Multithreshold progressive image sharing
with compact shadows
Lee Shu-Teng Chen Ja-Chen Lin
National Chiao Tung University
Department of Computer Science and Information Engineering Hsinchu, Taiwan, 300
E-mail: [email protected]
Abstract. We propose a multithreshold progressive reconstruction method. The image is encoded three times using Joint Photographic Experts Group (JPEG): first with a low-quality factor, then with a medium-quality factor, and last with a high-quality factor. Huffman coding is employed to encode the difference between the important image and the high-quality JPEG decompressed image. The three JPEG codes and the Huffman code are shared, respectively, ac-cording to four prespecified thresholds. The n-generated equally im-portant shadows can be stored or transmitted using n channels in parallel. Cooperation among these generated shadows can progres-sively reconstruct the important image. The reconstructed image is loss-free when the number of collected shadows reaches the largest threshold. Each shadow is very compact and so can be hidden suc-cessfully in the JPEG codes of cover images to reduce the probabil-ity of being attacked when transmitted in an unfriendly environment. Comparisons with other image sharing methods are made. The con-tributions, such as easiness to apply to scalable Moving Picture Ex-perts Group (MPEG) video transmission or resistance to differential attack, are also included.© 2010 SPIE and IS&T.
关DOI: 10.1117/1.3295710兴
1 Introduction
Secret sharing is an approach for protecting data.1–9 Blakley1and Shamir2were the first to propose the idea of a 共t,n兲 threshold sharing scheme. Polynomials were used to share a secret among n participants. Any t of the n partici-pants共t艋n兲 could reconstruct the secret, but t−1 partici-pants could not. Thien and Lin4 extended the work of Shamir by sharing a secret image and generated n shadows. The size of each shadow in their work is only 1/t of that of the original secret image, and so the storage space and transmission time are kept low. To reduce the size of each shadow further, Tso9 elegantly quantized the secret image and then shared it. Lin and Tsai7also transformed the secret image into the frequency domain and then shared the first discrete cosine transform 共DCT兲 coefficient, which was used as a seed in a random number generator to yield a sequence of numbers that were then used to rearrange the values of the second to tenth DCT coefficients in each transformed block. In almost all sharing methods, since
each generated shadow looks like noise, hiding methods10–16 may be employed to hide each noise-like shadow in a cover image.
The missing-allowable feature makes image sharing methods useful to the distributed storage of a secret image. Specifically, the n shadows of an image can be stored in n places. Later, to reconstruct the image, the n shadows are grabbed over n distinct channels. Some of the n communi-cation channels or the n storage disks may be out of service temporarily or deliberately if the owner of a shadow refuses to cooperate, but neither case will affect the reconstruction, as long as the number of missing shadows is not more than n − t. The potential problem of losing an image forever is thus erased using image sharing. Additionally, collecting fewer than t shadows yields nothing but noise, and this feature increases security.
Conventional image sharing methods reveal either the entire secret image共when any t of the n shadows are col-lected兲 or nothing 共when fewer than t shadows are collected兲. This all-or-nothing property is useful when the image being shared is top secret. However, not all images in daily life are top secret. In many circumstances, the shared image might be sensitive in some way and yet not top secret. Restated, although an image must not be viewed by only a minority of the participants, the reconstruction of the image can still involve certain quality levels, such as low quality, middle quality, and high quality, based on whether the number of collected shadows reaches the cor-responding thresholds. Such sharing is called progressive sharing: the sensitive image is reconstructed with improv-ing quality, as determined by the number of the collected shadows in the decoding meeting.
Progressive sharing has a range of applications. Con-sider, for example, image searching in an antiterrorism in-telligence office or a witness-protection program, when an authorized officer searches for a sensitive image from a missing-allowable database system with n distributed stor-age places. If the shadows of each imstor-age have been formed earlier by a traditional all-or-nothing sharing scheme, a user must wait for the entire image to be downloaded共by col-lecting t out of the n shadows兲 and then check whether the reconstructed image is useful. In contrast, using shadows in a progressive manner can reduce searching time: in the ear-lier stage of the reconstruction, people can obtain a rough version of the image by collecting a smaller number共t1兲 of Paper 09029RR received Mar. 9, 2009; revised manuscript received Dec.
1, 2009; accepted for publication Dec. 4, 2009; published online Feb. 9, 2010.
shadows共t1⬍t艋n兲; they can then abort further transmis-sion as soon as the rough vertransmis-sion indicates that the candi-date image is absolutely not the sought image. Meanwhile, fewer than t1shadows reveal nothing but noise, providing a certain degree of protection of a sensitive image.
Another example involves the increasing use of mobile devices or computers to browse the web. Providing pro-gressive versions of an image allows each authorized cus-tomer or team member to have more choices. Moreover, the number of downloadable shadows, which control the qual-ity of the reconstructed versions, can be determined by the level of the paid membership 共or authorized rank兲 of the downloader. In particular, if the image is too offensive or violent or allowed to be inspected only by a particular po-lice team or intelligence squad, then controlling the number of shadows in a progressive manner can yield flexible de-sign benefits or facilitate management of the system 共as members of a single team but with different ranks are au-thorized to download different numbers of the n created shadows兲. An example of the third application has been presented elsewhere:17assume that the leader of a research team wants to prohibit any employee from selling high-quality sensitive pictures or blueprints on the black market, but the leader still wants the employees to cooperate every day—for example, to improve a design in blueprints, prepa-rations for surgery, or a body-guarding program, which may be directed at people shown in the images. The leader keeps some of the n generated shadows, and each of the employees has one of the remaining shadows. If the em-ployees want to take a closer look, they have to ask the leader for permission. The leader can lend them one or more shadows to increase the clarity of the image. If an employee takes the shadows to an enemy, the remaining employees can still seek permission from the leader when they want to view the images with great clarity.
Recently, various progressive image sharing schemes have been proposed.17–20However, in Fang’s scheme,18the size of each shadow was expanded to four times larger than the input image. To avoid expansion, people may use approaches17,19,20that were based on the sharing scheme of Thien and Lin.4 Chen and Lin19 adopted a bit-plane scan-ning procedure to rearrange the input image pixels; the re-arranged data were then shared. Wang and Shyu20elegantly decomposed the input image using spatial and depth infor-mation simultaneously and then shared the decomposed im-age. Hung et al.17shared the quantized DCT coefficients of the input image. Although the shadows in their work were much smaller than those in the preceding three works,18–20 the reconstruction of their image was not lossless when all n shadows were collected.
This work offers a new design with all of the advantages of lossless reconstruction 共when most of the shadows are collected兲, compact size, and progressive sharing. All n products 共or n shadows兲 of the input image are compact and so can be hidden in stego media easily without exces-sively affecting the image quality of the cover media. These compact shadows are equally significant, meaning that the reconstructed version of the image depends only on the number of collected shadows.共Any of the shadows could be missing, so the sender or the receiver need not worry about which shadows are sent or collected first. This in-creases the probability of success of the decoding meeting.兲
The proposed scheme is easier than the progressive image sharing schemes17–20 to apply to scalable Moving Picture Experts Group 共MPEG兲 video transmission,21,22 and the shadows herein can resist differential attack.23–25
The rest of this work is organized as follows: Section 2 reviews related works. Section 3 presents the proposed scheme. Section 4 presents experimental results and makes comparisons with other methods. Section 5 discusses secu-rity. Last, Sec. 6 summarizes the contributions of this work. 2 Related Works
2.1 JPEG
Joint Photographic Exports Group 共JPEG兲26 is an interna-tional image compression standard that is used commonly on the Internet. A given image is divided into several blocks of 8⫻8 pixels each. The 8⫻8 pixels of each block are transformed using DCT, and the transformed 8⫻8 co-efficients are quantized using a quantization table. The quantized 8⫻8 coefficients are then scanned in zigzag or-der for entropy coding. After all of the blocks have been sequentially processed, the JPEG code is generated. A pa-rameter called the quality factor QF 共between 0 and 100兲 controls the quality of the JPEG decompressed image. A higher QF corresponds to higher quality of the JPEG de-compressed image共and a larger created JPEG file兲.
2.2 Thien and Lin’s Image Sharing Method
Thien and Lin4propose a共t,n兲 threshold method for split-ting a grayscale secret image into n shadows. First, all of the gray values between 251 and 255 in the secret image must be truncated to 250 because the arithmetic operations in Eq.共1兲are modulo 251. They then use a key to permute the pixels of the secret image; the permuted image is then partitioned into several sectors of t pixels each. For each not-yet-processed sector, define a polynomial:
f共z兲 = a0+ a1z + a2z2+ . . . + a
t−1zt−1 共mod 251兲, 共1兲 where a0 to at−1 are the t pixel values. The n values f共1兲, f共2兲,..., and f共n兲 are calculated and then attached to the n shadows. After all sectors have been processed, the n shad-ows are created. Since every t pixels in the secret image contribute a single pixel to each of the n generated shad-ows, each shadow size is 1/t of the secret image size.
In collecting at least t shadows, Thien and Lin take the first not-yet-used pixel from each of the t shadows and use these t pixel values f共z1兲, f共z2兲,..., and f共zt兲 to evaluate the t coefficients a0 to at−1 in Eq. 共1兲 for the first sector by reconstructing the共t−1兲–deg polynomial f共z兲 as
f共z兲 = f共z1兲 共z − z2兲共z − z3兲 ... 共z − zt兲 共z1− z2兲共z1− z3兲 ... 共z1− zt兲 + f共z2兲 共z − z1兲共z − z3兲 ... 共z − zt兲 共z2− z1兲共z2− z3兲 ... 共z2− zt兲 + . . . + f共zt兲 共z − z1兲共z − z2兲 ... 共z − zt−1兲 共zt− z1兲共zt− z2兲 ... 共zt− zt−1兲 共mod 251兲. 共2兲 By processing all pixels of the obtained t shadows in order, they obtain the permuted image, which is then depermuted to reveal the secret image.
An example is presented in the following. To divide t = 2 pixel values 100 and 200 into n = 3 shadows, Eq.共1兲is used to compute the three shadows: f共1兲=共100+200 ⫻1兲mod 251=49; f共2兲=共100+200⫻2兲mod 251=249; and f共3兲=共100+200⫻3兲mod 251=198. Later, if two shadows f共1兲=49 and f共3兲=198 are received, Eq.共2兲 is used to re-construct the polynomial as
f共z兲 ⬅ f共1兲 ⫻ 共z − 3兲/共1 − 3兲 + f共3兲 ⫻ 共z − 1兲/共3 − 1兲 ⬅ 49 ⫻ 共z + 248兲/249 + 198 ⫻ 共z + 250兲/2 ⬅ 49 ⫻ 共z + 248兲 ⫻ 125 + 198 ⫻ 共z + 250兲 ⫻ 126 ⬅ 100 + 200z 共mod 251兲.
The original pixel values, 100 and 200, are thus obtained.
2.3 Chen et al.’s JPEG Data Hiding Method
Chen et al.27 propose a reversible JPEG steganography method for hiding secret data in a JPEG compression code. First, the JPEG compression code of the cover image is entropy decoded to obtain all quantized 8⫻8 blocks 兵F其 and an 8⫻8 quantization table Q. Next, to embed the se-cret in the quantized coefficient F共i, j兲 of each block F, the scaling factor Q共i, j兲 of the original quantization table Q is modified to a smaller factor Q
⬘
共i, j兲 that satisfied Q共i, j兲/Q⬘
共i, j兲艌2, enabling the two nearby noninteger points F共i, j兲−0.5 and F共i, j兲+0.5 of F共i, j兲 to be mapped to two integer points M共i, j兲 and N共i, j兲, respectively, by M共i, j兲 = 关F共i, j兲 − 0.5兴 ⫻ Q共i, j兲/Q⬘
共i, j兲, 共3兲 N共i, j兲 = 关F共i, j兲 + 0.5兴 ⫻ Q共i, j兲/Q⬘
共i, j兲, 共4兲 where 0艋i, j⬍8. After the values of M共i, j兲 and N共i, j兲 are determined, based on the to-be-hidden secret digit, an inte-ger point in the half-open interval关M共i, j兲,N共i, j兲兲 is inden-tified as the stego quantized coefficient F⬘
共i, j兲. For ex-ample, let F⬘
共i, j兲 be M共i, j兲 if the to-be-hidden secret is 0; let F⬘
共i, j兲 be M共i, j兲+1 if the to-be-hidden secret is 1; let F⬘
共i, j兲 be M共i, j兲+2 if the to-be-hidden secret is 2; and so on. Last, each embedded quantized block F⬘
is entropy en-coded, and the modified quantization table Q⬘
is included in the JPEG file header to yield the JPEG stego code. The original quantization table Q and generated JPEG stego code are transmitted to the receiver.When the original quantization table Q and the gener-ated JPEG stego code are received, the secret can be com-pletely extracted and the original JPEG code can also be reconstructed. First, the JPEG stego code is entropy de-coded to obtain all stego quantized blocks 兵F
⬘
其 and the 8⫻8 modified quantization table Q⬘
. Next, based on the stego quantized coefficient F⬘
共i, j兲 of each block F⬘
, the Q⬘
共i, j兲–to–Q共i, j兲 inverse scaling is employed to recon-struct the original quantized coefficient F共i, j兲 using F共i, j兲 = Round关F⬘
共i, j兲 ⫻ Q⬘
共i, j兲/Q共i, j兲兴, 共5兲 where 0艋i, j⬍8. After the value of F共i, j兲 is reconstructed, use Eq.共3兲to compute M共i, j兲, which is then used to extract the decimal equivalent of the hidden data Z共i, j兲 asZ共i, j兲 = F
⬘
共i, j兲 − M共i, j兲. 共6兲An example of the embedding and extraction processes is as follows. Assume that the quantized coefficient is F共0,0兲=12, and Q共0,0兲=16 is the original quantizer that was used by JPEG. Let Q
⬘
共0,0兲=4 be the modified quan-tizer. According to Eqs.共3兲and共4兲, the values M共0,0兲 and N共0,0兲 are computed usingM共0,0兲 = 共12 − 0.5兲 ⫻ 16/4 = 46,
N共0,0兲 = 关共12 + 0.5兲 ⫻ 16/4兴 = 50.
Accordingly, in the half-open interval 关M共0,0兲=46, N共0,0兲=50兲, the value F
⬘
共0,0兲=46 is the stego quantized coefficient after 0 is embedded.关F⬘
共0,0兲=47 after 1 is em-bedded; F⬘
共0,0兲=48 after 2 is embedded; F⬘
共0,0兲=49 af-ter 3 is embedded.兴 Laaf-ter, assume that the obtained stego quantized coefficient is F⬘
共0,0兲=47. To reconstruct the quantized coefficient F共0,0兲 from F⬘
共0,0兲=47, use Eq.共5兲 to evaluateF共0,0兲 = Round共47 ⫻ 4/16兲 = 12.
共Notably, 12 is the original quantized coefficient. Hence, the JPEG hiding method27 is called reversible.兲 Then use Eq.共3兲 to compute
M共0,0兲 = 共12 − 0.5兲 ⫻ 16/4 = 46.
Last, from Eq. 共6兲, the decimal equivalent of the hidden data is extracted as
Z共0,0兲 = F
⬘
共0,0兲 − M共0,0兲 = 47 − 46 = 1.2.4 Galois Field
A Galois field 共GF兲 is a finite field of pk elements with addition共⫹兲 and multiplication 共⫻兲 operations that satisfy commutative, associative, and distributive laws where p is a prime number and k is a positive integer. In general, the arithmetic over GF共p兲 is the same as modulo p, and thus Thien and Lin4use a Galois field with pk= p1= p = 251 ele-ments. The proposed method employs a Galois field with 2k elements, and the arithmetic over GF共2k兲 is based on the representation of each element in GF共2k兲. An element in GF共2k兲 is generally represented using a polynomial-basis representation, as a binary polynomial of degree less than k. The k-tuple of coefficients of the binary polynomial corre-sponds to the binary representation of an integer between 0 and 2k− 1.
Let A =共ak−1. . . a1a0兲2 and B =共bk−1. . . b1b0兲2 be two k-bits binary elements in GF共2k兲. In the polynomial-basis representation, A and B are A共X兲=ak−1Xk−1+ . . . +a
1X + a0 and B共X兲=bk−1Xk−1+ . . . +b1X + b0, respectively. Define the addition of A and B as
A + B =
兺
i=0 k−1ai丣bi, 共7兲
where 丣 is the exclusive-or 共XOR兲 operator. For the sub-traction, because each element in GF共2k兲 is its own additive inverse, the subtraction of B from A is thus defined as
A − B = A +共− B兲 = A + B =
兺
i=0 k−1ai丣bi. 共8兲
The multiplication and division involve a primitive polyno-mial H共X兲, where H共X兲 is a k-deg irreducible polynomial 共i.e., it has no nontrivial factors兲. To multiply A by B, the remainder C共X兲=ck−1Xk−1+ . . . +c
1X + c0 is computed in a long division, defined by
C共X兲 = A共X兲B共X兲 mod H共X兲, 共9兲
where the operations of the binary coefficients in the poly-nomial multiplication and in the mod H共X兲 operations are all modulo 2 such that all the resulting coefficients are still in兵0, 1其 and therefore binary. After the binary polynomial C共X兲=ck−1Xk−1+ . . . +c
1X + c0 has been determined using Eq.共9兲, the multiplication of the two k-bits binary elements A and B is defined as
A⫻ B = C = 共ck−1. . . c1c0兲2. 共10兲 Last, to divide A by B, Eqs.共9兲 and共10兲are used to mul-tiply A by B−1, where B−1is the unique element E in GF共2k兲 such that关B共X兲E共X兲兴mod H共X兲=1.
3 Proposed Method
3.1 Encoding
The quality factor QF苸兵0,1... ,100其 in JPEG is used to control image quality. To reconstruct an important image s with various quality levels based on the number of received JPEG stego codes, a low-quality factor QFL苸兵0,1, ... ,5其, a medium-quality factor QFM苸兵10,11, ... ,25其, and a high-quality factor QFH苸兵55,56, ... ,85其 can be used to generate, respectively, a low-quality JPEG code c1, a medium-quality JPEG code c2, and a high-quality JPEG code c3. The quality levels of the JPEG images r1, r2, and r3decompressed from codes c1, c2, and c3, respectively, are around 18 to 28 dB, 30 to 34 dB, and 36 to 40 dB. To re-construct the image s error-freely, a difference image d is created by subtracting from the image s the high-quality JPEG image r3 that is decompressed from the high-quality JPEG code c3. The difference image d is compressed using Huffman coding to generate the Huffman code c4. Last, based on the five user-defined integer parameters 1⬍t1 ⬍t2⬍t3⬍t4艋n, the generated codes c1, c2, c3, and c4 are shared according to Eq.共11兲. As shown in Fig.1, the pro-posed 关共t1, t2, t3, t4兲,n兴 threshold scheme comprises four phases: 共1兲 codes generation, 共2兲 sharing, 共3兲 shares com-bining, and 共4兲 data hiding. The encoding algorithm is given here:
Input: An important image s; five positive integer pa-rameters t1, t2, t3, t4, and n, where 1⬍t1⬍t2⬍t3⬍t4 艋n; and n cover images.
Output: The n JPEG stego codes
Step 1a: Compress the important image s using JPEG three times共with quality factors of QFL, QFM, and QFH, respectively兲, yielding a low-quality JPEG code c1of s, a medium-quality JPEG code c2of s, and a high-quality JPEG code c3 of s.
Step 1b: Compute the difference image d by subtracting from the image s its high-quality JPEG image r3 that is decompressed from the JPEG code c3. Compress the dif-ference image d using Huffman coding to generate the Huffman code c4of the image d.
Step 2: For each code ci 共i=1,2,3,4兲, use Eq.共11兲 to split the code ciinto n shares.
Step 3: For each x = 1 , 2 , . . . , n, the x’th shadow is formed by binding together the x’th share of c1, the x’th share of c2, the x’th share of c3, and the x’th share of c4. Step 4: Use the JPEG hiding method27 to hide the n shadows in the n JPEG codes of the n cover images, respectively. 共This generates n desired JPEG stego codes, and the cooperation of several stego codes can view the important image s at certain quality levels.兲 Note: In step 2, the code ci is divided into sectors of ti bytes each. Each byte is treated as a number between 0 and 255. Our share-generating polynomial is
g共x兲 = b0+ b1x + b2x2+ ¯ + bt i−1x
ti−1关over GF共256兲兴, 共11兲 where b0 to bti−1 are the ti numbers of the sector, and the computations in Eq.共11兲 are over GF共256兲. Then, g共1兲 to g共n兲 are sequentially assigned to n shares. Since each sector of ti bytes contributes only a single byte 关a value in the range 0 to 255 and determined by Eq.共11兲兴 to each gener-ated share, the size of each share of the code ciis ti times smaller than that of the code ci. In step 3, the size of each
Important image s
Compress s by JPEG using the quality
factor QFL
Compress s by JPEG using the quality
factor QFH Compress s by JPEG
using the quality factor QFM Use modified (t = t4, n) sharing to share Use modified (t = t1, n) sharing to share Use modified (t = t3, n) sharing to share Use modified (t = t2, n) sharing to share
For QFH, get the difference image d by
subtracting from image s the JPEG decompressed image related to QFH Hide n shadows in n JPEG compressed images, respectively. Output n JPEG stego codes Combine the four x'th shares to form the x'th shadow for x=1,2,…,n. Huffman coding
shadow is 兺i=14 兩ci兩/ti, where 兩ci兩 denotes the length of the code ci. In step 4, to avoid attracting the attention of attack-ers, the n shadows are hidden in n JPEG codes.
3.2 Decoding
When any t1of the n JPEG stego codes are received, the t1 shadows can be extracted from the t1 JPEG stego codes by inverse hiding. For each x = 1 , 2 , . . . , t1, the x’th shadow is partitioned to yield the x’th share of each code ci共1艋i 艋4兲. The t1 shares of the code c1 are then used to recon-struct the low-quality JPEG code c1 in inverse sharing, which can be done either by the matrix multiplication method共detailed in the following兲 or by the Lagrange in-terpolation method used in Thien and Lin4共The two meth-ods are with similar computation loads.兲 The reconstructed JPEG code c1 is decompressed to yield the low-quality JPEG image r1, which is an approximate version of the original important image s. When t2 共or t3兲 JPEG stego codes are available, the reconstruction process is similar to that described earlier, and the reconstructed image r2共or r3兲 will be of medium共or high兲 quality.
Last, if at least t4JPEG stego codes are received, the t4 shadows can also be extracted from the t4 JPEG stego codes by inverse hiding. Then, for each x = 1 , 2 , . . . , t4, the x’th shadow is divided to generate the x’th share of each code ci共1艋i艋4兲. The obtained t4shares of the code c4are used in inverse sharing to reconstruct the Huffman code c4 by matrix multiplication or Lagrange interpolation, and the reconstructed code c4is then decompressed to generate the difference image d. Adding the image d to the image r3 yields the error-free image s.
The use of matrix multiplication to reconstruct the code cifrom any tiout of the n shares共1艋i艋4兲 is described in the following. Recall that the sharing process uses Eq.共11兲 to generate the n pixel values g共1兲 to g共n兲. These n values can also be computed using
关b0b1 . . . bti−1兴
冤
1 1 . . . 1 1 2 . . . n . . . . 1 2ti−1 . . . nti−1冥
=关g共1兲 g共2兲 ... g共n兲兴 关over GF共256兲兴. 共12兲 Accordingly, in the inverse-sharing process, when any tiof the n shares are obtained 共and without loss of generality, suppose that the first ti shares are obtained兲, the first ti not-yet-used pixels g共1兲, g共2兲,..., and g共ti兲 are taken from the ti shares, and the ti coefficients b0 to bti−1 of the first sector are reconstructed using关g共1兲 g共2兲 ... g共ti兲兴
冤
1 1 . . . 1 1 2 . . . ti . . . . 1 2ti−1 . . . t i ti−1冥
−1 =关b0b1 . . . bti−1兴 关over GF共256兲兴. 共13兲 The arithmetic computations in Eq. 共13兲 are still overGF共256兲. Code ciis reconstructed by sequentially process-ing all pixels of the obtained ti shares.
3.3 Example of Sharing and Inverse-Sharing
Processes Based on GF(256)
An example of the sharing and inverse-sharing processes based on GF共28= 256兲 and H共X兲=X8+ X4+ X3+ X + 1 is pre-sented in the following. To partition ti= 2 numbers兵100 and 200其 of 8 bits each into n=3 shares, Eq. 共11兲 is used to compute: g共1兲=100+200⫻1关over GF共256兲兴=172; g共2兲 = 100+ 200⫻2关over GF共256兲兴=239; and g共3兲=100+200 ⫻3关over GF共256兲兴=39. In obtaining the two shares g共1兲 = 172 and g共3兲=39, the inverse matrix of
关
11 31兴
is computed over GF共256兲 as冋
1 1 1 3册
−1 =冋
140 141 141 141册
关over GF共256兲兴. Then Eq.共13兲yields关b0b1兴 = 关172 39兴
冋
140 141
141 141
册
关over GF共256兲兴,where b0= 172⫻140+39⫻141关over GF共256兲兴=100, and b1= 172⫻141+39⫻141关over GF共256兲兴=200. Notably, the two numbers 100 and 200 can also be revealed by the Lagrange interpolation approach, as
g共z兲 ⬅ g共1兲 ⫻ 共z − 3兲/共1 − 3兲 + g共3兲 ⫻ 共z − 1兲/共3 − 1兲 ⬅ 172 ⫻ 共z + 3兲/共1 + 3兲 + 39 ⫻ 共z + 1兲/共3 + 1兲 ⬅ 172 ⫻ 共z + 3兲 ⫻ 141 + 39 ⫻ 共z + 1兲 ⫻ 141 ⬅ 100 + 200z 关over GF共256兲兴.
4 Experiments and Comparisons
4.1 Experimental Results
The inequalities共t1= 3兲⬍共t2= 4兲⬍共t3= 5兲⬍共t4= 6兲 and the irreducible polynomial H共X兲=X8+ X4+ X3+ X + 1 are used to generate n = 6 shadows of the important image. The JPEG source code that is used in the experiments is taken from the fourth public release of the Independent JPEG Group’s free software.28The quality of an image is measured by the peak signal-to-noise ratio共PSNR兲.
In the first experiment, the 512⫻512 grayscale impor-tant image s Lena, displayed in Fig.2, is encoded by JPEG with three quality factors QFL= 5, QFM= 25, and QFH= 85. The four codes c1, c2, c3, and c4have lengths 5750, 13,787, 45,972, and 115,458 bytes, respectively. The six cover im-ages Peppers, Jet, Boat, Lake, Baboon, and Zelda, shown in Fig.3, are all encoded using JPEG with QF= 75 to hide the six shadows and thus obtain the six JPEG stego codes. Figure 4 displays the n = 6 images that are decompressed from our six JPEG stego codes without any extraction of the hidden shadows, and the PSNRs of them are 37.42, 37.41, 36.40, 34.39, 32.73, and 38.67 dB, respectively. When different numbers of JPEG stego codes are received, the reconstructed versions r1, r2, r3, and r4 of Lena are as plotted in Fig. 5, and the respective PSNRs are 27.32, 33.67, 39.35, and⬁ dB.
In the second experiment, the important image is the 512⫻512 grayscale image Tiffany. The six shadows are generated and then remain hidden in the six JPEG codes that are generated in the first experiment. The PSNRs of the decompressed images from these six JPEG stego codes are 37.75, 37.70, 36.65, 34.61, 32.97, and 38.96 dB, respec-tively.共These values are a little better than the 37.42, 37.41, 36.40, 34.39, 32.73, and 38.67 dB values obtained in the first experiment.兲 For Tiffany, the PSNRs of the versions reconstructed using any 3, 4, and 5 JPEG stego codes are 28.37, 34.12, and 39.79 dB, respectively.共These values are a little better than those, 27.32, 33.67, and 39.35 dB, for Lena.兲 When six JPEG stego codes are obtained, the recon-structed Tiffany is identical to the original Tiffany.
Last, Table1shows the bit rates关bits per pixel 共bpp兲兴 of the JPEG-Q75 codes 共created using JPEG with QF=75兲 before and after hiding our shadows. The bit rate will in-crease significantly after hiding a large-size secret. How-ever, the bit rate of the JPEG stego code herein still falls in
the reasonable range of JPEG, i.e., the bit rate of the stego code herein is smaller than that of the JPEG code generated using QF= 95, as shown in Table 1. This alleviates the problem of code length. If the shadows of other image-sharing methods4,8,9,18–20 had been used, then the problem would have been worse.关The reason for using QF=95 as the upper bound to examine the bit rate of the JPEG stego codes is that, as stated in Kim et al.’s work,29 the general quality factors 共QFs兲 that are used in digital cameras are between 90 and 95.兴
Fig. 2 Original 512⫻512 important image Lena used in the first experiment.
(a) (b) (c)
(d) (e) (f)
Fig. 3 Six 512⫻512 cover images Peppers, Jet, Boat, Lake, Ba-boon, and Zelda, which are utilized to cover the important image Lena.
(a) (b) (c)
(d) (e) (f)
Fig. 4 Six 512⫻512 images decompressed from six output JPEG stego codes. 共The six decompressions are done independently, without any extraction of the hidden important image Lena.兲 PSNRs of共a兲 to 共f兲 are 37.42, 37.41, 36.40, 34.39, 32.73, and 38.67 dB, respectively.
(a) (b)
(c) (d)
Fig. 5 Four versions of important image Lena reconstructed from various numbers of received JPEG stego codes:共a兲 from any three JPEG stego codes 共PSNR=27.32 dB兲; 共b兲 from any four JPEG stego codes共PSNR=33.67 dB兲; 共c兲 from any five JPEG stego codes 共PSNR=39.35 dB兲; and 共d兲 from the six JPEG stego codes 共and identical to the original important image Lena兲.
4.2 Comparisons
Table2compares other sharing schemes4,8,9,17–20with ours in terms of shadow-size expansion, progressive ability, and lossless reconstruction ability. Each shadow in two of the related works8,18 is four times larger than the original im-portant image, indicating that size expansions had occurred. In contrast, each shadow in all of the associated works4,9,17,19,20and ours is smaller than the original impor-tant image. Although Thien and Lin,4 Tso,9 and Hung et al.17all shared the image without size expansion, Thien and Lin4 and Tso9 could not reconstruct the image progres-sively, whereas Hung et al.17 could not reconstruct the im-age in an error-free manner. Only Chen and Lin,19 Wang and Shyu,20 and ourselves have achieved reconstruction with all three desired characteristics. Among these three methods, as presented in Table3, each shadow size herein 共12.89% of 512⫻512 bytes兲 is smaller than those in Chen and Lin’s method19 共22.22%兲 and Wang and Shyu’s20 共50%兲. Hence, the transmission time in this work is less, and the survival rate in an unfriendly environment, in which the network connection time is unstable among the n channels used to store the n shadows, is increased. Equiva-lently, in this work, the storage space in a distributed stor-age system is most reduced. The smaller size of the shad-ows also facilitates the hiding of shadshad-ows in stego media.
The construction of Table 3, which compares the shadow sizes among nonexpanded schemes, is explained in the following. All data 共except those obtained herein兲 are directly taken from the aforementioned works.4,9,17,19,20For fairness of comparison, the shadow sizes in Table3are all measured before hiding: all are shadow sizes, and none is a stego media size. This action eliminates the size-altering effects of particular post-processing 共hiding兲 approaches. Assume that the given important image is the 512⫻512 grayscale image Lena, and the共largest兲 threshold value is set to 6 for all schemes, except that Hung et al.’s scheme17 uses 5 as the largest threshold value because their scheme did not provide a version with a threshold value being 6. For each x = 1 , 2 , . . . , n, the four x’th shares are combined to form the x’th shadow, and thus each shadow herein has size 兺i=14 兩ci兩/ti=共5750/3兲+共13,787/4兲 +共45,972/5兲+共115,458/6兲=33,802 bytes 共which is
12.89% of the size of the 512⫻512 grayscale image Lena兲. According to Table3, the shadow sizes in the proposed method and Hung et al.’s17 are more economic than those in the related works.4,9,19,20 However, Hung et al.’s method17is not lossless when all shadows are collected. In fact, if the original important image can be satisfactorily reconstructed with some loss, then our step 1b can be omit-ted, such that no Huffman code c4is generated. Then, each of our shadows can be reduced to 兺i=13 兩ci兩/ti=共5750/3兲 +共13,787/4兲+共45,972/5兲=14,559 bytes 共which is 5.55% of the size of the 512⫻512 grayscale important image Lena兲. Restated, the size of each shadow in this lossy ver-sion is about half of that in Hung et al.’s scheme.17 More-over, in this lossy version, the total shadow size herein is 14,559⫻6=87,354 bytes, which is still smaller than 30,723⫻5=153,615 obtained by Hung et al.17
When the five shadows are collected, the 39.35-dB Lena关identical to that in Fig.5共c兲兴 is reconstructed, better than the 37.04-dB Lena revealed by Hung et al.17Notably, Tso9reconstructed Table 1 Bit rates共bpp兲 of the JPEG codes of cover images.
Cover image bpp of the 共no-hiding兲 JPEG-Q75 code bpp after hiding a shadow of size 14,559 bytes in the JPEG-Q75 code bpp after hiding a shadow of size 33,802 bytes in the JPEG-Q75 code bpp of the 共no-hiding兲 JPEG-Q95 code Peppers 0.94 1.62 2.33 2.52 Jet 0.94 1.64 2.37 2.47 Boat 1.02 1.73 2.46 2.74 Lake 1.25 1.95 2.64 3.44 Baboon 1.89 2.63 3.25 4.09 Zelda 0.81 1.48 2.21 2.54
Table 2 Comparisons among image sharing methods共Refs.4,8,9, and17–20兲. Scheme Nonexpanded size of each shadow Progressive ability Lossless reconstruction using all shadows
Lin and Lin8 3
Fang18 3 3
Thien and Lin4 3 3
Tso9 3 3
Chen and Lin19 3 3 3
Wang and Shyu20 3 3 3
Hung et al.17 3 3
the image with very high PSNR共45.1 dB兲 when collecting shadows of size 27,307 bytes. Although only the 39.35-dB image can be reconstructed herein with the collection of shadows of size 14,559 bytes, the image can be recon-structed without any loss when shadows of size 33,802 bytes 共which are smaller than those of size 43,691 bytes obtained by Tso’s nonprogressive lossless ap-proach兲 are collected.
Last, the size of each shadow in the proposed 关共t1, t2, t3, t4兲,n兴 threshold scheme is 兺i=14 兩ci兩/ti. Therefore, it is suggested that the readers set the largest threshold t4to n, in order to save space. However, if the readers want to have more freedom, they may use their own choice of a
threshold t4⬍n, at the price of wasting space for the shad-ows. When t4⬍n, a simulation is done in the following. Assume that n, the number of cover images, is at least 6. In general, the smallest threshold t1 cannot be 1 because the purpose of sharing is that no participant alone can be trusted. Hence, 兵1⬍t1= 2⬍t2= 3⬍t3= 4⬍t4= 5⬍n=6其 is used to generate n = 6 shadows, the size of which is then compared to those in the image-sharing schemes4,9,17,19,20 when the 共largest兲 threshold value is set to 5 for all these works. The comparison results are given in Table 4. It is observed that each shadow herein is still smaller than those Table 3 Comparison of shadow sizes in nonexpanded methods.4,9,17,19,20The共largest兲 threshold in all
works is set to 6. But Hung et al.’s method17 uses 5 as the largest threshold value because their method did not provide a version with a threshold value of 6.
Scheme
Each shadow size共bytes兲, and the quality of the reconstructed image
Each shadow size over 512⫻512 共given image size兲 Thien and Lin4
共nonprogressive兲 43,691; lossless 16.67%
Tso9
共nonprogressive兲 43,691 if losslessversion with 45.1-dB image quality共or 27,307 in lossy兲
16.67% 共10.42% in lossy version兲
Chen and Lin19 58,254; lossless 22.22%
Wang and Shyu20 131,072; lossless 50%
Hung et al.17 30,720 in lossy version with 37.04-dB image quality
11.72% in lossy version
Our scheme 33,802 if lossless共or 14,559 in lossy version with 39.35-dB image quality兲
12.89% 共5.55% in lossy version兲
Table 4 Same as Table3except that the共largest兲 threshold value in all works 共Refs.4,9,17,19, and
20兲 is set to 5 instead of 6.
Scheme
Each shadow size共bytes兲, and the quality of the reconstructed image
Each shadow size over 512⫻512 共given image size兲 Thien and Lin4
共nonprogressive兲 52,429; lossless 20%
Tso9
共nonprogressive兲 52,429 if losslesswith 45.1-dB image quality共or 32,768 in lossy version兲
20%
共12.5% in lossy version兲
Chen and Lin19 74,898; lossless 28.57%
Wang and Shyu20 131,072; lossless 50%
Hung et al.17 30,720 in lossy version with 37.04-dB
image quality
11.72% in lossy version
Our scheme 42,056 if lossless共or 18,964 in lossy version with 39.35-dB image quality兲
16.04% 共7.23% in lossy version兲
in the related works,4,9,19,20 and each shadow in our lossy approach is also smaller than that in Hung et al.’s lossy approach.17
5 Security Discussion
The code ci 共1艋i艋4兲 cannot easily be revealed if fewer than ti shadows are intercepted. To determine the coeffi-cients b0 to bti−1 in Eq. 共11兲, ti equations are required. If only ti− 1 equations are available关and without loss of gen-erality, suppose that g共1兲, g共2兲,..., and g共ti− 1兲 are inter-cepted兴, then the following ti− 1 equations can be con-structed:
冦
g共1兲 = 共b0+ b1+ . . . + bti−1兲 关over GF共256兲兴, g共2兲 = 共b0+ 2b1. . . + 2ti−1bti−1兲 关over GF共256兲兴, . . .
g共ti− 1兲 = 关b0+共ti− 1兲bi兴
关+ . . . +共ti− 1兲ti−1b
ti−1兴 关over GF共256兲兴
冧
.The preceding ti− 1 equations are solved for the tiunknown coefficients. The set of possible solutions has 256 members, so the probability of guessing the right solution is 1/256. Since兩ci兩/tipolynomials exist for the code ci, the probabil-ity of obtaining the right code ci is 共1/256兲兩ci兩/ti. For
ex-ample, for a low-quality JPEG code ci of size 5000 bytes, the number of sectors is 2500 if ti is 2. The probability of obtaining the correct JPEG code c1 is only 共1/256兲2500= 10−6020.
If the security of the shadows is to be increased, a seed may be used in a random number generator to generate a random sequence for each shadow, and then XOR opera-tions can be applied between the random sequence and the shadow. Each row of the XOR-encrypted shadows of the code cican then be circularly shifted by a certain number of bytes. Similar operations are applied to each column. This will transform the shadows to their safer versions. In this process, each seed that is used to generate a random se-quence is based on the creation time and shadow index. The seed can then be kept by all n participants or held by the company leader if the leader insists on attending the decoding meeting.
As stated in three works,23–25 attackers may slightly change the plaintext and then observe the change in the ciphertext. This is so-called differential attack, which the related progressive image-sharing methods18–20 cannot handle. The attackers may try to find a relationship between the plaintext and its ciphertext. Therefore, to ensure high security, the change in the ciphertext should cover a very large area if change occurs over a small area in the plain-text. To check this, the number of pixels change rate 共NPCR兲 is used to measure the number of pixels that are changed in the ciphertext when only one pixel is changed in the plaintext. To define NPCR, let X and Y be two cipher-texts of size W⫻H, where the plaincipher-texts of X and Y differ by only one pixel. Let X共i, j兲 and Y共i, j兲 be the pixel values at position共i, j兲 in X and Y, respectively. Define
NPCR =
兺
i,j D共i, j兲
W⫻ H ⫻ 100%, 共14兲
where D共i, j兲 is defined as D共i, j兲 =
再
0, X共i, j兲 = Y共i, j兲1, X共i, j兲 ⫽ Y共i, j兲
冎
. 共15兲The unified average changing intensity共UACI兲 is used to measure the average intensity of the differences between two ciphertexts X and Y. It is defined as
UACI = 1
W⫻ H
冋
兺
i,j兩X共i, j兲 − Y共i, j兲兩
255
册
⫻ 100%. 共16兲For random images, the expected values of NPCR and UACI are 99.609375% and 33.46354%, respectively, ac-cording to Kwok and Tang’s work,25 which is an image encryption method rather than an image-sharing method. The NPCR values herein are between 99.54% and 99.60% 共very close to 99.609375%兲, indicating that each XOR-enhanced shadow is very sensitive to a change in a single byte in the code c1; the UACI values are between 33.36% and 33.45%共very close to 33.46354%兲, suggesting that the change of each XOR-enhanced shadow that is associated with a single-byte change in the code c1is very large. Simi-lar observations are made when the code c1is replaced by the code c2, c3, or c4. Hence, the enhanced version can resist differential attack. Notably, to achieve this ability to resist differential attack, the design is based on simple XOR operations, unlike other designs.23–25 Also, this XOR-enhanced version does not increase the shadow size.
Last, since the shadows are hidden using Chen et al.’s JPEG hiding method,27 the security after hiding is discussed in the following. Possible attack due to visual inspection is avoided. As presented at the end of Sec. 4.2, in the 兵1⬍t1= 2⬍t2= 3⬍t3= 4⬍t4= 5其 and 兵1⬍t1= 3⬍t2= 4⬍t3= 5⬍t4= 6其 experiments, each shadow has size 兺i=14 兩ci兩/ti=共5750/2兲+共13,787/3兲+共45,972/4兲 +共115,458/5兲=42,056 bytes and 兺i=14 兩ci兩/ti=共5750/3兲 +共13,787/4兲+共45,972/5兲+共115,458/6兲=33,802 bytes, re-spectively. If other sets of兵1⬍t1⬍t2⬍t3⬍t4其 are used to generate n shadows, then each still has size smaller than 42,056 bytes. Hence, 42,056 bytes is the largest possible shadow size for all possible combinations of 兵1⬍t1⬍t2 ⬍t3⬍t4其. Now, each shadow of size 42,056 bytes can be hidden in a JPEG-Q65 code of a cover image after a JPEG compression with QF= 65. Figure 6 shows the six images decompressed from the six created JPEG stego codes.共The hidden secret is left untouched when the JPEG decompres-sion is performed.兲 Visual quality of these decompressed images is acceptable, reducing the probability that the codes get attacked when the attackers use visual inspection to find suspicious images.
Suspicious JPEG code length is avoided. Besides the evidence shown in Table1, Table5lists the bit rates of the JPEG-Q65 codes before and after hiding the largest shadow of size 42,056 bytes. The bit-rates of the JPEG codes cor-responding to QF= 95 are also listed to observe whether the bit rates of our JPEG stego codes are reasonable. From
Table 5, it is observed that even after hiding the largest shadow of size 42,056 bytes, the bit rate herein is still be-low that of the plain JPEG-Q95 code. Hence, the attackers will not be suspicious about the length of our stego codes. The proposed method can pass the Chi-square30 and StegSpy31analyses, which are tools to determine whether a secret is hidden in an image or a JPEG code. For the im-ages Peppers or Jet, Figs.7共a兲–7共d兲display the results after Guillermito’s Chi-square analysis tool is utilized to exam-ine each pixel of the JPEG images mentioned there. The red curve indicates the probability that pairs of values follow a random distribution, and the green one represents the aver-age value of all least significant bits共LSBs兲 in one block of pixels. The green curves in Figs.7共a兲–7共d兲 suggest to the attackers that there is nothing strange in these JPEG images because all four green curves are around 共0+1兲/2=0.5. Also, after a sort of latency, all red curves are flat at zero, indicating that the possibility of the existence of the hidden
secret is very low.共color online only.兲 A similar phenom-enon also holds when the image Peppers or Jet is replaced by other images Boat, Lake, etc. As for StegSpy analysis, let OriginalPeppers.jpg be the JPEG-Q65 code of the gray-scale image Peppers. A secret共either our shadow or a ran-domly secret of size 3000 bytes兲 is then embedded in Origi-nalPeppers.jpg by the Hiderman31and JPegX32hiding tools to generate two JPEG stego codes: Stego-of-Hiderman.jpg and Stego-of-JPegX.jpg. Let our six created JPEG stego codes, the decompressed images of which are in Fig.6, be ourstegoPeppers.jpg, ourstegoJet.jpg, etc. Figure8displays the results after the StegSpy analysis tool inspects these nine JPEG codes. The tool found that some data are hidden in Stego-of-Hiderman.jpg and Stego-of-JPegX.jpg, but not in our stego codes, and hence the hidden data will be ig-nored by the attackers.
6 Summary
This work proposes a 关共t1, t2, t3, t4兲,n兴 threshold progres-sive image-sharing scheme. The contributions of this work are as follows.
• The proposed scheme has progressive ability. 共Those in other works4,9 do not have progressive ability,
(a) (b) (c)
(d) (e) (f)
Fig. 6 Same as Fig.4except that the to-be-hidden shadows are generated by the proposed关共t1= 2, t2= 3, t3= 4, t4= 5兲, n=6兴
thresh-old scheme. PSNRs of 共a兲 to 共f兲 are 35.73, 35.57, 34.65, 33.14, 30.94, and 35.94 dB, respectively.
Table 5 Same as Table1except that the JPEG codes used in hiding are created using QF= 65.
Cover image bpp of the 共no-hiding兲 JPEG-Q65 code bpp after hiding a shadowof size 42,056 bytes in the JPEG-Q65 code bpp of the (no-hiding) JPEG-Q95 code Peppers 0.76 2.40 2.52 Jet 0.77 2.45 2.47 Boat 0.83 2.52 2.74 Lake 1.02 2.65 3.44 Baboon 1.57 3.16 4.09 Zelda 0.65 2.30 2.54 (a) (b) (c) (d)
Fig. 7 Results of Chi-square analysis:共a兲 is for the original JPEG cover image Peppers共no-hiding兲; 共b兲 is for our JPEG stego image Peppers in Fig.6共a兲;共c兲 is for the original JPEG cover image Jet 共no-hiding兲; and 共d兲 is for our JPEG stego image Jet in Fig.6共b兲.
whereas the progressive methods,17,19,20 which also use the sharing algorithm of Thien and Lin4 to gener-ate shadows, either have shadows that are larger than ours or cannot achieve lossless reconstruction.兲 Tradi-tional image sharing schemes are suitable for sharing top-secret images because of their all-or-nothing prop-erty. Progressive schemes provide a flexible means of viewing sensitive images progressively at certain qual-ity levels. As indicated in Table3, each of the shadows herein is smaller than those in the related works4,9,19,20 共and about half the size of that in Hung et al.,17
as determined by comparing their lossy approach with our lossy approach兲. This improvement reduces stor-age space and transmission time and facilitates the hiding of shadows. Therefore, the proposed method is better suited to transmit an image though limited com-munication channels.
• Easier to apply to scalable MPEG video transmission. The proposed scheme can also be adopted in the trans-mission of scalable MPEG video. Scalable MPEG video transmission depends on the adapting of video compression bit streams with a range of quality levels to meet various network environments or different end-user requirements. Since an MPEG video encoder also uses quality factors to control the quality of de-coded video, MPEG video codes of various quality levels can be generated with various quality factors, and then these MPEG codes can be shared. Hence, the proposed scheme conveniently provides a scalable video transmission system 共and still with much
smaller shadows than those of progressive schemes17–20; the reasoning for being much smaller is skipped in order to reduce paper length兲.
• Each of the shadows in the proposed scheme can resist differential attack, whereas the image-sharing methods4,9,19,20 in Table 3 cannot. Simple XOR and circular-shift operations are adopted to enhance the security of noiselike shadows, to enable them to resist differential attack.
• Unlike in several works,4,5,8,9,12,14,18–20 the to-be-shared data herein are the compression code rather than the raw file. The use of the compression code has at least the following two advantages:共1兲 since com-pression disturbs the correlation between adjacent pix-els of an image, the permutation process that is em-ployed elsewhere4,9,19,20 before the image is shared can be omitted; 共2兲 after inverse-sharing reconstruc-tion, the compression code requires less storage space than the raw file used in several works;4,5,8,9,12,14,18–20 yet the benefit of lossless reconstruction when most of the shadows are collected is retained.
• Arithmetic operations are performed over GF共28兲 which can be replaced by GF共2k兲 for any positive in-teger k, rather than GF共p兲, which is used in the image sharing schemes4,5,7,12,14,17,19 for a prime number p. This increases the convenience of sharing digital data, which are often in binary form, regardless of whether they are pixel values or bit streams.
Acknowledgments
The work was supported by National Science Council, Tai-wan, under NSC Grant No. 97-2221-E-009-120-MY3. The authors thank the reviewers for valuable suggestions. References
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Lee Shu-Teng Chen received his BS de-gree in computer science from National Chiao Tung University共NCTU兲, Taiwan, in 1999, and his MS degree in computer sci-ence and information engineering from Na-tional Taiwan University, Taiwan, in 2001. He has been in the PhD program since 2004 and is currently a PhD candidate in the Department of Computer Science and Information Engineering at NCTU. His cur-rent research interests include data hiding and image sharing.
Ja-Chen Lin received his BS degree and MS degree from NCTU, Taiwan. In 1988, he received his PhD degree in mathemat-ics from Purdue University, West Lafayette, Indiana. He joined the Department of Com-puter and Information Science at NCTU and became a professor there. His re-search interests include pattern recognition and image processing. Dr. Lin is a member of the Phi-Tau-Phi Scholastic Honor Soci-ety.
