A novel scheme for lossless authenticated multiple secret images sharing using polynomials and extended visual cryptography

IET Information SecurityVolume 15, Issue 1 p. 13-22 ORIGINAL RESEARCH PAPEROpen Access A novel scheme for lossless authenticated multiple secret images sharing using polynomials and extended visual cryptography Krishnaraj Bhat, Corresponding Author Krishnaraj Bhat krrishvaradha08@gmail.com orcid.org/0000-0003-2448-898X Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, India Correspondence Krishnaraj Bhat, Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, 574110, India. Email: krrishvaradha08@gmail.comSearch for more papers by this authorUday Kumar Reddy K.R., Uday Kumar Reddy K.R. Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, IndiaSearch for more papers by this authorRanjan Kumar H.S., Ranjan Kumar H.S. Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, IndiaSearch for more papers by this authorDindayal Mahto, Dindayal Mahto Department of Information Technology, School of Computing, SASTRA Deemed to be University, Thanjavur, Tamil Nadu, IndiaSearch for more papers by this author Krishnaraj Bhat, Corresponding Author Krishnaraj Bhat krrishvaradha08@gmail.com orcid.org/0000-0003-2448-898X Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, India Correspondence Krishnaraj Bhat, Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, 574110, India. Email: krrishvaradha08@gmail.comSearch for more papers by this authorUday Kumar Reddy K.R., Uday Kumar Reddy K.R. Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, IndiaSearch for more papers by this authorRanjan Kumar H.S., Ranjan Kumar H.S. Department of Computer Science and Engineering, NMAM Institute of Technology, Nitte, Karnataka, IndiaSearch for more papers by this authorDindayal Mahto, Dindayal Mahto Department of Information Technology, School of Computing, SASTRA Deemed to be University, Thanjavur, Tamil Nadu, IndiaSearch for more papers by this author First published: 28 December 2020 https://doi.org/10.1049/ise2.12001AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Researchers have put forward many variations of schemes for secret image sharing on grounds of visual cryptography and polynomials. The authors of this paper put forward a novel scheme for sharing multiple secret images with perfect reconstruction and authentication for cheating prevention. The perfect reconstruction of all shared secret images is obtained by applying polynomial-based encryption and decryption, and one-time authentication is done for all shared secret images by stacking the intelligible shares created by using extended visual cryptography. Properties' analysis, experimental results and comparisons are given to exhibit the potency of the scheme. 1 INTRODUCTION Naor and Shamir [1] proposed the (v, z) – visual cryptography scheme (VCS) – where a secret image is divided into z noisy images called shares or shadows, and issued to z participants. A minimum of v shares from v participants are stacked to visually decrypt the original secret image. The beauty of this scheme is that it does not require a key to perform encryption or decryption, and even though z–v shares are absent, the secret images can be revealed. As the shares are unintelligible, they get the attention of attackers and become vulnerable to attacks while being transmitted in an open network. To overcome this issue, an extended visual cryptography scheme (EVCS) [2] is proposed in which shares are made intelligible besides reconstructing the original secret using the intelligible shares. Cheating prevention methods in (v, z)–VCS with respect to different kinds of cheating and cheaters are proposed in [3-5]. Hu and Tzeng [3] proposed three cheating approaches in VCS and EVCS, and a general method for transforming a VCS into a different VCS which has got the cheating prevention feature is also proposed. Each participant contains a verification share for authenticating other participants' shares. They also proposed betterments on previous cheating prevention schemes. Ren et al. [4] proposed a cheating prevention method where each participant has a verification share created by using Latin square to authenticate other participants' shares supporting both deterministic VCS and probabilistic VCS. Kumar et al. [5] proposed cheating prevention by using remedial number system (RNS) and mixed radix representation (MRR) for encryption and decryption, respectively, supporting only (v, v)–image secret sharing (ISS). It does not discuss how the moduli values used in RNS and MRR are shared among participants. Two-in-one ISS (TiOISS) schemes, where low-quality secret image is revealed in the first phase of decoding and high-quality secret image is recreated in the second phase of decoding, are proposed in [6-13]. Schemes in which stacking is used in the first phase of decoding and Lagrange's interpolation in the second phase are proposed in [6-9, 11, 13]. Li et al. [10] proposed the first-phase decoding using stacking and the second-phase decoding using few XOR operations. The secret image has to be resized if its half-toned image does not have adequate black pixels. The size of shares depends on the number of black pixels, pixel expansion and number of grey values greater than 253 in the secret image. Chen et al. [11] combined polynomial image secret sharing (PISS) and random grid-based visual secret sharing (RGVSS), where the size of shares is the same as that of secret image. Sridhar [12] proposed a scheme for sharing multiple secret images in which low-quality secret images are recreated in the first phase of decoding by stacking the circular shares which are rotated in counter-clockwise direction at different angles, respectively. A truncation of PISS pixels is done before embedding them into visual multiple-secret sharing (VMSS) to improve share's visual quality due to which revealed secret images after the second phase of decoding is not the same as original secret images. There is a constraint on the size of the secret images in which PISS shadows have to be created with respect to threshold v and radius of circular shares. Also, there is an extra burden of carrying a number of PISS pixels and authentication bits in shares. Wu and Yang [13] proposed a scheme in which a greyscale secret image is converted into p-radix image from which z p-radix shares are obtained by (v, z)–PSIS under mod p operation. Then these generated p-radix shares are embedded into colour shares using colour shares generation algorithms which are needed to recreate the secret image in lossless form after decryption. Again a secret image is half-toned and shared using z colour shares formed by applying (v, z) colour black white (CBW)–VCS which can be decoded by stacking. Among these, only Srividhya et al. [9, 12] and Sridhar and Sudha [9, 12] support authentication of intelligible shares with respect to a malicious attacker from outside. Schemes for sharing multiple secrets at a time are proposed in by Chang et al. [14], Nitharwal and Verma [15], Ovhal et al. [16], Deshmukh et al. [17], Sasaki and Watanabe [18], Kabirirad and Eslami [19], Kapadiya et al. [20], and Prasetyo and Guo [21]. Chang et al. [14] proposed sharing multiple secret images using the Chinese Remainder Theorem (CRT) and Lagrange's interpolation. Each shared pixel produced can have a value greater than 255 which needs more than a byte memory to get accommodated, and no mean for identifying these shared pixel values greater than 255 is proposed. Nitharwal and Verma [15] and Ovhal et al. [16] proposed sharing multiple secret images supporting only (z, z) ISS where the number of shares z produced is equal to the number of secret images shared. Nitharwal and Verma [15] used XOR and bit-reversal operations, whereas Ovhal et al. [16] used modulo addition and subtraction, XOR and bit-reversal operations. Deshmukh et al. [17] utilised CRT and XOR operations to share multiple secret images where the number of shares z produced is equal to the number of secret images shared. When the number of secret images to be shared is odd, it is not able to reconstruct recovered secret images correctly. Sasaki and Watanabe [18] provided adequate conditions to be satisfied by encryption of visual secret sharing scheme actualising a general form of access structure for multiple secrets and two constructions of such scheme with encryption fulfilling the condition. Kabirirad and Eslami [19] proposed sharing multiple secret images using XOR, reverse and cyclic shift operations, where v consecutive shares are required to reconstruct the secret images. The number of secret images that can be shared at a time is v, where v being the threshold in (v, z) – multiple secret image sharing scheme. Kapadiya et al. [20] proposed sharing multiple secret images using XOR operation. To share z secret images, z + 1 shares are generated among which one is a universal share, and shares are created in order starting from the universal share. To retrieve all z secret images, all z + 1 shares are required. But, to retrieve secret images individually, all shares are not required. To recover the first secret image, the universal share and the second share are sufficient. To recover the second secret image, the universal share, the second share and third share are required, and so on. Exact recovery of the secret images is not done. But, they are visually alike original ones. Prasetyo and Guo [21] overcome the limitation of Deshmukh et al. [17] with respect to odd numbers of secret images. It is done by introducing symmetric and transferred masking coefficients to produce a set of shared images. To further improve security, hyper-chaotic scrambling is used to transmute the secret images ahead of generating shared images. The existing schemes briefed so far have the following limitations: (i) identification and management of shares are difficult due to their noisy nature [1, 4-8, 10, 11, 13-19, 21]; (ii) extra verification shares are generated for authentication which consume extra storage space [14, 15, 17]; (iii) difficult to find the reference points of rotation for shares carrying multiple secret images data [12]; (iv) shares have to be circumvolved in a specific way to reconstruct each of the multiple secret images [12]; (v) support only (v, v) – ISS for v ≥ 2 [5, 15-17, 20, 21]; (vi) the number of shares to be created is dependent on the number of secret images to be shared [15-17, 20, 21]; and (vii) need of ordered consecutive shares to recover the secret images [15-17, 19-21]. In addition to overcoming the above-mentioned limitations, the proposed scheme has other advantages as follows: (i) maximum of m/2 secret images are shared using z shares, m being the pixel expansion, (ii) lossless recovery of all m/2 secret images using at least v shares; (iii) only one-time authentication for the recovering of all m/2 shared secret images; and (iv) resistant to different kinds of cheating and cheaters mentioned in [3]. The rest of the article is organized as follows. Section 2 briefs preliminaries utilised in the scheme proposed. Sections 3 and 4 describe the scheme proposed and provide analysis on the properties of the scheme, respectively. Section 5 delivers the results of experiments and comparison details. Section 6 provides conclusion and future work details, and references, respectively. 2 PRELIMINARIES 2.1 (v, z)–Visual Cryptography Scheme A secret image is segregated into z shares by the dealer by using (v, z)–VCS, 2 ≤ v ≤ z. To reveal the secret image, at least v shares have to be stacked. Any less than v shares cannot disclose the secret image. A (v, z)–VCS is defined as follows 1: Definition 1.(v, z)–VCS comprises two accumulations Q0 and Q1 of z × m Boolean matrices, m is the pixel expansion. If the following two requirements hold, then the scheme is considered legitimate: By stacking any v rows in W ∈ Q0, the count of white (black) pixels in the resulting vector should be greater (lesser) than the count of white (black) pixels in the resulting vector that is got by stacking any v rows in B ∈ Q1. By stacking any j rows, 1 ≤ j < v, in W ∈ Q0, the count of white (black) pixels in the resulting vector should be the same as the count of white (black) pixels in the resulting vector that is got by stacking any j rows in B ∈ Q1. Requirement 1 is in respect of the contrast of the revealed secret image and requirement 2 is in respect of the security of the scheme that makes sure that by stacking any less than v shares, only unintelligible image is revealed. Figure 1 depicts the (2, 2)–VCS. It can be seen that shares 1 and 2 are noisy, revealing no information about secret. But on stacking, secret is revealed. ‘+’ denotes stacking operation. FIGURE 1Open in figure viewerPowerPoint (2, 2)–VCS: (a) secret image, (b) share 1, (c) share 2 and (d) revealed secret image (share 1 + share 2) 2.2 (v, z)–extended visual cryptography scheme Definition 2.Let { ( Q w b 1 , … , b z , Q b b 1 , … , b z ) } b 1 , … , b z ϵ { w , b } be a couple of 2z collections of z × m Boolean matrices forming a (v, z) – EVCS, if there exist values rF, rS and t satisfying the following: For b 1 , … , b z ϵ { w , b } , the threshold t and the relative difference rF are such that, for any M ∈ Q w b 1 , … , b z , the OR K of any v of the z rows satisfies H(K) ≤ t–rF × m; whereas, for any M ∈ Q b b 1 , … , b z , H(K) ≥ t. For any b 1 , … , b z ϵ { w , b } and for any subset {x1, …, xp} ∈ {1, …, z} with p < v, the two collections of p × m matrices R d b 1 , … , b z with d ∈ {w, b} obtained by restricting each z × m matrix in Q d b 1 , … , b z to rows x1, …, xp contain the identical matrices with the same number of occurrences. For any x ∈ {1, …, z} and any b 1 , … , b x − 1 , b x + 1 , … , b z ϵ { w , b } , it results that min H ( M x ) M ∈ μ b – max H ( M x ) M ∈ μ w ≥ rS × m, where μ b = ∪ b 1 , … , b x − 1 , b x + 1 , … , b z ϵ { w , b } Q w b 1 , … , b x − 1 b b x + 1 , … , b z μ w = ∪ b 1 , … , b x − 1 , b x + 1 , … , b z ϵ { w , b } Q b b 1 , … , b x − 1 w b x + 1 , … , b z The values rF and rS are relative difference of the revealed image and relative difference of the shares, respectively; m is the pixel expansion; and H(K) is the vector K's hamming weight. Requirement 1 is in respect of contrast of the revealed secret image and requirement 2 is in respect of the security of the scheme where less than v shares cannot reveal any details about secret on stacking. Requirement 3 implies that the meaning of original image in the share is not modified. Figure 2 depicts the (2, 2)–EVCS, where it is shown that shares 1 and 2 are intelligible, revealing no details about the secret image individually. FIGURE 2Open in figure viewerPowerPoint (2, 2) – EVCS: (a) secret image, (b) share 1, (c) share 2, and (d) revealed secret image (share 1 + share 2) 2.3 (v, z)–polynomial image secret sharing scheme Shamir [22] first proposed (v, z) PISSS for dividing a secret into z shares and obtain the secret back by using v shares. Thien and Lin [23] proposed an improvement on [22], where the size of shares is v times smaller than that of secret image. The operations are carried out under mod 251. As pixel values scale from 0 to 255, the values above 250 are truncated to 250. This caused loss of information in the reconstructed secret image. But, this problem is overcome by performing operations under GF (28), where GF stands for the Galois field. Zhou et al. [24] took two adjacent pixels in the secret image as one unit which can have the values in the range [0, 65,535]. If 65,536 is obtained as the value of shared pixel by Lagrange's interpolation under prime modulus 65,537, then encryption operation is repeated unless a value in the range [0, 65,535] is obtained. Also, it requires that the number of pixels in secret image to be even. We use GF (28) operations in our proposed scheme. Polynomial equations (1) and (2) are used for encryption and decryption, respectively, and Equation (2) is Lagrange's interpolation formula: L ( u ) = ( s 0 + s 1 u 1 + s 2 u 2 + … + s k − 1 u v − 1 ) mod GF ( 2 8 ) , (1) L ( u ) = ( ∑ i = 1 v ( c i × h i ) ) mod GF ( 2 8 ) , (2)where h i = ∏ 1 ≤ j ≤ v , i ≠ j ( u − u j u i − u j ) . In Equation (1), L(u) is the polynomial of degree v−1 and si, 0 ≤ i ≤ v−1, are v secret values. In Equation (2), ci, 1 ≤ i ≤ v, are the v cipher values obtained for v different u values in Equation (1). It is possible to obtain more than v cipher values for the same v secret values. But, at least, v cipher values are required to retrieve the original v secret values. 2.4 Digital half-toning Digital half-toning is the transformation of a greyscale image into a binary image containing black and white pixels only. As most of the VCS and EVCS schemes work with black and white pixels, secret images in the greyscale form are converted into black and white images. In our proposed scheme, we apply the widely used Floyd–Steinberg error diffusion technique for half-toning. In this technique, 43.75%, 18.75%, 31.25% and 6.25% of the error calculated at a pixel position ( f, g) are transferred to unprocessed pixels at positions ( f, g + 1), ( f + 1, g−1) ( f + 1, g) and ( f + 1, g + 1), respectively. Figure 3 shows half-toned version of a colour image obtained by applying the Floyd–Steinberg half-toning to each red, green and blue (RGB) component of a rainbow colour image. FIGURE 3Open in figure viewerPowerPoint Half-toning: (a) original image and (b) half-toned image 3 PROPOSED (v, z)–LAMSIS SCHEME In this section, the proposed (v, z)–LAMSIS (lossless authenticated multiple secret images sharing) scheme consists of two parts: encryption and decryption with an example is described. The (v, z)–LAMSIS scheme can be used on binary, greyscale and colour images. The encryption and the decryption are done by the dealer. 3.1 Encryption Input: r secret images SI1, SI2, …, SIr, one authentication image A and z cover images C1, C2, …, Cz. Output: z meaningful shares S1, S2, …, Sz. Step 1: Half-tone A, and each Ci to get HA and HCi, 1 ≤ i ≤ z, HA and HCi are the half-toned authentication image and half-toned cover images, respectively. Step 2: Using (v, z) – EVCS, obtain the intermediate shares S1, S2, …, Sz by using HCi, 1 ≤ i ≤ z as cover images and HA as the secret image. Step 3: Read v consecutive pixels from each SIj and store them in sij for each SIj, 1 ≤ i ≤ v and 1 ≤ j ≤ r. Step 4: Substitute these v secret values sij in (1) and compute z cipher values cxj for each SIj, 1 ≤ x ≤ z. Step 5: Express cxj as cxj = qxj × 16 + pxj, where qxj and pxj are the quotient and the remainder obtained on dividing cxj by 16, respectively. Step 6: Read corresponding block of m pixels from each share Sx, 1 ≤ x ≤ z and find the positions of the first r black pixels and r free pixels. Step 7: Store qxj, 1 ≤ j ≤ r, values in least four bits of r black pixels and pxj values in least four bits of r free pixels in order. Step 8: Iterate from Step 3 to Step 7 unless all the pixels in r secret images are processed. Here, the authentication image is known only to the dealer and each participant chooses his/her own cover image. Free pixels are those pixels in a block other than the first r black pixels and r = m/2. 3.2 Decryption Input: v intelligible shares S1, S2, …, Sv. Output: r reconstructed secret images SI1, SI2, …, SIr on successful authentication. Step 1: Stack the v shares Si, 1 ≤ i ≤ v, to check whether the authentication image is reconstructed. If yes, proceed to Step 2. Else, cheating is detected and reconstruction of secret images is stopped. Step 2: Read block of m pixels from each share Sx, 1 ≤ x ≤ v, and find the positions of the first r dark grey pixels and remaining r pixels. Step 3: Retrieve qxj, 1 ≤ j ≤ r, values from the least four bits of first r dark grey pixels and pxj values from the least four bits of the remaining r pixels. Step 4: Compute cipher values cxj as cxj = qxj × 16 + pxj. Step 5: Substitute these v cxj values in Equation (2) to get v corresponding sxj secret values for each j, 1 ≤ j ≤ r which are stored in corresponding locations of secret images being reconstructed. Step 6: Iterate from Step 2 to Step 5 unless all the blocks in the v shares are processed. 3.3 Example We illustrate our proposed scheme for one block with v = 2 and z = 2. Here, pixel expansion m = 4 and r = m/2 = 2. Let two corresponding secret pixels in the first and second secret images are {100, 200} and {50, 150}, respectively, in a component. Let the corresponding blocks in shares 1 and 2 generated after Step 2 in encryption are {0, 255, 255, 0} and {0, 255, 0, 255}, respectively, which correspond to share block generation for white pixels in both cover images and white pixel in authentication image, where 0 is the black pixel and 255 is the white pixel in the RGB colour model. Using Equation (1), secret pixels of secret image 1 and secret image 2 are converted into cipher values {172, 239} and {164, 5}, respectively; 172 = (10 × 16) + 12, 239 = (14 × 16) + 15, 164 = (10 × 16) + 4 and 5 = (0 × 16) + 5. Values 172 and 164 are stored in the block of share 1, and values 239 and 5 are stored in the block of share 2. After encryption, modified block contents in shares 1 and 2 are {10, 252, 244, 10} and {14, 255, 0, 245}, respectively. During decryption, original cipher values are obtained after Step 4 as {172, 164} from share 1's block and {239, 5} from share 2's block. Applying Equation (2) on {172, 239} and {164, 5}, we get back the original secret pixels {100, 200} and {50, 150}, respectively. 4 PROPERTIES ANALYSIS In this section, the (v, z)–LAMSIS scheme is proven to be a valid one by Theorems 1–3, having properties such as security, contrast and cheating prevention. We have also analysed about the size of shares, cover images and authentication image for general values of v and z, and about the quality of shares. 4.1 Security and contrast properties Theorems 1 and 2 are used to prove the security and contrast properties of the (v, z)–LAMSIS scheme, respectively. Theorem 1.The (v, z)–LAMSIS scheme is secure. Proof:.The original z shares are generated by using (v, z)–EVCS which is already proven to be secure and each block of m pixels is embedded with cipher values of m/2 secret images computed using (v, z)–PISSS which is also already proven to be secure. Secure means less than v shares cannot disclose information about the secret image. Hence, the (v, z)–LAMSIS scheme is as strong as (v, z)–EVCS and (v, z)–PISSS. Theorem 2.The secret images can be disclosed by using t (t ≥ v) shares in the (v, z)–LAMSIS scheme. Proof:.As the cipher values generated using (v, z)–PISSS with respect to each secret image are stored in shares generated using (v, z)–EVCS, a minimum of v in t shares are sufficient to recover the secret image pixels and, hence, recover all the secret images. 4.2 Cheating prevention Authentication for cheating prevention in the (v, z)–LAMSIS scheme is implemented using the stacking operation carried out by the dealer. A successful cheating devolves on the correct supposition of the authentication image pixels by stacking the shares of victim(s) and cheater(s), and finding all pixel values of secret images correctly. We consider two types of cheaters: malicious participant (MP) and malicious outsider (MO) as discussed in 3. MP is a cheater who is a licit participant and MO is a cheater who is not in the set of licit participants. Theorem 3.The (v, z)–LAMSIS scheme is cheat immune to t', 1 ≤ t' ≤ z−1 cheaters. Proof:.According to distinct number t' and kinds of cheaters, the proof for cheat immunity of the proposed scheme is explained with respect to four cases as follows: (i) an MO has no details about the shares of the licit participants. The only thing he can carry out is make some fake shares which can reveal fake secret images using Equation (2) and give it to a dealer for decryption. But, cheating will be caught in Step 1 of decryption. (ii) An MP has only information about his share and cover image. The only thing he can do is to create some extra fake shares for revealing fake secret images of his choice and deliver it to a dealer for decryption. But again, cheating will be noticed in Step 1 of decryption. (iii) For t' < v MPs, they do not get information about authentication image as authentication image is shared using (v, z)–EVCS. So, they also fail during authentication in Step 1 of decryption. (iv) For v ≤ t' ≤ n−1 MPs, they have information about authentication image obtained by stacking their shares. They create fake shares in such a way that on stacking them with victim's share, authentication image is obtained. But, as they do not have information about victim's share, they cannot recover the secret images using Equation (2) that they wanted to show to the victim after decryption. This is because the cipher values embedded in the victim's share generated using Equation (1) are unique for original secrets. When these cipher values are used with the ciphers values of fake shares in Equation (2), wrong secret pixel values are generated, failing to cheat the victim. 4.3 Size of shares, cover images and authentication image To share secret images, each of size f × g using (v, z)–LAMSIS scheme, cover images and authentication image, each of size f × (g/v) are used, where f is the height and g is the width of a secret image. If m is the pixel expansion of the scheme, then the size of each share becomes f × ((g/v) × m). Using these facts, we will prove that the overall size of shares produced using the proposed scheme is lesser than or equal to the overall size of shares produced using any general (v, z)–VCS with pixel expansion 1 for sharing r = m/2 secret images. Lemma 1.The total size of the shares produced using (v, z)–LAMSIS scheme for sharing r secret images, each of size f × g, is lesser than or equal to the total size of shares generated using any general (v, z)–VCS with pixel expansion 1 sharing one secret image at a time. Proof:.For the proposed scheme, we know that the size of each share is f × ((g/v) × m) which carries information about r = m/2 (i.e. m = 2r) secret images. Therefore, the total size of shares TS(v, z)–LAMSIS is z × f × ((g/v) × 2r). For general (v, z)–VCS with pixel expansion 1, the size of each share is f × g. Therefore, the total size of shares TSgeneral (v, z) is z × f × g × r for sharing r secret images. Without loss of generality, as v ≥ 2, z × f × (g/v) × 2r ≤ z × f × g × r, that is, TS(v, z)–LAMSIS ≤ TSgeneral (v, z). To be exact, for v = 2, TS(v, z)–LAMSIS = TSgeneral (v, z) and for v > 2, TS(v, z)–LAMSIS < TSgeneral (v, z). For example, in (2, 3)–LAMSIS scheme, m = 6. It means the number of secret images r that can be shared is m/2 = 6/2 = 3. Let the size of each secret image be 100 × 90. Then, the size of each cover image and authentication image is 100 × (90/2). From the proof of Lemma 1, TS(2, 3)–LAMSIS = 3 × 100 × (90/2) × 6 = 81,000 and TSgeneral (2, 3) = 3 × 100 × 90 × 3 = 81,000, that is, TS(2, 3)–LAMSIS = TSgeneral (2, 3). Similarly, in (3, 3)–LAMSIS scheme, m = 6. So, r = 3. By keeping the size of each secret image as the same as before, the size of each cover image and authentication image is 100 × (90/3). Now, TS(3, 3)–LAMSIS = 3 × 100 × (90/3) × 6 = 54,000 and TSgeneral(3, 3) = 3 × 100 × 90 × 3 = 81,000 that is TS(3, 3)– < TSgeneral(3, 3). 4.4 Quality of shares Visual quality of shares is assessed using peak-signal-to-noise ratio (PSNR). The higher the PSNR value, the higher the quality is. PSNR of the revealed image in respect of the original image is given by PSNR = 10 log 10 ( 255 2 MSE ) (3)where MSE is the mean square error computed by MSE = 1 f × g ∑ i = 1 f ∑ j = 1 g ( x i j − y i j ) 2 (4) In Equation (4), f is the height and g is the width of the image, xij and yij are the original pixel value and transformed pixel value at the same position, respectively. We give the lowest bound on the PSNR value of a share generated using the proposed scheme after embedding the cipher values that is at the end of encryption, with respect to the corresponding share before embedding cipher values that is, after Step 2 of encryption, in Lemma 2. Lemma 2.The minimum PSNR value of a share generated using the proposed (v, z)–LAMSIS scheme is 24.608 dB with respect to the corresponding share before embedding the cipher values. Proof: The maximum difference that can be obtained between original and transformed pixel values at a position is 15. If this maximum difference is considered to be the same for all pixel positions, then this will be the scenario where the PSNR value will be