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Aspect ratio invariant visual secret sharing schemes with minimum pixel expansion
Pattern Recognition Letters 26 (2005) 193–206 www.elsevier.com/locate/patrec
Aspect ratio invariant visual secret sharing schemes with minimum pixel expansion Ching-Nung Yang *, Tse-Shih Chen Department of Computer Science and Information Engineering, National Dong Hwa University, #1, Da Hsueh Rd, Sec. 2, Shou-Feng, Hualien, Taiwan, ROC Received 19 July 2004
Abstract Visual secret sharing (VSS) scheme is a perfect secure method to hide a secret image by breaking it into shadow images and one can decode it easily by the human visual system. We use m, the pixel expansion, subpixels to represent a pixel. Suppose the secret image is a circle and m is not a square value, i.e., the aspect ratio is changed. After performing the VSS technique, the circle will be changed to an ellipse and consequently lead to the loss of information. To avoid distorting the image, the dummy subpixels are added to keep the aspect ratio unchanged. In this paper, we propose a novel method to dramatically reduce the number of extra subpixels to construct the aspect ratio invariant VSS schemes. 2004 Elsevier B.V. All rights reserved. Keywords: Visual secret sharing scheme; Visual cryptography; Secret sharing scheme; Aspect ratio
1. Introduction VSS scheme is a perfect secure method to share the secret (Naor and Shamir, 1995; Droste, 1996; Ateniese et al., 1996; Verheul and Van Tilborg, 1997; Eisen and Stinson, 2002; Tzeng and Hu, 2002; Chang and Chuang, 2002; Hofmeister et al., 2000). The decoder is human visual system and we can easily recover the secret by using the *
Corresponding author. Tel.: +886 3 8634025/8662500; fax: +886 3 8634010/8662781. E-mail address: [email protected] (C.-N. Yang).
eyes of human beings without the help of any computing devices. For a (k, n) VSS scheme, the secret image is divided into n different shadows and each pixel is represented as m subpixels. We can recover the secret by stacking any k (k 6 n) shadows, but k 1 or fewer shadows will get no information. All the VSS schemes, except the image size invariant VSS schemes (Yang, 2004; Ito et al., 1999), will have pixel expansion. For example, the values of m for (2, 2), (2, n), (3, n) and optimal (k, k) Naor–Shamir VSS schemes are 2, n, 2n 2, and 2k1 respectively. For the (2, 2) Naor–Shamir VSS schemes, two subpixels (1B1W) are used to represent one
0167-8655/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2004.08.025
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pixel in the original secret image and 2B (resp. 1B1W) is black (resp. white) color in the stacked result. However it will change the aspect ratio from aL:aW to 2aL:aW and distort the image. So, we often use four subpixels 2B2W to represent a pixel and 4B (resp. 2B2W) is black (resp. white) color in the recovered image. The pixel expansion m is now increased to 4. In this paper, we propose a new aspect ratio invariant VSS scheme to reduce the pixel expansion by transforming every 2 · 2 pixels (four pixels in a square) to m 0 · m 0 subpixels. For making the pixel expansion as small as possible, we need to find the minimum m 0 such that every pixel is represented as m subpixels and others are extra dummy subpixels to keep the aspect ratio, i.e., (m 0 )2 = 4 · m + (the number of extra subpixels). How to properly put the corresponding subpixels for each pixel in an m 0 · m 0 square block to hold the clearness of the recovered image is the major issue in the paper. This paper is organized as follows. In Section 2, we introduce the conventional VSS scheme. In Section 3, the new aspect ratio invariant VSS schemes are proposed and the compared results are also given. Experimental results are shown in Section 4. Section 5 concludes the paper.
2. The basic VSS scheme and motivation 2.1. The basic VSS scheme A formal definition of the basic VSS scheme is given in (Naor and Shamir, 1995). For generalization, we herein use the VSS scheme with a general access structure (CQual; CForb) to describe the concept of VSS scheme. Each pixel is divided into m black and white subpixels in n shadows (printed on transparencies) given to n participants in a set P = {1, 2, . . ., n}. A certain subset X 2 CQual can get the shared secret by stacking their shadows and X 2 CForb will not get any information of the secret image. Note that X P, CQual 2P and CForb 2P, where CQual \ CForb = /. A (k, n) VSS scheme is with the forbidden set CForb including all the sets of (k 1) or fewer participants and the qualified sets CQual = 2P CForb. The VSS
scheme can be described by n · m Boolean matrix S = [sij], where sij = 1 if and only if the jth subpixel in the ith shadow is black, otherwise sij = 0. When shadows i1, i2, . . ., ir in a set X 2 CQual are stacked, we see a recovered secret whose black subpixels are represented by the Boolean ‘‘or’’ of rows i1, i2, . . . ir in S. The gray level of this recovered image is proportional to the Hamming weight of the ‘‘or’’ed m-vector V. For the fixed threshold 1 5 d 5 m and relative difference a > 0, if H(V) = d, this gray level is interpreted by the users visual system as black, and if H(V) 5 d am, the result is interpreted as white. Definition. A VSS scheme with a general access structure (CQual; CForb) can be shown as two collections of n · m Boolean function matrices B0 and B1. When sharing a white (resp. black) pixel, the dealer randomly chooses one row of the Boolean matrix B0 (resp. B1) to a relative shadow. The chosen matrix defines the gray level of the m subpixels in every one of the n shadows. A VSS scheme is considered valid if the following conditions are met: 1. (Contrast condition) For any S in B0 (resp. B1), the ‘‘or’’ed V of rows i1, i2, . . ., ir in a set X = {i1, i2, . . ., ir} 2 CQual satisfies H(V) 5 d am (resp. H(V) = d). 2. (Security condition) For any subset X = {i1, i2, . . ., ir} 2 CForb, the two collections of r · m matrices obtained by restricting each n · m matrices in Bi, i 2 {0, 1}, to rows i1, i2, . . ., ir are not visual in the sense that they contain the same matrices with the same frequencies.
Example 1. For a VSS scheme with P = {1, 2}, CQual = {{1, 2}}, and CForb = {{1}, {2}}, i.e., (2, 2) VSS scheme, the basis matrices B0 and B1 can be 10 10 designed as B0 ¼ , B1 ¼ . From Fig. 10 01 1(a) and (b), it is observed that the aspect ratios of the original image and the recovered image are not same. To keep the aspect ratio unchanged, we may use m = 4 and design the basis matrices as
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Fig. 1. (2, 2): The recovered images for the (2, 2) VSS schemes with m = 2 and 4: (a) The original secret; (b) The aspect ratio is changed (m = 2); (c) The aspect ratio is unchanged (m = 4).
1100 1100 the following: B0 ¼ , B1 ¼ , or 1100 0011 1000 1000 B0 ¼ , B1 ¼ (all-zero columns are 1000 0100 1011 1011 added), or B0 ¼ , B1 ¼ (all-one 1011 0111 columns are added). Fig. 1(c) shows that the recovered image has the same aspect ratio by using the 1011 1011 basis matrices B0 ¼ and B1 ¼ . 1011 0111
2.2. Motivation Initial motivation for keeping the aspect ratio unchanged is to avoid the distortion of recovered image. For example, if the shape of secret image is our information and the original image is a circle (Fig. 2(a)). The recovered image for m = 2 (Fig. 2(b)) is an ellipse and the distortion of image will compromise our secret and create a new motivation for designing the aspect ratio invariant VSS schemes. The existing method is to add minimum extra subpixels into the black and white matrices
Fig. 2. The distortion due to the variant aspect ratio: (a) The original secret; (b) (2, 2) VSS scheme with m = 2; (c) (2, 2) VSS scheme with m = 4.
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B0 and B1 to let the number of columns, m, be a square value and do not violate the contrast and security conditions. It is obvious pffiffiffiffi 2 that the minimum number of extra bits is d me m, where dme is the smallest integer no less than m. For example, the pffiffiffi minimum number of extra bits is d 2e2 2 ¼ 2 for a (2, 2) VSS scheme. The recovered image by adding two extra black subpixels (Fig. 2(c)) is still a circle. But how to further reduce the added subpixels and simultaneously hold the clearness of recovered image is an interesting problem. The work presented in this paper addresses this problem.
3. Design of the VSS schemes with aspect ratio invariant Before describing our construction method, we describe a trivial method by adding extra subpixels directly into the basis matrices. First give the following definition. The trivial method is induced from Naor and Shamir (1995) (the authors recommended to add the extra subpixels for avoiding the distortion of the aspect ratio of the original image) and the observations that adding the column vectors consisting n 1s or 0s in the matrices of the original VSS scheme also forms a VSS scheme (Eisen and Stinson, 2002). The formal description is shown in the following definition and theorem. Definition 1. Let B0 and B1 be n · m white and black matrices for the basic (k, n) VSS scheme. Let 1 and 0 be all-one and all-zero n · 1 column matrices, respectively. Let C0 and C1 be the matrices obtained by concatenating the matrices as the following where is a concatenation operation: C 0 ¼ B0 1 1 0 0; and |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffi 2 x d me mx ð1Þ C 1 ¼ B1 1 1 0 0 : |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffi 2 x d me mx Theorem 1. The matrices C0 and C1, obtained in (1), are the white and black matrices for a (k, n) VSS scheme with the invariant paspect ffiffiffiffi 2 ratio and the minimum pixel expansion is d me .
Proof. pffiffiffiffi 2 It is obvious that concatenating the same d me m n · 1 column matrices to B0 and B1 will not violate the contrast and security conditions. The number of columns of Ci, the pffiffiffi ffi 2 pffiffiffipixel ffi 2 expansion, is m þ x þ p d ffiffiffiffim e m x ¼ d m e , i.e., the square of d m e. Therefore, the aspect ratio is unchanged. Because dae ispthe ffiffiffiffi smallest integer no less than a, the value d m e2 is the minimum square value greater than or equal to m. h 3.1. Construction method for an aspect ratio invariant VSS scheme with the original expansion m=2 We now describe the construction of an aspect ratio invariant (2, 2) VSS scheme with pixel expansion m = 2 such that we can easily understand the basic concept of our proposed construction. Sup 10 pose the basis matrices are B0 ¼ and 10 10 B1 ¼ . To keep the aspect ratio unchanged, 01 we process a four-pixeled square block each time, as shown in Fig. 3(a), by using B0 and B1 to get 10101010 10101010 and B01 ¼ . Then B00 ¼ 10101010 01010101 add one black (or white) subpixels to let the total number of subpixels be 9 (=32). Without loss of generality, use the black subpixel as the dummy subpixel, i.e., the new white and black matrices are 2 3 10 10 10 10 1 D0 ¼ B00 1 ¼ 4 10 10 |{z} 10 |{z} 10 |{z} 1 5 |{z} |{z} R1
R2
R3
R4
10 01 |{z}
10 01 |{z}
10 01 |{z}
d0
and 2
10 D1 ¼ B01 1 ¼ 4 01 |{z} R1
R2
R3
R4
3 1 1 5; |{z} d0
where the two subpixels in R1 region are used to represent the pixel 1 . Regions R2 , R3 and R4 are for the pixels 2 , 3 and 4 , respectively. The d0 black subpixel is used as the dummy subpixel, not for contrast and security, but for the aspect
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Fig. 3. Nine-subpixeled pattern for the (2, 2) VSS scheme with the aspect ratio invariant: (a) four-pixeled square block; (b) regular nine-subpixeled pattern; (c) four partially regular nine-subpixeled patterns.
ratio. The nine subpixels are then put as the pattern shown in Fig. 3(b) such that Ri and i have the almost same geographic distribution, position and direction. Although the shadows will have some subpixels ( d0 ) at regular position, it will not compromise the security; however the shadows are not completely random and may cause some suspicions. But when randomly permute the columns of D0 and D1, the recovered image will lose the clearness at the edge of the secret image. To let the shadows be more random and still hold the clearness of the recovered image, we randomly use the four partially regular nine-subpixeled patterns shown in Fig. 3(c), i.e., the d0 black subpixel is included in Ri region, i = 1, 2, 3, 4, with the same probability. These four patterns of arranged subpixels have the almost same correspondence between the pixel i and the subpixels in Ri . Fig. 4 shows the recovered images by using regular (Fig. 3(b)), partially regular (Fig. 3(c)), and totally random (randomly permute D0 and D1) nine-subpixeled patterns. We use the crossed lines as the original secret image (Fig. 4(a)) to test the performance of these three patterns. It is observed that the partially regular pattern (Fig. 4(c)) has the advantages over other two patterns (Fig. 4(b) and (d)), the more random shadow and the clearness of the recovered image. The pixel expansion is 9 ¼ 2:25 for our aspect ratio invariant (2, 2) VSS 4 scheme.
The formal construction for our aspect ratio invariant (2, 2) VSS scheme with the original pixel expansion m = 2 is described in Algorithm 1 (using regular nine-subpixeled pattern), Algorithm 2 (using partially regular nine-subpixeled pattern) and Algorithm 3 (using totally random nine-subpixeled pattern). Algorithm 1. (Processing every four (2 · 2) pixels in the original image) Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S 1 and S 2. Step 1: For i = 1 to 4 do Break the pixel i in B into two subpixel by using the matrices B0 and B1 and put them in region Ri ; Step 2: Put R1 , R2 , R3 , R4 and a dummy black subpixel d0 according the pattern in Fig. 3(b) to create two square blocks B1 and B2 . Step 3: Deliver the two blocks and S 2.
B1
and
B2
to S 1
Algorithm 2. (Processing every four (2 · 2) pixels in the original image)
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Fig. 4. The recovered image for the aspect ratio invariant VSS scheme by using different patterns of nine arranged subpixels: (a)–(c) partially regular pattern; (d) totally random pattern.
Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S 1 and S 2. Step 1: Randomly choose j from {1, 2, 3, 4}. Step 2: Let
Rj
=
Rj
[
d0
.
Step 3: For i = 1 to 4 do If (i = j) then Break the pixel i in B into three subpixel by using the matrices B0 1, B1 1 and then put them in region Ri ; else Break the pixel i in B into two subpixel by using B0 and B1 and put them in region Ri ; Step 4: Put
R1
,
R2
,
R3
and
R4
according the jth
pattern in Fig. 3(c) to create two square blocks B1 and B2 .
Step 5: Deliver the two blocks and S 2.
B1
and
B2
to S 1
Algorithm 3. (Processing every four (2 · 2) pixels in the original image) Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S1 and S2. Step 1: For i = 1 to 4 do Break the pixel i in B into two subpixel by using B0 and B1; Step 2: Randomly put the above eight subpixels and a dummy black subpixel in a 3 · 3 square block to create two square blocks B1 and B2 . Step 3: Deliver the two blocks B1 and B2 to S1 and S2.
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Our proposed construction uses the same basis matrices of conventional VSS scheme and adds the minimum extra black subpixels to keep the aspect ratio unchanged. Using the same approach, we can extend the construction method from m = 2 to m > 2.
199
pffiffiffiffiffiffi Theorem 3. The pixel expansion, m2 (=d 4me2 =4), for an aspect ratio invariant (k, n) VSS scheme in Definition pffiffiffiffi 2 is no larger than the pixel expansion, m1 (=d me2 ), for an aspect ratio invariant (k, n) VSS scheme in Definition 1. Proof
3.2. Construction method for an aspect ratio invariant VSS scheme with the original expansion m>2 3.2.1. Construction method The following definition and theorem shows how to construct an aspect ratio invariant VSS scheme with original expansion m > 2. Definition 2. Let B0 and B1 be n · m white and black matrices for the basic (k, n) VSS scheme. Let 1 be all-one n · 1 column matrix. Let D0 and D1 be the matrices obtained by concatenating the matrices as the following: D 0 ¼ B0 B0 B0 B0 1 1 ; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffiffi 2 d 4me 4m
D 1 ¼ B1 B1 B1 B1 1 1 : |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffiffi 2
and ð2Þ
d 4me 4m
Theorem 2. The matrices D0 and D1 obtained in (2), are the white and black matrices for a (k, n) VSS scheme with the invariant paspect ratio and ffiffiffiffi the minimum pixel expansion is d me2 =4. Proof. It is obvious that B0 B0 B0 B0 and B1 B1 B1 B1 are the basis matrices of (k, n) VSS scheme with pixel expansion 4m. Concatenate pffiffiffiffiffiffi 2 the same d 4me 4m n · 1 1 column matrices to B0 B0 B0 B0 and B1 B1 B1 B1 will not violate the contrast and security conditions. pffiffiffiffiffiffiThe numberpffiffiffiffiffiffi of columns of Di is 4m þ d 4me2 2 4m ¼ d 4me . Therefore, the aspect ratio is unchanged. Due to the definition of d Æ e, the value pffiffiffiffiffiffi 2 d 4me is a minimum square value greater than or equal pffiffiffiffi 2to 4m. So the minimum pixel expansion is d me =4. h The following theorem shows that our proposed method has the better pixel expansion than the method defined in Definition 1.
pffiffiffiffiffiffi 2 pffiffiffiffi 2 m2 ¼ d 4me =4 ¼ d2 me =4: pffiffiffiffi pffiffiffiffi Since d2 me 6 2d með* da be 6 dae dbeÞ; pffiffiffiffi 2 pffiffiffiffi 2 so m2 ¼ d2 me =4 6 22 d me =4 pffiffiffiffi 2 ¼ d me ¼ m1 : From Theorem 3, our proposed method will have the less pixel expansion. Table 1 shows the compared results for the original expansion m 6 20. The upper row is the method in Definition 1 (denoted as Method 1) and the lower row is our proposed method (denoted as Method 2) for each m. The parameters in Table 1 are described as follows: the value m is the original pixel expansion of (k, n) VSS scheme. Notation ns represents the number of subpixels used for a square block (one pixel (Method 1) and four pixels (Method 2)), then ns = m or 4m. The minimum number of the extra subpixels ne is added to make nt = ns + ne square, where nt is the total number of all subpixels. The pixel expansions for the aspect ratio invariant VSS schemes are m1, m2 for Method 1 and Method 2, respectively. We define an Improvement Factor, IF ¼ mm12 , in the last column. From Table 1, it is shown that our method always has the optimum pixel expansion. We have improvements for m = 5, 6, 10, 11, 12, 17, 18, 19, 20. For m = 5 and 6, 25-subpixeled square block is used. For m = 10, 11 and 12, 49-subpixeled square block is used. For m = 17, 18, 19 and 20, 81-subpixeled square block is used. Let us proceed to the next chief consideration of our pffiffiffiffiffifficonstruction method. How to arrange the d 4me2 subpixels in a square block to maintain the two characteristics mentioned in the above section, the random shadow and the clearness of the recovered image.
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Table 1 Pixel expansions m1 and m2 for the aspect ratio invariant VSS scheme m 6 20 Pixel expansion m
Number of subpixels used for a square block nsa
Minimum number of added extra subpixels ne to make ns + ne squareb
Total number of subpixels nt = ns + nec
Pixel expansion m1 (Method 1) and m2 (Method 2)d
2
2 8
2 1
4 9
4 9/4
3
3 12
1 4
4 16
4 4
4
–
5
5 20
6
–
Improvement factor IFe 1.78 1
–
–
–
4 5
9 25
9 25/4
1.44
6 24
3 1
9 25
9 25/4
1.44
7
7 28
2 8
9 36
9 9
1
8
8 32
1 4
9 36
9 9
1
9
–
10
10 40
11
–
–
–
6 9
16 49
16 49/4
1.31
11 44
5 5
16 49
16 49/4
1.31
12
12 48
4 1
16 49
16 49/4
1.31
13
13 52
3 12
16 64
16 16
1
14
14 56
2 8
16 64
16 16
1
15
15 60
1 4
16 64
16 16
1
16
–
–
–
–
–
17
17 68
8 13
25 81
25 81/4
1.23
18
18 72
7 9
25 81
25 81/4
1.23
19
19 76
6 5
25 81
25 81/4
1.23
20
20 80
5 1
25 81
25 81/4
1.23
a b c d e
–
ns = m (Method 1) and 4m (Method 2). pffiffiffiffiffiffi pffiffiffiffi ne ¼ d me2 m (Method 1) and d 4me2 4m (Method 2). pffiffiffiffiffiffi pffiffiffiffi nt ¼ d me2 (Method 1) and d 4me2 (Method 2). pffiffiffiffiffiffi 2 pffiffiffiffi 2 m1 ¼ d me and m2 ¼ d 4me =4. pffiffiffiffi m1 4d me2 IF ¼ ¼ pffiffiffiffiffiffi 2 . m2 d 4me
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pffiffiffiffiffiffi 2 3.2.2. d 4me -subpixeled patterns for the (k, n) VSS scheme with expansion m Three patterns of arranged subpixels are proposed in this section. The first is regular pattern; the second is partially regular pattern; the third is totally random pattern. A. Regular Pattern (Pa): Put the dummy black subpixels d i , i = 0, . . ., 4, as a cross in the center of the pattern. B. Partially Regular Pattern (Pb): Put the dummy black subpixels d i i = 1, . . ., 4, into the region Ri and then randomly put d0 into Ri for processing the square block each time. C. Totally Random Pattern (Pc): Randomly permute the columns of D0 and D1. Patterns Pa are designed as follows: 25-subpixeled square pattern (m = 5 and 6), 49-subpixeled square pattern (m = 10, 11 and 12) and 81-subpixeled square pattern (m = 17, 18, 19 and 20) are shown in Figs. 5–7, respectively. Four Pb patterns for 25-subpixeled square block (m = 5 and 6) are shown in the following figure. The pattern in Fig. 8(a) is R1 = R1 [ d1 [ d0 and Ri = Ri [ d i , i = 2, 3, and 4; the pattern in Fig. 8(b) is R2 = R2 [ d2 [ d0 and Ri = Ri [ d i , i = 1, 3, and 4; the pattern in Fig. 8(c) is R3 = R3 [ d3 [ d0 and Ri = Ri [ d i , i = 1, 2, and 4; the pattern in Fig. 8(d) is R4 = R4 [ d4 [ d0 and Ri = Ri [ d i , i = 1, 2, and 3. Other Pb patterns for different m can be designed by using the same approach.
Fig. 5. Pa pattern for m = 5 and 6: (a) m = 5, ns = 20, ne = 5, nl = 25; (b) m = 6, ns = 24, ne = 1, nt = 25.
4. Experimental results In this section, we present two experimental results to illustrate the performance of our construction by using three patterns of arranged subpixels: Pa, Pb and Pc. The first is (2, 2) VSS scheme with m = 2, m1 = 4 and m2 ¼ 94; the second is Naor–Shamir (2, 5) VSS scheme with m = 5, m1 = 9 and m2 254. Example 2. For a (2, 2) VSS scheme with the basis 10 10 matrices B0 ¼ and B1 ¼ . From Eq. 10 01 (2), the white and black matrices D0 and D1 of our proposed scheme are D0 ¼ B0 B0 B0 B0 1 2 10 10 10 10 ¼ 4 10 1 0 1 0 1 0 |{z} |{z} |{z} |{z} R1
R2
R3
R4
1
3
1 5; |{z}
and
d0
Fig. 6. Pa pattern for m = 10, 11 and 12: (a) m = 10, ns = 40, ne = 9, nt = 49; (b) m = 11, ns = 44, ne = 5, nt = 49; (c) m = 12, ns = 48, ne = 1, nt = 49.
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Fig. 7. Pa pattern for m = 17, 18, 19 and 20: (a) m = 17, ns = 68, ne = 13, nt = 81; (b) m = 18, ns = 72, ne = 9, nt = 81; (c) m = 19, ns = 76, ne = 5, nt = 81; (d) m = 20, ns = 80, ne = 1, nt = 81.
Fig. 8. Pb patterns for m = 5 and 6: (a)
D 1 ¼ B1 B1 B1 B1 1 2 10 10 10 10 4 ¼ 01 |{z} 01 |{z} 01 |{z} 01 |{z} R1
R2
R3
R4
1
3
1 5: |{z} d0
d0
in
R1
; (b)
d0
in
R2
; (c)
d0
in
R3
; (d)
d0
in
R4
.
Suppose a test secret image is a printed text VSS and LENA image. After performing three patterns Pa, Pb and Pc, the revealed secret images are shown in Fig. 9. It is observed that using Pa (Fig. 9(a)) has the best clearness of
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203
Fig. 9. The aspect ratio invariant (2, 2) VSS scheme: (a) Pa pattern is used; (b) Pb pattern is used; (c) Pc pattern is used.
recovered image; however the shadow is not completely random but a little regular. One can see that from the white areas of the recovered image VSS . When Pb pattern is used, the clearness of recovered image shown in Fig. 9(b) is almost the same as Fig. 9(a) and the shadow has more randomness. When using Pc pattern, the clearness of recovered image is compromised.
Example 3. For the Naor–Shamir (2, 5) VSS scheme with the basis matrices 2
10000
3
7 6 6 10000 7 7 6 7 B0 ¼ 6 6 10000 7 7 6 4 10000 5 10000
2 and
10000
3
7 6 6 01000 7 7 6 7 B1 ¼ 6 6 00100 7: 7 6 4 00010 5 00001
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Fig. 10. The aspect ratio invariant (2, 5) VSS schemes: (a) the original secret; (b) the proposed method with pattern Pa is used; (c) the proposed method with pattern Pb is used; (d) the proposed method with pattern Pc is used; (e) construction method defined in Definition 1.
From Eq. (2), the white and black matrices D0 and D1 of our proposed scheme are D 0 ¼ B0 B0 B0 B0 1 1 1 1 1 3 2 10000 10000 10000 10000 11111 7 6 6 10000 10000 10000 10000 11111 7 7 6 6 10000 10000 10000 10000 11111 7 7 6 ¼6 7; 6 10000 10000 10000 10000 11111 7 7 6 7 6 4 10000 10000 10000 10000 11111 5 |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} R1
R2
R3
R4
d0
and
Fig. 10(a) is the test secret image, a thin printed text VSS . Fig. 10(b)–(d) show the revealed images and their corresponding shadows by using Pa, Pb and Pc patterns, respectively, and the expansion is 254 ¼ 6:25. Fig. 10(e) is the result by using the method defined in Definition 1 with the basis matrices 2
100001111
7 6 6 100001111 7 7 6 7 6 7; C0 ¼ 6 100001111 7 6 7 6 6 100001111 7 5 4 100001111
D 1 ¼ B1 B1 B1 B1 1 1 1 1 1 3 2 10000 10000 10000 10000 11111 7 6 6 01000 01000 01000 01000 11111 7 7 6 6 00100 00100 00100 00100 11111 7 7 6 ¼6 7: 6 00010 00010 00010 00010 11111 7 7 6 7 6 4 00001 00001 00001 00001 11111 5 |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} R1
R2
R3
R4
d0
3
2
100001111
3
7 6 6 010001111 7 7 6 7 6 7 C1 ¼ 6 001001111 7 6 7 6 6 000101111 7 5 4 000011111
(from Eq. (1)) and the pixel expansion is 9. The shadow pattern in Fig. 10(b) is very regular and the revealed image has the good quality. Although the shadow is completely random in Fig. 10(d), the clearness of recovered image is poor. It is obvious that Fig. 10(c) is the proper choice, since the shadow in Fig. 10(c) is random like Fig. 10(d) and the contrast of revealed image
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205
Fig. 10 (continued)
is good enough to ‘‘see’’ the secret. Fig. 10(e) shows that the shadow has large pixel expansion and the contrast of revealed image is similar to Fig. 10(c) and worse than Fig. 10(b).
5. Conclusion In this paper we propose the aspect ratio invariant VSS scheme for avoiding the distor-
tion of recovered image. We show how to reppffiffiffiffiffiffi 2 resent four pixels in a square block by d 4me subpixels in a square block such that the aspect ratio is unchanged. Three patterns of arranged subpixels, Pa, Pb and Pc are proposed. We can choose one of them to implement the aspect ratio invariant VSS scheme to achieve our demand. From the experimental results, it is observed that pattern Pb is a reasonable choice.
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References Ateniese, G., Blundo, C., De Santis, A., Stinson, D.R., 1996. Visual cryptography for general access structures. ECCC, Electronic Colloquium on Computational Complexity (TR96-012), via WWW using
cryptography. Theoretical Computer Science 240 (2), 471– 485. Ito, R., Kuwakado, H., Tanaka, H., 1999. Image size invariant visual cryptography. IEICE Trans. Fundamentals E 82-A (10), 481–494. Naor, M., Shamir, A., 1995. Visual cryptography. Advances in Cryptology—EUROCRYPT94, Lecture Notes in Computer Science, vol. 950, pp. 1–12. Verheul, E.R., Van Tilborg, H.C.A., 1997. Constructions and properties of k out of it n visual secret sharing schemes. Designs, Codes and Cryptography 11 (2), 179– 196. Tzeng, W.G., Hu, C.M., 2002. A New approach for visual cryptography. Designs, Codes and Cryptography 27 (3), 207–227. Yang, C.N., 2004. New visual secret sharing schemes using probabilistic method. Pattern Recognition Letters 25 (4), 481–494.
Aspect ratio invariant visual secret sharing schemes with minimum pixel expansion Ching-Nung Yang *, Tse-Shih Chen Department of Computer Science and Information Engineering, National Dong Hwa University, #1, Da Hsueh Rd, Sec. 2, Shou-Feng, Hualien, Taiwan, ROC Received 19 July 2004
Abstract Visual secret sharing (VSS) scheme is a perfect secure method to hide a secret image by breaking it into shadow images and one can decode it easily by the human visual system. We use m, the pixel expansion, subpixels to represent a pixel. Suppose the secret image is a circle and m is not a square value, i.e., the aspect ratio is changed. After performing the VSS technique, the circle will be changed to an ellipse and consequently lead to the loss of information. To avoid distorting the image, the dummy subpixels are added to keep the aspect ratio unchanged. In this paper, we propose a novel method to dramatically reduce the number of extra subpixels to construct the aspect ratio invariant VSS schemes. 2004 Elsevier B.V. All rights reserved. Keywords: Visual secret sharing scheme; Visual cryptography; Secret sharing scheme; Aspect ratio
1. Introduction VSS scheme is a perfect secure method to share the secret (Naor and Shamir, 1995; Droste, 1996; Ateniese et al., 1996; Verheul and Van Tilborg, 1997; Eisen and Stinson, 2002; Tzeng and Hu, 2002; Chang and Chuang, 2002; Hofmeister et al., 2000). The decoder is human visual system and we can easily recover the secret by using the *
Corresponding author. Tel.: +886 3 8634025/8662500; fax: +886 3 8634010/8662781. E-mail address: [email protected] (C.-N. Yang).
eyes of human beings without the help of any computing devices. For a (k, n) VSS scheme, the secret image is divided into n different shadows and each pixel is represented as m subpixels. We can recover the secret by stacking any k (k 6 n) shadows, but k 1 or fewer shadows will get no information. All the VSS schemes, except the image size invariant VSS schemes (Yang, 2004; Ito et al., 1999), will have pixel expansion. For example, the values of m for (2, 2), (2, n), (3, n) and optimal (k, k) Naor–Shamir VSS schemes are 2, n, 2n 2, and 2k1 respectively. For the (2, 2) Naor–Shamir VSS schemes, two subpixels (1B1W) are used to represent one
0167-8655/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.patrec.2004.08.025
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pixel in the original secret image and 2B (resp. 1B1W) is black (resp. white) color in the stacked result. However it will change the aspect ratio from aL:aW to 2aL:aW and distort the image. So, we often use four subpixels 2B2W to represent a pixel and 4B (resp. 2B2W) is black (resp. white) color in the recovered image. The pixel expansion m is now increased to 4. In this paper, we propose a new aspect ratio invariant VSS scheme to reduce the pixel expansion by transforming every 2 · 2 pixels (four pixels in a square) to m 0 · m 0 subpixels. For making the pixel expansion as small as possible, we need to find the minimum m 0 such that every pixel is represented as m subpixels and others are extra dummy subpixels to keep the aspect ratio, i.e., (m 0 )2 = 4 · m + (the number of extra subpixels). How to properly put the corresponding subpixels for each pixel in an m 0 · m 0 square block to hold the clearness of the recovered image is the major issue in the paper. This paper is organized as follows. In Section 2, we introduce the conventional VSS scheme. In Section 3, the new aspect ratio invariant VSS schemes are proposed and the compared results are also given. Experimental results are shown in Section 4. Section 5 concludes the paper.
2. The basic VSS scheme and motivation 2.1. The basic VSS scheme A formal definition of the basic VSS scheme is given in (Naor and Shamir, 1995). For generalization, we herein use the VSS scheme with a general access structure (CQual; CForb) to describe the concept of VSS scheme. Each pixel is divided into m black and white subpixels in n shadows (printed on transparencies) given to n participants in a set P = {1, 2, . . ., n}. A certain subset X 2 CQual can get the shared secret by stacking their shadows and X 2 CForb will not get any information of the secret image. Note that X P, CQual 2P and CForb 2P, where CQual \ CForb = /. A (k, n) VSS scheme is with the forbidden set CForb including all the sets of (k 1) or fewer participants and the qualified sets CQual = 2P CForb. The VSS
scheme can be described by n · m Boolean matrix S = [sij], where sij = 1 if and only if the jth subpixel in the ith shadow is black, otherwise sij = 0. When shadows i1, i2, . . ., ir in a set X 2 CQual are stacked, we see a recovered secret whose black subpixels are represented by the Boolean ‘‘or’’ of rows i1, i2, . . . ir in S. The gray level of this recovered image is proportional to the Hamming weight of the ‘‘or’’ed m-vector V. For the fixed threshold 1 5 d 5 m and relative difference a > 0, if H(V) = d, this gray level is interpreted by the users visual system as black, and if H(V) 5 d am, the result is interpreted as white. Definition. A VSS scheme with a general access structure (CQual; CForb) can be shown as two collections of n · m Boolean function matrices B0 and B1. When sharing a white (resp. black) pixel, the dealer randomly chooses one row of the Boolean matrix B0 (resp. B1) to a relative shadow. The chosen matrix defines the gray level of the m subpixels in every one of the n shadows. A VSS scheme is considered valid if the following conditions are met: 1. (Contrast condition) For any S in B0 (resp. B1), the ‘‘or’’ed V of rows i1, i2, . . ., ir in a set X = {i1, i2, . . ., ir} 2 CQual satisfies H(V) 5 d am (resp. H(V) = d). 2. (Security condition) For any subset X = {i1, i2, . . ., ir} 2 CForb, the two collections of r · m matrices obtained by restricting each n · m matrices in Bi, i 2 {0, 1}, to rows i1, i2, . . ., ir are not visual in the sense that they contain the same matrices with the same frequencies.
Example 1. For a VSS scheme with P = {1, 2}, CQual = {{1, 2}}, and CForb = {{1}, {2}}, i.e., (2, 2) VSS scheme, the basis matrices B0 and B1 can be 10 10 designed as B0 ¼ , B1 ¼ . From Fig. 10 01 1(a) and (b), it is observed that the aspect ratios of the original image and the recovered image are not same. To keep the aspect ratio unchanged, we may use m = 4 and design the basis matrices as
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195
Fig. 1. (2, 2): The recovered images for the (2, 2) VSS schemes with m = 2 and 4: (a) The original secret; (b) The aspect ratio is changed (m = 2); (c) The aspect ratio is unchanged (m = 4).
1100 1100 the following: B0 ¼ , B1 ¼ , or 1100 0011 1000 1000 B0 ¼ , B1 ¼ (all-zero columns are 1000 0100 1011 1011 added), or B0 ¼ , B1 ¼ (all-one 1011 0111 columns are added). Fig. 1(c) shows that the recovered image has the same aspect ratio by using the 1011 1011 basis matrices B0 ¼ and B1 ¼ . 1011 0111
2.2. Motivation Initial motivation for keeping the aspect ratio unchanged is to avoid the distortion of recovered image. For example, if the shape of secret image is our information and the original image is a circle (Fig. 2(a)). The recovered image for m = 2 (Fig. 2(b)) is an ellipse and the distortion of image will compromise our secret and create a new motivation for designing the aspect ratio invariant VSS schemes. The existing method is to add minimum extra subpixels into the black and white matrices
Fig. 2. The distortion due to the variant aspect ratio: (a) The original secret; (b) (2, 2) VSS scheme with m = 2; (c) (2, 2) VSS scheme with m = 4.
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B0 and B1 to let the number of columns, m, be a square value and do not violate the contrast and security conditions. It is obvious pffiffiffiffi 2 that the minimum number of extra bits is d me m, where dme is the smallest integer no less than m. For example, the pffiffiffi minimum number of extra bits is d 2e2 2 ¼ 2 for a (2, 2) VSS scheme. The recovered image by adding two extra black subpixels (Fig. 2(c)) is still a circle. But how to further reduce the added subpixels and simultaneously hold the clearness of recovered image is an interesting problem. The work presented in this paper addresses this problem.
3. Design of the VSS schemes with aspect ratio invariant Before describing our construction method, we describe a trivial method by adding extra subpixels directly into the basis matrices. First give the following definition. The trivial method is induced from Naor and Shamir (1995) (the authors recommended to add the extra subpixels for avoiding the distortion of the aspect ratio of the original image) and the observations that adding the column vectors consisting n 1s or 0s in the matrices of the original VSS scheme also forms a VSS scheme (Eisen and Stinson, 2002). The formal description is shown in the following definition and theorem. Definition 1. Let B0 and B1 be n · m white and black matrices for the basic (k, n) VSS scheme. Let 1 and 0 be all-one and all-zero n · 1 column matrices, respectively. Let C0 and C1 be the matrices obtained by concatenating the matrices as the following where is a concatenation operation: C 0 ¼ B0 1 1 0 0; and |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffi 2 x d me mx ð1Þ C 1 ¼ B1 1 1 0 0 : |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffi 2 x d me mx Theorem 1. The matrices C0 and C1, obtained in (1), are the white and black matrices for a (k, n) VSS scheme with the invariant paspect ffiffiffiffi 2 ratio and the minimum pixel expansion is d me .
Proof. pffiffiffiffi 2 It is obvious that concatenating the same d me m n · 1 column matrices to B0 and B1 will not violate the contrast and security conditions. The number of columns of Ci, the pffiffiffi ffi 2 pffiffiffipixel ffi 2 expansion, is m þ x þ p d ffiffiffiffim e m x ¼ d m e , i.e., the square of d m e. Therefore, the aspect ratio is unchanged. Because dae ispthe ffiffiffiffi smallest integer no less than a, the value d m e2 is the minimum square value greater than or equal to m. h 3.1. Construction method for an aspect ratio invariant VSS scheme with the original expansion m=2 We now describe the construction of an aspect ratio invariant (2, 2) VSS scheme with pixel expansion m = 2 such that we can easily understand the basic concept of our proposed construction. Sup 10 pose the basis matrices are B0 ¼ and 10 10 B1 ¼ . To keep the aspect ratio unchanged, 01 we process a four-pixeled square block each time, as shown in Fig. 3(a), by using B0 and B1 to get 10101010 10101010 and B01 ¼ . Then B00 ¼ 10101010 01010101 add one black (or white) subpixels to let the total number of subpixels be 9 (=32). Without loss of generality, use the black subpixel as the dummy subpixel, i.e., the new white and black matrices are 2 3 10 10 10 10 1 D0 ¼ B00 1 ¼ 4 10 10 |{z} 10 |{z} 10 |{z} 1 5 |{z} |{z} R1
R2
R3
R4
10 01 |{z}
10 01 |{z}
10 01 |{z}
d0
and 2
10 D1 ¼ B01 1 ¼ 4 01 |{z} R1
R2
R3
R4
3 1 1 5; |{z} d0
where the two subpixels in R1 region are used to represent the pixel 1 . Regions R2 , R3 and R4 are for the pixels 2 , 3 and 4 , respectively. The d0 black subpixel is used as the dummy subpixel, not for contrast and security, but for the aspect
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Fig. 3. Nine-subpixeled pattern for the (2, 2) VSS scheme with the aspect ratio invariant: (a) four-pixeled square block; (b) regular nine-subpixeled pattern; (c) four partially regular nine-subpixeled patterns.
ratio. The nine subpixels are then put as the pattern shown in Fig. 3(b) such that Ri and i have the almost same geographic distribution, position and direction. Although the shadows will have some subpixels ( d0 ) at regular position, it will not compromise the security; however the shadows are not completely random and may cause some suspicions. But when randomly permute the columns of D0 and D1, the recovered image will lose the clearness at the edge of the secret image. To let the shadows be more random and still hold the clearness of the recovered image, we randomly use the four partially regular nine-subpixeled patterns shown in Fig. 3(c), i.e., the d0 black subpixel is included in Ri region, i = 1, 2, 3, 4, with the same probability. These four patterns of arranged subpixels have the almost same correspondence between the pixel i and the subpixels in Ri . Fig. 4 shows the recovered images by using regular (Fig. 3(b)), partially regular (Fig. 3(c)), and totally random (randomly permute D0 and D1) nine-subpixeled patterns. We use the crossed lines as the original secret image (Fig. 4(a)) to test the performance of these three patterns. It is observed that the partially regular pattern (Fig. 4(c)) has the advantages over other two patterns (Fig. 4(b) and (d)), the more random shadow and the clearness of the recovered image. The pixel expansion is 9 ¼ 2:25 for our aspect ratio invariant (2, 2) VSS 4 scheme.
The formal construction for our aspect ratio invariant (2, 2) VSS scheme with the original pixel expansion m = 2 is described in Algorithm 1 (using regular nine-subpixeled pattern), Algorithm 2 (using partially regular nine-subpixeled pattern) and Algorithm 3 (using totally random nine-subpixeled pattern). Algorithm 1. (Processing every four (2 · 2) pixels in the original image) Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S 1 and S 2. Step 1: For i = 1 to 4 do Break the pixel i in B into two subpixel by using the matrices B0 and B1 and put them in region Ri ; Step 2: Put R1 , R2 , R3 , R4 and a dummy black subpixel d0 according the pattern in Fig. 3(b) to create two square blocks B1 and B2 . Step 3: Deliver the two blocks and S 2.
B1
and
B2
to S 1
Algorithm 2. (Processing every four (2 · 2) pixels in the original image)
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Fig. 4. The recovered image for the aspect ratio invariant VSS scheme by using different patterns of nine arranged subpixels: (a)–(c) partially regular pattern; (d) totally random pattern.
Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S 1 and S 2. Step 1: Randomly choose j from {1, 2, 3, 4}. Step 2: Let
Rj
=
Rj
[
d0
.
Step 3: For i = 1 to 4 do If (i = j) then Break the pixel i in B into three subpixel by using the matrices B0 1, B1 1 and then put them in region Ri ; else Break the pixel i in B into two subpixel by using B0 and B1 and put them in region Ri ; Step 4: Put
R1
,
R2
,
R3
and
R4
according the jth
pattern in Fig. 3(c) to create two square blocks B1 and B2 .
Step 5: Deliver the two blocks and S 2.
B1
and
B2
to S 1
Algorithm 3. (Processing every four (2 · 2) pixels in the original image) Input: four pixels of a square block B in the original image O. Output: nine subpixels square blocks B1 and B2 in the shadow images S1 and S2. Step 1: For i = 1 to 4 do Break the pixel i in B into two subpixel by using B0 and B1; Step 2: Randomly put the above eight subpixels and a dummy black subpixel in a 3 · 3 square block to create two square blocks B1 and B2 . Step 3: Deliver the two blocks B1 and B2 to S1 and S2.
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Our proposed construction uses the same basis matrices of conventional VSS scheme and adds the minimum extra black subpixels to keep the aspect ratio unchanged. Using the same approach, we can extend the construction method from m = 2 to m > 2.
199
pffiffiffiffiffiffi Theorem 3. The pixel expansion, m2 (=d 4me2 =4), for an aspect ratio invariant (k, n) VSS scheme in Definition pffiffiffiffi 2 is no larger than the pixel expansion, m1 (=d me2 ), for an aspect ratio invariant (k, n) VSS scheme in Definition 1. Proof
3.2. Construction method for an aspect ratio invariant VSS scheme with the original expansion m>2 3.2.1. Construction method The following definition and theorem shows how to construct an aspect ratio invariant VSS scheme with original expansion m > 2. Definition 2. Let B0 and B1 be n · m white and black matrices for the basic (k, n) VSS scheme. Let 1 be all-one n · 1 column matrix. Let D0 and D1 be the matrices obtained by concatenating the matrices as the following: D 0 ¼ B0 B0 B0 B0 1 1 ; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffiffi 2 d 4me 4m
D 1 ¼ B1 B1 B1 B1 1 1 : |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pffiffiffiffi 2
and ð2Þ
d 4me 4m
Theorem 2. The matrices D0 and D1 obtained in (2), are the white and black matrices for a (k, n) VSS scheme with the invariant paspect ratio and ffiffiffiffi the minimum pixel expansion is d me2 =4. Proof. It is obvious that B0 B0 B0 B0 and B1 B1 B1 B1 are the basis matrices of (k, n) VSS scheme with pixel expansion 4m. Concatenate pffiffiffiffiffiffi 2 the same d 4me 4m n · 1 1 column matrices to B0 B0 B0 B0 and B1 B1 B1 B1 will not violate the contrast and security conditions. pffiffiffiffiffiffiThe numberpffiffiffiffiffiffi of columns of Di is 4m þ d 4me2 2 4m ¼ d 4me . Therefore, the aspect ratio is unchanged. Due to the definition of d Æ e, the value pffiffiffiffiffiffi 2 d 4me is a minimum square value greater than or equal pffiffiffiffi 2to 4m. So the minimum pixel expansion is d me =4. h The following theorem shows that our proposed method has the better pixel expansion than the method defined in Definition 1.
pffiffiffiffiffiffi 2 pffiffiffiffi 2 m2 ¼ d 4me =4 ¼ d2 me =4: pffiffiffiffi pffiffiffiffi Since d2 me 6 2d með* da be 6 dae dbeÞ; pffiffiffiffi 2 pffiffiffiffi 2 so m2 ¼ d2 me =4 6 22 d me =4 pffiffiffiffi 2 ¼ d me ¼ m1 : From Theorem 3, our proposed method will have the less pixel expansion. Table 1 shows the compared results for the original expansion m 6 20. The upper row is the method in Definition 1 (denoted as Method 1) and the lower row is our proposed method (denoted as Method 2) for each m. The parameters in Table 1 are described as follows: the value m is the original pixel expansion of (k, n) VSS scheme. Notation ns represents the number of subpixels used for a square block (one pixel (Method 1) and four pixels (Method 2)), then ns = m or 4m. The minimum number of the extra subpixels ne is added to make nt = ns + ne square, where nt is the total number of all subpixels. The pixel expansions for the aspect ratio invariant VSS schemes are m1, m2 for Method 1 and Method 2, respectively. We define an Improvement Factor, IF ¼ mm12 , in the last column. From Table 1, it is shown that our method always has the optimum pixel expansion. We have improvements for m = 5, 6, 10, 11, 12, 17, 18, 19, 20. For m = 5 and 6, 25-subpixeled square block is used. For m = 10, 11 and 12, 49-subpixeled square block is used. For m = 17, 18, 19 and 20, 81-subpixeled square block is used. Let us proceed to the next chief consideration of our pffiffiffiffiffifficonstruction method. How to arrange the d 4me2 subpixels in a square block to maintain the two characteristics mentioned in the above section, the random shadow and the clearness of the recovered image.
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Table 1 Pixel expansions m1 and m2 for the aspect ratio invariant VSS scheme m 6 20 Pixel expansion m
Number of subpixels used for a square block nsa
Minimum number of added extra subpixels ne to make ns + ne squareb
Total number of subpixels nt = ns + nec
Pixel expansion m1 (Method 1) and m2 (Method 2)d
2
2 8
2 1
4 9
4 9/4
3
3 12
1 4
4 16
4 4
4
–
5
5 20
6
–
Improvement factor IFe 1.78 1
–
–
–
4 5
9 25
9 25/4
1.44
6 24
3 1
9 25
9 25/4
1.44
7
7 28
2 8
9 36
9 9
1
8
8 32
1 4
9 36
9 9
1
9
–
10
10 40
11
–
–
–
6 9
16 49
16 49/4
1.31
11 44
5 5
16 49
16 49/4
1.31
12
12 48
4 1
16 49
16 49/4
1.31
13
13 52
3 12
16 64
16 16
1
14
14 56
2 8
16 64
16 16
1
15
15 60
1 4
16 64
16 16
1
16
–
–
–
–
–
17
17 68
8 13
25 81
25 81/4
1.23
18
18 72
7 9
25 81
25 81/4
1.23
19
19 76
6 5
25 81
25 81/4
1.23
20
20 80
5 1
25 81
25 81/4
1.23
a b c d e
–
ns = m (Method 1) and 4m (Method 2). pffiffiffiffiffiffi pffiffiffiffi ne ¼ d me2 m (Method 1) and d 4me2 4m (Method 2). pffiffiffiffiffiffi pffiffiffiffi nt ¼ d me2 (Method 1) and d 4me2 (Method 2). pffiffiffiffiffiffi 2 pffiffiffiffi 2 m1 ¼ d me and m2 ¼ d 4me =4. pffiffiffiffi m1 4d me2 IF ¼ ¼ pffiffiffiffiffiffi 2 . m2 d 4me
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201
pffiffiffiffiffiffi 2 3.2.2. d 4me -subpixeled patterns for the (k, n) VSS scheme with expansion m Three patterns of arranged subpixels are proposed in this section. The first is regular pattern; the second is partially regular pattern; the third is totally random pattern. A. Regular Pattern (Pa): Put the dummy black subpixels d i , i = 0, . . ., 4, as a cross in the center of the pattern. B. Partially Regular Pattern (Pb): Put the dummy black subpixels d i i = 1, . . ., 4, into the region Ri and then randomly put d0 into Ri for processing the square block each time. C. Totally Random Pattern (Pc): Randomly permute the columns of D0 and D1. Patterns Pa are designed as follows: 25-subpixeled square pattern (m = 5 and 6), 49-subpixeled square pattern (m = 10, 11 and 12) and 81-subpixeled square pattern (m = 17, 18, 19 and 20) are shown in Figs. 5–7, respectively. Four Pb patterns for 25-subpixeled square block (m = 5 and 6) are shown in the following figure. The pattern in Fig. 8(a) is R1 = R1 [ d1 [ d0 and Ri = Ri [ d i , i = 2, 3, and 4; the pattern in Fig. 8(b) is R2 = R2 [ d2 [ d0 and Ri = Ri [ d i , i = 1, 3, and 4; the pattern in Fig. 8(c) is R3 = R3 [ d3 [ d0 and Ri = Ri [ d i , i = 1, 2, and 4; the pattern in Fig. 8(d) is R4 = R4 [ d4 [ d0 and Ri = Ri [ d i , i = 1, 2, and 3. Other Pb patterns for different m can be designed by using the same approach.
Fig. 5. Pa pattern for m = 5 and 6: (a) m = 5, ns = 20, ne = 5, nl = 25; (b) m = 6, ns = 24, ne = 1, nt = 25.
4. Experimental results In this section, we present two experimental results to illustrate the performance of our construction by using three patterns of arranged subpixels: Pa, Pb and Pc. The first is (2, 2) VSS scheme with m = 2, m1 = 4 and m2 ¼ 94; the second is Naor–Shamir (2, 5) VSS scheme with m = 5, m1 = 9 and m2 254. Example 2. For a (2, 2) VSS scheme with the basis 10 10 matrices B0 ¼ and B1 ¼ . From Eq. 10 01 (2), the white and black matrices D0 and D1 of our proposed scheme are D0 ¼ B0 B0 B0 B0 1 2 10 10 10 10 ¼ 4 10 1 0 1 0 1 0 |{z} |{z} |{z} |{z} R1
R2
R3
R4
1
3
1 5; |{z}
and
d0
Fig. 6. Pa pattern for m = 10, 11 and 12: (a) m = 10, ns = 40, ne = 9, nt = 49; (b) m = 11, ns = 44, ne = 5, nt = 49; (c) m = 12, ns = 48, ne = 1, nt = 49.
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Fig. 7. Pa pattern for m = 17, 18, 19 and 20: (a) m = 17, ns = 68, ne = 13, nt = 81; (b) m = 18, ns = 72, ne = 9, nt = 81; (c) m = 19, ns = 76, ne = 5, nt = 81; (d) m = 20, ns = 80, ne = 1, nt = 81.
Fig. 8. Pb patterns for m = 5 and 6: (a)
D 1 ¼ B1 B1 B1 B1 1 2 10 10 10 10 4 ¼ 01 |{z} 01 |{z} 01 |{z} 01 |{z} R1
R2
R3
R4
1
3
1 5: |{z} d0
d0
in
R1
; (b)
d0
in
R2
; (c)
d0
in
R3
; (d)
d0
in
R4
.
Suppose a test secret image is a printed text VSS and LENA image. After performing three patterns Pa, Pb and Pc, the revealed secret images are shown in Fig. 9. It is observed that using Pa (Fig. 9(a)) has the best clearness of
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Fig. 9. The aspect ratio invariant (2, 2) VSS scheme: (a) Pa pattern is used; (b) Pb pattern is used; (c) Pc pattern is used.
recovered image; however the shadow is not completely random but a little regular. One can see that from the white areas of the recovered image VSS . When Pb pattern is used, the clearness of recovered image shown in Fig. 9(b) is almost the same as Fig. 9(a) and the shadow has more randomness. When using Pc pattern, the clearness of recovered image is compromised.
Example 3. For the Naor–Shamir (2, 5) VSS scheme with the basis matrices 2
10000
3
7 6 6 10000 7 7 6 7 B0 ¼ 6 6 10000 7 7 6 4 10000 5 10000
2 and
10000
3
7 6 6 01000 7 7 6 7 B1 ¼ 6 6 00100 7: 7 6 4 00010 5 00001
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Fig. 10. The aspect ratio invariant (2, 5) VSS schemes: (a) the original secret; (b) the proposed method with pattern Pa is used; (c) the proposed method with pattern Pb is used; (d) the proposed method with pattern Pc is used; (e) construction method defined in Definition 1.
From Eq. (2), the white and black matrices D0 and D1 of our proposed scheme are D 0 ¼ B0 B0 B0 B0 1 1 1 1 1 3 2 10000 10000 10000 10000 11111 7 6 6 10000 10000 10000 10000 11111 7 7 6 6 10000 10000 10000 10000 11111 7 7 6 ¼6 7; 6 10000 10000 10000 10000 11111 7 7 6 7 6 4 10000 10000 10000 10000 11111 5 |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} R1
R2
R3
R4
d0
and
Fig. 10(a) is the test secret image, a thin printed text VSS . Fig. 10(b)–(d) show the revealed images and their corresponding shadows by using Pa, Pb and Pc patterns, respectively, and the expansion is 254 ¼ 6:25. Fig. 10(e) is the result by using the method defined in Definition 1 with the basis matrices 2
100001111
7 6 6 100001111 7 7 6 7 6 7; C0 ¼ 6 100001111 7 6 7 6 6 100001111 7 5 4 100001111
D 1 ¼ B1 B1 B1 B1 1 1 1 1 1 3 2 10000 10000 10000 10000 11111 7 6 6 01000 01000 01000 01000 11111 7 7 6 6 00100 00100 00100 00100 11111 7 7 6 ¼6 7: 6 00010 00010 00010 00010 11111 7 7 6 7 6 4 00001 00001 00001 00001 11111 5 |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} R1
R2
R3
R4
d0
3
2
100001111
3
7 6 6 010001111 7 7 6 7 6 7 C1 ¼ 6 001001111 7 6 7 6 6 000101111 7 5 4 000011111
(from Eq. (1)) and the pixel expansion is 9. The shadow pattern in Fig. 10(b) is very regular and the revealed image has the good quality. Although the shadow is completely random in Fig. 10(d), the clearness of recovered image is poor. It is obvious that Fig. 10(c) is the proper choice, since the shadow in Fig. 10(c) is random like Fig. 10(d) and the contrast of revealed image
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Fig. 10 (continued)
is good enough to ‘‘see’’ the secret. Fig. 10(e) shows that the shadow has large pixel expansion and the contrast of revealed image is similar to Fig. 10(c) and worse than Fig. 10(b).
5. Conclusion In this paper we propose the aspect ratio invariant VSS scheme for avoiding the distor-
tion of recovered image. We show how to reppffiffiffiffiffiffi 2 resent four pixels in a square block by d 4me subpixels in a square block such that the aspect ratio is unchanged. Three patterns of arranged subpixels, Pa, Pb and Pc are proposed. We can choose one of them to implement the aspect ratio invariant VSS scheme to achieve our demand. From the experimental results, it is observed that pattern Pb is a reasonable choice.
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References Ateniese, G., Blundo, C., De Santis, A., Stinson, D.R., 1996. Visual cryptography for general access structures. ECCC, Electronic Colloquium on Computational Complexity (TR96-012), via WWW using
cryptography. Theoretical Computer Science 240 (2), 471– 485. Ito, R., Kuwakado, H., Tanaka, H., 1999. Image size invariant visual cryptography. IEICE Trans. Fundamentals E 82-A (10), 481–494. Naor, M., Shamir, A., 1995. Visual cryptography. Advances in Cryptology—EUROCRYPT94, Lecture Notes in Computer Science, vol. 950, pp. 1–12. Verheul, E.R., Van Tilborg, H.C.A., 1997. Constructions and properties of k out of it n visual secret sharing schemes. Designs, Codes and Cryptography 11 (2), 179– 196. Tzeng, W.G., Hu, C.M., 2002. A New approach for visual cryptography. Designs, Codes and Cryptography 27 (3), 207–227. Yang, C.N., 2004. New visual secret sharing schemes using probabilistic method. Pattern Recognition Letters 25 (4), 481–494.