New Size-Reduced Visual Secret Sharing ... - Semantic Scholar
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.2 FEBRUARY 2006
620
LETTER
New Size-Reduced Visual Secret Sharing Schemes with Half Reduction of Shadow Size Ching-Nung YANG†a) , Member and Tse-Shih CHEN† , Nonmember
SUMMARY The Visual Secret Sharing (VSS) scheme proposed by Naor and Shamir is a perfectly secure scheme to share a secret image. By using m sub pixels to represent one pixel, we encrypt the secret image into several noise-like shadow images. The value of m is known as the pixel expansion. More pixel expansion increases the shadow size and makes VSS schemes impractical for real application. In this paper, we propose new size-reduced VSS schemes and dramatically decrease the pixel expansion by a half. key words: visual secret sharing, visual cryptography
1.
Introduction
The so-called (k, n) visual secret sharing (VSS) scheme encrypts an image by dividing a pixel into m sub pixels in n shadows [1]–[4]. When stacking k or more shadows, we can “see” the secret image without the need for complicated cryptographic operations; however k − 1 or fewer shadows will not get any information. Some approaches to reduce the pixel expansion based on the existing VSS schemes were proposed [5]–[9]. Kuwakado and Tanaka [5] proposed a size-reduced VSS scheme, using the deletion of some columns in the black and white matrices, but the scheme did not work for reducing the shadow size of (k, k) VSS schemes. An extreme size-reduced VSS scheme with no pixel expansion was first discussed by Ito et al. [6]. Yang gave a general description for this size-invariant scheme using the probabilistic concept [7]. Both VSS schemes have no pixels expansion but recover the poor-quality secret image. Generalizations of the probabilistic model with any pixel expansion were also given in [8], [9] for trading the pixel expansion with the contrast. In [1], the authors mentioned that handling larger groups of pixels, instead of each pixel separately, yields better results. However, no concrete scheme has been shown. Here we process a two-pixeled block each time to achieve the half reduction of shadow size. The rest of this paper is organized as follows. Section 2 describes basic VSS schemes. In Sect. 3, constructions for the proposed sizereduced VSS schemes are proposed; also, analyses of the contrast and the recognition of the edge are given. Comparison with the previous size-reduced schemes and experiManuscript received June 20, 2005. Manuscript revised July 21, 2005. Final manuscript received October 24, 2005. † The authors are with the Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, 974, Taiwan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.2.620
mental results are shown in Sect. 4. Finally, conclusions are drawn in Sect. 5. 2.
k-out-of-n VSS Schemes with the Pixel Expansion m
We use the notation (k, n, m) VSS scheme to denote a (k, n) VSS scheme with the pixel expansion m. A (k, n, m) VSS Scheme can be described by the black and white n × m Boolean matrices where the element si j = 1 if and only if the jth sub pixel in the ith shadow is black, otherwise si j = 0. C1 and C0 are their corresponding black and white sets including all matrices obtained by permuting the columns of B1 and B0 , respectively. When one black (resp. white) pixel is shared, we randomly select one matrix from the set C1 (resp. C0 ) and then choose one row of this matrix to a relative shadow. When shadows i1 , i2 ,. . . , ir (r ≥ k) are stacked, these r rows in Bi are OR-ed. The pixel color of the recovered image is proportional to H(V) (the Hamming weight of this OR-ed m-vector V). If H(V) ≥ dB , this gray level is interpreted by the user’s vision as Black and if H(V) ≤ dW the result is interpreted as White, where dB > dW . For r < k, the two collections of r × m matrices obtained from C1 and C0 are the same in the sense that they contain the same matrices with the same frequencies. The contrasts represented by dB and dW were defined in [1], [3], [4]. The first definition was given by Naor and Shamir as αNS = (dB − dW )/m [1]. Others were introduced, αVV = (dB − dW )/(m (2m − dB − dW ))(by Verheul and Van Tilborg) and αES = (dB − dW )/(2m − dB )(by Eisen and Stinson), to further contend with the real contrast of the recovered image [3], [4]. 3.
The Proposed Size-Reduced VSS Schemes
In [1], Naor and Shamir had mentioned that one can handle several pixels to reduce the pixel expansion. In Sect. 3.1, we first introduce a trivial method using the existing C1 and C0 but processing two-pixeled block each time. Finally, the pixel expansion is reduced by half; however this construction brings the line disappearance problem. A refined construction in Sect. 3.2 is proposed to address the problem. The basic concept is to process a two-pixeled block (i.e., , , and ) each time, by four corresponding sets C11 , C00 , C10 and C01 . The first two sets contribute the contrast of the recovered image, while the last two sets determine the clearness of edges between black and white areas. For our new two-pixeled VSS schemes, the formal
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
LETTER
621
contrast and security conditions are redefined as follows. C1(Contrast Condition): For any S of the matrix in C11 (resp. C00 ), the OR-ed V of rows i1 , i2 , . . . , ir , for r ≥ k, satisfies H(V) ≥ dB (resp. ), where dB > dW . H(V) ≤ dW C2(Edge Precision Condition): For any S of the matrix in C10 (resp. C01 ), the OR-ed V of rows i1 , i2 , . . . , ir , for r ≥ k, satisfies (Vn /|C10 |) ≥ PBW (resp. (Vn /|C01 |) ≥ PBW ) where the symbol Vn is defined as follows and 0 < PBW ,PBW ≤ 1. All possible vectors V are w-out-ofm m-tuples, 0 < w < m, and the symbol Vn is used to denote the maximum number of the same w-out-of-m pattern in all possible vectors V. C3(Security Condition): For r < k, the four collections of r × m matrices obtained from C11 , C00 , C11 and C00 are the same in the sense that they contain the same matrices with the same frequencies. 3.1 Construct the Size-Reduced (k, n, m/2) VSS Schemes Using (k, n, m) VSS Schemes Construction 1: Let C1 and C0 be the black and white sets for (k, n, m) VSS schemes. Then the sets C11 = C1 and C00 = C0 , and the sets C10 and C01 can be constructed as the following nine methods: 1) C10 = C01 = C1 ∪ C0 ; 2) C10 = C01 = C1 ; 3) C10 = C01 = C0 ; 4) C10 = C1 and C01 = C0 ; 5) C10 = C0 and C01 = C1 ; 6) C10 = C1 ∪ C0 and C01 = C1 ; 7) C10 = C1 ∪ C0 and C01 = C0 ; 8) C10 = C1 and C01 = C1 ∪ C0 ; 9) C10 = C0 and C01 = C1 ∪ C0 . Theorem 1: The scheme from Construction 1 is a (k, n, m/2) VSS scheme. Proof: First, we prove Construction 1-1. It is obvious that the pixel expansion is now m/2 due to the two-pixeled operation. Since C11 = C1 and C00 = C0 , then dB and in the (k, n, m/2) VSS scheme are dB = dB and dW And, C10 = C1 ∪ C0 , so PBW = (Vn /|C10 |) = dW = dW . m m Max m! , m! 2 dB d W m m ×m!) = Max 1 2 ,1 2 . PW B = d dW B m m Max 1 2 ,1 2 Thus, conditions C1 and dB dW C2 are met. To prove the security condition C3, it is observed that every row of these four sets is the same to the row in C1 and C0 . Thus, the scheme satisfies C3 condition. The proofs of Construction 1-2 ∼ 1-9 are similar. Some measurements, the contrast, the average edge precision and the distinction of line, are defined to evaluate our size-reduced VSS schemes. We herein adopt the concept of Naor-Shamir contrast [1] and the average contrast definition in[5] to define our contrast as αYC =
1 |C11 |
H(V)−
AllS inC 11
m
1
|C00 |
AllS inC 00
H(V)
.
Also, we define the average edge precision P MEAN = (PBW + PW B )/2. The distinctionof black line in white area is DBIW = DC10 −C00 + DC01 −C00 2 (i.e. the distinction between C10 and C00 , and the distinction between C01 and C00 ) and DCi j −Ckl = Ci j − Ci j ∩ Ckl Ci j where “−” is a set difference; the distinction
of white line in black area DWIB = DC10 −C11 + DC01 −C11 2. The average distinction is defined as D MEAN = (DBIW + DWIB )/2. The value αYC shows the contrast between black and white areas. The average edge precision P MEAN determines the clearness of the edge (Notice that if the stacked patterns V are not the same then it will lack consistency and the edge is irregular.). The distinction D MEAN is the value to measure whether we can distinguish the thin lines in the recovered images. The values αYC , P MEAN and D MEAN for Construction 1-1 are calculated as follows: αYC =
1 |C1 |
1
|C11 |
H(V)−
AllS inC11
H(V)−
1
|C00 |
m
1 |C0 | AllS inC 0
AllS inC00
H(V)
H(V)
1 = m (dB −dW ) = m = αNS . (PBW +PW B ) P MEAN = 2 = m m ,1 2 . Max 1 2 dW dB DC10 −C00 = DC01 −C00 = DC10 −C11 = DC01 −C11 (m!) = (2×m!) = 1/2; = 1/2; DBIW = DWIB = (1/2+1/2) 2 (DBIW +DWIB ) = 1/2. D MEAN = 2 AllS inC
All the values PBW , PW B , P MEAN , DBW , DW B and D MEAN for a size-reduced (2, 2, 1) VSS scheme are calculated and shown in Table 1. From Table 1, we successfully reduce the pixel expansion of (2, 2, 2) VSS scheme to 1 with the same contrast as αYC = αNS = 1/2. P MEAN = 1/2 (Construction 1-1, 1-3, 1-7 and 1-9) means that the edge of black and white areas has 50% precision; the value D MEAN = 1/2 shows that we can distinguish the thin lines (line width less than 3 pixels) with 50% precision. Some constructions either improve the precision of the
Table 1
(2, 2, 1) VSS schemes.
PBW PW B P MEAN DBIW DWIB D MEAN Cons. 1-1 Cons. 1-2 Cons. 1-3 Cons. 1-4 Cons. 1-5 Cons. 1-6 Cons. 1-7 Cons. 1-8 Cons. 1-9
1/2 1 1/2 1 1/2 1/2 1/2 1 1/2
1/2 1 1/2 1/2 1 1 1/2 1/2 1/2
1/2 1 1/2 3/4 3/4 3/4 1/2 3/4 1/2
1/2 1 0 1/2 1/2 3/4 1/4 3/4 1/4
1/2 0 1 1/2 1/2 1/4 3/4 1/4 3/4
1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
αYC = αNS 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.2 FEBRUARY 2006
622
Fig. 1
The recovered images for nine size-reduced (2, 2, 1) VSS schemes.
edge or the distinction of the thin lines. Constructions 12, 1-4, 1-5, 1-6 and 1-8 improved the average precision of edge (larger than 50% precision), i.e. the recovered images will have more regular edges. Moreover, Constructions 1-2, 1-6 and 1-8 have the better recognition of the black line on the white area since DBIW > DWIB ; Constructions 1-3, 1-7 and 1-9 have the better recognition of the white line on the black area since DBIW < DWIB ; Constructions 1-1, 1-4 and 1-5 have half distinction on both black and white areas. All of these nine constructions have 50% average distinction. We use a test pattern, four verticals and one horizontal of one pixel in width, to test the size-reduced (2, 2, 1) VSS scheme; the left background is white and the right is black. Figures 1(a)–(i) are the recovered images for Construction 1-1∼Construction 1-9, respectively. It is observed that the recovered images are consistent with the values PBW , PW B , P MEAN , DBW , DW B , and D MEAN . Figure 1(a) shows that the verticals always display because the two-pixeled blocks, (,), are represented as () or () with 50% probability. Figures 1(c)–(e), (g) and (i) show that the black verticals may disappear since at least one of two-pixeled blocks, (,), is represented as () for Construction 1-3∼1-5, Construction 1-7 and Construction 1-9. In Fig. 1(d), the black (resp. white) verticals will disappear when the dealer only just processes () (resp. ()) and the vertical lines will remain when processing () (resp. ()). For observing the edge irregularity, Fig. 1(b) has the best regularity because the average edge precision P MEAN = 1. To confirm our definition of contrast, it is observed that the left white region and the right black region for all the recovered images, have the same contrast like the original (2, 2, 2) VSS scheme. From Fig. 1, Construction 1-1 does not have the line
disappearance problem but it has the problem of edge irregularity. The next section shows how to retain the line display and meantime improve the edge regularity by adding an extra sub pixel. 3.2 A Solution for the Thin Line Disappearance Problem for the Size-Reduced VSS Schemes In this section, we propose the size-reduced (2, 2, (m + 1)/2) VSS schemes, in order to solve the thin line disappearance problem (see Fig. 1). The enhanced construction is shown below. Construction 2: Let C1 and C0 be the sets for (2, 2, m) VSS scheme. Define the operation ⊗ as appending a column vector L to all matrices in a set C from left (L ⊗ C) or right (C ⊗ L). Then the sets for the (2, 2, (m + 1)/2 ) VSS scheme are constructed as follows: 1 0 1 0 ⊗ C1 , ⊗ C 1 , C1 ⊗ , C1 ⊗ ; C11 = 0 1 0 1 1 0 1 0 C00 = ⊗ C0 , ⊗ C 0 , C0 ⊗ , C0 ⊗ ; 1 0 1 0 1 0 1 0 C10 = ⊗ C0 , ⊗ C 0 , C1 ⊗ , C1 ⊗ ; 0 1 1 0 1 0 1 0 C01 = ⊗ C1 , ⊗ C 1 , C0 ⊗ , C0 ⊗ . 1 0 0 1 Theorem 2: The scheme from Construction 2 is a sizereduced (2, 2, (m + 1)/2)VSS scheme with the contrast (dB − dW − 1/2)/(m + 1); meantime it can solve the thin line disappearance problem. Proof: It is easy to verify that C1, C2 and C3 conditions are
LETTER
623
met. To prove that this scheme can solve the thin line disappearance problem, there is always one black sub pixel in the left of the stacked result in C10 set for two-pixeled block (), and one black sub pixel in the right of the stacked result in for two-pixeled block (). Thus, we do not have the line disappearance. The pixel expansion is obviously (m + 1)/2. By the contrast definition, we have
αYC = =
(dB +1)×4|C1 | dW ×2|C0 |+(dW +1)×2|C0 | − 4|C 1 | 4|C0 | (m+1) (dB − dW − 1/2)/(m + 1).
The proof is completed.
(a) The recovered image. Fig. 2
Example 1: Construct a (2, 2, 1.5) VSS scheme from Naor-Shamir (2, 2, 2) VSS scheme, by using Construction 2. From Construction 2, we have 1 10 01 0 10 ⊗ , , ⊗ , C11 = 0 01 10 1 01 01 10 01 1 10 01 , , ⊗ , , 10 01 10 0 01 10 0 110 101 010 001 ⊗ = , , , , 1 001 010 101 110 101 011 100 010 , , , . 010 100 011 101 Three other sets are obtained as follows: 110 101 010 001 101 C00 = , , , , , 110 101 010 001 101 011 100 010 , , ; 011 100 010 110 101 010 001 101 C10 = , , , , , 010 001 110 101 011 011 100 010 , , ; 101 010 100 110 101 010 001 101 C01 = , , , , , 101 110 001 010 100 011 100 010 , , . 010 101 011 The values αYC , P MEAN and D MEAN are calculated as follows: = 1/2; αYC = ((3×8)/8−(2×4+1×4)/8) 3 PBW = PW B = 1/2, P MEAN = (1/2+1/2) = 1/2; 2 (6/8+6/8) = 5/8, D = = 6/8, D MEAN = DBIW = (5/8+5/8) WIB 2 2 (5/8+6/8) = 11/16. 2 Figure 2(a) is the recovered image for the (2, 2, 1.5) VSS schemes using Construction 2. When compared with Figs. 1(b)–(i), the black and white verticals always display (better than Construction 1-2∼1-9) because Construction 2 can solve the line disappearance problem. The reason is described as follows. Figure 2(b) shows all possible stacked patterns for Example 1. There is always one black sub pixel in the left of
(b) All possible stacked patterns.
The size-reduced (2, 2, 1.5) VSS scheme.
the stacked result for two-pixeled block () and one black sub pixel in the the right of the stacked result for two-pixeled block (); this will make the verticals clearer than the Construction 1-1 and does not have the line disappearance problem. Meantime, the stacked results for two-pixeled block () are all three black (3B) sub pixels and the stacked results for two-pixeled block () are four 1B2 W and four 2B1 W patterns. Thus, the contrast of the recovered image is assured. Also, since the value D MEAN = 11/16 is improved, and thus the clearer edge is better than Construction 1. 4.
Experimental Results and Comparison
4.1 Experimental Results One standard image House (from USC-SIPI image database) is used to test the effects of edges and image qualities (see Fig. 3). Because the sets C10 and C01 in Construction 1 are constructed by C1 , C0 or their union, thus the qualities of the recovered image are compromised when the processed patterns are () and (). So, there is no definite answer which method is the best way to share the secret image; it depends on the pattern of secret image. For example, from these figures, it is observed that Construction 1-5 has the best image quality. 4.2 Comparison In this section we compare three size-reduced VSS schemes, Kuwakado-Tanaka scheme [5], Ito-Kuwakado-Tanaka (or Yang) scheme [6], [7]. Note that Ito-Kuwakado-Tanaka scheme in [6], in fact, is Yang’s probabilistic VSS scheme. Suppose that our (k, n, mYC ) VSS scheme, (k, n, mKT ) Kuwakado-Tanaka VSS scheme and (k, n, mIKT ) ItoKuwakado-Tanaka VSS scheme are all constructed from the traditional (k, n, m) VSS scheme. For our proposed scheme and Kuwakado-Tanaka VSS scheme, they have the same contrasts αYC = αKT = αNS . For the probabilistic VSS scheme, the contrast is αIKT = (p0 − p1 ) (dB /m) − (dW /m) = ((dB − dW )/m) = αNS when use Naor-Shamir approach, where p0 and p1 are defined in [7]. Thus all three contrasts are equal to αNS . The comparison is shown in Table 2 and our scheme is better than Kuwakado-Tanaka scheme, i.e. the pixel expansion of our scheme is no larger than that in Kuwakado-
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.2 FEBRUARY 2006
624
Fig. 3 Table 2
Comparison of three size-reduced VSS schemes. m
n=3 n=4 n=4 n=5 n=5 n=5 n=5
k=2 k=2 k=3 k=2 k=2 k=3 k=4
The recovered images for the (2, 2, 1) and (2, 2, 1.5) schemes.
3 6 6 5 10 8 15
αYC = αKT = αIKT mKT [5] mIKT [6], mYC = m/2 [7] (The proposed = αNS scheme) 1/3 2 1 1.5 1/3 3 1 3 1/6 3 1 3 1/5 4 1 2.5 3/10 5 1 5 1/8 5 1 4 1/15 8 1 7.5
a few simple calculations now. In fact, the idea that uses the simple computation to get the perfect reconstruction is also analogous to the VSS schemes with the perfect reconstruction [10], [11]. Because the judgment of image contrast is very subjective, we performed an experiment with these schemes to compare the images more fairly. The recovered image was evaluated by 10 people and gave a mean opinion score (MOS) to measure the image quality. Considering the situations of “calculation” and “MOS”, we give a more reasonable compared table, Table 3, for these three size-reduced VSS schemes. 5.
Tanaka schemes. Moreover, our scheme can be constructed from any (k, n, m) VSS schemes to achieve further reduction, but Kuwakado-Tanaka size-reduced VSS scheme does not work for reducing the shadow size of (k, k, m) VSS schemes. The probabilistic scheme has no pixels expansion; however it has a problem in the recognition of the small areas (see the detail analysis in [7]). Figure 4 shows the recovered images for (2, 3, 2) Kuwakado-Tanaka scheme, (2, 3, 1) Ito-Kuwakado-Tanaka scheme and our proposed (2, 3, 1.5) VSS scheme (the scheme of Sect. 3.1). All of them are constructed based on the Shamir-Naor (2, 3, 3) VSS scheme. It seems that our recovered image has the better visual image quality than other two schemes. For our proposed scheme and the probabilistic scheme [6], [7], some pixels in the secret image are lost permanently. But consider examining the pixel patterns in the shadows, the secret pixels can be correctly recovered for Kuwakado-Tanaka VSS scheme. However, it needs
Concluding Remarks
In this paper, two new size-reduced VSS schemes, the (k, n, m/2) scheme and the (2, 2, (m + 1)/2) scheme, from the traditional (k, n, m) scheme and (2, 2, m) scheme are proposed. When compared to the Kuwakado-Tanaka scheme [5] and the probabilistic scheme [6], [7], our size-reduced scheme works better either according to the pixel expansion or the image quality. It is shown that the edges in our recovered images are a little irregular but the recovered images have the same contrast as the Naor-Shamir scheme. It is indefinite that which method in Construction 1 has the best performance, i.e. the clear edge and the high contrast, because it depends on the arrangement of black and white pixels in the secret image. So how to transform a gray image to a black and white secret image for our construction needs careful consideration.
LETTER
625
Fig. 4 Table 3
The recovered images for three size-reduced VSS schemes. Comparison of three size-reduced (2, 3) VSS schemes.
K-T scheme [5] I-K-T scheme [6], [7] The proposed scheme Pixel expansion 2 1 1.5 The visual image quality of recovered im➂ ➁ ➀ age* Perfect reconstruction of the secret pixel YES NO NO Weakness Does not reduce the (k, k, The problem of recogniz- The problem of thin line m) VSS schemes. ing the small areas. disappearance. ∗:The recovered image was evaluated by 10 people and gave a mean opinion score to measure the image quality; ➀ means that it has the superior MOS, i.e. the best quality, and ➂ has the less MOS.
Acknowledgments The authors would like to express their deep gratitude to the reviewers for the valuable and usefull comments which improved the paper. References [1] M. Naor and A. Shamir, “Visual cryptography,” Advances in Cryptology-EUROCRYPT’94, Lect. Notes Comput. Sci., no.950, pp.1–12, 1995. [2] G. Ateniese, C. Blundo, A. De Santis, and D.R. Stinson, “Visual cryptography for general access structures,” ECCC, Electronic Colloquium on Computational Complexity, TR96-012, 1996. [3] E.R. Verheul and H.C.A. Van Tilborg, “Constructions and properties of k out of n visual secret sharing schemes,” Des. Codes Cryptogr., vol.11, no.2, pp.179–196, 1997. [4] P.A. Eisen and D.R. Stinson, “Threshold visual cryptography schemes with specified whiteness,” Des. Codes Cryptogr., vol.25,
no.1, pp.15–61, 2002. [5] H. Kuwakado and H. Tanaka, “Size-reduced visual secret sharing scheme,” IEICE Trans. Fundamentals, vol.E87-A, no.5, pp.1193– 1197, May 2004. [6] R. Ito, H. Kuwakado, and H. Tanaka, “Image size invariant visual cryptography,” IEICE Trans. Fundamentals, vol.E82-A, no.10, pp.2172–2177, Oct. 1999. [7] C.N. Yang, “New visual secret sharing schemes using probabilistic method,” Pattern Recognit. Lett., vol.25, no.4, pp.481–494, 2004. [8] S. Cimato, R. De Prisco, and A. De Santis, “Probabilistic visual cryptography schemes,” The Computer Journal, to appear, available at http://comjnl.oxfordjournals.org. [9] C.N. Yang and T.-S. Chen, “Size-adjustable visual secret sharing schemes,” IEICE Trans. Fundamentals, vol.E88-A, no.9, pp.2471– 2474, Sept. 2005. [10] R. Lukac and K.N. Plataniotis, “Bit-level based secret sharing for image encryption,” Pattern Recognit., vol.38, no.5, pp.767–772, 2005. [11] R. Lukac and K.N. Plataniotis, “Colour image secret sharing,” Electron. Lett., vol.40, no.9, pp.529–530, 2004.
620
LETTER
New Size-Reduced Visual Secret Sharing Schemes with Half Reduction of Shadow Size Ching-Nung YANG†a) , Member and Tse-Shih CHEN† , Nonmember
SUMMARY The Visual Secret Sharing (VSS) scheme proposed by Naor and Shamir is a perfectly secure scheme to share a secret image. By using m sub pixels to represent one pixel, we encrypt the secret image into several noise-like shadow images. The value of m is known as the pixel expansion. More pixel expansion increases the shadow size and makes VSS schemes impractical for real application. In this paper, we propose new size-reduced VSS schemes and dramatically decrease the pixel expansion by a half. key words: visual secret sharing, visual cryptography
1.
Introduction
The so-called (k, n) visual secret sharing (VSS) scheme encrypts an image by dividing a pixel into m sub pixels in n shadows [1]–[4]. When stacking k or more shadows, we can “see” the secret image without the need for complicated cryptographic operations; however k − 1 or fewer shadows will not get any information. Some approaches to reduce the pixel expansion based on the existing VSS schemes were proposed [5]–[9]. Kuwakado and Tanaka [5] proposed a size-reduced VSS scheme, using the deletion of some columns in the black and white matrices, but the scheme did not work for reducing the shadow size of (k, k) VSS schemes. An extreme size-reduced VSS scheme with no pixel expansion was first discussed by Ito et al. [6]. Yang gave a general description for this size-invariant scheme using the probabilistic concept [7]. Both VSS schemes have no pixels expansion but recover the poor-quality secret image. Generalizations of the probabilistic model with any pixel expansion were also given in [8], [9] for trading the pixel expansion with the contrast. In [1], the authors mentioned that handling larger groups of pixels, instead of each pixel separately, yields better results. However, no concrete scheme has been shown. Here we process a two-pixeled block each time to achieve the half reduction of shadow size. The rest of this paper is organized as follows. Section 2 describes basic VSS schemes. In Sect. 3, constructions for the proposed sizereduced VSS schemes are proposed; also, analyses of the contrast and the recognition of the edge are given. Comparison with the previous size-reduced schemes and experiManuscript received June 20, 2005. Manuscript revised July 21, 2005. Final manuscript received October 24, 2005. † The authors are with the Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, 974, Taiwan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.2.620
mental results are shown in Sect. 4. Finally, conclusions are drawn in Sect. 5. 2.
k-out-of-n VSS Schemes with the Pixel Expansion m
We use the notation (k, n, m) VSS scheme to denote a (k, n) VSS scheme with the pixel expansion m. A (k, n, m) VSS Scheme can be described by the black and white n × m Boolean matrices where the element si j = 1 if and only if the jth sub pixel in the ith shadow is black, otherwise si j = 0. C1 and C0 are their corresponding black and white sets including all matrices obtained by permuting the columns of B1 and B0 , respectively. When one black (resp. white) pixel is shared, we randomly select one matrix from the set C1 (resp. C0 ) and then choose one row of this matrix to a relative shadow. When shadows i1 , i2 ,. . . , ir (r ≥ k) are stacked, these r rows in Bi are OR-ed. The pixel color of the recovered image is proportional to H(V) (the Hamming weight of this OR-ed m-vector V). If H(V) ≥ dB , this gray level is interpreted by the user’s vision as Black and if H(V) ≤ dW the result is interpreted as White, where dB > dW . For r < k, the two collections of r × m matrices obtained from C1 and C0 are the same in the sense that they contain the same matrices with the same frequencies. The contrasts represented by dB and dW were defined in [1], [3], [4]. The first definition was given by Naor and Shamir as αNS = (dB − dW )/m [1]. Others were introduced, αVV = (dB − dW )/(m (2m − dB − dW ))(by Verheul and Van Tilborg) and αES = (dB − dW )/(2m − dB )(by Eisen and Stinson), to further contend with the real contrast of the recovered image [3], [4]. 3.
The Proposed Size-Reduced VSS Schemes
In [1], Naor and Shamir had mentioned that one can handle several pixels to reduce the pixel expansion. In Sect. 3.1, we first introduce a trivial method using the existing C1 and C0 but processing two-pixeled block each time. Finally, the pixel expansion is reduced by half; however this construction brings the line disappearance problem. A refined construction in Sect. 3.2 is proposed to address the problem. The basic concept is to process a two-pixeled block (i.e., , , and ) each time, by four corresponding sets C11 , C00 , C10 and C01 . The first two sets contribute the contrast of the recovered image, while the last two sets determine the clearness of edges between black and white areas. For our new two-pixeled VSS schemes, the formal
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
LETTER
621
contrast and security conditions are redefined as follows. C1(Contrast Condition): For any S of the matrix in C11 (resp. C00 ), the OR-ed V of rows i1 , i2 , . . . , ir , for r ≥ k, satisfies H(V) ≥ dB (resp. ), where dB > dW . H(V) ≤ dW C2(Edge Precision Condition): For any S of the matrix in C10 (resp. C01 ), the OR-ed V of rows i1 , i2 , . . . , ir , for r ≥ k, satisfies (Vn /|C10 |) ≥ PBW (resp. (Vn /|C01 |) ≥ PBW ) where the symbol Vn is defined as follows and 0 < PBW ,PBW ≤ 1. All possible vectors V are w-out-ofm m-tuples, 0 < w < m, and the symbol Vn is used to denote the maximum number of the same w-out-of-m pattern in all possible vectors V. C3(Security Condition): For r < k, the four collections of r × m matrices obtained from C11 , C00 , C11 and C00 are the same in the sense that they contain the same matrices with the same frequencies. 3.1 Construct the Size-Reduced (k, n, m/2) VSS Schemes Using (k, n, m) VSS Schemes Construction 1: Let C1 and C0 be the black and white sets for (k, n, m) VSS schemes. Then the sets C11 = C1 and C00 = C0 , and the sets C10 and C01 can be constructed as the following nine methods: 1) C10 = C01 = C1 ∪ C0 ; 2) C10 = C01 = C1 ; 3) C10 = C01 = C0 ; 4) C10 = C1 and C01 = C0 ; 5) C10 = C0 and C01 = C1 ; 6) C10 = C1 ∪ C0 and C01 = C1 ; 7) C10 = C1 ∪ C0 and C01 = C0 ; 8) C10 = C1 and C01 = C1 ∪ C0 ; 9) C10 = C0 and C01 = C1 ∪ C0 . Theorem 1: The scheme from Construction 1 is a (k, n, m/2) VSS scheme. Proof: First, we prove Construction 1-1. It is obvious that the pixel expansion is now m/2 due to the two-pixeled operation. Since C11 = C1 and C00 = C0 , then dB and in the (k, n, m/2) VSS scheme are dB = dB and dW And, C10 = C1 ∪ C0 , so PBW = (Vn /|C10 |) = dW = dW . m m Max m! , m! 2 dB d W m m ×m!) = Max 1 2 ,1 2 . PW B = d dW B m m Max 1 2 ,1 2 Thus, conditions C1 and dB dW C2 are met. To prove the security condition C3, it is observed that every row of these four sets is the same to the row in C1 and C0 . Thus, the scheme satisfies C3 condition. The proofs of Construction 1-2 ∼ 1-9 are similar. Some measurements, the contrast, the average edge precision and the distinction of line, are defined to evaluate our size-reduced VSS schemes. We herein adopt the concept of Naor-Shamir contrast [1] and the average contrast definition in[5] to define our contrast as αYC =
1 |C11 |
H(V)−
AllS inC 11
m
1
|C00 |
AllS inC 00
H(V)
.
Also, we define the average edge precision P MEAN = (PBW + PW B )/2. The distinctionof black line in white area is DBIW = DC10 −C00 + DC01 −C00 2 (i.e. the distinction between C10 and C00 , and the distinction between C01 and C00 ) and DCi j −Ckl = Ci j − Ci j ∩ Ckl Ci j where “−” is a set difference; the distinction
of white line in black area DWIB = DC10 −C11 + DC01 −C11 2. The average distinction is defined as D MEAN = (DBIW + DWIB )/2. The value αYC shows the contrast between black and white areas. The average edge precision P MEAN determines the clearness of the edge (Notice that if the stacked patterns V are not the same then it will lack consistency and the edge is irregular.). The distinction D MEAN is the value to measure whether we can distinguish the thin lines in the recovered images. The values αYC , P MEAN and D MEAN for Construction 1-1 are calculated as follows: αYC =
1 |C1 |
1
|C11 |
H(V)−
AllS inC11
H(V)−
1
|C00 |
m
1 |C0 | AllS inC 0
AllS inC00
H(V)
H(V)
1 = m (dB −dW ) = m = αNS . (PBW +PW B ) P MEAN = 2 = m m ,1 2 . Max 1 2 dW dB DC10 −C00 = DC01 −C00 = DC10 −C11 = DC01 −C11 (m!) = (2×m!) = 1/2; = 1/2; DBIW = DWIB = (1/2+1/2) 2 (DBIW +DWIB ) = 1/2. D MEAN = 2 AllS inC
All the values PBW , PW B , P MEAN , DBW , DW B and D MEAN for a size-reduced (2, 2, 1) VSS scheme are calculated and shown in Table 1. From Table 1, we successfully reduce the pixel expansion of (2, 2, 2) VSS scheme to 1 with the same contrast as αYC = αNS = 1/2. P MEAN = 1/2 (Construction 1-1, 1-3, 1-7 and 1-9) means that the edge of black and white areas has 50% precision; the value D MEAN = 1/2 shows that we can distinguish the thin lines (line width less than 3 pixels) with 50% precision. Some constructions either improve the precision of the
Table 1
(2, 2, 1) VSS schemes.
PBW PW B P MEAN DBIW DWIB D MEAN Cons. 1-1 Cons. 1-2 Cons. 1-3 Cons. 1-4 Cons. 1-5 Cons. 1-6 Cons. 1-7 Cons. 1-8 Cons. 1-9
1/2 1 1/2 1 1/2 1/2 1/2 1 1/2
1/2 1 1/2 1/2 1 1 1/2 1/2 1/2
1/2 1 1/2 3/4 3/4 3/4 1/2 3/4 1/2
1/2 1 0 1/2 1/2 3/4 1/4 3/4 1/4
1/2 0 1 1/2 1/2 1/4 3/4 1/4 3/4
1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
αYC = αNS 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.2 FEBRUARY 2006
622
Fig. 1
The recovered images for nine size-reduced (2, 2, 1) VSS schemes.
edge or the distinction of the thin lines. Constructions 12, 1-4, 1-5, 1-6 and 1-8 improved the average precision of edge (larger than 50% precision), i.e. the recovered images will have more regular edges. Moreover, Constructions 1-2, 1-6 and 1-8 have the better recognition of the black line on the white area since DBIW > DWIB ; Constructions 1-3, 1-7 and 1-9 have the better recognition of the white line on the black area since DBIW < DWIB ; Constructions 1-1, 1-4 and 1-5 have half distinction on both black and white areas. All of these nine constructions have 50% average distinction. We use a test pattern, four verticals and one horizontal of one pixel in width, to test the size-reduced (2, 2, 1) VSS scheme; the left background is white and the right is black. Figures 1(a)–(i) are the recovered images for Construction 1-1∼Construction 1-9, respectively. It is observed that the recovered images are consistent with the values PBW , PW B , P MEAN , DBW , DW B , and D MEAN . Figure 1(a) shows that the verticals always display because the two-pixeled blocks, (,), are represented as () or () with 50% probability. Figures 1(c)–(e), (g) and (i) show that the black verticals may disappear since at least one of two-pixeled blocks, (,), is represented as () for Construction 1-3∼1-5, Construction 1-7 and Construction 1-9. In Fig. 1(d), the black (resp. white) verticals will disappear when the dealer only just processes () (resp. ()) and the vertical lines will remain when processing () (resp. ()). For observing the edge irregularity, Fig. 1(b) has the best regularity because the average edge precision P MEAN = 1. To confirm our definition of contrast, it is observed that the left white region and the right black region for all the recovered images, have the same contrast like the original (2, 2, 2) VSS scheme. From Fig. 1, Construction 1-1 does not have the line
disappearance problem but it has the problem of edge irregularity. The next section shows how to retain the line display and meantime improve the edge regularity by adding an extra sub pixel. 3.2 A Solution for the Thin Line Disappearance Problem for the Size-Reduced VSS Schemes In this section, we propose the size-reduced (2, 2, (m + 1)/2) VSS schemes, in order to solve the thin line disappearance problem (see Fig. 1). The enhanced construction is shown below. Construction 2: Let C1 and C0 be the sets for (2, 2, m) VSS scheme. Define the operation ⊗ as appending a column vector L to all matrices in a set C from left (L ⊗ C) or right (C ⊗ L). Then the sets for the (2, 2, (m + 1)/2 ) VSS scheme are constructed as follows: 1 0 1 0 ⊗ C1 , ⊗ C 1 , C1 ⊗ , C1 ⊗ ; C11 = 0 1 0 1 1 0 1 0 C00 = ⊗ C0 , ⊗ C 0 , C0 ⊗ , C0 ⊗ ; 1 0 1 0 1 0 1 0 C10 = ⊗ C0 , ⊗ C 0 , C1 ⊗ , C1 ⊗ ; 0 1 1 0 1 0 1 0 C01 = ⊗ C1 , ⊗ C 1 , C0 ⊗ , C0 ⊗ . 1 0 0 1 Theorem 2: The scheme from Construction 2 is a sizereduced (2, 2, (m + 1)/2)VSS scheme with the contrast (dB − dW − 1/2)/(m + 1); meantime it can solve the thin line disappearance problem. Proof: It is easy to verify that C1, C2 and C3 conditions are
LETTER
623
met. To prove that this scheme can solve the thin line disappearance problem, there is always one black sub pixel in the left of the stacked result in C10 set for two-pixeled block (), and one black sub pixel in the right of the stacked result in for two-pixeled block (). Thus, we do not have the line disappearance. The pixel expansion is obviously (m + 1)/2. By the contrast definition, we have
αYC = =
(dB +1)×4|C1 | dW ×2|C0 |+(dW +1)×2|C0 | − 4|C 1 | 4|C0 | (m+1) (dB − dW − 1/2)/(m + 1).
The proof is completed.
(a) The recovered image. Fig. 2
Example 1: Construct a (2, 2, 1.5) VSS scheme from Naor-Shamir (2, 2, 2) VSS scheme, by using Construction 2. From Construction 2, we have 1 10 01 0 10 ⊗ , , ⊗ , C11 = 0 01 10 1 01 01 10 01 1 10 01 , , ⊗ , , 10 01 10 0 01 10 0 110 101 010 001 ⊗ = , , , , 1 001 010 101 110 101 011 100 010 , , , . 010 100 011 101 Three other sets are obtained as follows: 110 101 010 001 101 C00 = , , , , , 110 101 010 001 101 011 100 010 , , ; 011 100 010 110 101 010 001 101 C10 = , , , , , 010 001 110 101 011 011 100 010 , , ; 101 010 100 110 101 010 001 101 C01 = , , , , , 101 110 001 010 100 011 100 010 , , . 010 101 011 The values αYC , P MEAN and D MEAN are calculated as follows: = 1/2; αYC = ((3×8)/8−(2×4+1×4)/8) 3 PBW = PW B = 1/2, P MEAN = (1/2+1/2) = 1/2; 2 (6/8+6/8) = 5/8, D = = 6/8, D MEAN = DBIW = (5/8+5/8) WIB 2 2 (5/8+6/8) = 11/16. 2 Figure 2(a) is the recovered image for the (2, 2, 1.5) VSS schemes using Construction 2. When compared with Figs. 1(b)–(i), the black and white verticals always display (better than Construction 1-2∼1-9) because Construction 2 can solve the line disappearance problem. The reason is described as follows. Figure 2(b) shows all possible stacked patterns for Example 1. There is always one black sub pixel in the left of
(b) All possible stacked patterns.
The size-reduced (2, 2, 1.5) VSS scheme.
the stacked result for two-pixeled block () and one black sub pixel in the the right of the stacked result for two-pixeled block (); this will make the verticals clearer than the Construction 1-1 and does not have the line disappearance problem. Meantime, the stacked results for two-pixeled block () are all three black (3B) sub pixels and the stacked results for two-pixeled block () are four 1B2 W and four 2B1 W patterns. Thus, the contrast of the recovered image is assured. Also, since the value D MEAN = 11/16 is improved, and thus the clearer edge is better than Construction 1. 4.
Experimental Results and Comparison
4.1 Experimental Results One standard image House (from USC-SIPI image database) is used to test the effects of edges and image qualities (see Fig. 3). Because the sets C10 and C01 in Construction 1 are constructed by C1 , C0 or their union, thus the qualities of the recovered image are compromised when the processed patterns are () and (). So, there is no definite answer which method is the best way to share the secret image; it depends on the pattern of secret image. For example, from these figures, it is observed that Construction 1-5 has the best image quality. 4.2 Comparison In this section we compare three size-reduced VSS schemes, Kuwakado-Tanaka scheme [5], Ito-Kuwakado-Tanaka (or Yang) scheme [6], [7]. Note that Ito-Kuwakado-Tanaka scheme in [6], in fact, is Yang’s probabilistic VSS scheme. Suppose that our (k, n, mYC ) VSS scheme, (k, n, mKT ) Kuwakado-Tanaka VSS scheme and (k, n, mIKT ) ItoKuwakado-Tanaka VSS scheme are all constructed from the traditional (k, n, m) VSS scheme. For our proposed scheme and Kuwakado-Tanaka VSS scheme, they have the same contrasts αYC = αKT = αNS . For the probabilistic VSS scheme, the contrast is αIKT = (p0 − p1 ) (dB /m) − (dW /m) = ((dB − dW )/m) = αNS when use Naor-Shamir approach, where p0 and p1 are defined in [7]. Thus all three contrasts are equal to αNS . The comparison is shown in Table 2 and our scheme is better than Kuwakado-Tanaka scheme, i.e. the pixel expansion of our scheme is no larger than that in Kuwakado-
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.2 FEBRUARY 2006
624
Fig. 3 Table 2
Comparison of three size-reduced VSS schemes. m
n=3 n=4 n=4 n=5 n=5 n=5 n=5
k=2 k=2 k=3 k=2 k=2 k=3 k=4
The recovered images for the (2, 2, 1) and (2, 2, 1.5) schemes.
3 6 6 5 10 8 15
αYC = αKT = αIKT mKT [5] mIKT [6], mYC = m/2 [7] (The proposed = αNS scheme) 1/3 2 1 1.5 1/3 3 1 3 1/6 3 1 3 1/5 4 1 2.5 3/10 5 1 5 1/8 5 1 4 1/15 8 1 7.5
a few simple calculations now. In fact, the idea that uses the simple computation to get the perfect reconstruction is also analogous to the VSS schemes with the perfect reconstruction [10], [11]. Because the judgment of image contrast is very subjective, we performed an experiment with these schemes to compare the images more fairly. The recovered image was evaluated by 10 people and gave a mean opinion score (MOS) to measure the image quality. Considering the situations of “calculation” and “MOS”, we give a more reasonable compared table, Table 3, for these three size-reduced VSS schemes. 5.
Tanaka schemes. Moreover, our scheme can be constructed from any (k, n, m) VSS schemes to achieve further reduction, but Kuwakado-Tanaka size-reduced VSS scheme does not work for reducing the shadow size of (k, k, m) VSS schemes. The probabilistic scheme has no pixels expansion; however it has a problem in the recognition of the small areas (see the detail analysis in [7]). Figure 4 shows the recovered images for (2, 3, 2) Kuwakado-Tanaka scheme, (2, 3, 1) Ito-Kuwakado-Tanaka scheme and our proposed (2, 3, 1.5) VSS scheme (the scheme of Sect. 3.1). All of them are constructed based on the Shamir-Naor (2, 3, 3) VSS scheme. It seems that our recovered image has the better visual image quality than other two schemes. For our proposed scheme and the probabilistic scheme [6], [7], some pixels in the secret image are lost permanently. But consider examining the pixel patterns in the shadows, the secret pixels can be correctly recovered for Kuwakado-Tanaka VSS scheme. However, it needs
Concluding Remarks
In this paper, two new size-reduced VSS schemes, the (k, n, m/2) scheme and the (2, 2, (m + 1)/2) scheme, from the traditional (k, n, m) scheme and (2, 2, m) scheme are proposed. When compared to the Kuwakado-Tanaka scheme [5] and the probabilistic scheme [6], [7], our size-reduced scheme works better either according to the pixel expansion or the image quality. It is shown that the edges in our recovered images are a little irregular but the recovered images have the same contrast as the Naor-Shamir scheme. It is indefinite that which method in Construction 1 has the best performance, i.e. the clear edge and the high contrast, because it depends on the arrangement of black and white pixels in the secret image. So how to transform a gray image to a black and white secret image for our construction needs careful consideration.
LETTER
625
Fig. 4 Table 3
The recovered images for three size-reduced VSS schemes. Comparison of three size-reduced (2, 3) VSS schemes.
K-T scheme [5] I-K-T scheme [6], [7] The proposed scheme Pixel expansion 2 1 1.5 The visual image quality of recovered im➂ ➁ ➀ age* Perfect reconstruction of the secret pixel YES NO NO Weakness Does not reduce the (k, k, The problem of recogniz- The problem of thin line m) VSS schemes. ing the small areas. disappearance. ∗:The recovered image was evaluated by 10 people and gave a mean opinion score to measure the image quality; ➀ means that it has the superior MOS, i.e. the best quality, and ➂ has the less MOS.
Acknowledgments The authors would like to express their deep gratitude to the reviewers for the valuable and usefull comments which improved the paper. References [1] M. Naor and A. Shamir, “Visual cryptography,” Advances in Cryptology-EUROCRYPT’94, Lect. Notes Comput. Sci., no.950, pp.1–12, 1995. [2] G. Ateniese, C. Blundo, A. De Santis, and D.R. Stinson, “Visual cryptography for general access structures,” ECCC, Electronic Colloquium on Computational Complexity, TR96-012, 1996. [3] E.R. Verheul and H.C.A. Van Tilborg, “Constructions and properties of k out of n visual secret sharing schemes,” Des. Codes Cryptogr., vol.11, no.2, pp.179–196, 1997. [4] P.A. Eisen and D.R. Stinson, “Threshold visual cryptography schemes with specified whiteness,” Des. Codes Cryptogr., vol.25,
no.1, pp.15–61, 2002. [5] H. Kuwakado and H. Tanaka, “Size-reduced visual secret sharing scheme,” IEICE Trans. Fundamentals, vol.E87-A, no.5, pp.1193– 1197, May 2004. [6] R. Ito, H. Kuwakado, and H. Tanaka, “Image size invariant visual cryptography,” IEICE Trans. Fundamentals, vol.E82-A, no.10, pp.2172–2177, Oct. 1999. [7] C.N. Yang, “New visual secret sharing schemes using probabilistic method,” Pattern Recognit. Lett., vol.25, no.4, pp.481–494, 2004. [8] S. Cimato, R. De Prisco, and A. De Santis, “Probabilistic visual cryptography schemes,” The Computer Journal, to appear, available at http://comjnl.oxfordjournals.org. [9] C.N. Yang and T.-S. Chen, “Size-adjustable visual secret sharing schemes,” IEICE Trans. Fundamentals, vol.E88-A, no.9, pp.2471– 2474, Sept. 2005. [10] R. Lukac and K.N. Plataniotis, “Bit-level based secret sharing for image encryption,” Pattern Recognit., vol.38, no.5, pp.767–772, 2005. [11] R. Lukac and K.N. Plataniotis, “Colour image secret sharing,” Electron. Lett., vol.40, no.9, pp.529–530, 2004.
