate the image S′11 according to the combination of all coefficient sequences calculated in step 4 7. Use key to decrypt S′11 and restore the image S11 |
Sharing algorithm of (2, 2) threshold random-grid VSS |
Input: binary image S2, with size of (H,W), threshold (2, 2) |
Output: 2 shadow images S2C1, S2C2, with size of (H,W) |
Algorithm: 1. Randomly select 0 or 1 to generate each pixel S2C1(h,w),1≤h≤H,1≤w≤W of the shadow image S2C1 2. Cycle to read pixels S2(h,w),(1≤h≤H,1≤w≤W) of S2 3. If S2(h,w)=1, thus S2C2(h,w)=¯S2C1(h,w) 4. If S2(h,w)=0, thus S2C2(h,w)=S2C1(h,w) 5. Generate another shadow image S2C2 from S2C2(h,w) |
Recovery algorithm of (2, 2) threshold random-grid VSS |
Input: 2 shadow images S2C1, S2C2, with size of (H,W) |
Output: binary image S′2, with size of (H,W) |
Algorithm: 1. For each pixel S′2(h,w),(1≤h≤H,1≤w≤W) of S′2, if stacking recovery (or operation), perform the following formula S′2(h,w)=S2C1(h,w)⊗S2C2(h,w) If lossless recovery (XOR operation), perform the following formula S′2(h,w)=S2C1(h,w)⊕S2C2(h,w) 2. Another shadow image S′2 will be generated from S′2(h,w). |
It can be seen from the above sharing and recovery methods that, when generating a shadow image, all pixels of one shadow can be randomly generated first, and then another shadow image can be generated according to the original binary secret image. In the recovery phase, logic OR operation or logical XOR operation can be used for recovery. If it is a logical OR operation, it is enough to overlay the shadow image visually. When the original image pixel is 1, the OR operation of the two subpixels must be 1; when the original image pixel is 0, the OR operation of the two subpixels may be 0 or 1. Therefore, OR operation can recover the original image with noisy background, which belongs to lossy recovery. If logical XOR operation is adopted, all pixels can be recovered accurately according to the same rule of 0 and 1, which can achieve the effect of lossless recovery. Figure 2 shows an example of constructing (2, 2) threshold image sharing using the above scheme.
According to the improvement and fusion of the above two types of schemes, we can construct an SIS scheme with the ability of shadow image verification.
This method improves the polynomial-based SIS scheme described in Section 3.1, and uses the generated verification information in Section 3.2 for lossless embedding. The framework of the proposed method is shown in Figure 3.
In Figure 3, the sharing phase starts from the given secret image S1. According to the size height and width (H, W) of S1, a binary image S2 with the same height and half width is first selected and generated. Then, two shadow images S2C1 and S2C2 are generated according to the random-grid VSS described in Section 3.2, in which S2C2 is used to verify the authenticity of the shadow image during recovery, and S2C1 is used to verify the authenticity of the shadow image. In the process of splitting S1 into n shadow images, lossless embedding is performed in the lowest two bits of each shadow image pixel. The specific method is detailed in Section 4.2. The sharing algorithm is given in "Sharing algorithm of the proposed method".
After sharing, the (k,n) threshold image can be transmitted independently by multi-channel. When secret image recovery is needed, the binary image S2C1 can be extracted from each received shadow image, with S2C2 XOR or superimposing can obtain image S2 to complete verification; then Lagrange interpolation method is used to calculate and restore the original secret image. When there are k or more shadow images that are verified as true, i.e., the binary image S2C1 can be generated, we can realize the lossless recovery of the secret image. The algorithm of recovery phase is given in "Recovery algorithm of the proposed method".
Sharing algorithm of the proposed method |
Input: grayscale secret image S1, with size of (H,W), threshold (k, n), n values of x, denoted by x1,…xn |
Output: n shadow images {SCi,1≤i≤n}, image size (H, W/2) and one verification shadow image S2C2 |
Algorithm: 1. According to S1, the binary image S2 with the size of (H, W/2) is selected to be generated 2. Using (2, 2) threshold random-grid algorithm for image S2 to generate shadow images S2C1 and S2C2 3. Select P = 257, set H = 0, j = 0 4. For pixels S1(h,2j−1)∈{S1(h,w)|1≤h≤H,1≤w≤W},j∈(1,W/2) construct k−1 degree polynomial f(x)=(a0+a1x+...+ak−1xk−1)modP Among them: a0=S1(h,2j - 1),ak−1=S1(h,2j), ai∈(0, 255) is random , i∈(1,k−2) 5. Calculate the pixels values of sc1=f(x1),sc2=f(x2),...,scn=f(xn) to be embedded to shadow image S2C1 according to the constraint conditions 6. Assign sc1,sc2,...,scn to SC1(h,j),SC2(h,j),...,SCn(h,j) 7. Select the next pair of pixels according to the current (h, j), and repeat steps 4-6 until h=H,j=W/2 8. Output n shadow images SC1,SC2,...,SCn |
In this scheme, two improvements have been made to the traditional Thien and Lin's scheme [3]. One is to choose to set the P value of the prime field to 257, so that the secret image can be restored without loss. As for the case of the invalid value 256, the limitation will be added to the pixel generation method of the shadow image described in Section 4.2. The second is to embed two secret pixels into the coefficient a0 and ak−1 each time, which can not only avoid the serious information leakage problem of the shadow image, but also avoid the high time complexity of the algorithm in the recovery phase.
In Thien and Lin's scheme [3], because k secret pixels are embedded each time, the loss of the randomness of coefficients will lead to serious information leakage, so additional encryption must be used to ensure the security. In addition, the time complexity of the scheme is high because it needs to calculate all k coefficients of the polynomial in the recovery phase. In our scheme, two secret pixels are embedded each time, keeping the randomness of coefficient setting. At the same time, considering the efficiency of Lagrange interpolation in the recovery phase, the computational complexity can be kept consistent with that of embedding only one secret pixel.
The solution formula of Lagrange interpolation method for known k points, denoted by (x1,y1),(x2,y2),...,(xk,yk), is in formula 4.1.
Recovery algorithm of the proposed method |
Input: any kxi1,xi2,...,xik and corresponding shadow images SCi1,SCi2,...,SCik,ik∈(1,n) size (H, W/2), verification shadow image S2C2 |
Output: recovered grayscale secret image S′1, with size of (H,W) |
Algorithm: 1. Read the lowest two bit values of each pixel for k shadow images in turn, use "XOR" to generate a binary image S2C′1 2. Calculate the "XOR" of S2C′1 and S2C2 to generate image S′2, and check whether it is the same as the original binary image S2, If not, the verification fails and the algorithm stops; 3. Set P = 257, h = 0, j = 0 4. Take out the pixels from the k-shadow images in turn denoted by SCi1(h,j),SCi2(h,j),...,SCik(h,j)(1≤h≤H,1≤j≤W/2) 5. Set f(xi1)=SCi1(h,j),f(xi2)=SCi2(h,j),...,f(xik)=SCik(h,j) Lagrange interpolation is used to solve the following equations: f(xi1)=(a0+a1xi1+...+ak−1xk−1i1)modP f(xi2)=(a0+a1xi2+...+ak−1xk−1i2)modP f(xik)=(a0+a1xik+...+ak−1xk−1ik)modP 6. Calculate the secret pixel values S′1(h,2j−1)=a0,S′1(h,2j)=ak−1 7. Select the next pair of pixels according to the current (h,j) value, and repeat steps 4-6 until h=H,j=W/2 8. Arrange the secret pixel values to generate the restored image S′1 |
L(x)=k∑i=1yili(x) | (4.1) |
li(x)=k∏j=1,j≠ix−xjxi−xj=x−x1xi−x1×...×x−xi−1xi−xi−1×x−xi+1xi−xi+1×...×x−xkxi−xk | (4.2) |
The time complexity to reconstruct the polynomial coefficients is mainly decided by the summation process of Eq. 4.1 and the multiplication times of Eq. 4.2. Substituting Eq. 4.2 into Eq. 4.1 we can get Eq. 4.3
L(x)=y1(x−x2xi−x2×...×x−xkxi−xk)+...+yi(x−x1xi−x1×...×x−xi−1xi−xi−1×x−xi+1xi−xi+1×...×x−xkxi−xk)+...yk(x−x1xi−x1×...×x−xkxi−xk) | (4.3) |
For the multiplication times in the above process, such as computing yi(x−x1xi−x1×...×x−xi−1xi−xi−1×x−xi+1xi−xi+1×...×x−xkxi−xk), given (x1,y1),(x2,y2)...,(xk,yk), yi(xi−x1)×...×(xi−xi−1)×(xi−xi+1)×...×(xi−xk) is a fixed item mainly decided by (x−x1)×...×(x−xi−1)×(x−xi+1)×...×(x−xk). According to the combination selection method, the calculation method of the constant coefficient a0 is as follows: C0k−1×(−x1)×...×(−xi−1)×(−xi+1)×...×(−xk), where C0k−1=1 and Ck−1k−1=1.
Since there is only one selected method for the generation of these two coefficients, there are k items in Eq. 4.3 and each item needs to calculate nearly a constant multiplication of k. Therefore, the time complexity of computing a0and ak−1 is O(k2), which is the same as that of embedding secret pixel only into a0.
Except these two coefficients, the calculation of any other coefficient of xi can not be completed within O(k2). For example, to solve the coefficient a1, since C1k−1=k−1 means that taking one x item from k−1 items, that is, x×(−x2)×...×(−xi−1)×(−xi+1)...×(−xk)+(−x1)×x×...×(−xi−1)×(−xi+1)×...×(−xk)+...+(−x1)×...×(−xi−1)×(−xi+1)×...×x,whose time complexity is O(k2). In Eq. 4.3, there are k items, and each item needs such time complexity, so the time complexity of the total secret pixels recovery calculation reaches O(k3).
Therefore, in our scheme, a0 and ak−1 embedded secret pixels can ensure that the time efficiency is consistent with that of only calculating a0. When k is greater than 3, better hiding effect can be achieved. Although the minimum value of k is limited to some extent, considering that more than three transmission paths are usually selected in the practical application of multi-channel transmission, this kind of comprehensive efficiency and the characteristics including compression rate are practically valuable.
It should be noted that since this method generates half size shadow image, when the width value of the secret image is not even, it needs to be even processed first, such as increasing 0. In the recovery process, if the last column of data is found to be special, then delete it.
One key step of our method is to embed the verification information into the shadow pixels. Through the improvement of the traditional polynomial-based algorithm in Section 4.1, when k is not less than 3, random variables will participate in the calculation process. For example, when k is 3, the polynomial can be constructed by selecting two given secret pixels s11 and s12 are in formula 4.4.
f(x)=(s11+a1x+s12x2)mod257 | (4.4) |
Among them, a1∈[0,255] is generated by random function.
Because the algorithm uses module 257, we can avoid the situation of f(x)=256 by changing the random value of a1 when generating the specific value of f(x), because 256 is beyond the range of grayscale image pixel value.
According to the above change idea, similarly, the value of f(x) can meet certain characteristics by adjusting a1. Here, we choose to set the XOR operation result of the lowest two bits of f(x) to a specific value, that is, in the process of generating the shadow images S1Ci(i∈[1,n]), we adjust the XOR operation result of the lowest two bits of S1Ci(h,w)(i∈[1,n],h∈[1,H],w∈[1,W/2]) to be the bit value of the verification shadowed pixel S2C1(h,w)(h∈[1,H],w∈[1,W/2]), so as to realize the lossless embedding of the verification information.
For the two consecutive secret pixel values s11 and s12 in S1, the corresponding bit value of s2c1 and (k,n) threshold generation algorithm is given in "The generation algorithm for shadow image with verification information embedded into the least two bits".
The generation algorithm for shadow image with verification information embedded into the least two bits |
Input: (k,n) threshold, x1,…xn, secret pixel values s11 and s12, and embedded bit value s2c1 |
Output: n image pixel values s1c1, s1c2, ..., s1cn |
Algorithm: 1. Set a=s11, ak−1=s12, randomly generate values of a1 to ak−2 from (0, 255) to construct polynomial f(x)=(a0+a1x+...+ak−1xk−1)modP 2. Initialize count index i=0 3. Set i=i+1, and calculate the value of f(xi) 4. If f(xi)>255, go to 1 5. If the XORed result of the lowest bit and the second lowest bit of f(xi)! =s2c1, go to 1 6. Assign the value of f(xi) to s1ci 7. If i<n, go to 3 8. Output the pixel values s1c1, s1c2, ..., s1cn of n shadow images |
Embedding verification information into the XORing result of the lowest two bits is mainly based on two considerations. First one is to hide the feature of the specific bit of the shadow image, so that each bit of the pixels in the same position of the generated n shadow image has no relevance, so as to prevent malicious users from generating the forged shadow image when obtaining some real shadow images by analyzing the bit relation. The other one is to keep good randomness to ensure the effective generation of shadow images. Under random condition, the probability of generating 0 or 1 is 50{%}. With the same randomness, the probability that the value generated by XORing result of the lowest two bits is 0 or 1 is also 50{%}. However the other binocular operations do not have such performance, such as "AND", "OR", etc. The weak randomness of the result will lead to a rapid increasing of repeated calculations. Please refer to section 5.2 for experimental verification.
In addition, the randomness retained by XOR operation can be extended to the multi object operation, that is, the XORing result of the lowest two bits can be extended to any multiple bits participating in the XORing operation. Under the condition of successfully generating the shadow bits, the shadow image still has good randomness, so that the proposed scheme has good expansion.
Because two kinds of secret sharing methods are integrated into the generation of shadow image, which has a certain impact on the original random distribution probability, so we need to analyze the security of embedding secret image.
Assuming that the secret image S1 is linearly independent of the verification image S2, the influence of this scheme on the verification image S2 is analyzed. we use the random-grid VSS for (2, 2) threshold to split S2 into S2C1 and S2C2. Each pixel of S2C1 is randomly generated, and each pixel value of S2C2 is changed according to the value of S2C1. Therefore, this scheme does not change the security of the original random-grid VSS scheme. Assuming that the number of pixels in S2 is m, the probability of brute-force cracking the original image S2 is 2m on the premise that any one of the shadow images is known. Here, we assume the m value is 128×64, which is far beyond the current limit that general-purpose computers can brutally crack.
For the secret image S1, the improved polynomial-based SIS scheme is used in our scheme. Since we share two secret pixels in a polynomial, the k value in the (k,n) threshold of this scheme must start from at least 3 so as to ensure the randomness of the sharing process. Thus at least one coefficient can be randomly selected in the polynomial construction process, such as a1 in formula 4.1. But at the same time, because the n pixels generated by each calculation of f(x) have the constraint, only one coefficient is used to calculate the value of the n pixels. The random value in [0, 255] may greatly reduce the randomness and even fail to meet the constraint conditions (see the experimental instructions in Section 5.3). Therefore, the actual k value in our scheme starts from 4, that is, the minimum degree form of the polynomial is:
f(x)=(s11+a1x+a2x2+s12x3)mod257 | (4.5) |
In this way, there are two coefficients randomly selected in [0, 255], which can ensure that when S11, S12 and x1,x2,…,xn are given, by randomly changing the values of a1 and a2, the n equal values of XORed result of the lowest two bits can be generated. It should be noted that the larger n is, the stronger the constraint conditions are. Even if k is 4, when n is very large, there may still be cased that we can not meet the constraint conditions. We will give the specific experimental results in the experimental part.
Based on the designed scheme in the previous section, we implement it with Python code. The grayscale image of 128×128 is selected as the secret image to be shared, and the binary image of 128×64 is set as the verification image. The experiments of SIS from (4, 4) threshold to (8, 10) thresholds are performed.
Among them, the (6, 8) threshold SIS experiment is shown in Figure 4. Figure 4 (a) is a secret image to be shared, and Figure 4 (b) is a verification image. The two verification shadow images generated by (2, 2) threshold random-grid VSS method are shown in Figure 4 (c) and Figure 4 (d). The 8 shadow images generated by integrating the bit pixels of Figure 4 (c) into the sharing process of Figure 4 (a) are shown in Figure 4 (e)-(l).
In the verification and recovery phase, XORed value of the least two bits is extracted from each shadow image participating in the recovery to generate the verification information, and the "XOR" operation is performed on Figure 4 (d) to verify the effectiveness of the shadow image, as shown in Figure 5.
Figure 5 shows eight shadow images lena1.bmp to lena8.bmp and a fake shadow image fake.bmp in the previous sharing process. The image on its right side is the corresponding generated verification image. It can be seen that the true shadow image can effectively recover the verification information, while the fake shadow image can not pass the verification.
After verification, the shadow image is calculated to restore the result image, and some of the recovered results are shown in Figure 6.
It can be seen from Figure 6 that when the number of shadow images is less than 6, the original secret image can not be recovered, and there is no secret leakage problem; when the number of shadow images is not less than 6, if all the shadow images are true, the lossless recovery of the original secret image can be realized, and if there are forged shadow images, the effective recovery cannot be achieved.
Because the verification information is embedded into the shadow image, the coefficients might not be randomly generated, and the required n shadow image pixels might not be generated, so the additional calculation will be increased when the scheme is implemented. In the experiment, the additional calculation times are compared and analyzed from two different aspects.
The first kind of comparative performance experiment is used to analyze the effect of the "XOR" in embedding verification information. In addition to the XOR mode, the binary logic operation of bit also has other modes such as AND and OR. In addition, this scheme is based on the improvement of the scheme in reference [16]. How about the performance compared with reference [16].
Let count denote the number of times that binomial coefficients are repeatedly selected for calculation when n shadow pixels are generated at one time and the constraint conditions are not satisfied. The constraint conditions include four cases: the lowest two bits of n shadow pixels have the same XORed value (this scheme), the lowest two bits OR operation, the lowest two bits have the same operation and the highest bit is the same (in [16]). For the original secret image with size of 128×64, generation average times ¯count of repeated calculation for each image pixel and different (k,n) thresholds, are shown in Table 1.
(k,n) | XOR Ours | AND | OR | Highest bit [16] |
(4, 4) | 15.19 | 236.71 | 236.89 | 15.35 |
(4, 5) | 31.37 | 31.36 | ||
(4, 6) | 66.73 | 65.16 | ||
(5, 5) | 31.46 | 524.25 | 520.34 | 31.54 |
(5, 6) | 63.09 | 2072.20 | 2090.10 | 63.83 |
(5, 7) | 129.25 | 11379.73 | 11307.26 | 128.56 |
(6, 6) | 63.46 | 2028.00 | 2146.88 | 63.71 |
(6, 7) | 129.48 | 8143.37 | 8498.96 | 130.05 |
(6, 8) | 262.07 | 32989.34 | 34368.82 | 256.86 |
It can be seen from table 1 that as the increase of n, the number of times of repeated calculation increases faster, but the value and growth rate of XOR mode are roughly consistent with that of high bit mode, which is significantly better than that of AND and OR mode. Because there is the same random probability between the operation value of XOR and the method of highest bit condition, that is, the probability of generating constraint value of 0 or 1 when binomial coefficient is randomly selected is 50%. This random selection mechanism with equal probability plays an important role in finding the values satisfying the constraint condition quickly. When "AND" and "OR" are used, the probability of generating constraint value 0 or 1 when binomial coefficient is randomly selected is biased. For "AND" mode, only when both values are 1, the result is 1, that is, the probability of generating 1 is 25%, and the probability of generating 0 is 75%. For "OR" mode, the biased probability leads to more times of repeated calculation with faster growth rate. When k is small, the feasible solution may not be generated due to the small random range, as shown in Table 1 (4, 5) and (4, 6) thresholds.
The second type of comparative experiment is to quantitatively analyze the scale of additional calculation times generated by the scheme itself with the change of (k,n) values. When k is not less than 4, we take the average value of multiple calculations to get the experimental results for (k,n) threshold from (4, 4) to (8, 10) as shown in Table 2.
n | k | 4 | 5 | 6 | 7 | 8 |
4 | 15.19 | |||||
5 | 31.37 | 31.46 | ||||
6 | 66.73 | 63.09 | 63.46 | |||
7 | 140.85 | 129.25 | 129.48 | 130.43 | ||
8 | 309.32 | 266.70 | 262.07 | 259.49 | 258.42 | |
9 | 756.24 | 536.18 | 531.68 | 528.90 | 520.78 | |
10 | 1061.84 | 1060.83 | 1053.71 | 1053.51 |
It can be seen from Table 2 that as the increase of (k,n) threshold, the number of times of repeated calculation shows a certain rule with the change of k and n. When n is fixed, the number of times of repeated calculation changes little with the increase of k, but when k is small, the number of times of repeated calculation will increase due to the small range of random value; while when k is fixed, the number of times of repeated calculation increases obviously with the increase of n, and the more image values that meet the constraint conditions need to be generated at the same time, showing a growth scale of about twice if n is too large. If the value of k is large and the value of k is small, such as (4, 10) threshold, there may also be the case that the threshold of n is larger than that of k.
Because the scheme adopts the mechanism of compressed shadow image, that is, embedding two secret pixels into one shadow image pixel each time, and limiting the random value condition of coefficient, it is necessary to carry out security experiment analysis on whether there is image leakage in the scheme.
In the experiment, we first try to generate a (3, 4) threshold SIS example. Because only one coefficient can change randomly when k=3, we can not find a feasible solution in the process of pixel generation of some shadow images due to the constraints. Therefore, we mainly analyze whether there is information leakage in the sharing process when k=4. Figure 7 and Figure 8 respectively show the recovered results of (4, 4) threshold and the distribution of shadow image pixel values.
It can be seen from Figure 7 and Figure 8 that in the minimum threshold experiment of this scheme, there is a weak non random situation only when three shadow images are used for recovery, but the recovery results when k is less than 4 cannot be effectively analyzed.
In a word, we can see that this scheme can effectively realize the compressible SIS process with the ability of shadow image verification on the basis of adding a certain amount of repeated calculation. Compared with the reference [16], the advantages of this scheme are mainly reflected in two aspects. One is to improve the highest embedding verification bit to the lowest two bits "XOR" value to set the verification bit, so that there is no linear relationship between any bits of the generated shadow image, hiding the embedding rule of verification information, and improving the security from the perspective of anti-shadow image analysis; the other is to set the "XOR" value of the lowest two bits as the verification bit. On the one hand, two secret pixels are used to calculate the pixel value of the shadow image at one time. Although the number of repetitions of the calculation is equivalent to the highest embedding due to the limitation of the lowest two XORed values each time, the overall generation efficiency is doubled because the generated shadow image is 1/2 times of the shadow image in [16].
In order to realize the verification of secret image sharing (SIS) and shadow image, based on the current polynomial-based SIS scheme and the (2, 2) threshold random-grid VSS with superposing in the "XOR" way, by improving the existing scheme strategy in the process of polynomial-based SIS mod 257, a scheme of compressed SIS with the ability of shadow image verification is implemented. The scheme has the ability of shadow image verification, pixel compression, loss tolerance and lossless recovery. Compared with the existing similar schemes, this scheme has some advantages in hiding pixel relation and high generating efficiency.
Based on the existing research situation, in the future we will continue to carry out research to improve the efficiency of shadow image generation: first, to study whether it can improve the current process of completely selecting coefficients in a random way, and reduce the number of repeated calculations to generate effective shadow image pixels; second, to study other embedding methods of verification information, so as to make the embedding of verification information more efficient.
The authors would like to thank the anonymous reviewers for their valuable comments. This work was supported in part by the National Natural Science Foundation of China (grant number: 61602491), and the Key Program of the National University of Defense Technology (grant number: ZK-17-02-07).
The authors declared that they have no conflicts of interest to this work.
[1] |
Barrilleaux B, Phinney DG, Prockop DJ, et al. (2006) Review: ex vivo engineering of living tissues with adult stem cells. Tissue Eng 12: 3007–3019. doi: 10.1089/ten.2006.12.3007
![]() |
[2] |
Hutmacher DW (2000) Scaffolds in tissue engineering bone and cartilage. Biomaterials 21: 2529–2543. doi: 10.1016/S0142-9612(00)00121-6
![]() |
[3] |
Lee J, Guarino V, Gloria A, et al. (2010) Regeneration of Achilles' tendon: the role of dynamic stimulation for enhanced cell proliferation and mechanical properties. J Biomat Sci-Polym E 21: 1173–1190. doi: 10.1163/092050609X12471222313524
![]() |
[4] |
Veronesi F, Giavaresi G, Guarino V, et al. (2015) Bioactivity and bone healing properties of biomimetic porous composite scaffold: in vitro and in vivo studies. J Biomed Mater Res A 103: 2932–2941. doi: 10.1002/jbm.a.35433
![]() |
[5] |
Guarino V, Galizia M, Alvarez-Perez MA, et al. (2015) Improving surface and transport properties of macroporous hydrogels for bone regeneration. J Biomed Mater Res A 103: 1095–1105. doi: 10.1002/jbm.a.35246
![]() |
[6] |
Guarino V, Ambrosio L (2010) Temperature-driven processing techniques for manufacturing fully interconnected porous scaffolds in bone tissue engineering. P I Mech Eng H 224: 1389–1400. doi: 10.1243/09544119JEIM744
![]() |
[7] |
Liu X, Ma PX (2009) Phase separation, pore structure, and properties of nanofibrous gelatin scaffolds. Biomaterials 30: 4094–4103. doi: 10.1016/j.biomaterials.2009.04.024
![]() |
[8] |
Gong S, Wang H, Sun Q, et al. (2006) Mechanical properties and in vitro biocompatibility of porous zein scaffolds. Biomaterials 27: 3793–3799. doi: 10.1016/j.biomaterials.2006.02.019
![]() |
[9] | Chiu YC, Larson JC, Isom Jr. A, et al. (2010) Generation of porous poly(ethylene glycol) hydrogels by salt leaching. Tissue Eng C 16: 905–912. |
[10] | De Nardo L, Bertoldi S, Cigada A, et al. (2012) Preparation and characterization of shape memory polymer scaffolds via solvent casting/particulate leaching. J Appl Biomater Func 2: 119–126. |
[11] |
Intranuovo F, Gristina R, Brun F, et al. (2014) Plasma modification of PCL porous scaffolds fabricated by solvent-casting/particulate-leaching for tissue engineering. Plasma Process Polym 11: 184–195. doi: 10.1002/ppap.201300149
![]() |
[12] |
Hou Q, Grijpma DW, Feijen J (2003) Porous polymeric structures for tissue engineering prepared by a coagulation, compression moulding and salt leaching technique. Biomaterials 24: 1937–1947. doi: 10.1016/S0142-9612(02)00562-8
![]() |
[13] |
Yoon JJ, Song SH, Lee DS, et al. (2004) Immobilization of cell adhesive RGD peptide onto the surface of highly porous biodegradable polymer scaffolds fabricated by a gas foaming/salt leaching method. Biomaterials 25: 5613–5620. doi: 10.1016/j.biomaterials.2004.01.014
![]() |
[14] |
Kim TK, Yoon JJ, Lee DS, et al. (2006) Gas foamed open porous biodegradable polymeric microspheres. Biomaterials 27: 152–159. doi: 10.1016/j.biomaterials.2005.05.081
![]() |
[15] |
Kim BS, Kim EJ, Choi JS, et al. (2014) Human collagen-based multilayer scaffolds for tendon-to-bone interface tissue engineering. J Biomed Mater Res A 102: 4044–4054. doi: 10.1002/jbm.a.35057
![]() |
[16] |
Sultana N, Wang M (2008) Fabrication of HA/PHBV composite scaffolds through the emulsion freezing/freeze-drying process and characterisation of the scaffolds. J Mater Sci-Mater M 19: 2555–2561. doi: 10.1007/s10856-007-3214-3
![]() |
[17] |
Xiong Z, Yan Y, Zhang R, et al. (2001) Fabrication of porous poly(L-lactic acid) scaffolds for bone tissue engineering via precise extrusion. Scripta Mater 45: 773–779. doi: 10.1016/S1359-6462(01)01094-6
![]() |
[18] |
Seck TM, Melchels FPW, Feijen J, et al. (2010) Designed biodegradable hydrogel structures prepared by stereolithography using poly(ethylene glycol)/poly(d,l-lactide)-based resins. J Control Release 148: 34–41. doi: 10.1016/j.jconrel.2010.07.111
![]() |
[19] |
Elomaa L, Teixeira S, Hakala R, et al. (2011) Preparation of poly(ε-caprolactone)-based tissue engineering scaffolds by stereolithography. Acta Biomater 7: 3850–3856. doi: 10.1016/j.actbio.2011.06.039
![]() |
[20] |
Yan M, Tian X, Peng G, et al. (2017) Hierarchically porous materials prepared by selective laser sintering. Mater Design 135: 62–68. doi: 10.1016/j.matdes.2017.09.015
![]() |
[21] | Liverani L, Guarino V, La Carrubba V, et al. (2017) Porous biomaterials and scaffolds for tissue engineering, In: Narayan R, Encyclopedia of Biomedical Engineering. |
[22] |
Granados-Hernández MV, Serrano-Bello J, Montesinos JJ, et al. (2018) In vitro and in vivo biological characterization of poly(lactic acid) fiber scaffolds synthesized by air jet spinning. J Biomed Mater Res B 106: 2435–2446. doi: 10.1002/jbm.b.34053
![]() |
[23] |
Guarino V, Ambrosio L (2016) Electrofluidodynamics: exploring a new toolbox to design biomaterials for tissue regeneration and degeneration. Nanomedicine 11: 1515–1518. doi: 10.2217/nnm-2016-0108
![]() |
[24] |
Blaker JJ, Knowles JC, Day RM (2008) Novel fabrication techniques to produce microspheres by thermally induced phase separation for tissue engineering and drug delivery. Acta Biomater 4: 264–272. doi: 10.1016/j.actbio.2007.09.011
![]() |
[25] | Manferdini C, Guarino V, Zini N, et al. (2010) Mineralization occurs faster on a new biomimetic hyaluronic acid-based scaffold. Biomaterials 31: 3986–3996. |
[26] |
Guarino V, Lewandowska M, Bil M, et al. (2010) Morphology and degradation properties of PCL/HYAFF11® composite scaffolds with multi-scale degradation rate. Compos Sci Technol 70: 1826–1837. doi: 10.1016/j.compscitech.2010.06.015
![]() |
[27] |
Salerno A, Guarino V, Oliviero O, et al. (2016) Bio-safe processing of polylactic-co-caprolactone and polylactic acid blends to fabricate nanofibrous porous scaffolds for tissue engineering. Mat Sci Eng C-Mater 63: 512–521. doi: 10.1016/j.msec.2016.03.018
![]() |
[28] | Luciani A, Guarino V, Ambrosio L, et al. (2019) Solvent and melting induced microspheres sintering techniques : a comparative study of morphology and mechanical properties. J Mater Sci-Mater M 22: 2019–2028. |
[29] |
Guarino V, Causa F, Salerno A, et al. (2008) Design and manufacture of microporous polymeric materials with hierarchal complex structure for biomedical application. Mater Sci Technol 24: 1111–1117. doi: 10.1179/174328408X341799
![]() |
[30] |
Whang K, Healy KE (1995) A novel method scaffolds to fabricate bioabsorbable. Polymer 36: 837–842. doi: 10.1016/0032-3861(95)93115-3
![]() |
[31] | Ma PX, Zhang R (1998) Synthetic nano-scale fibrous extracellular matrix. J Biomed Mater Res 46: 60–72. |
[32] |
Sohn DG, Hong MW, Kim YY, et al. (2015) Fabrication of dual-pore scaffolds using a combination of wire-networked molding (WNM) and non-solvent induced phase separation (NIPS) techniques. J Bionic Eng 12: 565–574. doi: 10.1016/S1672-6529(14)60146-3
![]() |
[33] |
Shin KC, Kim BS, Kim JH, et al. (2005) A facile preparation of highly interconnected macroporous PLGA scaffolds by liquid–liquid phase separation II. Polymer 46: 3801–3808. doi: 10.1016/j.polymer.2005.02.114
![]() |
[34] | Li S, Chen X, Li M (2011) Effect of some factors on fabrication of poly(L-lactic acid) microporous foams by thermally induced phase separation using N,N-dimethylacetamide as solvent. Prep Biochem Biotech 41: 53–72. |
[35] |
Billiet T, Vandenhaute M, Schelfhout J, et al. (2012) A review of trends and limitations in hydrogel-rapid prototyping for tissue engineering. Biomaterials 33: 6020–6041. doi: 10.1016/j.biomaterials.2012.04.050
![]() |
[36] |
Forbes SJ, Rosenthal N (2014) Preparing the ground for tissue regeneration : from mechanism to therapy. Nat Med 20: 857–869. doi: 10.1038/nm.3653
![]() |
[37] |
Hollister SJ (2005) Porous scaffold design for tissue engineering. Nat Mater 4: 518–524. doi: 10.1038/nmat1421
![]() |
[38] |
Skoog SA, Goering PL, Narayan RJ (2014) Stereolithography in tissue engineering. J Mater Sci-Mater M 25: 845–856. doi: 10.1007/s10856-013-5107-y
![]() |
[39] |
Ko SH, Pan H, Grigoropoulos CP (2007) All-inkjet-printed flexible electronics fabrication on a polymer substrate by low-temperature high-resolution selective laser sintering of metal nanoparticles. Nanotechnology 18: 345202. doi: 10.1088/0957-4484/18/34/345202
![]() |
[40] |
Hutmacher DW (2001) Scaffold design and fabrication technologies for engineering tissues-state of the art and future perspectives. J Biomat Sci-Polym E 12: 107–124. doi: 10.1163/156856201744489
![]() |
[41] | Khademhosseini A, Bong GC (2009) Microscale technologies for tissue engineering. 2009 IEEE/NIH Life Science Systems and Applications Workshop, Bethesda, MD, USA, 56–57. |
[42] |
Fischbach C, Chen R, Matsumoto T, et al. (2007) Engineering tumors with 3D scaffolds. Nat Methods 4: 855–860. doi: 10.1038/nmeth1085
![]() |
[43] |
Rutz AL, Hyland KE, Jakus AE, et al. (2015) A multimaterial bioink method for 3D printing tunable, cell-compatible hydrogels. Adv Mater 27: 1607–1614. doi: 10.1002/adma.201405076
![]() |
[44] |
Guarino V, Cirillo V, Altobelli R, et al. (2015) Polymer-based platforms by electric field-assisted techniques for tissue engineering and cancer therapy. Expert Rev Med Devic 12: 113–129. doi: 10.1586/17434440.2014.953058
![]() |
[45] |
Guaccio A, Guarino V, Alvarez-Perez MA, et al. (2011) Influence of electrospun fiber mesh size on hMSC oxygen metabolism in 3D collagen matrices: Experimental and theoretical evidences. Biotechnol Bioeng 108: 1965–1976. doi: 10.1002/bit.23113
![]() |
[46] |
Cirillo V, Guarino V, Alvarez-Perez MA, et al. (2014) Optimization of fully aligned bioactive electrospun fibers for "in vitro" nerve guidance. J Mater Sci-Mater M 25: 2323–2332. doi: 10.1007/s10856-014-5214-4
![]() |
[47] |
Pires LR, Guarino V, Oliveira MJ, et al. (2016) Ibuprofen-loaded poly(trimethylene carbonate-co-ε-caprolactone) electrospun fibres for nerve regeneration. J Tissue Eng Regen M 10: E154–E166. doi: 10.1002/term.1792
![]() |
[48] | Alvarez-Perez MA, Guarino V, Cirillo V, et al. (2012) In vitro mineralization and bone osteogenesis in poly(ε-caprolactone)/gelatin nanofibers. J Biomed Mater Res A 100: 3008–3019. |
[49] |
Cirillo V, Clements BA, Guarino V, et al. (2014) A comparison of the performance of mono- and bi-component electrospun conduits in a rat sciatic model. Biomaterials 35: 8970–8982. doi: 10.1016/j.biomaterials.2014.07.010
![]() |
[50] |
Fasolino I, Guarino V, Cirillo V, et al. (2017) 5-Azacytidine-mediated hMSC behavior on electrospun scaffolds for skeletal muscle regeneration. J Biomed Mater Res A 105: 2551–2561. doi: 10.1002/jbm.a.36111
![]() |
[51] |
Guarino V, Altobelli R, Cirillo V, et al. (2015) Additive electrospraying: a route to process electrospun scaffolds for controlled molecular release. Polym Advan Technol 26: 1359–1369. doi: 10.1002/pat.3588
![]() |
[52] | Guarino V, Cruz-Maya I, Altobelli R, et al. (2017) Antibacterial platforms via additive electrofluidodynamics for oral treatments. Nanotechology 28: 505303. |
[53] |
Slowing II, Vivero-Escoto JL, Wu C, et al. (2008) Mesoporous silica nanoparticles as controlled release drug delivery and gene transfection carriers. Adv Drug Deliver Rev 60: 1278–1288. doi: 10.1016/j.addr.2008.03.012
![]() |
[54] |
Vivero-Escoto JL, Slowing II, Trewyn BG, et al. (2010) Mesoporous silica nanoparticles for intracellular controlled drug delivery. Small 6: 1952–1967. doi: 10.1002/smll.200901789
![]() |
[55] |
Belmoujahid Y, Bonne M, Scudeller Y, et al. (2015) SBA-15 mesoporous silica as a super insulating material. Eur Phys J Special Topics 224: 1775–1785. doi: 10.1140/epjst/e2015-02498-3
![]() |
[56] |
Vargas-Osorio Z, González-Gómez MA, Piñeiro Y, et al. (2017) Novel synthetic routes of large-pore magnetic mesoporous nanocomposites (SBA-15/Fe3O4) as potential multifunctional theranostic nanodevices. J Mater Chem B 5: 9395–9404. doi: 10.1039/C7TB01963G
![]() |
[57] | Vargas-Osorio Z, Chanes-Cuevas OA, Pérez-Soria A, et al. (2017) Physicochemical effects of amino- or sulfur-functional groups onto SBA-15 sol-gel synthesized mesoporous ceramic material. Phys Status Solidi C 14: 1600099. |
[58] |
Kresge CT, Leonowicz ME, Roth WJ, et al. (1992) Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism. Nature 359: 710–712. doi: 10.1038/359710a0
![]() |
[59] |
Vartuli JC, Schmitt KD, Kresge CT, et al. (1994) Effect of surfactant/silica molar ratios on the formation of mesoporous molecular sieves: inorganic mimicry of surfactant liquid-crystal phases and mechanistic implications. Chem Mater 6: 2317–2326. doi: 10.1021/cm00048a018
![]() |
[60] |
Kresge CT, Roth WJ (2013) The discovery of mesoporous molecular sieves from the twenty year perspective. Chem Soc Rev 42: 3663–3670. doi: 10.1039/c3cs60016e
![]() |
[61] |
Feliczak-Guzik A, Jadach B, Piotrowska H, et al. (2016) Synthesis and characterization of SBA-16 type mesoporous materials containing amine groups. Micropor Mesopor Mat 220: 231–238. doi: 10.1016/j.micromeso.2015.09.006
![]() |
[62] | Gonzalez G, Sagarzazu A, Cordova A, et al. (2017) Comparative study of two silica mesoporous materials (SBA-16 and SBA-15) modified with a hydroxyapatite layer for clindamycin controlled delivery. Micropor Mesopor Mat 256: 251–261. |
[63] |
Chang JS, Chang KLB, Hwang DF, et al. (2007) In vitro cytotoxicitiy of silica nanoparticles at high concentrations strongly depends on the metabolic activity type of the cell line. Environ Sci Technol 41: 2064–2068. doi: 10.1021/es062347t
![]() |
[64] |
Soler-Illia GJAA, Sanchez C, Lebeau B, et al. (2002) Chemical strategies to design textured materials: from microporous and mesoporous oxides to nanonetworks and hierarchical structures. Chem Rev 102: 4093–4138. doi: 10.1021/cr0200062
![]() |
[65] |
Eliaz N, Metoki N (2017) Calcium phosphate bioceramics: a review of their history, structure, properties, coating technologies and biomedical applications. Materials 10: 334. doi: 10.3390/ma10040334
![]() |
[66] | Ambrosio L, Guarino V, Sanginario V, et al. (2012). Injectable calcium phosphate based composites for skeletal bone treatments. Biomed Mater 7: 024113. |
[67] |
Guarino V, Ambrosio L (2013) Thermoset composite hydrogels for bone/intervertebral disc interface. Mater Lett 110: 249–252. doi: 10.1016/j.matlet.2013.08.046
![]() |
[68] | Vallet-Regí M, Manzano-García M, Colilla M (2012) Biocompatible and bioactive mesoporous ceramics, In: Vallet-Regí M, Manzano-García M, Colilla M, Biomedical Applications of Mesoporous Ceramics: Drug Delivery, Smart Materials and Bone Tissue Engineering, Boca Raton: CRC Press, 1–66. |
[69] |
Samavedi S, Whittington AR, Goldstein AS (2013) Calcium phosphate ceramics in bone tissue engineering: a review of properties and their influence on cell behavior. Acta Biomater 9: 8037–8045. doi: 10.1016/j.actbio.2013.06.014
![]() |
[70] | Ishikawa K (2014) Calcium phosphate cement, In: Ben-Nissan B, Advances in Calcium Phosphate Biomaterials, Springer, 199–227. |
[71] |
Ambard AJ, Mueninghoff L (2006) Calcium phosphate cement: review of mechanical and biological properties. J Prosthodont 15: 321–328. doi: 10.1111/j.1532-849X.2006.00129.x
![]() |
[72] |
Coti KK, Belowich ME, Liong M, et al. (2009) Mechanised nanoparticles for drug delivery. Nanoscale 1: 16–39. doi: 10.1039/b9nr00162j
![]() |
[73] |
Vallet-Regi M, Rámila A, del Real RP, et al. (2001) A new property of MCM-41: drug delivery system. Chem Mater 13: 308–311. doi: 10.1021/cm0011559
![]() |
[74] |
Balas F, Manzano M, Horcajada P, et al. (2006) Confinement and controlled release of bisphosphonates on ordered mesoporous silica-based materials. J Am Chem Soc 128: 8116–8117. doi: 10.1021/ja062286z
![]() |
[75] |
Vallet-Regí M, Ruiz-González L, Izquierdo-Barba I, et al. (2006) Revisiting silica based ordered mesoporous materials: medical applications. J Mater Chem 16: 26–31. doi: 10.1039/B509744D
![]() |
[76] |
Mourino V, Boccaccini AR (2010) Bone tissue engineering therapeutics: controlled drug delivery in three-dimensional scaffolds. J R Soc Interface 7: 209–227. doi: 10.1098/rsif.2009.0379
![]() |
[77] |
Werner J, Sa S (2008) Hierarchical pore structure of calcium phosphate scaffolds by a combination of gel-casting and multiple tape-casting methods. Acta Biomater 4: 913–922. doi: 10.1016/j.actbio.2008.02.005
![]() |
[78] |
Vallet-Regí M (2008) Current trends on porous inorganic materials for biomedical applications. Chem Eng J 137: 1–3. doi: 10.1016/j.cej.2007.10.015
![]() |
[79] |
Baeza A, Izquierdo-Barba I, Vallet-Regí M (2010) Biotinylation of silicon-doped hydroxyapatite: a new approach to protein fixation for bone tissue regeneration. Acta Biomater 6: 743–749. doi: 10.1016/j.actbio.2009.09.004
![]() |
[80] |
Dorozhkin SV (2015) Calcium orthophosphate-containing biocomposites and hybrid biomaterials for biomedical applications. J Funct Biomater 6: 708–832. doi: 10.3390/jfb6030708
![]() |
[81] | Perez RA, Kim HW, Ginebra MP (2012) Polymeric additives to enhance the functional properties of calcium phosphate cements. J Tissue Eng 3: 2041731412439555. |
[82] |
Deb P, Deoghare AB, Borah A, et al. (2018) Scaffold development using biomaterials : A review. Mater Today Proc 5: 12909–12919. doi: 10.1016/j.matpr.2018.02.276
![]() |
[83] |
Zarrin A, Moztarzadeh F (2018) Synthesizing and characterizing of gelatin-chitosan-bioactive glass (58s) scaffolds for bone tissue engineering. Silicon 10: 1393–1394. doi: 10.1007/s12633-017-9616-z
![]() |
[84] | Wingender B, Bradley P, Saxena N, et al. (2016) Biomimetic organization of collagen matrices to template bone-like microstructures. Matrix Biol 52–54: 384–396. |
[85] |
Kang Z, Zhang X, Chen Y, et al. (2017) Preparation of polymer/calcium phosphate porous composite as bone tissue scaffolds. Mat Sci Eng C-Mater 70: 1125–1131. doi: 10.1016/j.msec.2016.04.008
![]() |
[86] |
Xu Y, Gao D, Feng P, et al. (2017) A mesoporous silica composite scaffold : Cell behaviors, biomineralization and mechanical properties. Appl Surf Sci 423: 314–321. doi: 10.1016/j.apsusc.2017.05.236
![]() |
[87] |
Mondal S, Hoang G, Manivasagan P, et al. (2018) Nano-hydroxyapatite bioactive glass composite scaffold with enhanced mechanical and biological performance for tissue engineering application. Ceram Int 44: 15735–15746. doi: 10.1016/j.ceramint.2018.05.248
![]() |
[88] |
Guarino V, Scaglione S, Sandri M, et al. (2014) MgCHA particles dispersion in porous PCL scaffolds: in vitro mineralization and in vivo bone formation. J Tissue Eng Regen M 8: 291–303. doi: 10.1002/term.1521
![]() |
[89] |
Scaglione S, Guarino V, Sandri M, et al. (2012) In vivo lamellar bone formation in fibre coated MgCHA–PCL-composite scaffolds. J Mater Sci-Mater M 23: 117–128. doi: 10.1007/s10856-011-4489-y
![]() |
1. | Jingju Liu, Lei Sun, Jinrui Liu, Xuehu Yan, Fake and dishonest participant location scheme in secret image sharing, 2021, 18, 1551-0018, 2473, 10.3934/mbe.2021126 | |
2. | Fei Hu, Weihai Li, Nenghai Yu, (k, n) threshold secret image sharing scheme based on Chinese remainder theorem with authenticability, 2023, 83, 1573-7721, 40713, 10.1007/s11042-023-17270-0 |
(k,n) | XOR Ours | AND | OR | Highest bit [16] |
(4, 4) | 15.19 | 236.71 | 236.89 | 15.35 |
(4, 5) | 31.37 | 31.36 | ||
(4, 6) | 66.73 | 65.16 | ||
(5, 5) | 31.46 | 524.25 | 520.34 | 31.54 |
(5, 6) | 63.09 | 2072.20 | 2090.10 | 63.83 |
(5, 7) | 129.25 | 11379.73 | 11307.26 | 128.56 |
(6, 6) | 63.46 | 2028.00 | 2146.88 | 63.71 |
(6, 7) | 129.48 | 8143.37 | 8498.96 | 130.05 |
(6, 8) | 262.07 | 32989.34 | 34368.82 | 256.86 |
n | k | 4 | 5 | 6 | 7 | 8 |
4 | 15.19 | |||||
5 | 31.37 | 31.46 | ||||
6 | 66.73 | 63.09 | 63.46 | |||
7 | 140.85 | 129.25 | 129.48 | 130.43 | ||
8 | 309.32 | 266.70 | 262.07 | 259.49 | 258.42 | |
9 | 756.24 | 536.18 | 531.68 | 528.90 | 520.78 | |
10 | 1061.84 | 1060.83 | 1053.71 | 1053.51 |
(k,n) | XOR Ours | AND | OR | Highest bit [16] |
(4, 4) | 15.19 | 236.71 | 236.89 | 15.35 |
(4, 5) | 31.37 | 31.36 | ||
(4, 6) | 66.73 | 65.16 | ||
(5, 5) | 31.46 | 524.25 | 520.34 | 31.54 |
(5, 6) | 63.09 | 2072.20 | 2090.10 | 63.83 |
(5, 7) | 129.25 | 11379.73 | 11307.26 | 128.56 |
(6, 6) | 63.46 | 2028.00 | 2146.88 | 63.71 |
(6, 7) | 129.48 | 8143.37 | 8498.96 | 130.05 |
(6, 8) | 262.07 | 32989.34 | 34368.82 | 256.86 |
n | k | 4 | 5 | 6 | 7 | 8 |
4 | 15.19 | |||||
5 | 31.37 | 31.46 | ||||
6 | 66.73 | 63.09 | 63.46 | |||
7 | 140.85 | 129.25 | 129.48 | 130.43 | ||
8 | 309.32 | 266.70 | 262.07 | 259.49 | 258.42 | |
9 | 756.24 | 536.18 | 531.68 | 528.90 | 520.78 | |
10 | 1061.84 | 1060.83 | 1053.71 | 1053.51 |