Hindawi Security and Communication Networks Volume 2021, Article ID 6629726, 14 pages https://doi.org/10.1155/2021/6629726

Research Article A Verifiable -Based Secret Image Sharing Scheme in 5G Networks

Shiyue Qin ,1 Zhenhua Tan,2 Fucai Zhou ,2 Jian Xu,2 and Zongye Zhang2

1School of Computer Science and Engineering, Northeastern University, Shenyang, China 2Software College, Northeastern University, Shenyang, China

Correspondence should be addressed to Fucai Zhou; [email protected]

Received 17 November 2020; Revised 11 December 2020; Accepted 29 December 2020; Published 28 January 2021

Academic Editor: Jinwei Wang

Copyright © 2021 Shiyue Qin et al. -is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. With the development and innovation of new techniques for 5G, 5G networks can provide extremely large capacity, robust integrity, high bandwidth, and low latency for multimedia image sharing and storage. However, it will surely exacerbate the privacy problems intrinsic to image transformation. Due to the high security and reliability requirements for storing and sharing sensitive images in the 5G network environment, verifiable steganography-based secret image sharing (SIS) is attracting increasing attention. -e verifiable capability is necessary to ensure the correct image reconstruction. From the literature, efficient cheating verification, lossless reconstruction, low reconstruct complexity, and high-quality stego images without pixel expansion are summarized as the primary goals of proposing an effective steganography-based SIS scheme. Compared with the traditional underlying techniques for SIS, cellular automata (CA) and matrix projection have more strengths as well as some weaknesses. In this paper, we perform a complimentary of these two techniques to propose a verifiable secret image sharing scheme, where CA is used to enhance the security of the secret image, and matrix projection is used to generate shadows with a smaller size. From the steganography perspective, instead of the traditional least significant bits replacement method, matrix encoding is used in this paper to improve the embedding efficiency and stego image quality. -erefore, we can simultaneously achieve the above goals and achieve proactive and dynamic features based on matrix projection. Such features can make the proposed SIS scheme more applicable to flexible 5G networks. Finally, the security analysis illustrates that our scheme can effectively resist the collusion attack and detect the shadow tampering over the persistent adversary. -e analyses for performance and comparative demonstrate that our scheme is a better performer among the recent schemes with the perspective of functionality, visual quality, embedding ratio, and computational efficiency. -erefore, our scheme further strengthens security for the images in 5G networks.

1. Introduction and organizations to interact with others more actively and exchange more information in a greater open network Nowadays, a huge amount of data is available in multimedia environment [2]. However, it may exacerbate the privacy forms, such as images and videos. With the growing pop- problems intrinsic to multimedia transfer in networks as a ularity of multimedia data, multimedia sharing has become result. As image sharing is one of the basic multimedia one of the most popular Internet services. Meanwhile, the sharing services, this paper aims to achieve secure image emergence of fifth-generation networks (5G) brings many share and storage in 5G networks. advantages to supply power to such multimedia sharing Based on the 5G technique advantages, it is foreseen that services. Compared to the actual 4G technologies, 5G will be 5G will have great potential to handle challenges in various more flexible and will provide significantly higher band- fields, such as smart cities [3], healthcare [4], and industrial width, as well as robust integrity, more capacity, and very control [5], and traffic congestion [6]. In such scenarios, low latency [1]. Such features are fundamental for facilitating medical images, design drawings, military images, and fi- information flow and timeliness of data, thus improving nancial reports [7] always contain massive sensitive infor- sharing services quality. 5G will undoubtedly drive people mation. Hence, the sharing and storing of these images can 2 Security and Communication Networks be more vulnerable towards several kinds of attacks, such as shadow distribution again. Such a feature is necessary for an eavesdropping and tampering. Secret image sharing (SIS) SIS scheme to adapt the data flow rate in 5G networks. [8, 9] developing from the traditional is a However, these features are attracted very little attention to promising way to ensure the privacy and integrity of the SIS. secret image for distributed storing and sharing in the -erefore, in this paper, we are willing to propose a network. However, ordinary SIS will generate n shadows in verifiable steganography-based secret image sharing scheme the form of noise-like images [10, 11] which might arise for 5G networks that achieves the following goals: (1) high- suspicious of the invaders [12] during the shadow dissem- quality stego images without pixel expansion; (2) an efficient ination. -erefore, steganography-based SIS will further cheating verification ability to handle the dishonest par- improve the security of SIS systems in a more open 5G ticipant; (3) lossless reconstruction and low reconstruct network. complexity; and (4) proactive feature and dynamic feature. A general (k, n)-threshold steganography-based SIS To meet the design goals described above, we combine Two- scheme enables a dealer to distribute its secret image among Dimensional Reversible Linear Cellular Automata with a set of untrusted participants via the public channel. Memory (2D-RLCAM) and matrix projection together to Firstly, the dealer generates n noise-like shadows for the design the scheme. 2D-RLCAM is a kind of cellular secret image. -en, n shadows are embedded into the automata additionally with memory and reversible abilities. preselected meaningful images (called cover images) so as Our contributions are as follows: to generate n meaningful stego images. Finally, each of the (i) Security enhanced: during the secret distribution participants is allocated a stego image. When someone procedure, CA is used to preprocess the secret requests the secret’s reconstruction, they need to extract image. -en, the matrix projection-based secret enough shadows from the stego images. Only the combi- sharing scheme, as introduced in the typical work of nation of at least k shadows can perform the lossless re- Bai and Zou [11], is employed with some changes to construction, and less than k shadows give no clues about generate small shadows. Increased by CA’s security, the secret image in contrast. -erefore, steganography- the secret image security is significantly improved to based SIS can protect the secret image with higher security solve the matrix projection’s weaknesses. because the eavesdroppers cannot detect secret information from meaningful images. However, most of the existed (ii) Needless of the consecutive shadows for recon- steganography-based SIS schemes have a problem with the struction: the consecutive requirement for reversing pixel expansion, for example, the stego images generated in in CA is embedded into the secret image’s sub- the schemes of [13–15] are expanded four times the original matrices, but not directly for the secret image. secret image to realize high-quality steganography. Here, Hence, the consecutive shadows in this paper are the challenge of proposing steganography-based SIS needless for secret reconstruction to overcome CA’s schemes is to create high-quality stego images so that the weaknesses. modifications are not visually perceptible. (iii) High-quality stego images without pixel expansion: Moreover, the verification ability is necessary to check matrix encoding is using in this paper for steg- the validity of the stego images for the secret image re- anography. Our scheme obtains enhancement in construction. If a fake stego image is submitted, the both the security of secret image and the visual reconstructed image will not be exactly the same as originally quality of the stego image, benefiting from smaller outsourced. Cheating detection is one of the methods to shadows and the comparatively slight modification locate a set of potential invalid shadows. It is easy to design to the cover image. SIS with verification ability by using hash codes [16–18]. (iv) Feature optimization for 5G scenarios: for the However, most steganography-based SIS with cheating verifiable feature, a separate verification matrix is detection ability needs to embed these extra authentication used for cheating detection. It can achieve a rela- bits into the cover image, which unfortunately leads to high tively low verification complexity based on XOR recovery (authentication) complexity and pixel expansion. operations. For proactive and dynamic features, -erefore, an efficiency verification mechanism is needed to they can be realized with only a few times of matrix be designed. operations. Additionally, in 5G network scenarios, a natural ex- (v) Lossless reconstruction and low reconstruct com- tension of such schemes is to consider the attached proactive plexity: the experiments over the test images feature and dynamic feature. -e proactive feature is es- demonstrate the effectiveness and efficiency of the sential in some circumstances that the secret needs to be kept proposed scheme via several indices. for a very long time, such as medical image records [19]. It can resist persistent adversaries by updating the shadows -e rest of this paper is organized as follows: the related periodically. -e secret cannot be discovered even from work is given in Section 2. -e preliminaries of 2D-RLCAM, enough shadows, which are mixed with past and present the typical work of Bai and Zou [11], and matrix encoding shadows. Especially in 5G network environments, the per- are introduced in Section 3. -e procedures of the proposed sistent adversary will be more general who can stay in the SIS scheme are discussed in detail in Section 4. -e security network all the time. Moreover, the dynamic feature is also analyses for persistent adversaries are studied in Section 5. practical which supports efficient secret updating without -e experimental results based on the real images and Security and Communication Networks 3 comparative analysis are given in Section 6. Finally, the the security and functionality of matrix projection-based SIS paper ends with a conclusion in Section 7. need to be improved. In this paper, we combine Two-Dimensional Reversible 2. Related Work Linear Cellular Automata with Memory (2D-RLCAM) and matrix projection together to avoid the weaknesses of these 2.1. SIS. From the literature, Shamir’s secret sharing [20] is techniques but keeping the advantages. one of the underlying techniques used to build SIS schemes mostly, but always has the pixel loss problem [21, 22]. In 2.2.Steganography-Based SIS. -e least significant bits (LSB) order to solve this problem, several new techniques have method is one of the most common solutions for steg- been used to devise a new (k, n)-threshold SIS scheme, such anography. -is method directly replaces the least signifi- as cellular automata (CA) [23, 24] and matrix projection cant bits of some pixels in the cover image with the shadow, [25]. such as in Yang et al.’s Shamir-based scheme [13], Chang Compared with Shamir-based SIS, CA-based SIS et al.’s Chinese remainder-based scheme [14], and Eslami [12, 15, 26, 27] has more advantages: (1) the computation et al.’s linear cellular automata- (LCA-) based scheme [15]. complexity of CA-based SIS is only O(n), which is more -ese schemes enable the shadow embedding and concealed efficient while Shamir-based SIS needs O(n log n); (2) shadow extraction in a highly efficient way, without any the consecutive properties in CA provide a stronger security cryptographic computation. However, the above schemes guarantee for the secret image. Nevertheless, most CA-based have a problem with the pixel expansion. One of the SIS schemes also exist in some problems: (1) only consec- reasons is that the generated shadows’ size is relatively big. utive shadows can reveal the secret image [15, 26, 27] which Another reason is that the requirement of the steganography limited the availability and (2) the pixel expansion problem space is relatively big for performing LSB so that the stego still exists. -e scheme of Eslami et al. [15] is the first CA- image generated in the above schemes is expanded four based SIS, which improved tamper detection. However, in times the original secret image to realize high-quality this scheme, the shadow’s size is the same as the original steganography. secret image. If the steganography of these shadows is Matrix encoding (ME) method is another steganography performed, the pixel expansion problem will be unavoidable. method mainly based on some XOR operations, rather than -e scheme of Zarepour-Ahmadabadi et al. [12] is a mul- direct replacement as in LSB. Hence, the information of the tistage multisecret image sharing scheme based on CA and shadow does not directly appear in the pixels of the cover Shamir-based sharing algorithm that each secret image has image, which increases the security of the concealed in- its own access structure. In this scheme, the secrets can be formation. Moreover, to hide a 3-bit length secret, at most reconstructed based on their own prespecified order, which only 1 bit of a pixel is needed to be changed via the ME can apply to more general and flexible scenarios. However, method, but usually 3 bits are needed in the LSB method. this scheme executed the Shamir-based sharing algorithm Hence, ME can realize the comparatively slight modification multiple times during the secret distribution and recon- to the cover image. Compared with LSB, ME needs more struction, which still made a higher computation computational cost because of some simple XOR operations, complexity. but it can obtain enhancement in both the security of secret Matrix projection-based SIS [11, 16, 28] can significantly data and the visual quality of stego images. -erefore, we use reduce the size of the shadows. In the basic matrix pro- ME in our paper to hide the shadow into a cover image. jection-based (k, n) secret sharing scheme [10], for a m × m Moreover, Yan et al. [29] introduced the requirement of secret matrix, the generated shadows are vectors with the lossless reconstruction of the cover images in an SIS scheme. size of only m × 1. Based on it, Bai and Zou [11] construct a -e reversible cover images are needed in several scenarios, proactive secret sharing scheme. -e generated shadows in such as law enforcement, experimental investigations, and their scheme are (m + k) × 1 vectors. -e size of the shadows remote sensing [30]. However, the traditional SIS ap- is still small. -erefore, matrix projection-based SIS can proaches have some limitations for the cover image re- effectively reduce the modifications over the cover images construction, such as the restrictions of the cover image and even create high-quality stego images by avoiding the formats and kinds, or loss reconstruction of the cover image. pixel expansion problem. However, the existing matrix In the scheme of Yan et al. [29], the lossless reconstruction is projection-based SIS schemes only generate noise-like image realized, but the format of cover images is restricted to the shadows but do not consider combining steganography. binary image. In this paper, we generate a submatrix for the Additionally, there are two problems with the existed matrix grayscale cover image to recover it losslessly. projection-based SIS schemes: (1) matrix projection-based (k, n) secret sharing scheme is only a strong ramp secret sharing but not perfect secret sharing [25], which may ex- 2.3. Verifiable SIS. It is easy to design SIS with verification pose information about the secret proportional to the ability by using hash codes [16–18] or information hiding number of the obtained shadows till enough, and (2) without [15, 26, 31]. Chang et al. [14] used the Chinese remainder the verification ability [11, 28]. Although the scheme of theorem (CRT) to generate four authentication bits by using Azzahra and Sugeng [16] realized cheating detection with the hash function in order to obtain better authentication the help of an additional authentication image, the dealer ability. Recently, Liu et al. [17] divided each stego image into must be involved in the reconstruction process. -erefore, nonoverlapping blocks with size 1 × 3 firstly and then 4 Security and Communication Networks

generated two authentication bits for each stego block. -is with the cell space of Sb � {}0, 1, ... , 7 (b � 8) and with scheme also achieved good visual quality and authentication transfer function F depending on Moore neighborhood N. ability. However, most of the above schemes need to embed 2D Reversible Linear Cellular Automata with Memory (2D- these extra authentication bits into the cover image, which RLCAM) is a kind of CA additionally with memory and unfortunately leads to high recovery (authentication) com- reversible abilities. (t) plexity and pixel expansion. Besides the above solutions, the For a r × s 2D-CA, if ai,j ∈ Sb is taken to denote the state double verification mechanism [15], the random grid [32], and of one cell at the position (i, j) at a certain time t, the state of the extra authentication image [16] were also used to provide a N of it can be represented as follows: powerful verification of the fake stego images. In the scheme of a(t) a(t) a(t) Azzahra and Sugeng [16], the dealer kept an extra authenti- ⎜⎛ i− 1,j− 1 i− 1,j i− 1,j+1 ⎟⎞ ⎜ ⎟ cation image privately. After the secret reconstruction, the (t) ⎜ (t) (t) (t) ⎟ Ni,j �⎜ ai,j− 1 ai,j ai,j+1 ⎟. (1) dealer performed the cheating detection by checking if the ⎝⎜ ⎠⎟ (t) (t) (t) authentication image is recovered correctly. However, this ai+1,j− 1 ai+1,j ai+1,j+1 scheme suffers from drawbacks in that even enough shadows cannot reconstruct the secret image because the dealer must be Denote C(t) as the CA’s configuration of time t which (t) (0) involved in the reconstruction process. To some degree, the consisted of all of the ai,j (1 ≤ i ≤ r, 1 ≤ j ≤ s). C is called the participation of the dealer makes this scheme diverge from the initial configuration. When dealing with the iteration of CA, (t) (t) original intention of the secret sharing. the periodic boundary condition ai,j � ax,y(i ≡ x mod r, In this paper, we embed a verification matrix into the j ≡ y mod s) is used in this paper. public known information by XOR operations. It can avoid In 2D-RLCAM of capacity k (2D-k-RLCAM), the state of the problems of high recovery complexity and pixel ex- the configuration C(t+1) evolves as follows: pansion. Moreover, the valid requester can perform an ef- C(t+1) � FC� (t),C(t− 1), ... ,C(t− k+1)�. (2) ficient cheating detection via some simple XOR operations but without the additional help from the dealer. (t+1) (t+1) More concretely, the state of each cell ai,j ∈ C can be computed as (3). Note that k denotes the capability of 3. Preliminaries memory here but not the threshold value of SIS. In this 3.1. Reversible Linear Cellular Automata with Memory. paper, 2D-3-RLCAM is used. Cellular automaton (CA) is a deterministic and discrete system. -is paper deals with two-dimensional CA (2D-CA)

a(t+1) ��f �N(t) � + f �N(t− 1) � + · · · + f �N(t− k+2) � + f �N(t− k+1) �� b i,j w i,j w i,j wk− i,j wk i,j mod 1 2 1 (3) ��f �N(t) � + f �N(t− 1) � + · · · + f �N(t− k+2) � + a(t− k+1)� mod b. w1 i,j w2 i,j wk− 1 i,j i,j

f �N(t) � � � a(t) � a(t) + a(t) + a(t) . (t) (t) (t− 1) (t− 1) (t− k+1) w i,j λα,β ∗ i+α,j+β i− 1,j+1 i+1,j− 1 i+1,j+1 In equation (3), Ni,j ∈ C ,Ni,j ∈ C , ... ,Ni,j (t− k+1) (5) ∈ C are the Moore neighborhood of the cell ai,j in the t, t − , ... , t − k + �w , ... , w � � (t) (t) time 1 1, respectively. Besides, 1 k Let C denote a reversion state of time t and a�i,j denote F f � (t) is the rule number of the transfer function and w is the state of one cell in it. Ni,j denotes the neighborhood of it. computed by (4). -erefore, the value of w can be repre- During the reversion of 2D-k-RLCAM after n iterations, the (k+n− 1) sented as (w)10 � (λ− 1,− 1, λ− 1,0, λ− 1,1, λ0,− 1, λ0.0, λ0,1, λ1,− 1, final configuration C is taken as the initial reversion (0) λ , λ ) , where w ∈ [0, 29]. Notice that, if f (N(t− k+1)) � state C� . -erefore, the configuration C(k+n− 1− t) is taken as 1,0 1,1 2 wk i,j (t) (t+1) (t− k+1) the reversion state C� . -e state reverses as C� � F′ a is satisfied, w should be equal to 16. (t− k+1) (t− k+2) (t) i,j k (C� , C� , ... , C� ) by

(t+ ) (t) (t− 1) (t− k+2) (t) (t) 1 � � � a�i,j � − �fw �Ni,j � + fw �Ni,j � + · · · + fw �Ni,j �� fw�Ni,j � � � λα,β ∗ ai+α,j+β, k− 1 k− 2 1 + a�(t− k+1) mod b. · �where α, β �{} − 1, 0, 1 and λα,β ∈ {}0, 1 �. i,j (4) (6) A simple example for 2D-3-RLCAM: assume n � 5 and A simple example for w � 69 of (4): we can obtain assume the set of �C(0),C(1),C(2)� is the initial configuration. (3) (7) (69)10 � (001000101)2. -erefore, only the values of -erefore, C is the state after the first iteration, and C is λ− 1,1, λ1,− 1, λ1,1 are equal to 1 and the others are equal to 0. the final state after 5 iterations after computing (7). During (t) Hence, we can compute fw(Ni,j ) as follows: the reversible process, �C(7),C(6),C(5)� will be taken as the Security and Communication Networks 5

(0) (1) (2) R � (P − S) initial reversion state �C� , C� , C� �, and C(4) will be (iii) A3: compute the remainder matrix A mod p, where PA is the LT × LT projection matrix of (3) recovered by C� � C(4) as in (8): A. -e remainder matrix R is also with the size of LT × LT. C(7) � f �C(6) � + f �C(5) � + C(4), (7) w1 w2 (iv) A4: distribute shadow vectors θi(i�1,2,..n) to partici- pants and make the remainder matrix R publicly � (3) � (1) � (2) � (0) C � − �fw �C + fw �C � + C known. 1 2 (8) � − f �C(6) � − f �C(5) � + C(7) � C(4). Update shadows at the beginning of t + 1(t ≥ 0): w1 w2 (i) B1: generate an k × k orthonormal matrix LM by (9) 3.2. Bai and Zou’s Proactive Secret Sharing Scheme. Bai and as the update matrix, and distribute it via a secure Zou’s (k, n)-threshold PSS scheme [11] can periodically channel to each participant. renew n shadows without modifying the secret. -e secrets (t) (t+1) (ii) B2: each participant updates its shadow θi to θi cannot be discovered from enough shares, which are mixed by (10) and erases the variables θ(t) and LM: with past and present shares. -e scheme is based on the i matrix projection method, and the proactive feature is �θ(t) � (t+1) ⎢⎡ i 1: m ⎥⎤ achieved through Pythagorean triples. θi �⎣ ⎦, (10) �(t) � Matrix projection: let A be an m × k matrix of rank LM ∗ θi m+1: m+k T − 1 T k(0 ≤ k ≤ m) and PA � A(A A) A is the m × m projection [ (t)] (t) matrix of it. Give k linearly independent k × 1 vectors xi and θi 1: m denotes the top part of θi which contains the first m elements and [θ(t)] is the lower compute θi � Axi. -e matrix B � [θ1, θ2, ... , θk] can satisfy i m+1: m+k T − 1 T k PB � B(B B) B � PA. part which contains the remaining elements. ∗ Pythagorean triples: randomly choose {}a, b (a, b ∈ Z , S 2 Reconstruction of the secret matrix 0: a > b > 0), and compute the Pythagorean triples �Z1 � a − 2 3 2 2 2 2 2 (i) C1: compute the matrix B � [θ , θ , ... , θ ] based on b ,Z2 � 2ab, Z � a + b } where Z1 + Z2 � Z3. Based on 1 2 k randomly chosen �g, h �(1 ≤ g ≠ h ≤ k), an orthonormal k × k any k valid shadows. − 1 T matrix LM can be constructed by (9) where LM � LM . (ii) C2: the secret S0 � [PB − R]UL: m×m mod p. LMi,j denotes the element of position (i, j) in LM: When the current time period t > 0, there exists PA ≠ PB Z ⎪⎧ 1 because the shadows are updated. However, equation (11) is ⎪ mod p, if i � j � g, ⎪ Z3 still holding based on the matrix projection and the or- ⎪ ⎪ thonormal update matrix LM. -e detailed explanation can ⎪ ⎪ Z be seen in [11]. ⎪ 1 p, i � j � h, ⎪ mod if ⎪ Z3 �PA �UL: m×m ��PB �UL: m×m. (11) ⎪ ⎪ ⎨ -e proposed SIS scheme in this paper is constructed LMi,j �⎪ Z2 (9) ⎪ mod p, if i � g, j � h, based on Bai and Zou’s [11] scheme. -e detailed analysis of ⎪ Z ⎪ 3 the correctness and security of sharing and updating can be ⎪ ⎪ seen in [11]. ⎪ Z ⎪ 2 ⎪ − mod p, if i � h, j � g, ⎪ Z3 ⎪ ⎪ 3.3. Matrix Encoding. Matrix encoding (ME) is a technology ⎩⎪ for steganography. Generally, for a secret S, a ME process 0, otherwise. can be defined as a 3-tuple �Cmax, n, k �. It denotes that no k Now suppose the dealer wants to share a secret m × m more than Cmax bits can be changed among n � 2 − 1 modifiable bits when hiding a k-bit length secret S. matrix S0. -en, Bai and Zou’s [11] proactive scheme based on matrix projection method can be constructed in the Let the set of �a1, a2, ... , an� denote the steganography following three phases. An,Bn, and Cn denote the n-th step cell, where ai is a modifiable bit. Compute f � (a1 ∗ 1) in each phase A, B, C. ⊕(a2 ∗ 2) ⊕ ··· ⊕(an ∗ n) and p � f ⊕ S. -e cell �a , a , ... , an� is changed into �a′, a′, ... , an′� by (12). In the Construction of shadows for a secret matrix S0 (with the 1 2 1 2 size of m × m) at the time period t � 0, where m > 2k − 3: secret extraction phase, the secret S can be directly extracted by S′ � (a1′ ∗ 1) ⊕ (a2′ ∗ 2) ⊕ ··· ⊕(an′ ∗ n) � S. (i) A1: filling S0 into a matrix S with the size of LT × LT ⎧⎨ if p � 0: a′i � ai, i � 1, ... , n, via random values, where LT � m + k. Take S0 � [S] m × m S ⎩ UL: m×m to denote that matrix 0 can be if p ≠ 0: ap′ � 1 − ap, a′i � ai, i � 1, ... , n(i ≠ p). extracted from the upper left corner of the matrix S. (12) (ii) A2: construct a set of LT × 1 shadow vectors θi(i�1,2,..n) � Axi based on a random LT × k matrix A (1, 7, 3)-ME is used in this paper. It means that the secret and n linearly independent k × 1 vectors xi. value is should be satisfied S ∈ [0, 7]. For example, assume 6 Security and Communication Networks

we want to hide S � (6)10 � (100)2 into a pixel with the shadow vectors θ Set � �θ1, θ2, ... , θn�. -e size of each value of (105)10 � (1101001)2 � �a1, a2, a3, a4, a5, a6, a7� shadow vector θi is LT × 1, where LT � m + k. during a (1, 7, 3)-ME process. Compute f � 0 and p � 6 first. -en, change a6′ � 1 − a6 � 1 and obtain the pixel value � �a1′, a2′, a3′, a4′, a5′, a6′, a7′� � (1101011)2 � (107)10 that 4.1.2. Generation of Stego Images via Matrix Encoding and embeds with the secret information. Finally, we can recover Distribution. In this step, we generate the stego image GIi by the secret S′ � 6 � S. hiding the shadow vector θi into the cover image CIi. Let -erefore, the secret S is hidden in a pixel, but not mc × nc denote the size of the cover image CIi. We restrict appears directly in this pixel. It shows stronger security than 3LT ≤ mcnc because the cover image is needed to be large the standard least significant bit (LSB) method. Moreover, it enough to hide the LT-length shadow vector θi via at most changes 1 bit to hide a 3-bit length secret in a (1, 7, 3)-ME in this paper. (1, 7, 3)-ME process. However, at most 7 bits need to be First of all, construct a chaotic mapping [33] based on changed in a standard least significant bit (LSB) method. CIi to select the steganography cells uniformly. -e -erefore, the ME method can achieve a slighter modifi- primary value of the chaotic mapping can be calculated 5 (8∗i) cation than the LSB method for the same case. as X0 � �i�1 pi/2 where p1, p2, p3, p4, and p5 are the Moreover, in order to achieve a better image steg- first five pixels of CIi and X0 ∈ [0, 1]. -en, compute a a l -length chaotic sequence X � �X ,X , ... ,X � based on anography, the changed bit p should be controlled among 1 1 2 l1 the least significant bits of a pixel. -erefore, the steg- chaotic mapping function and X0, where l1 � mcnc − 5.A anography cell is assigned into three pixels X, Y, Z by X � pseudorandom sequence X′ is the ascending order of X. �x1, x2, .., x6, a1, a2�,Y � �y1, y2, .., y6, a3, a4�, and Z � �z1, Every three elements in X′ is the positions of a steg- z2, .., z5, a5, a6, a7}, and take {}X, Y, Z ⟶ S to denote the anography cell and a (1,7,3)-ME for θi can be performed embedding process in this paper. as in Section 3.3. Finally, after embedding the shadow vector θi into the cover image CIi, the stego image GIi is 4. Proposed Scheme generated. A simple case of embedding the vector θ1 � [v1, v2] into -e proposed (k, n)-threshold verifiable SIS scheme is a 4 × 4 cover image CI is shown in Figure 2 (assume the ∗ constructed based on 2D-3-RLCAM and matrix projection, chaotic mapping function is Xi+1 � α Xi ∗ (1 − Xi) and the ( , , ) and the steganography is realized based on 1 7 3 -ME. -e parameter α � 4 here, and assume X1 � 0.63 directly). -e system model consists of three entities: a dealer D, a set of five pixels in the grey blocks are used to compute the primary P � P ,P , ... ,P participants Set � 1 2 n�, and a set of requesters value X0 for the chaotic mapping. -e generated pseudo- �Q1,Q2, ... � (request for reconstructing the secret image). random sequence X′ is the position for the steganography Moreover, there is a concept of “time period t” for that the cells. To skip the first five pixels, the pixel no. should be entire lifetime of the secret is divided into many short time computed as X′ + 5. Finally, the embedding is performed as periods determined by the system-wide clock. -e proposed �p14, p8, p15 � ⟶ v1 and �p12, p6, p10 � ⟶ v2. scheme contains four procedures: the secret distribution After generating the stego image sets GI Set for the procedure, the shadow update procedure, the secret re- shadow vectors θ Set, distribute each stego image to each construction procedure, and the secret update procedure. participant via a public communication channel. -e overall procedure of the secret distribution and the From the description above, we can observe the fol- secret reconstruction is listed in Figure 1. -e serial numbers lowing facts of the stego images: (n) in Figure 1 are corresponding to the serial numbers (n) in each section. (i) Slight and decentralized modification: the steganog- raphy cells are chosen uniformly and decentralized over the cover image based on the sensitive chaotic 4.1. Secret Distribution Procedure. -e dealer D owns an mapping. On this basis, only a trifle statistical dif- original secret image SI for distributed storage, which is an ference of cover images will be made via the com- m × m grayscale image. D also owns a set of grayscale paratively slight and decentralized modification of the meaningful cover images CI Set � �CI1, CI2, ... , CIn�. pixels by using the ME steganography method. D generates shadow vectors θ Set � �θ1, θ2, ... , θn� and (ii) -e constraint for the cover image: the shadow size embeds them into preselected CI Set so as to generate a set of is only m + k, so we can assume the cover image is stego images GI Set � �GI , GI , ... , GI �. -en, D allocates 1 2 n smaller than the secret image (mc, nc ≤ m) because each participant Pi with a unique stego image GIi in the there is no pixel expansion in our scheme. -ere- public channel and makes the public information publicly fore, the larger cover image is not necessary for this known. paper. (iii) Extract the shadow independently: the positions of 4.1.1. Generation of Shadows via Matrix Projection. the steganography pixels can be directly obtained Firstly, Bai and Zou’s scheme [11] is adopted to generate n from the stego image (by using the grey blocks in shadow vectors via matrix projection as described in Section Figure 2). Consequently, the requesters can perform 3.2 (the step of A2). Construct LT × k matrix A and n linearly the reconstruction, and the participants can per- independent vectors x1, x2, ... , xn, and then compute the form the shadow update by themselves. Security and Communication Networks 7

Secret distribution procedure Secret reconstruction procedure Intermediate matrices Remainder matrix A Remainder matrix for CIi R R R R R R R 0, 1, 2, 3, 4 Publish L L L L L L ( T × T) ( T × T) ( T × T) Stop the reconstruction

(2) (6) Remainder matrices Fail RCI , RCI ,..., RCIn Projection matrix Filling matrices 1 2 ——— —— Intermediate matrices P (u) (u+1) (u+2) Cheating A C , C , C ,CIM ,VM (6) R R R R R Pass L L i Public board 0, 1, 2, 3, 4 detection ( T × T) (L × L ) L L T T ( T × T)

(6) (3) (6) (6) Verification matrix Random matrix VM Reconstructed Reconstructed Reconstructed A m m Verification matrix submatrix secret matrices ( × ) (2) R R R (u) R (u+1) R (u+2) L k VM CIMi C , C , C ( T × ) (m × m) (mc × nc) (m × m) Submatrix Secret matrices (u) (u+1) (u+2) CIMi C C C (5) , , (5) (4) Random vectors m n m m x , x ,..., x ( c × c) ( × ) 1 2 n (k × 1) B L k Matrix ( T × ) (5) & projection matrix P L L (1) B ( T × T) Reconstructed submatrices Cover image Submatrices SIM' , SIM' , SIM' SIM , SIM , SIM 1 2 3 CIi 1 2 3 (m × m) m m (2) Shadow vectors (mc × nc) ( × ) θ , θ ,..., θ 1 2 n L Cover image Shadow vectors ( T × 1) (3) CI , CI ,..., CIn θ , θ ,..., θ Reconstructed 1 2 1 2 k Reconstructed L Cover images Secret images ( T × 1) Secret images SI CIi' = CIi SI' = SI m n m m (2) (m × m) ( c × c) ( × ) Participants (1)

P , P , ..., Pn Stego images 1 2 Stego images , ,..., Distribution with , ,..., GI1 GI2 GIn GI1 GI2 GIk GI , GI ,..., GIn 1 2

Figure 1: Overall procedure of secret distribution and secret reconstruction.

12345678910 11 X Compute 0 X 0.63 0.930.26 0.77 0.71 0.82 0.59 0.97 0.12 0.980.42 9 3410 781256 11 p p p p 1 5 9 13 (1,7,3)– Ascending 0.120.26 0.42 0.59 0.63 0.770.71 0.82 0.93 0.97 0.98 p p p p p p p ME v 2 6 10 14 14 8 15 1 123 45678 9 10 11 (1,7,3)– p p p p 3 7 11 15 p p p ME v X' 9310 74 15 628 11 12 6 10 2 p p p p 123 45678 9 10 11 4 8 12 16 Pixel no. 14 8 15 12 6 10 9 11 137 16 (X' + 5) CI Figure 2: An example of embedding in a stego image via (1,7,3)-ME.

4.1.3. Preprocessing of the Secret Image. For the secret image 4.1.5. Generation of the Separate Verification Part. For each SI, convert it into three m × m submatrices SIM1, SIM2, cover image CIi, the independent m × m verification matrix 2 and SIM3 where SIM1 � SI/8 , SIM2 � SI/8, and SIM3 � VM is generated as in (13). CIMi � CI mod 8 is the sub- (u) (u+1) SI mod 8, respectively. -e submatrices SIM1, SIM2, matrix for CIi. Here, we call the matrices C ,C , 2 1 (u+2) and SIM3 can satisfy that SIM1 ∗ 8 + SIM2 ∗ 8 + SIM3 C , CIMi, VM are the secret matrices. 0 ∗ 8 � SI. (u) (u+1) (u+2) VM ��C ⊕ C ⊕ C ⊕ CIMi� mod 8. (13)

4.1.4. Generation of Secret Matrices via RLCAM. Construct a 2D-3-RLCAM with the discrete state set 4.1.6. Generation of the Public Information. As in Bai and Sb(b � 8). Firstly, define the iteration value u and the rule Zou’s scheme [11], a publicly known remainder matrix is number �w1, w2, w3 � of it (notice that w3 � 16 in RLCAM). needed to help with the following secret reconstruction. -en, take the three submatrices SIM1, SIM2, and SIM3 as Notice that we have to additionally verify the secret image the initial configuration C(0),C(1),C(2), and obtain the states and the cover images, so we need to generate a remainder of C(u),C(u+1),C(u+2) after u times of iteration by (3). matrix for each cover image. To simplify the notation, take R -erefore, the secret image SI is encrypted into three secret denotes the remainder matrix for an arbitrary cover image. matrices C(u),C(u+1),C(u+2). -e same number of the remainder matrices is needed to be Moreover, make �w1, w2, w3 � and u to be privately generated as the cover images actually. known by each requester, which can be seemed like the First of all, convert the secret matrices into the LT × LT (u) (u+1) (u+2) internal secret key of the system. filling matrices C , C , C , CIMi, VM by filling with 8 Security and Communication Networks

(t) (t) (t) (u) (u) � , , ... , I � random values as (14). In (14), C � [C ]UL: m×m repre- images GI1 GI2 G k ∈ GI Set during a single time sents that m × m matrix C(u) can be extracted from the upper period as follows. left corner of the matrix C(u): C(u) � C(u) , 4.3.1. Extraction of the Shadow Vectors via Matrix Encoding. � �UL: m×m -e requester computes the steganography cells firstly and (u+1) (t) C ��C(u+1) � , then extracts the embedding shadow vectors �θ1 , UL: m×m (t) (t) (t) θ2 , ... , θk } ∈ θ Set from the stego images by themselves C(u+2) ��C(u+2) � , (14) as in Section 3.3. We omit the details here due to space UL: m×m limitations.

CIMi ��CIMi � , UL: mc×nc VM �[VM] . 4.3.2. Reconstruction of the Secret Matrices via Matrix UL: m×m Projection. -e requester selects a remainder matrix R from

-en, compute a projection matrix PA of matrix A (A is the public board which corresponds to one of the obtained already generated in Step 1), where PA is a LT × LT matrix. stego images GIi. -e secret image SI and the corresponding Calculate intermediate matrices R0,R1,R2,R3,R4 by per- cover image CIi can be simultaneously reconstructed. (u) (u+1) R R ,R ,R ,R ,R forming subtraction between PA and C , C , Split back into submatrices 0 1 2 3 4 and P (t), (t), C, CIMi, VM as in the following equation: generate a project matrix B over shadow vectors �θ1 θ2 ... , (t)} (u) θk . -en, perform the subtraction as steps C1 and C2 R � PA − C , 0 in Section 3.2 between PB and R0,R1,R2,R3,R4. Finally, (u+ ) extract the reconstructed secret matrices R (u) ,R (u+1) ,R (u+2) , R � P − C 1 , C C C 1 A R ,R CIMi VM from the obtained matrices as follows: (u+ ) ( ) R � P − C 2 , 15 2 A RC(u) � �PB − R0 �UL: m×m, R � P − , 3 A CIMi RC(u+1) � �PB − R1 �UL: m×m, R � P − . 4 A VM RC(u+2) � �PB − R2 �UL: m×m, (16) R � �P − R � , Finally, the remainder matrix R is computed via CIMi B 3 UL: mC×nC 4 4− i R � � Ri ∗ 8 , and make the remainder matrices for all i�0 R � �PB − R � . cover images publicly known. VM 4 UL: m×m As shown in Section 3.2, the equation [PB]UL: m×m � [PA]UL: m×m is satisfied if there do not exist fake shadows. 4.2. Shadow Update Procedure. Let �θ(t), θ(t), ... , θ(t)� de- 1 2 n -erefore, the equation RC(u) � [PB − R0]UL: m×m � note the shadow vectors during time period t. -e update [P − R ] � C(u) will also be satisfied. (t+1) (t+1) (t+1) A 0 UL: m×m process to generate �θ1 , θ2 , ... , θn � is similar to Bai and Zou’s scheme [11] in Section 3.2. At the beginning of time period t + 1, the dealer D 4.3.3. Verification for Cheating Detection. -e separate generates a k × k orthonormal matrix LM and distributes it cheating detection can be performed by the following via a to each participant. After receiving LM, (t) equation: each participant Pi extracts the shadow vectors θi from its (t+ ) 1 R ��R (u) ⊕ R (u+1) ⊕ R (u+2) ⊕ R � mod 8. (17) stego image and derives its new shadow vector θi by (10). VM C C C CIMi (t+1) -en, Pi embeds θi into its stego image again and erases all the variables except its current stego image. Because If (17) holds, the requester proceeds with the following steganography is based on the computationally efficient steps or stops in this step. -e effectiveness of the verification XOR operation, the shadow updating is efficient even over mechanism is proved in -eorem 3. the stego images. Note that the correctness of the fact that the update shadow vectors generated during the same time period can 4.3.4. Reconstruction of the Secret Image via RLCAM. reconstruct the secret matrix is illustrated clearly in Bai and After passing the cheating detection, the requester can re- �R (u) ,R (u+1) ,R (u+2) � Zou’s scheme [11]. Moreover, a mixture of the shadows with cover the secret image with C C C via 2D-3- (0) (1) (2) different time periods cannot recover the secret, while RLCAM. Take it as the initial state �C� , C� , C� � for the equation (11) in Section 3.2 cannot be obtained. -e ex- � (u) planation of this is also described in detail in Bai and Zou’s reversion process, and the configurations of �C � SIM3′, scheme [11]. � (u+1) � (u+2) C � SIM2′, C � SIM1′} can be obtained by (6) after u iterations. Finally, the reconstructed secret image SI′ � SIM′ ∗ 82 + 4.3. Secret Reconstruction Procedure. A requester Qi can 1 ′ 1 ′ 0 reconstruct SI as long as obtaining enough valid stego SIM2 ∗ 8 + SIM3 ∗ 8 can be obtained. Security and Communication Networks 9

4.3.5. Reconstruction of the Cover Image. As described unqualified group� θ1, θ2, ... , θk− 1� of shadow vectors, but the previously, the last three bits of some pixels in GIi are adversary still cannot explicitly learn about the secret image. changed for steganography, and the secret matrix CIMi just maintains them. -erefore, the cover image C′i can be re- Proof 1. As proved in Bai and Zou’s scheme [11], the secret C′ � [ − ( )] + R covered as i GIi GIi mod 8 CIMi. sharing using matrix projection is a strong ramp secret Notice that, in most of the existed traditional steg- sharing (RSS) with k access levels, but not a perfect secret anography-based SIS schemes, there have some limitations sharing (PSS) such as Shamir’s scheme. Strong RSS also does in the reconstruction of the cover images: (1) they input the not reveal any part of the secret explicitly from any k − 1 same cover image for embedding all of the shadows [14, 34], shadows. However, the secret’s exposed information is which results in shares with similar content; (2) they cannot proportional to the size of the unqualified group of the losslessly reconstruct the cover image; and (3) the cover shadows. recovery method requires more computation. Such limita- As an improvement, the matrix projection is performed tions make these schemes inapplicable to some scenarios. over the secret matrices of SI after the iterations of RLCAM. For example, losslessly reconstructing the cover image can -e secret matrices can seem like the of SI with help to share searching with image recognition since a hash the RLCAM. Hence, the privacy-protecting for recovering SI function-based searching method is generally sensitive to is further protected under the irreversible property of cover image content including even a slight distortion [29]. RLCAM when without rule number, and the consecutive requirement of the configurations (secret matrices in this 4.4. Secret Updating Procedure. When a new secret image is paper) with the correct order. -erefore, the exposed information cause by the matrix to be shared, the dealer D only needs to update the LT × LT public remainder matric R for each cover image. Intuitively projection is only relative to secret matrices. Based on the in the secret distribution part in Figure 1, only steps (1), (5), security of RLCAM, the configuration after iterations leaks and (6) are needed but without the requirement for steps (3) nothing about the SI. Consequently, the adversary cannot and (4). -erefore, secret updating is very efficient because get any information about the original secret image from any k − communication between the dealer and the participants is arbitrarily 1 shadows, much less the recovery. not required. Theorem 2 (persistent collusion attack). During an adjacent 5. Security Analysis time periods, the adversary compromises k − 1 participants during each time period, but the adversary still cannot ex- In the proposed scheme, we consider the dealer and the plicitly learn about the secret image. requesters are fully trusted. Assume parts of the participants can be compromised by a statistical and persistent adversary. Proof 2. Assume P1′ denotes the set of k1 participants A statistical and persistent adversary: in 5G network compromised only in period t − 1, P2′ denotes the set of k2 scenarios, the network environments are more open and participants compromised in both of the period t − 1 and t, flexible. -erefore, we need to consider a stronger potential and P3′ denotes the set of k3 participants compromised only adversary instead of an ordinary adversary. -e ordinary in period t, where k1 + k2 � k2 + k3 � k − 1. adversary at most can corrupt up tok − 1 shadows during the Additionally, the unqualified group of shadow vectors entire lifetime of the secret. In this paper, we consider the obtained during t − 1 from P′ and P′ is denoted as Θ(t− 1) � adversary can stay on the networks for several time periods 1 2 1 (t− 1) (t− 1) (t− 1) �θi � and Θ2 � �θi � . -e unqualified and move among the objects. i�1,...,k1 1,...,k2 -e adversary attempts to reconstruct or destroy the group of shadow vectors obtained during t from the set P2′ secret image illegally during several time periods. Within a P′ (t) � (t) (t) � and 3 is denoted as Θ2 �θi �i� ,...,k and Θ3 t k − 1 2 time period , the adversary at most can corrupt up to 1 (t) n n k − �θi � . participants among participants (restrict that ≤ 2 1). If i�1,...,k3 a participant is corrupted, the adversary can learn or tamper (t− 1) (t) Based on Θ2 and Θ2 , we can construct an equation with its shadow vector during this time period. At the be- for the unknown updating matrix LM as follows: ginning of the next time period, all participants will be “rebooted” to the safe state, and their shadows are erased and LM ∗ � θ(t− 1) , ... , θ(t− 1) � �√��√� 1[m+1: m+k] k2[m+1: m+k] alternated by the new ones. Because the adversary can k×k �√√√√√√√√√√√√√��√√√√√√√√√√√√√� M1: k×k2 corrupt up to k − 1 participants during each time period, it (18) has more power than the ordinary adversary so that it can (t) (t) �� θ [m+ : m+k], ... , θk [m+ : m+k] � . corrupt some participants multiple times throughout the �√√√√√√√√√√√√√��√√√√√√√√√√√√√�1 1 2 1 entire lifetime of the secret image. M2: k×k2 However, updating matrix LM cannot be determined by 5.1. Collusion Attack (18) because the matrices M1 and M2 are not full rank matrices (the rank of M1 and M2 is k2 but not k). Since LM (t− 1) (t) Theorem 1 (collusion attack). During a time period, the cannot be determined, Θ1 cannot be turned into Θ1 and (t) (t− 1) adversary compromises k − 1 participants to obtain an also Θ3 cannot be turned into Θ3 . -ereby, enough 10 Security and Communication Networks

shadows θi for the same time period cannot be inferred and images (CI01, CI02, and CI03). -e experimental threshold also the matrix B which is necessary for secret reconstruction is set as (2, 3). cannot be determined. -e results of the generated stego images are shown as Consequently, the persistent adversary cannot obtain ste01, ste02, and ste03. It can be seen that there exists no enough shadows even for multiple time periods, and the significant difference between the stego image and its cor- privacy of the secret image can be effectively guaranteed. responding cover image. -e indices of PSNR and SSIM of the stego images are good enough (the detailed analyses of PSNR and SSIM are shown in Section 6.3). -erefore, the 5.2. Shadow Tampering stego images generated in our proposed scheme have a high quality. Theorem 3. Assume a requester obtains a set of shadow -e reconstruction results are also shown in Table 1 vectors� θ1, θ2, ... , θk�. If each of the shadow vectors is valid, (assume we use the remainder matrix of CI01). Table 1 shows (17) will be explicitly holding, or holding in a negligibility the reconstructed secret image Rec-SI for the secret image SI probability if there exists at least one invalid shadow. and the reconstructed cover image Rec-CI01 for the cover image CI01. -ese reconstructed images are lossless (where Proof 3. First of all, consider the case that all of the shadow PSNR � ∞ and SSIM � 1). vectors are valid. As proof in [11], there exist Moreover, we also test if there is a tampered shadow for [PB]UL: m×m � [PA]UL: m×m, and the remainder matrix R is reconstruction. As shown in the image Rec-SI (×), the secret left to be publicly known which cannot be modified by image recovery will be totally failed with a tampered shadow. anyone. Hence, the reconstructed secret matrices will be equal to the original secret matrices, obviously. -erefore, equation (17) can be explicitly holding. 6.3. Visual Quality. -e general criteria to check the visual In contrast, if there exists at least one invalid shadow quality of the stego images are the peak-signal-to-noise ratio vector (whether an older one or a tampered one), (PSNR) and structural similarity (SSIM). Table 2 lists the [PB]UL: m×m will be a meaningless matrix, which has no mean value of PSNR and SSIM produced by the proposed relevance with [PA]UL: m×m. -erefore, the reconstructed scheme for different sizes of the cover images with threshold R (u) ,R (u+1) ,R (u+2) ,R ,R k � 2, 3, 7, 15, respectively. secret matrices C C C CIMi VM can also seem like random meaningless matrices. -erefore, we should PSNR has explicit physical meanings, which evaluates discuss if the probability Prob(VM) of holding (17) is nearly the similarity between the original image and the output � + zero, in this case. For each element x1, x2, x3, x4, x5 ∈ [0, 7] image. -e value of PSNR ∞ for completely identical in each reconstructed secret matrices, there exists images. From Table 2, PSNR is larger than 55 dB for 96 × 96 × x1⊕x2⊕x3⊕x4? � x5 as the same form of (17). -e proba- and 256 256 cover images, which is almost indistin- bility of holding this equation is 1/8. Hence, Prob � guishable from the human visual system (HVS). 2 (VM) (1/8)m which is effective for verification. For a 10 × 10 Additionally, the details of the SSIM algorithm can be 300 secret image, Prob(VM) � 1/2 is small enough to be ig- found in [37]. SSIM is used to estimate the dependencies nored. -erefore, the proposed scheme can effectively detect between neighboring pixels. SSIM � 0 for extremely dis- the tampered shadows. similar, and SSIM � 1 for completely identical. From Table 2, SSIM shows 90% structural similarity between the original 6. Experimental Results and cover image and stego image of the proposed scheme, which is very close to the upper bound. Performance Evaluation 6.1. Implementation. -e scheme is implemented in Visual C# 2017 on a computer with an Intel Core i7 1.8 GHz CPU 6.4. Embedding Ratio (ER) and Detection Ratio (DR). processor and 8 GB RAM that runs a Windows 10 operating Embedding ratio (ER) shows the amount of information system. Different sizes of cover images are considered in this embedded in each cover pixel, which is the ratio of the paper, which includes 30 × 30 images chosen from the number of share bits to the number of all pixels of a cover CIFAR-10 dataset [35], 96 × 96 images chosen from the image. For resistance against steganography, ER should be × STL-10 dataset [36], and the 256 × 256 standard images less than 0.25 bpp. For 30 30 cover images, we can obtain � k � � k � × Baboon, Lena, and Pepper. Besides, a 256 × 256 standard ER 0.2867 ( 2) and ER 0.2922 ( 7). For 96 96 � k � � k � image Boat is considered as the secret image. As described cover images, ER 0.0280 ( 2) and ER 0.0285 ( 7). above, we do not discuss the case for the larger cover images. -erefore, ER of the proposed scheme is effective enough. Detection ratio (DR) shows the detecting ability of in- valid shares, which is the ratio of detected tampered pixels to 6.2. Simulation Experiments. First of all, the effectiveness of all tampered pixels. As described above, the verification of the proposed scheme is shown in Table 1, where Diff denotes our proposed scheme is based on a separate m × m matrix, the different pixels between the output image and its cor- which is fully covering the stego image. -erefore, as the responding original one. -e 256 × 256 standard image Boat scheme of Eslami et al. [15], DR in this paper is approxi- is taken as the secret image (SI). -e 256 × 256 standard mately 256/256, which is effective enough for cheating images Lena, Pepper, and Baboon are taken as the cover detection. Security and Communication Networks 11

Table 1: -e stego images and the reconstructed images.

Secret image Cover image

SI CI01 CI02 CI03

Stego image

ste01 ste02 ste03 PSNR 66.40 67.03 66.71 SSIM 0.999996 0.999997 0.999996 Diff 227/65536 223/65536 215/65536

Rec image

Rec-SI Rec-CI01 Rec-SI (×) PSNR ∞ ∞ — SSIM 1 1 — Diff 0/65536 0/65536 65420/65536

Table 2: PSNR and SSIM of different threshold and cover image. PSNR SSIM 30 96 256 30 96 256 k � 2 47.21 57.29 66.60 0.9996 0.99997 0.999996 k � 3 47.39 57.29 66.35 0.9996 0.99997 0.999995 k � 7 47.92 57.88 65.92 0.9997 0.99997 0.999995 k � 15 48.10 57.97 65.63 0.9997 0.99996 0.999995

6.5. Comparative Analysis. In order to further illustrate the (5) -e underlying techniques for steganography. performance of the proposed scheme, we compared it with (6) -e size of the shadows typical works ([10] (2006) and [15] (2010)) and some recent (7) -e average number of modified bits per pixel schemes ([16] (2018), [17] (2017), and [12] (2018)). -e works described in [10, 15] are the classical schemes (8) PSNR designed based on matric projection method and CA, re- (9) SSIM spectively. -e comparative analysis is listed in Table 3. Here in this part, the threshold k � 3. As shown in -e first five indices in Table 3 are the characteristics of Table 3, our scheme achieves a clear advantage in the per- these schemes: spective of almost all indices. (1) If the shares are meaningful images Compared with the MP-based SIS schemes [10, 16], we (2) If it is a verifiable scheme achieve more functions. Meanwhile, we certainly not expand the shadows’ size and even accomplish a smaller shadow (3) If it has the features of proactive and dynamic than such schemes. -e scheme of Eslami et al. [15] is the (4) -e underlying techniques for SIS first steganography-based SIS constructed by using 1D-CA. 12 Security and Communication Networks

Table 3: -e comparative analysis. (1) (2) (3) (4) (5) (6) (7) (8) (9) [10] × × × MP and polynomial — m2/(k + m) — —— [16] × √ × MP and polynomial — m2/(k + m) — —— [15] √ √ × 1D-CA LSB m2 1/4 47.12 0.99815 2 [17] √ √ × Polynomial ME LH/3k LH/(12 km ) 49.47 — [12] √ √ × 2D-CA and polynomial LSB (Dynamic) (Dynamic) 54.55 0.99996 Ours √ √ √ 2D-CA and MP ME m + k (m + k)/4 m2 66.35 0.999996

It can achieve a higher computation complexity than our network environments with efficient secret distribution and proposed scheme because we have the extra MP’s opera- secret updating. In contrast, the above schemes need to tions. However, viewed from the indices listed in Table 3, we reprocess the distribution for the secret update. Secondly, for can achieve a better performance of the shadow size and the the potential adversary in the large-scale complex network stego image quality than it. Moreover, this scheme realizes environment, our scheme prevents the eavesdroppers via the the secret protection based on CA’s security, but only higher quality steganography. Our scheme can also protect consecutive shadows can reveal the secret image that limits the secret against the persistent adversary through the ef- the practicality. As an improvement, our proposed scheme ficient shadows update. We consider the stronger adversary also benefits from the security provided by CA but avoids than the above schemes. this weakness. In the scheme of Liu et al. [17], LH is the length of the 7. Conclusion Huffman code of the difference image of the secret image, 2 where LH � m for the straightforward binary image. Hence, In this paper, a verifiable secret image sharing scheme is only when m2/3k − (m + k) > 0 holding, its generated proposed by employing the 2D-RLCAM and the matrix shadow will be smaller than ours. -e solution of m2/3k − projection, which aims to provide a secure multimedia (m + k) > 0 is restricted by m ≤ 4k. It means that our scheme sharing and storing service in 5G network scenarios. In this will get the stego images with higher quality when m ≤ 4k paper, 2D-RLCAM and the matrix projection are combined when both of us using (1, 7, 3)-ME for steganography. together creatively in handling some existing weaknesses. -e scheme of Zarepour-Ahmadabadi et al. [12] is a -e combination enhances the security of the secret image multisecret image sharing scheme. Each secret image con- and the visual quality of stego image. Furthermore, a matrix tains a specific threshold value, and the size of shadows is encoding-based steganography approach is used to make dynamically relative to it. So that the access structure in [12] our scheme more secure when sharing in a public network is greater than ours. However, it needs to combine the hash channel. -e security analysis and the performance analysis values in each shadow, which helps to arrange the shadows demonstrate it. -erefore, the proposed scheme can si- according to the desired order to recover the secret image via multaneously support verifiable, dynamical, and proactive 2D-CA correctly. -ese values need a relatively larger space features, which is superior to most existing SIS schemes. -e for embedding. efficient secret and shadow updating procedures make the Moreover, besides the scheme of Eslami et al. [15], all the proposed scheme more suitable for deployment in the 5G other schemes may increase the computational complexity due platform. We aim to optimize the size of public information to several times of Lagrange interpolation operations (Lagrange for a more efficiently secret update in the future. interpolation is the main operation of polynomial-based secret sharing). In our proposed scheme, the computation complexity Data Availability is mainly based on matrix operations. Specifically, during the secret distribution procedure and the secret reconstruction -e image data used to support the findings of this study procedure, we mainly have the matrix subtraction operations, have been deposited in the Cifar-10 dataset and Stl-10 the matrix XOR operations, and computing the projection dataset repository (http://www.cs.toronto.edu/kriz/cifar. matrix. Compared with the calculation of the projection ma- html and https://cs.stanford.edu/acoates/stl10/). trix, the computation complexity of the matrix subtraction and the matrix XOR can be ignored because they are lightweight. Conflicts of Interest -e calculation of the projection matrix is the heaviest com- -e authors declare that there are no conflicts of interest putation in our scheme. However, we only need to compute the regarding the publication of this paper. projection matrix PA only once during the secret distribution procedure and compute the projection matrix PB only once Acknowledgments during the secret reconstruction procedure. -erefore, our proposed scheme has a lower computation complexity than the -is work was partly supported by the National Natural other schemes. Science Foundation of China under Grant nos. 62072090, In conclusion, compared with the above schemes, our 61872069, 61772127, 61703088, 61402097, and 61772125, the scheme is more suitable for 5G network environments. First Fundamental Research Funds for the Central Universities of all, our scheme can adapt to the rapid data changes in 5G under Grant no. N171704005, CERNET Innovation Project Security and Communication Networks 13 under Grant no. 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