Visual secret sharing schemes for plural secret images

Abstract

Visual secret sharing schemes w i t h q plural secret images, for short VSS-q-PI schemes, are studied for general access structures and gray-scale a n d / o r color secret images.Let N = { 1 , 2 , . . . , n } and 2N be the set of n shares and the family of all the subsets of N, respectively. We suppose that all secret images are encrypted at once into n shares. Each secret image is denoted by SZ('), i = 1 , 2 , . . . , q, which has the same size. Let l?g:al, i = 1 , 2 , . . . , q, be the family of qualzfied sets for the i-th secret zmage, and let Forb be the family of forbidden sets. Then, any set in r:ial can decrypt the i-th secret image SI(^) while any set in r F o r b cannot gain any information of any secret image. w e call r = ({r$;a,);=l, r F o r b ) an access structure for q secret amages. Note that each satisfy the following monotonaczty.Therefore, for each I'gial, the mammal qualified sets of the i-th secret image can be defined as follows.It is worth noting that the VSS-1-PI scheme with the access structure (rd:)al, r F o r b ) coincides with the usual VSS scheme for one secret image. For a given set X & Af, the set of indices of the secret images that can be decr pted from X , say Z ( X ) , is represented as Z ( X ) = { i : X E r:;,,, 1 5 i 5 q } . For instance, Z ( N ) = { 1 , 2 , . . . , q } , and Z ( F ) = 0 for any 3 E r F o r b . Furthermore, we consider two categories A1 and A2 for the access structures of the VSS-q-PI schemes. Definition 1 Define that N(2) = UP(i))-Er(L)-Q(+, QWl (i) A n access structure r is in A 1 i f it satisfies Q(a)-n Ncz', # 0 for any i and i' such that Q(a)-E r$;al and (ii) A n access structure r is in A 2 i f it satisfies that Q ( 2 ) -n 0Let E be the set of colors used in all shares. In this paper, colors in E are expressed by lowercase san-serif fonts and the subtractive mixture of colors which corresponds to overlapping the colors printed on transparencies is represented by U. For example, the fact that the subtractive mixture of yellow and cyan generates green is represented as y U c = g. Furthermore, in the framework of the VSS-q-PI scheme, the colors of decrypted images are represented by the set of m subpixels which expresses the gray-scales or colors. For mdimensional row vectors of colors ?! = [ x1 x2". xm 1, y = [ y1 y p ... y m ] where x,,yi E E , we define an operation U as z U g = [ X I U y1 x2 U y2 ... xm U ym 1, which i' E T(Q(+)\{i} l . N(2') # 0 for any i and i'(# i). -m m 'For a set S C N , s means the complement of S. represents the subtractive mixtures of two pixels with m subpixels. For a matrix T = ' [ z 1 z 2 ... zn] E E"", where t means the transpose of the matrix, and an arbitrary set X = { U I , U Z , . .. ,U,-} C N , an 1x1 x m matrix TUX] is defined as TBX] = t[z,,z,, . . . z U T ] E E l X t m . Then, the colors obtained by stacking the %-th shares, i = 1,2,. . . , r , are represented by the mapping ,O : EIXlm + E" defined by Now we define a color matrix. Let V(') be the set of grayscales or colors on the secret image SI(') represented by m s...

Visual Cryptography Schemes with Dihedral Group Access Structure for Many Images

“…Using this example, we explain the rule of determining the color of each subpixel. In order to do so easily, we rearrange all the subregions so that each orbit consists of the subregions lying on the same position (see (2)). Moreover, these orbits are partitioned into four subsets S(1), S(α), S(β), S(βα).…”

“…Moreover, these orbits are partitioned into four subsets S(1), S(α), S(β), S(βα). Of course, there is a bijection between the subregions of (1) and those of (2), and thus if we determine the colors of subregions of (2), the colors of original subregions are determined. Let τ ∈ D 4 .…”

“…For any element ρ ∈ D 4 , choose one subregion sub(ρ, x, τ ) from ρ(x) ∩ S(τ ) so that the four subregions sub(ρ, x, τ ), ρ ∈ {1, α, β, βα}, are contained in four distinct orbits of S(τ ). Furthermore, for any fixed ρ, we can choose sub(ρ, x, τ ) so that it is placed at the same position in every S(τ ), τ ∈ D 4 (see (2)). …”

“…, n}, and {Image(A) | A ∈ F} be a set of |F | different secret images. Then we can make n shares S(1), S (2), . .…”

“…, S(n) so that for any element A ∈ F, a stack of the shares in {S(i) | i ∈ A} reconstructs the secret image Image(A), and we cannot get any information on Image(B) from the set of transparencies if B ⊆ A. This kind of VCS for many secret images has been studied in some papers including [3], [6], [2].…”