A Shamir

CC Chang

CC Lin

CC Thien

GR Blakley

L Bai

R Lukac

RZ Wang

SK Chen

TH Chen

EURASIP Journal on Advances in Signal Processing

English

Kuang-Shyr Wu

Crossref

Wu EURASIP Journal on Advances in Signal Processing 2013, 2013:49http://asp.eurasipjournals.com/content/2013/1/49RESEARCH Open AccessA secret image sharing scheme for light imagesKuang-Shyr WuAbstractIn this study, a new (r, n)-threshold secret image sharing scheme with low information overhead for images isprovided, which has a low distortion rate, and is more applicable for light images. A secret image is encoded into nnoise-like shadow images to satisfy the condition that any r of the n shares can be used to reveal the secret image,and no information on the secret can be revealed from any r – 1 or fewer shares. The size of the shadow images isrelatively small. The experimental results show the effectiveness of the proposed scheme.Keywords: Secret sharing, Image sharing1. IntroductionRecently, digital-image sharing has played an importantrole because of the increasing requirements for imagetransmission. Effective and secure protection for import-ant messages is a primary concern in commercial andmilitary applications [1]. Numerous techniques, such asimage hiding and watermarking, were developed to in-crease the security of the secret.For the reliability issue, particularly for large-sized im-ages, such as satellite or medical images, the secret imagesharing (SIS) approaches are useful for protecting sensitiveinformation [1]. The basic idea of secrete sharing is totransform an image into n shadow images that are trans-mitted and stored separately. The original image can bereconstructed only if the shadow images that participatedin the revealing process form a qualified set. To avoid thesingle-point failure, for example, the encoded content iscorrupted during transmission, the (r, n)-threshold imagesharing schemes were therefore developed. In these devel-oped schemes, the original image can be revealed if r ormore of these n shadow images are obtained; however,users who with complete knowledge of r – 1 shares can-not obtain the original image.Currently, numerous approaches have been developedfor image sharing. Blakley [2] and Shamir [3] independ-ently proposed original concepts of secret sharing in1979. The proposed (r, n)-threshold scheme encodes theinput data D into n shares, which are then distributedCorrespondence: keithwu@uch.edu.twDepartment of Computer Science and Information Engineering, Chien HsinUniversity of Science and Technology, No. 229, Jianxing Rd, Zhongli City,Taoyuan County 32097, Taiwan© 2013 Wu; licensee Springer. This is an OpenLicense (http://creativecommons.org/licenses/bmedium, provided the original work is properlyamong n recipients. As previously mentioned, D can bereconstructed by anyone who obtains a predefined num-ber r, where 2 ≤ r ≤n, of the images.In 2002, Thien and Lin [1] proposed an elegant SISscheme based on the (r, n)-threshold. In the scheme, thesize of generated shadow images is only 1/r of that ofthe original image, which is advantageous in subsequentstorage and transmission. Following the work of Thienand Lin, certain image sharing schemes [4-6] have beenproposed to reduce the size of the shadow images.Lin and Tsai [4] transformed the secret image to fre-quency domain, and shared the first ten coefficients ofeach block. Wang and Su [5] designed an SIS method ap-plying the image difference and the algorithm of Huffmancoding in the sharing process. Chang et al. [6] proposed amethod for color images sharing with smaller shadowimages.For the concern of flexibility in various applications,many improved image sharing schemes were explored.Lin and Tsai [7] incorporated digital watermarking tech-nique with the image sharing technique to have additionalcapabilities of steganography and authentication. Thienand Lin [8] developed a method to make the shadow im-ages look like portraits. Chen and Lin [9] applied the shar-ing concept to build a fault-tolerant progressive imagetransmission approach. Lukac and Plataniotis [10] utilizedthe concept of bit-level decomposition to share colorimages. Bai [11] classified the sharing schemes into twocategories, the first called perfect secret sharing remainsthe same requirement as the original idea [1], and theother called ramp secrete sharing (RSS) has the propertythat the exposed information is proportional to the size ofAccess article distributed under the terms of the Creative Commons Attributiony/2.0), which permits unrestricted use, distribution, and reproduction in anycited.Wu EURASIP Journal on Advances in Signal Processing 2013, 2013:49 Page 2 of 5http://asp.eurasipjournals.com/content/2013/1/49the unqualified group. Bai [11] also proposed an SISscheme by using a combination of matrix projection andShamir’s method. For the RSS, recently, Wang et al. [12]designed an incrementing visual cryptography schemeusing random grids; Chen and Tsao [13] investigated athreshold RG-based visual secret sharing scheme aimingat providing the wide-use version.Although numerous secret sharing schemes have beendeveloped, the Thien–Lin scheme [1] remains the firstchoice among various applications because of its simpli-city and efficiency. In particular, for embedded systemsand hand-held devices, numerous operations of SISschemes proposed by other researchers, such as GaloisField GF(2m) and inverse matrices, might not be visibleor require extra computation power. These complicatedoperations would increase the power consumption orchip area for the chip design process.In the Thien–Lin scheme, the input pixel values mustbe truncated into 0 to 251. The light pixel values greaterthan 251 of the secret image are lost. This drawbackmight not be suitable for light images. Therefore, theproposed method mainly focuses on the improvement ofthe Thien–Lin scheme.The remainder of the article is organized as follows.Section 2 reviews the Shamir and Thien–Lin’s schemes.The proposed SIS method is introduced in Section 3,and experimental results and theoretical analysis areshown in Section 4. Finally, conclusions are summarizedin Section 5.2. ReviewImage sharing is a critical means in the field of digitalimage security and digital image hiding. The basic ideain image sharing is to divide secret data into pieces fordistribution to different persons, and certain subsets ofthese people can recover the entire secret. In general,image sharing can be described as dividing a secretimage into n sub images, which have a certain degree ofviability, but do not contain each other’s information.Through the application of an indeterminate equation,the gray values of images are divided in order to achieveimage sharing and encryption. The secret image can bereconstructed through the complete knowledge of r subimages, where r ≤ n.A simple and effective algorithm for image sharingbased on above description is addressed in Section 2.1.2.1. The Shamir (r, n) SIS schemeShamir [3] developed an (r, n)-threshold-based secretsharing scheme for 2 ≤ r ≤ n, n is the number of shadowimages. The secret can be reconstructed by obtainingthe predefined number r of n shares.The scheme is to construct a polynomial function asf xð Þ ¼ a0 þ a1xþ a2x2 þ . . . þ ar1xr1 modp;ð1Þwhere p is a prime number, a0 is the secret data and theremaining coefficients a1, a2,. . ., ar–1 are randomly chosen(known values) from the integer range [0. . .p – 1].For each secret data, a secret share is a pair of (xi, yi)whereyi ¼ f xið Þ; 1 £ i £n; and 0 < x1 < x2 < . . . < xn < p:(2)According to Equations (1) and (2), if we acquire anyr or more pairs of the n shares, then at least r equationsyi = f(xi) can be held in hand, the secret data a0 istherefore resolved.On the other hand, the secret data a0 can also easilybe obtained by using Lagrange’s interpolation [3].2.2. The Thien–Lin (r, n) SIS schemeThien and Lin [1] extended Shamir’s idea, and proposedan SIS scheme based on the (r, n)-threshold scheme, inwhich each generated shadow image is 1/r the size of thesecret image in 2002. In their method, the arithmetic op-erations are evaluated in the prime Galois Field GF(251).Consequently, a preprocessing to truncate pixel values lar-ger than 250 is necessary. They also applied a permutationstep on the original image before performing a sharingprocess to hide the correlations among neighboring pixels.Consider a secret image O, comprising m pixels, to en-code O to n shadow images S1, S2, S3,. . ., Sn, the sharingsteps of the Thien–Lin (r, n) SIS scheme, where 2 ≤ r ≤ n,are summarized below.Step 1. Truncate the pixel values in O greater than 250(251 to 255) to 250, O’ denotes the image aftertruncation.Step 2. Generate a permutation sequence with a secretkey to permute the pixels of O’, the permutatedimage is expressed as Q.Step 3. Set the current processing section number j to 1.Step 4. Sequentially take r non-processed pixels, a0, a1,a2,…, ar–1, of Q to form a section j, and create apolynomial of degree r – 1 as follows:fj xð Þ ¼ a0 þ a1xþ a2x2 þ…þ ar−1xr−1 mod251:ð3ÞStep 5. Generate n pixels:fj 1ð Þ; fj 2ð Þ; fj 3ð Þ;…; fj nð Þ; ð4Þand sequentially assign them to the n shadowimages S1, S2, S3,…,Sn.Figure 2 The secret image after permutationWu EURASIP Journal on Advances in Signal Processing 2013, 2013:49 Page 3 of 5http://asp.eurasipjournals.com/content/2013/1/49Step 6. Increase j by 1.Step 7. Repeat Steps 4 through 6 until all pixels of Qare processed.This method uses all the coefficients of Equation (3)to share the secret pixels so that the size of the shadowimages can be 1/r. This process is a lossy SIS scheme, noprediction exists to completely recover the originalimage O although the visual quality is good.For the sharing phase of the Thien–Lin (r, n)-SISscheme, without loss of generality, we assume the rshares are S1, S2, S3,…, Sr, and the following steps can beused to reveal the secret image O’ using any r (2 ≤ r ≤ n)of the shadow images.Step 1. Set the current processing section number j to 1.Step 2. Take one non-processed pixel from each of ther shadow images.Step 3. Use these r pixels, fj(1), fj(2), fj(3),…, fj(r), andLagrange’s interpolation to solve thecoefficients a0, a1, a2,…, ar–1 in Equation (3).They are the corresponding r pixel values ofthe jth section in Q.Step 4. Increase j by 1.Step 5. Repeat Steps 2 through 4 until all pixels of theshadow images S1, S2, S3, …, Sr are processed.Step 6. Apply the inverse-permutation operation to thepermuted image Q to recover the secret imageto O’.3. The proposed methodFor the SIS of light images, the Thien–Lin method mightnot be sufficient to conduct the light areas because ofFigure 1 The secret image (original image for sharing) Lenathe truncation. In this study, the first step of the pro-posed method is to change the prime number 251 ofThien–Lin’s method to 257, in which the remainders arein the range of [0. . .256].The purpose of extension of the prime number is toprotect the light image pixels from truncation errors,but it appears impossible to store a number greater than255 into an 8-bit pixel. However, it works correctly whenthe pixel values of fj(x) in Equation (3) are in the rangeof [0. . .255], the only exception we are required to sat-isfy is “256.”The major difference between the proposed method andthe Thien–Lin method is that we use prime number 257instead of 251, this change makes the truncation errordisappeared, but numerous details still remain crucial.To encode an image O into n shadow images, S1, S2,S3,. . ., Sn, the sharing steps of the proposed method aresummarized below.Step 1. Generate a permutation sequence with a secretkey to permute the pixels of O, Q depicts thepermutated image.Step 2. Set the current processing section number j to 1.Step 3. Sequentially, take r not-processed yet pixels, a0,a1, a2,…, ar–1, of Q to form a sharing section j.Without loss of generality, let the currentprocessing section be the jth section of theimage, and create a polynomial of degree r – 1as follows:fj xð Þ ¼ a0 þ a1xþ a2x2 þ…þ ar−1xr−1 mod 257:ð5ÞFigure 3 The four noise-like shadow images of the secret imageFigure 4 The reconstructed image by using theproposed methodWu EURASIP Journal on Advances in Signal Processing 2013, 2013:49 Page 4 of 5http://asp.eurasipjournals.com/content/2013/1/49Step 4. Generate the n shadow pixels:fj 1ð Þ; fj 2ð Þ; fj 3ð Þ;…; fj nð Þ; ð6Þset fj(x) to zero if fj(x) is equal to 256.Step 5. Assign fj(1), fj(2), fj(3),…, fj(n) in Equation (6) tothe jth pixel of the n shadow images accordingly.Step 6. Increase j by 1.Step 7. Repeat Steps 3 through 6 until all pixels of Qare processed.We set fj(x) to zero for the specific case of “256” inStep 4 to solve the overflow problem, but the normalpixel values of zero are also affected. If this case occurs,the sharing step is not lossless, and the decoding resultwill trend to be unacceptable. Therefore, we need to de-velop a decoding scheme to solve this problem.The proposed decoding scheme uses a similarity calcu-lation process of the surrounding pixels to avoid the dis-tortion problem.The following steps can be used to reveal the secretimage O’.Step 1. Set the current processing section j to 1.Step 2. Take a not-processed yet pixel of position j fromeach of the r shadow images.Step 3. If any of the r pixels is equal to zero, recordposition j into a file, otherwise run Step 3.1.Step 3.1 Use these r pixels fj(1), fj(2), fj(3),…, fj(r) andLagrange’s interpolation to solve thecoefficients a0, a1, a2,…,ar–1 in Equation (5).They are exactly the r pixel values of the jthsection in Q.Step 4. Increase j by 1.Step 5. Repeat Steps 2 through 4 until all pixels of theshadow images are processed.Step 6. Apply the inverse-permutation operation to thepermuted image Q to get the secret image to Q’.Step 7. For each j (recorded in Step 3), run Step 3.1 bydifferent combinations of zero and 256 for thezero pixels of the shadow image (the otherpixels remain the same values), then we haveseveral candidate solution sets (the number ofcombinations) in hand. Calculate each pixel’ssquared Euclidean distances with its knownneighboring pixels (at most eight) and sumthem up for every solution set, the solution setwith minimal distance is the solution we needfor Section j.Wu EURASIP Journal on Advances in Signal Processing 2013, 2013:49 Page 5 of 5http://asp.eurasipjournals.com/content/2013/1/49Store each pixel value of Section j in Q’. The newimage O’ is the reconstructed secret image.The proposed method has three benefits. First, byusing the prime number 257, the preprocessing of thetruncation is no longer necessary, and computation timeis saved. Second, the problem we are facing in ourscheme is no longer the truncation but the overflow.Therefore, the distortion can be minimized by the use ofcorrelation calculation of adjacent pixels.Third, the only drawback of the Thien–Lin method isthat the light images might not be good enough. Becausethe proposed method has no truncation process neededso that the light images are applicable to the proposedmethod.4. ExperimentsThe experimental results of the proposed method areprepared in this section to show the (2, 4)-threshold SISscheme. The 256 × 256 gray-level secret image “Lena” isshown in Figure 1. The permuted image is shown inFigure 2, and the four shadow images are shown inFigure 3. The size of each shadow image is only 1/2 ofthat of the secret image. Figure 4 is the reconstructedimage obtaining any two out of four in Figure 3, inwhich the size of the reconstructed image is identical tothe secret image.For security, the following is a theoretical analysis. Inthis experiment, we used a 256 × 256 gray-level secretimage. For this image, because each section is formed ofr pixels, there are (256 × 256)/r sections, that is (256 ×256)/r polynomials. To solve the r pixels (coefficients) ofthe polynomial, r equations should be acquired. If ahacker acquires r – 1 shadow images, he can constructonly r – 1 equations. The possibility of guessing theright solution is then only 1/256. Hence, if there are(256 × 256)/r polynomials, the possibility of obtainingthe correct image is only (1/256)(256×256)/r. In the Thien–Lin method, the possibility is (1/251)(256×256)/r [1].5. ConclusionsThis proposed method gives us an insight into SIS. Theconcept is to use prime number 257 to replace 251 inThien–Lin’s method [1]. The proposed method has the fol-lowing properties: (i) a secret image can be reconstructedfrom any r shadow image nearly perfectly or without anyloss; (ii) the method reduces the size of the shadow imagesfor further storage or transmission; (iii) the proposedmethod can protect the secret image if any (r – 1) or fewershadow images are stolen, and the possibility of guessingright is low; and (iv) this method can be used on light im-ages. The experimental results and theoretically analysisshow that the proposed scheme performs well.Competing interestsThe author declares that he has no competing interests.Received: 14 November 2012 Accepted: 21 February 2013Published: 15 March 2013References1. CC Thien, JC Lin, Secret image sharing. Compt. Graph. 26(5), 765–770 (2002)2. GR Blakley, Safeguarding cryptographic keys, in Proceedings of AFIPS 1979National Computer Conference, New Jersey. USA 48, 313–317 (1979)3. A Shamir, How to share a secret. Commun. ACM 22(11), 612–613 (1979)4. CC Lin, WH Tsai, Secret image sharing with capability of share datareduction. Opt. Eng 42, 2340–2345 (2005)5. RZ Wang, CH Su, Secret image sharing with smaller shadow images. PatternRecognit. Lett. 27(6), 551–555 (2006)6. CC Chang, CC Lin, YH Chen, A novel secret image sharing scheme in colorimages using small shadow images. Inf. Sci 178, 2433–2447 (2008)7. CC Lin, WH Tsai, Secret image sharing with steganography andauthentication. J. Syst. Softw. 73(3), 405–414 (2004)8. CC Thien, JC Lin, An image sharing method with user-friendly shadowimages. IEEE. Trans. Circuits Syst. Video Technol. 13(12), 1161–1169 (2003)9. SK Chen, JC Lin, Fault-tolerant and progressive transmission of images.Pattern Recognit. 38(12), 2466–2471 (2005)10. R Lukac, KN Plataniotis, Bit-level based secret sharing for image encryption.Pattern Recognit. 38(5), 767–772 (2005)11. L Bai, A reliable (k, n) image secret sharing scheme with low informationoverhead. Int. J. Compt. Appl. 32(1), 9–14 (2010)12. RZ Wang, YC Lan, YK Lee, SY Huang, SJ Shyu, TL Chia, Incrementing visualcryptography using random grids. Opt. Commun 283, 4242–4249 (2010)13. TH Chen, KH Tsao, Threshold visual secret sharing by random grids. J. Syst.Softw 84, 1197–1208 (2011)doi:10.1186/1687-6180-2013-49Cite this article as: Wu: A secret image sharing scheme for light images.EURASIP Journal on Advances in Signal Processing 2013 2013:49.Submit your manuscript to a journal and beneﬁ t from:7 Convenient online submission7 Rigorous peer review7 Immediate publication on acceptance7 Open access: articles freely available online7 High visibility within the ﬁ eld7 Retaining the copyright to your article    Submit your next manuscript at 7 springeropen.com

A secret image sharing scheme for light images

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A secret image sharing scheme for light images

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