- The classic hash function, public key cryptography and visual cryptography are technically fused in the developed scheme. - In the shadow distribution phase, each participant can verify the received shadow using his private key. - In the restora- tion phase, for the case of dealer attendance, he can verify each shadow received using his secret key. - However, in the above two traditional mechanisms, although the secret is not leaked, in general the secret is not available when part of the data is lost or destroyed. - Thus, in the transmission, it is important to losslessly retrieve the original secret data when a certain amount of data is distorted or lost. - As a result, secret sharing has been introduced and image secret shar- ing (ISS) has further been introduced as well to protect secret image data.. - nỡ-threshold, the dealer splits a secret image into n shadows, a : k : a shadow images or shares. - In the collection and restoration phase, there are two cases of dealer attendance and dealer nonattendance. - The secret image can be restored after collecting any k or more shad- ows, even at most n k shadows that are lost, which is the so-called loss-tolerant feature of ISS [37]. - The secret image is restored from any k or more shadows by stacking them and recognizing them with only the naked human eye. - If an attacker steals any k 1 or less shadows, he cannot restore the secret image even with high computational power. - To restore a secret image with high quality, by randomly building a đk 1ỡ-degree polynomial, Shamir [28] originally designed the first polynomial-secret sharing for the đk . - In polynomial-ISS, the dealer first splits a secret image into n shadows and then distributes them to n participants. - In the restora- tion phase, for the case of dealer attendance, as in Fig. - 1(b), the dealer will fail to restore the secret image if there is a fake shadow among the k collected shadows, and he is also unable to distinguish the fake one. - However, the scheme can only verify the shadow in the restoring phase for the case of dealer attendance. - Unfortunately, it is only for the case of dealer attendance in the restoration phase and has possible information leakage with larger values of n k. - Unfortunately, it cannot verify the shadow in the distribution phase and may fail to verify the shadow with a larger value of n k.. - Due to the problem of commercial interests and possible security risks in the transmission of informa- tion, three kinds of verification are needed. - Finally, the above MVISS may be utilized in the scenario.. - A grayscale secret image and a binary verification image are input to generate n grayscale shadows. - The classic hash function, public key cryptography and VSS are technically fused in the developed scheme by using a screening operation. - In the splitting phase, by using a random number generator with a secret key, the dealer first splits two coordinates for each participant. - In the shadow distribution phase, each participant can verify the shadow using his private key. - In the restoration phase, for the case of dealer atten- dance, he can verify each shadow received using his secret key. - In addition, the secret image can be losslessly restored with any k or more shad- ows without auxiliary permutation. - Section 2 focuses on used preliminaries for the developed MVISS. - The notations used in the article are shown in Appendix A.. - ShamirỖs original polynomial-ISS scheme is shown in the algorithm of ShamirỖs original polynomial-ISS.. - In the restoring phase, by Lagrange interpolation, the coefficients of fđxỡ will be solved and then set s Ử f đ0ỡ when given any k shadow pixels. - We further analyze the selection of P in the original polynomial-ISS as follows. - In the developed MVISS, P Ử 257 and S 1 C i đh . - nỡ-threshold RGVSS [38] are in the Algorithm of đ2 . - This feature of public key cryptography is very suitable for shadow verification in the shadow distribution phase. - Developed MVISS 3.1. - The shadow(s) can be verified in the shadow distribution phase.. - The dealer can verify the received shadow(s) in the restoration phase.. - A participant can verify the received shadow(s) in the restoration phase.. - the denominator means the total bits in the final shadows.. - The developed MVISS. - The design idea of the developed MVISS is given in Fig. - The developed multiparty verification in image secret sharing (MVISS) Input: S 1 and S 2 . - go to next secret image position, for i Ử 1 . - wỡ, go to the next secret image position, for i Ử 1 . - w ỡ ỡ Ử HV i đ ỡ, where w i Ử H h ợ 1, go to the next secret image position, for i Ử 1 . - Design concept of the developed multiparty verification in image secret sharing (MVISS).. - Verification and restoration in the developed MVISS.. - Step 1: In the shadow distributing phase, when receiving S 1 C i , participant i extracts Z 0i x by using the XOR4LBs operation on the i-th line of S 1 C i , for x Ử h 1 . - Step 2: In the secret image restoring phase, for the case of dealer attendance, when receiving S 1 C i , the dealer uses the random number generator and his private key k D to generate Z i h 1 . - In the secret image restoring phase, for the case of dealer nonattendance, for participant i, prior to sending S 1 C i to participant j, exchange XOR4LBsđS 1 C i ỡ and XOR4LBsđS 1 C j ỡ first, and obtain the decrypted binary verification image S 0 2 through stacking XOR4LBsđS 1 C i ỡ and XOR4LBsđS 1 C j ỡ. - The performance and security analyses of the developed MVISS will be presented.. - S 1 C q k in the restoring phase.. - The developed scheme is a đk . - We have N R Ử P k1 256 P n 1 2 in the developed scheme. - In the shadow distribution phase, using S 1 C i and k i s , participant i can verify whether S 1 C i is fake. - In the restoration phase, for the case of dealer attendance, using S 1 C i and k D , the dealer can verify whether S 1 C i is fake for i Ử 1 . - W 2 possible coordinates to be hashed, i.e., the start- ing and ending plain coordinates ( Z i h. - Similarly, in the restoration phase, for the case of dealer attendance, the dealer with k D can generate Z i h 1 . - In the restoration phase, for the case of dealer nonattendance, in Step 8 of Algorithm 1, XOR4LBs S đ 1 C i đ h . - S 2 C n , the binary secret image S 2 is visually recognized. - Since there is no pixel expansion in the developed ISS, all the tested images have the same size of 256 256. - More thresh- olds are performed to indicate the feasibility of the developed MVISS. - 3 exhibits the results of the developed MVISS, where k Ử 2 . - 3(cỜd) present the 2 shadows S 1 C 1 and S 1 C 2 , which have the same size as the original secret image. - Thus, the developed MVISS has no pixel expansion and no auxiliary information. - 3(f) shows the secret image restored with the 2 shadows using Lagrange interpolation, indicating that the secret image S 1 is losslessly restored. - 4 exhibits the results of the developed MVISS, where k Ử 3 . - 4(iỜl) show the secret image restored with some 2 or more shadows using Lagrange inter-. - Experiments of the developed MVISS for a đ2. - 4(iỜl), the secret image restored with any 3 or more shadows is lossless, while nothing of the secret image with 2 or fewer shadows is leaked. - The secret image is losslessly restored with any k or more shadows.. - The developed MVISS will be compared with that of Yan et al. - Experiments of the developed MVISS for a đ3. - 5(h) shows the secret image restored with the first 2 shadows by Lagrange interpolation. - 5(h), the restored secret image with any 2 or more shadows is lossless.. - 6 exhibits the results of the developed MVISS with the same parameters, where k 1 p Ử f5 . - 6(g) shows the secret image restored with the 2 shadows using Lagrange interpolation, showing the secret image S 1 is losslessly restored.. - Experiments of the developed MVISS for the đ2;3ỡ-threshold.. - 7(i) shows the secret image restored with the first 2 shadows by Lagrange interpolation. - 7 (i), the restored secret image with any 2 or more shadows is lossless. - Actually, Jiang et al.Ỗs work [17] has the verification ability in the restoration phase for the cases of both dealer attendance and dealer nonattendance. - Yan et al.Ỗs scheme and Jiang et al.Ỗ scheme can verify the shadow only in the restoring phase, while our MVISS has this ability in both distributing and restoring phases for the cases of both dealer attendance and dealer nonattendance, i.e., multiparty verification.. - We can extend the developed MVISS in the following ways.. - Some other advantageous public key cartographies and hash functions can be implemented in the developed MVISS.. - We can apply some other ISS principles in the developed MVISS, such as the Chinese remainder theorem-ISS.. - Through fusing the classic hash function, public key cryp- tography and VSS, the developed MVISS is suitable for both the shadow distribution phase and the restoration phase for the case of dealer attendance and dealer nonattendance. - In the future, we will further analyze the security and use other ISS principles, hash functions and public key cryptography in our MVISS to obtain more features, such as the Chinese remainder theorem-based ISS.. - Alhamid, Distortion less secret image sharing scheme for internet of things system, Cluster Computing (2017), https://doi.org/10.1007/s y.. - Main notations in the article.. - S 1 A grayscale secret image. - t The number of collected shadows in the restoring phase. - Restored grayscale secret image S 0 1 by shadows S 1 C i 1 ;S 1 C i 2 . - Zhou, Essential secret image sharing scheme with the same size of shadows, Digital Signal Processing Ờ60, https://doi.. - Tsai, Secret image sharing with steganography and authentication, Journal of Systems and Software Ờ414, https://doi.. - Yang, Scalable secret image sharing scheme with essential shadows, Signal Processing: Image Communication Ờ55.. - Yang, (k, n) secret image sharing scheme capable of cheating detection, EURASIP Journal on Wireless Communications and Networking . - Lin, Secret image sharing, Computers &. - Nabiyev, Secret image sharing scheme with adaptive authentication strength, Pattern Recognition Letters Ờ. - Liu, Secret image sharing with separate shadow authentication ability, Signal Processing: Image Communication https://doi.org/10.1016/j.image URL: http://www.sciencedirect.com/science/article/pii/S . - Lu, Robust secret image sharing resistant to noise in shares, ACM Transactions on Multimedia Computing, Communications, and Applications (2020) (accepted).
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