A {k, n}-secret Sharing Scheme for Color Images

Transcription

1 A {k, n}-seret Sharing Sheme for Color Images Rastislav Luka, Konstantinos N. Plataniotis, and Anastasios N. Venetsanopoulos The Edward S. Rogers Sr. Dept. of Eletrial and Computer Engineering, University of Toronto, 10 King s College Road, Toronto, M5S 3G4, Canada {lukar,kostas,anv}@dsp.utoronto.a Abstrat. This paper introdues a new {k, n}-seret sharing sheme for olor images. The proposed method enrypts the olor image into n olor shares. The seret information is reovered only if the k (or more) allowed shares are available for deryption. The proposed method utilizes the onventional {k, n}-seret sharing strategy by operating at the bit-levels of the deomposed olor image. Modifying the spatial arrangements of the binary omponents, the method produes olor shares with varied both the spetral harateristis among the RGB omponents and the spatial orrelation between the neighboring olor vetors. Sine enryption is done in the deomposed binary domain, there is no obvious relationship in the RGB olor domain between any two olor shares or between the original olor image and any of the n shares. This inreases protetion of the seret information. Inverse ryptographi proessing of the shares must be realized in the deomposed binary domain and the proedure reveals the original olor image with perfet reonstrution. 1 Introdution Visual ryptography [2],[3],[4],[8],[11] is a popular ryptographi tool used for protetion of sanned douments and natural digital images whih are distributed via publi networks. These tehniques are based on the priniple of sharing seret information among a group of partiipants. The shared seret an be reovered only when a oalition of willing partiipants are polling their enrypted images, the so-alled shares, together. Seret sharing shemes are usually termed visual sine the seret (original) information an be diretly revealed from staked shares (e.g realized as transparenies) through simple visual inspetion, without any omputer-guided proessing [9],[11]. A {k, n}-threshold visual ryptography sheme [5],[6],[11] often alled {k, n}- visual seret sharing (VSS), is used to enrypt an input image by splitting the original ontent into n, seemingly random, shares. To reover the seret information, k (or more) allowed shares must be staked together. M. Bubak et al. (Eds.): ICCS 2004, LNCS 3039, pp , Springer-Verlag Berlin Heidelberg 2004

2 A {k, n}-seret Sharing Sheme for Color Images 73 (a) (b) () Fig. 1. Visual ryptography: (a) original binary image, (b,) share images, derypted, output image 2 {k, n}-seret Sharing Sheme Due to its algorithmi nature, onventional visual ryptography shemes operate on a binary input (Fig. 1) [10]. Assuming a K 1 K 2 binary image (blak and white image with 0 values denoting the blak and 1 values denoting the white), eah binary pixel r (i,j) determined by spatial oordinates i =1, 2,..., K 1 and j =1, 2,..., K 2 is replaed via an enryption funtion f e ( ) with a m 1 m 2 blok of blak and white pixels in eah of the n shares. Repeating the proess for eah input pixel, a K 1 K 2 input binary image is enrypted into n binary shares eah one with a spatial resolution of m 1 K 1 m 2 K 2 pixels. Sine the spatial arrangement of the pixels varies from blok to blok, the original information annot be revealed without aessing a predefined number of shares. Let as assume a basi {2, 2}-threshold struture whih is the basi ase designed within the {k, n}-vss framework [1],[7]. Assuming for simpliity a basi struture with 2 2 bloks s 1 =[s (2i 1,2j 1),s (2i 1,2j),s (2i,2j 1),s (2i,2j) ] S 1 and s 2 =[s (2i 1,2j 1),s (2i 1,2j),s (2i,2j 1),s (2i,2j) ] S 2, the enryption proess is defined as follows: f e (r (i,j) )= { [s1, s 2 ] T C 0 for r (i,j) =0 [s 1, s 2 ] T C 1 for r (i,j) =1 (1) where C 0 and C 1 are the sets obtained by permuting the olumns of the n m 1 m 2 basis matries A 0 and A 1, respetively [10]. Sine m 1 m 2 represents the fator by whih eah share is larger than the original image, it is desirable to make m 1 m 2 as small as possible. In the ase of the {2, 2}-VSS the optimal hoie m 1 and m 2 leads to m 1 = 2 and m 2 = 2 resulting in 2 2 bloks s 1 and s 2.

3 74 R. Luka, K.N. Plataniotis, and A.N. Venetsanopoulos (a) (b) (e) () Fig. 2. Halftoning-based seret sharing of olor images: (a) original olor image, (b) halftone image obtained using Floyd-Steinberg filter [12], (,d) share images, (e) derypted, output image Assuming the {2, 2}-VSS the sets { } C 0 = [ 0,1,0,1 1,0,1,0 0,0,1,1 1,1,0,0 1,0,0,1 0,1,1,0 1,0,1,0 0,1,0,1 1,1,0,0 0,0,1,1 0,1,1,0 1,0,0,1 ] (2) { } C 1 = [ 0,1,0,1 1,0,1,0 0,0,1,1 1,1,0,0 1,0,0,1 0,1,1,0 0,1,0,1 1,0,1,0 0,0,1,1 1,1,0,0 1,0,0,1 0,1,1,0 ] (3) inlude all matries obtained by permuting the olumns of the 2 4 basis matries A 0 and A 1, respetively [10],[11]. The basi matries onsidered here are defined as follows: [ ] [ ] A 0 =,A = (4) 0101 If a seret pixel is white, i.e. r (i,j) = 1, then eah pixel in s 1 is equivalent to eah pixel in s 2, and thus, [s 1, s 2 ] T an be any member of set C 1. If a seret pixel is blak, i.e. r (i,j) = 0, then eah pixel in s 1 should omplement eah pixel in s 2 and thus, [s 1, s 2 ] T should be seleted from set C 0. The hoie of [s 1, s 2 ] T is guided by a random number generator, whih determines the random harater of the shares. The derypted blok is produed through a deryption funtion f d ( ). In the ase of the {2, 2}-sheme based on the basis matries of (4), f d ( ) an be defined as follows: { s1 for s y 2 2 = f d (s 1, s 2 )= 1 = s 2 (5) [0, 0, 0, 0] for s 1 s 2

4 A {k, n}-seret Sharing Sheme for Color Images 75 (a) (b) (e) () Fig. 3. Halftoning-based seret sharing of olor images: (a) original olor image, (b) halftone image obtained using Floyd-Steinberg filter [12], (,d) share images, (e) derypted, output image where s 1 =[s (u,v),s (u,v+1),s (u+1,v),s (u+1,v+1) ] S 1 and s 2 =[s (u,v),s s (u+1,v),s (u,v+1), (u+1,v+1) ] S 2, for u = 1, 3,..., 2K 1 1 and v = 1, 3,..., 2K 2 1, are 2 2 share bloks whih are used to reover the output blok y 2 2 = y (u,v),y (u,v+1),y (u+1,v),y (u+1,v+1) as s (u,v) or blak pixels desribed as [0, 0, 0, 0]. The appliation of a onventional {k, n}-vss sheme to a K 1 K 2 natural image requires halftoning [7],[10]. The image is first transformed into a K 1 K 2 halftone image by using the density of the net dots to simulate the intensity levels [12]. Applying the proedure for eah olor hannel of the original image ( Fig. 2a) independently, eah olor hannel of the halftone image (Fig. 2b) is a binary image and thus appropriate for the VSS. Assuming {2, 2}-VSS, the two olor shares obtained by the proedure are depited in Fig. 2,d. Figure 2e shows the 2K 1 2K 2 derypted image (result) obtained by staking the two shares together.

5 76 R. Luka, K.N. Plataniotis, and A.N. Venetsanopoulos Visual inspetion of both the original image ( Fig. 1a and Fig. 2a) and the reovered image (Fig. 1d and Fig. 2e) indiates that: i) the derypted image is darker, and ii) the input image is of quarter size ompared to the derypted output. Moreover, the derypted olor image depited in Fig. 2e ontains a number of olor artifats due to nature of the algorithm. To end this, the onventional {k, n}-threshold visual ryptography i) annot provide perfet reonstrution, either in terms of pixel intensity or spatial resolution, and ii) is not appropriate for real-time appliations. Figure 3 shows the images obtained using the onventional {2, 2}-seret sharing sheme applied to the image with the different olor senario ompared to Fig. 2. It an be again observed that the derypted image depited in Fig. 3e ontains shifted olors whih often prohibit orret pereption of fine image details. Is has to be mentioned that the halftoning-based {k, n}-visual seret sharing shemes are the most popular hoie for natural image seret sharing. Another seret sharing approah for olor images is based on mean olor-mixing [8]. However, this method is not appropriate for pratial appliations due to signifiant redution of olor gamut and the extreme inrease in the spatial resolution of the shares. Other works, e.g. [9],[13] deals with analytial onstrution of the seret sharing shemes for olor images. 3 {k, n}-color Seret Sharing Let x : Z 2 Z 3 be a K 1 K 2 Red-Green-Blue (RGB) olor image representing a two-dimensional matrix of the three-omponent olor vetors (pixels) x (i,j) = [x (i,j)1,x (i,j)2,x (i,j)3 ] loated at the spatial position (i, j), for i =1, 2,..., K 1 and j =1, 2,..., K 2. Assuming that desribes the olor hannel (i.e. = 1 for Red, = 2 for Green, and = 3 for Blue) and the olor omponent x (i,j) is oded with B bits allowing x (i,j) to take an integer value between 0 and 2 B 1, the olor vetor x (p,q) an be equivalently expressed in a binary form as follows: x (i,j) = B b=1 xb (i,j) 2B b (6) where x b (i,j) =[xb (i,j)1,xb (i,j)2,xb (i,j)3 ] {0, 1}3 denotes the binary vetor at the b-bit level, with b = 1 denoting the most signifiant bits (MSB). 3.1 Enryption If the -th omponent of the binary vetor x b (i,j) is white (xb (i,j) = 1), enryption is performed through [s 1, s 2 ] T C 1 replaing x b (i,j) by binary bloks s 1 and s 2 in eah of the two shares. Otherwise, the referene binary omponent is blak (x b (p,q) = 0), and enryption is defined via [s 1, s 2 ] T C 0. This forms an enryption funtion defined as follows: f e (x b (i,j) )= { [s1, s 2 ] T C 0 for x b (i,j) =0 [s 1, s 2 ] T C 1 for x b (i,j) =1 (7)

6 A {k, n}-seret Sharing Sheme for Color Images 77 (a) (b) () Fig. 4. Proposed {2, 2}-seret sharing sheme for olor images: (a) original olor image, (b,) share images, derypted, output image By replaing the binary omponents x b (i,j) with binary bloks s 1 and s 2 for one partiular b, the proess generates two 2K 1 2K 2 vetor-valued binary shares S1 b and S2, b respetively. A random number generator guides the hoie of [s b 1, s b 2] T and determines the random harater of S1 b and S2. b Thus, the proess modifies both the spatial orrelation between spatially neighboring binary vetors s b (u,v) = [s b (u,v)1,s b (u,v)2,s b (u,v)3 ] Sb 1 or s b (u,v) = [s b (u,v)1,s b (u,v)2,s b (u,v)3 ] Sb 1, for u = 1, 2,..., 2K 1 and v =1, 2,..., 2K 2, and the spetral orrelation among omponents s b (u,v) or s b (u,v), for =1, 2, 3, of the individual binary vetors s b (u,v) or s b (u,v), respetively. Bit-level staking of the enrypted bit-levels produes the olor vetors s (u,v) S 1 and s (u,v) S 2 as s (u,v) = B b=1 s b (u,v) 2B b and s (u,v) = B b=1 s b (u,v) 2B b (8) Due to random proessing taking plae at the bit-levels, S 1 and S 2 ontain only random, olor noise like information (Fig. 4b,). Sine enryption is realized in the deomposed binary vetor spae, no detetable relationship between the original olor vetors x (p,q) and the olor noise of S 1 or S 2 an be found in the RGB olor domain. This onsiderably inreases seurity and prevents unauthorized deryption through brute-fore enumeration. 3.2 Deryption The deryption proedure is designed to satisfy the perfet reonstrution property. The original olor data must be reovered from the olor shares S 1 and S 2 using inverse algorithmi steps. Therefore, the deryption proedure is applied to the deomposed binary vetor arrays of the olor shares. Assuming that (i, j), for i =1, 2,..., K 1 and j =1, 2,..., K 2, denotes the spatial position in the original image and denotes the olor hannel, the orresponding 2 2 binary share bloks are s b = {s b (2i 1,2j 1),s b (2i 1,2j),s b (2i,2j 1),s b (2i,2j) } and s b = {s b (2i 1,2j 1),s b (2i 1,2j),s b (2i,2j 1),s b (2i,2j)}. Based on the arrangements of the basis matries A 0 and A 1 in (4) used in the proposed {2, 2}-seret sharing

7 78 R. Luka, K.N. Plataniotis, and A.N. Venetsanopoulos (a) (b) () Fig. 5. Proposed {2, 2}-seret sharing sheme for olor images: (a) original olor image, (b,) share images, derypted, output image sheme, if both bloks are onsistent, i.e. s b = s b, the derypted original bit x b (i,j) is assign white, i.e. xb (i,j) = 1. Otherwise, the bloks are inonsistent, i.e. s b s b and the original bit is reovered as blak, i.e. x b (i,j) omparison forms the following deryption funtion x b (i,j) = f d(s b, s b )= { 1 for s b 0 for s b = s b s b = 0. This logial whih is used to restore the binary vetors x b (i,j). The proedure ompletes with the bit-level staking (6) resulting in the original olor vetor x (i,j). Figure 4d shows the derypted olor output. Sine the proposed method satisfies the perfet reonstrution property, the output image is idential to the original depited in Fig 4a. Note that perfet reonstrution is demonstrated also in Fig 5, whih depits two full olor shares (Fig 5b,) and the derypted output ( Fig 5d) obtained by the proposed {2, 2}-seret sharing sheme applied to the test image Atlas ( Fig 5a). It has to be mentioned that (9), whih is defined for a {2, 2}-sheme, an be more generally desribed as follows: { 1 for [s b, s b ] T C 1 o b (i,j) = f d(s b 1, s b 2)= 0 for [s b, s b ] T (10) C 0 This onept an be further generalized for the share bloks {s b, s b, s b,...} defined in the speifi {k, n}-threshold shemes. 4 Conlusion A {k, n} seret sharing sheme with perfet reonstrution of the olor inputs was introdued. The method ryptographially proesses the olor images replaing (9)

8 A {k, n}-seret Sharing Sheme for Color Images 79 the bit omponents with a blok of bits for eah of the n shares. By separate enrypting eah bit plane of the deomposed olor image, the method produes olor shares with varied spetral and spatial harateristis. Sine enryption is realized in the deomposed binary domain, the proedure inreases protetion against attaks performed in the RGB olor domain. This makes the method attrative for seure transmission over the untrusted publi hannel. Moreover, the perfet reonstrution property allows to reveal the original olor image without visual impairments or olor shifts. Referenes 1. Adhikari, A., Sikdar, S.: A new (2,n)-visual threshold sheme for olor images. Pro. INDOCRYPT 2003, LNCS 2904 (2003) Ateniese, G., Blundo, C, de Santis, A., Stinson, D.G.: Visual ryptography for general aess strutures. Information and Computation 129 (1996) Ateniese, G., Blundo, C, de Santis, A., Stinson, D.G.: Extended apabilities for visual ryptography. Theoretial Computer Siene 250 (2001) Droste, S.: New results on visual ryptography. Pro. Advanes in Cryptology - CRYPTO 96, LNCS 1109 (1996) Eisen, P.A., Stinson, D.R.: Threshold visual ryptography shemes with speified levels of reonstruted pixels. Design, Codes and Cryptography 25 (2002) Hofmeister, T., Krause, M., Simon, H.U.: Contrast optimal k out of n seret sharing shemes in visual ryptography. Theoretial Computer Siene 240 (2000) Hou, J.C.: Visual ryptography for olor images. Pattern Reognition 36 (2003) Ishihara, T., Koga, H.: A visual seret sharing sheme for olor images based on meanvalue-olor mixing. IEICE Trans. Fundamentals E86-A (2003) Koga, H., Iwamoto, M., Yakamoto, H.: An analyti onstrution of the visual seret sharing sheme for olor images. IEICE Trans. Fundamentals E84-A (2001) Lin, C.C., Tsai, W.H.: Visual ryptography for gray-level images by dithering tehniques. Pattern Reognition Letters 24 (2003) Naor, M., Shamir, A.: Visual Cryptography. Pro. EUROCRYPT 94, LNCS 950 (1994) Ulihney, R.A.: Dithering with blue noise. Proeedings of the IEEE 76 (1988) Yang, C.N.: A note on olor visual enryption. Journal of Information Siene and Engineering 18 (2002)