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Optics & Laser Technology 44 (2012) 538–548
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology
0030-39
doi:10.1
n Corr
E-m
journal homepage: www.elsevier.com/locate/optlastec
A secret image sharing based on neighborhood configurationsof 2-D cellular automata
Jun Jin n, Zhi-hong Wu
College of Computer Science, Sichuan University, Chengdu 610065, China
a r t i c l e i n f o
Article history:
Received 7 March 2011
Received in revised form
23 August 2011
Accepted 26 August 2011Available online 16 September 2011
Keywords:
Cellular automata
Secret image sharing
Neighborhood configuration
92/$ - see front matter Crown Copyright & 2
016/j.optlastec.2011.08.023
esponding author.
ail address: [email protected] (J. Jin).
a b s t r a c t
The main goal of this work is to research how neighborhood configurations of two-dimensional cellular
automata (2-D CA) can be used to design secret sharing schemes, and then a novel (n, n)-threshold
secret image sharing scheme based on 2-D CA is proposed. The basic idea of the scheme is that the
original content of a 2-D CA can be reconstructed following a predetermined number of repeated
applications of Boolean XOR operation to its neighborhood. The main characteristics of this new scheme
are: each shared image has the same size as the original one; the recovered image is exactly the same as
the secret image, i.e., there is no loss of resolution or contrast; and the computational complexity is
linear. Simulation results and formal analysis demonstrate the correctness and effectiveness of the
proposed sharing scheme.
Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.
1. Introduction
Confidentiality of information is an essential necessity in theinformation era, and cryptography is one of its protecting tools.Nowadays images are widely used in multimedia applications,and sometimes, these image data contain private or confidentialinformation, therefore, it is important to protect them. Because ofhuge amount of data of image, the common cryptosystem havesome problems like being time-consuming. Cellular Automata [1](CA) are discrete dynamical systems, which simulate complexbehaviors by means of simple computational models. CA due tothe synchronous update mechanism for its individual compo-nents (cells) and ease hardware implementation is a goodcandidate for image cryptosystems. In 1985, Wolfram [2] firstused iterations of a CA with rule 30 to generate key stream instream cipher cryptography. In recent years, a lot of researcheshave been done in image encryption based on CA [3–7].
A secret sharing scheme is a method, which allows sharing asecret among a set of users in such a way that only qualifiedsubsets of these users can recover the secret. The basic idea insecret sharing schemes is to divide the secret into a fixed numberof pieces, called shares or shadows, which are distributed amongthe participants so that the pooled shares of certain subsetsof users allow the reconstruction of the secret. Secret sharingschemes were independently introduced by Shamir [8] andBlakley [9]. The most extended (k, n)-threshold schemes (krn),due to Shamir, which is based on polynomial interpolation, andBlakley, which is based on the intersection of affine hyperplanes.
011 Published by Elsevier Ltd. All
The best polynomial evaluation and interpolation algorithms havethe computational complexity O(nlog2n). The design of secretsharing schemes by means of cellular automata is a recent event,especially the secret image sharing [10–14]. However, all thesesharing schemes have the common ground that they all use aspecial type of CA—reversible cellular automata with memory. Inthis paper, a new sharing scheme for secret images using a generallinear 2-D CA without memory is proposed. The generation of theshares from the secret images is based on the neighborhood of CAand Boolean XOR operation. Application of Boolean XOR to theneighborhood configurations of a 2-D CA of R�R dimensions hasthe ability to reconstruct the 2-D CA after R/2or R repetitions. Theinvestigation in this paper is the first one that exploits theneighborhood configurations of a general linear 2-D CA in secretimage sharing. The proposed scheme has the same efficiency andapplicability as the schemes using reversible cellular automatawith memory, and has the main characteristics as follows:
(1)
right
Each share has the same size as the secret image.
(2) The recovered image coincides exactly with the secret image,i.e. there is no loss of resolution nor contrast.
(3) The computational complexity is linear.The rest of this paper is organized as follows: In Section 2, somebasic concepts regarding 2-D CA are introduced; related work iscovered in Section 3; In Section 4, the new secret sharing schemefor binary, grayscale and color images is presented; Some experi-mental results are shown in Section 5; Section 6 includes theanalysis of the security, efficiency and the statistical propertiesof the proposed scheme; and the conclusions are given outin Section 7.
s reserved.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548 539
2. 2-D cellular automata
Two-dimensional finite cellular automata (2-D CA) are discretedynamical systems formed by a finite two-dimensional array ofR�R identical objects called cells, such that each of them canassume a state. The state of each cell is an element of the finitestate set,
P. Throughout this paper we will consider
P¼Z. The
state of the (x,y)-th cell at time t is stx,yAS. The 2-D CA evolves
deterministically in discrete time steps, changing the states of allcells according to a local transition function
f : ðZÞn-Z
The updated state of each cell depends on the n variables ofthe local transition function, which are the previous states of a setof cells, including the cell itself, and constitute its neighborhood.Moore neighborhood with radius r¼1 of the (x,y)-th cell is
Vx,y ¼ fðx�1,y�1Þ,ðx�1,yÞ,ðx�1,yþ1Þ,ðx,y�1Þ,ðx,yÞ,ðx,yþ1Þ,
ðxþ1,y�1Þ,ðxþ1,yÞ,ðxþ1,yþ1Þg
Graphically it can be seen as Fig. 1.Matrix Ct ¼ fst
x,y9stx,yAS,1rx,yrRg is called the configuration
at time t of the 2-D CA, and C0 is the initial configuration of the CA.
3. Related works
In this section, we briefly review secret sharing schemes basedon cellular automata.
[10] uses one-dimensional memory cellular automata to sharea secret text; [11] proposes a secret sharing scheme for colorimages using two-dimensional reversible cellular automata withmemory; [12] proposes an image sharing scheme based on linearmemory cellular automata with steganographic properties; [13]proposes a color images sharing scheme by means of two-dimensional memory cellular automata; [14] proposes a multi-secret sharing scheme for color images based on the use ofbidimensional reversible cellular automata with memory.
All the previous sharing schemes have the common groundthat they all use a special type of CA—the reversible cellularautomata with memory (reversible MCA).
3.1. Reversible MCA
If the state of every cell on a CA at time tþ1 not only dependson the states of its neighborhood at time t, but also on the statesof other different groups of cells at times t�1, t�2, etc. This CA ismemory cellular automata (MCA) [15].
(x-1,y-1) (x,y-1) (x+1,y-1)
(x-1,y) (x+1,y)
(x-1,y+1) (x,y+1) (x+1,y+1)
(x,y)
Fig. 1. Moore neighborhood of the (x,y)-th cell.
k-th order reversible MCA [13] for which the local transitionfunction is of the following form:
aðtþ1Þx,y ¼
Xk�1
m ¼ 0
fmþ1ðVðt�mÞx,y ÞðmodcÞ
And satisfy
fkðVðt�kþ1Þx,y Þ ¼ aðt�kþ1Þ
x,y
The initial configuration of the k-th order reversible MCA isformed by k components, Cð0Þ,. . .,Cðk�1Þ, in order to initialize theevolution of the k-th order reversible MCA; whose inverse CA isanother MCA with local transition function
aðtþ1Þx,y ¼�
Xk�2
m ¼ 0
fk�m�1ðVðt�mÞx,y Þþaðt�kþ1Þ
x,y ðmodcÞ
3.2. The secret sharing schemes
Consider the secret image as the first component, C(0), of theinitial configuration for a k-th order reversible MCA, and the rest ofk�1 components, Cð1Þ,. . .,Cðk�1Þ, of the initial configuration are k�1random matrices. Start from the initial configurations,Cð1Þ,. . .,Cðk�1Þ, compute the (mþn�1)-th order evolution of theMCA: {C(0),y,C(k�1),C(k),y,C(m),y,C(mþn�1)}. The shares to be dis-tributed among the n participants are last n consecutive config-urations of the evolution of the MCA: S0 ¼ CðmÞ,. . .,Sn�1 ¼ Cðmþn�1Þ.Take ~S0 ¼ Cðmþaþk�1Þ,. . ., ~Sk�1 ¼ CðmþaÞ,ð0rarn�kÞ, and iteratemþaþk�1 times the inverse MCA, the secret initial configuration(the original image), C(0), is obtained.
4. The proposed secret image sharing scheme
In this section, the proposed (n, n) sharing scheme based on ageneral linear 2-D CA is described in detail.
4.1. Reasons for using cellular automata
The conventional encryption methods (block cipher), e.g. the3DES and AES methods, are incapable of encrypting data withpatterns, like images, when used as electronic codebooks(ECB) [3]. In order to overcome this problem, cipher blockchaining or cipher feedback and output feedback techniquesare usually applied.
CA is very powerful computational tools with synchronousupdate mechanism for their individual components (cells) and isspecially adopted for image related works. Since the basic modelof secret sharing being proposed, researchers have publishedmany related studies. Take Shamir’s polynomials for instance,which is one of the representative. In this application, we mustprovide distinct input values for Shamir’s secret polynomialwhereas we need no input values to start evolutions of CA;another drawback in Shamir-based sharing schemes is that weare forced to calculate ‘‘mod a prime number p’’ while CA do notimpose such restrictions. In addition, in calculating mod primes,we should perform a preprocessing on the secret image toproduce pixel values in proper range (i.e.0–251).This processresults in modified secret images whose size is greater than theoriginal image. Moreover, taking into consideration the imple-mentation of the proposed method in hardware, CA are perhapsthe computational structures best suited for a fully parallelhardware realization.
(x+1,y+1) (x-1,y+1) (x,y+1) (x+1,y+1) (x-1,y+1)
(x+1,y-1) (x-1,y-1) (x,y-1) (x+1,y-1) (x-1,y-1)
(x+1,y) (x-1,y) (x,y) (x+1,y) (x-1,y)
(x+1,y+1) (x-1,y+1) (x,y+1) (x+1,y+1) (x-1,y+1)
(x+1,y-1) (x-1,y-1) (x,y-1) (x+1,y-1) (x-1,y-1)
(x-1,y-1) (x-1,y-1) (x,y-1) (x+1,y-1) (x+1,y-1)
(x-1,y-1) (x-1,y-1) (x,y-1) (x+1,y-1) (x+1,y-1)
(x-1,y) (x-1,y) (x,y) (x+1,y) (x+1,y)
(x-1,y+1) (x-1,y+1) (x,y+1) (x+1,y+1) (x+1,y+1)
(x-1,y+1) (x-1,y+1) (x,y+1) (x+1,y+1) (x+1,y+1)
Fig. 2. Two boundary conditions of 2-D CA: (a) periodic boundary and
(b) reflecting boundary.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548540
4.2. Local transition function
In this paper, we consider a 2-D CA with the local transitionfunction which only adopts Boolean XOR operation.
Let Vx,y is the Moore neighborhood of the (x,y)-th cell, and stx,y
is the state of the (x,y)-th cell at time t. Choose odd number ofcells out of Vx,y, and take their states s as the variables of localtransition functionf, for example:
If 3 cells are chosen: (x�1, y�1), (x, y), (xþ1, yþ1), f will be
stþ1x,y ¼ st
x�1,y�1 � stx,y � st
xþ1,yþ1 ð1Þ
If 5 cells are chosen: (x�1, y), (x, y�1), (xþ1, y), (x, yþ1), (x, y),f will be
stþ1x,y ¼ st
x�1,y � stx,y�1 � st
xþ1,y � stx,yþ1 � st
x,y ð2Þ
If choose 7 cells are chosen: (x�1, y�1), (x, y�1), (xþ1, y�1),(x�1, yþ1), (x, y), (x, yþ1), (xþ1, yþ1), f will be
stþ1x,y ¼ st
x�1,y�1 � stx,y�1 � st
xþ1,y�1 � stx�1,yþ1 � st
x,y � stx,yþ1 � st
xþ1,yþ1
ð3Þ
If all 9 cells of Vx,y are chosen, f will be
stþ1x,y ¼ st
x�1,y�1 � stx,y�1 � st
xþ1,y�1 � stx�1,y � st
x,y � stxþ1,y
�stx�1,yþ1 � st
x,yþ1 � stxþ1,yþ1 ð4Þ
The states of the cells in Vx,y participate in f and their result isplaced in the central cell of the neighborhood. At every time stepall the cells of the CA recalculate their state concurrently accord-ing to f.
4.3. Reconstruction periodicity depended upon boundary conditions
As the number of cells of the 2-D CA is finite, boundaryconditions must be considered in order to assure the well-defineddynamics of the CA. Consider two boundary conditions: periodicboundary and reflection boundary. Periodic boundary can bevisualized as taping the left and right edges of the rectangle toform a tube, then taping the top and bottom edges of the tube toform a torus (doughnut shape); reflection boundary simulates aplane of symmetry, and is the same as a solid, slip wall boundarycondition. The two boundary conditions of 2-D CA are displayedin Fig. 2.
After performing many experiments on 2-D CA of the two kindsof boundary conditions, we find an interesting phenomenon:
Let R be the size of a 2-D CA, if R¼2k(kAZ), then, appropriatelychoose odd number of states out of Vx,y as the variables of f, andapply f t¼R/2 times for periodic boundary condition or t¼R timesfor reflecting boundary condition, the central initial state s0
x,y willbe obtained again. Take formula (1) for example, that is
s1x,y ¼ s0
x�1,y�1 � s0x,y � s0
xþ1,yþ1
s2x,y ¼ s1
x�1,y�1 � s1x,y � s1
xþ1,yþ1
. . .. . .
sR=2x,y ¼ s0
x,y ðFor periodic boundary conditionÞ
or,
sRx,y ¼ s0
x,y ðFor reflecting boundary conditionÞ
Additionally, the repetition of the rule per t¼R/2 for periodicboundary condition or t¼R for reflecting boundary condition,repeats the same result, confirming this periodicity of the attri-bute: a 2-D CA of size R¼2k(kAZ), if its boundary condition isperiodic, its reconstruction periodicity¼R/2; if its boundary con-dition is reflecting, then its reconstruction periodicity¼R.
It is worthy to point out that randomly choosing odd number ofcells cannot always have this periodical attribute; the cells must bechosen appropriately. We have tried a lot of experiments on
choosing cells to possess this attribute. We find that among thecells to be chosen, the central cell must appear, and the remaindercells must be of radial symmetry or rotational symmetry recipro-cally. As in Fig. 1, (x,y)-th cell must be chosen every time, and then,[(x�1,y�1)-th, (xþ1,yþ1)-th], [(x,y�1)-th, (x,yþ1)-th], [(x�1,y)-th, (xþ1,y)-th], [(xþ1,y�1)-th, (x�1,yþ1)-th] cells must be cho-sen in pairs respectively. We reckon this selectional restriction isresulted in by the calculated rule of Boolean XOR operation.
4.4. Neighborhood configurations
4.4.1. Representation of an image as a matrix
An image I¼{px,y9px,yAZ,1rx,yrR,R¼2k,kAZ} can be a binary,grayscale, or color image:
1.
If I is a binary image, the pixel value: px,yA[0,1]. 2. If I is a grayscale image, the pixel value: px,yA[0,255]. 3. If I is a color image, it consists of 3 layers ðIr ¼ fprx,y,1rx,yrRg,Ig ¼ fpg
x,y, 1rx,yrRg,Ib ¼ fpbx,y,1rx,yrRgÞ. The
pixel value: prx,y,pg
x,y,pbx,yA 0,255½ �.
4.4.2. Neighborhood configurations of 2-D CA
For the sake of simplicity, we consider a 4�4 image I shown inFig. 3. I¼{px,y,1rx,yr4}, it can be considered as a 4�4 2-D CAwith periodic boundary condition. px,y is the state of the (x,y)-thcell, in its Moore neighborhood Vx,y as shown in Fig. 1, px�1,y�1 isthe north-west neighborhood of px,y, px,y�1 is the north neighbor-hood of px,y, pxþ1,y�1 is the north-east neighborhood of px,y, px�1,y
is the west neighborhood of px,y, px�1,yþ1 is the south-west
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548 541
neighborhood of px,y, px,yþ1 is the south neighborhood of px,y,pxþ1,yþ1 is the south-east neighborhood of px,y, pxþ1,y is the eastneighborhood of px,y.
(1, 1) (2, 1) (3, 1) (4, 1)
(1, 2) (2, 2) (3, 2) (4, 2)
(1, 3) (2, 3) (3, 3) (4, 3)
(1, 4) (2, 4) (3, 4) (4, 4)
Fig. 3. A 4�4 image.
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
(4, 2) (1, 2) (2, 2)
(4, 3) (1, 3) (2, 3)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
(4, 2) (1, 2) (2, 2)
(4, 3) (1, 3) (2, 3)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
(4, 2) (1, 2) (2, 2)
(4, 3) (1, 3) (2, 3)
(4, 4) (1, 4) (2, 4)
(4, 1) (1, 1) (2, 1)
Fig. 4. Neighborhood configu
I is the initial configuration, say, C0¼ I. The Moore neighbor-
hoods of all the pixels in I constitute the 8 Moore neighborhoodmatrixes corresponding to C0. They are NW0, N0, NE0, W0, E0,SW0, S0, SE0.
NW0 ¼ fpx�1,y�1,1rx,yr4g, N0 ¼ fpx,y�1,1rx,yr4g,
NE0 ¼ fpxþ1,y�1,1rx,yr4g, W0 ¼ fpx�1,y,1rx,yr4g,
E0 ¼ fpxþ1,y,1rx,yr4g, SW0 ¼ fpx�1,yþ1,1rx,yr4g,
S0 ¼ fpx,yþ1,1rx,yr4g, SE0 ¼ fpxþ1,yþ1,1rx,yr4g
We call these matrixes neighborhood configurations, whichare all the same size of I. Graphically C0, NW0, N0, NE0,W0, E0, SW0, S0, SE0 related with I (Fig. 3) can be seen in Fig. 4.
4.5. Scheme
The scheme contains three phases: the setup phase, thesharing phase, and the recovery phase.
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(3, 2) (4, 2) (1, 2)
(3, 3) (4, 3) (1, 3)
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(3, 2) (4, 2) (1, 2)
(3, 3) (4, 3) (1, 3)
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(3, 2) (4, 2) (1, 2)
(3, 3) (4, 3) (1, 3)
(3, 4) (4, 4) (1, 4)
(3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 2) (1, 2) (2, 2) (3, 2) (4, 2) (1, 2)
(4, 3) (1, 3) (2, 3) (3, 3) (4, 3) (1, 3)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
(4, 4) (1, 4) (2, 4) (3, 4) (4, 4) (1, 4)
(4, 1) (1, 1) (2, 1) (3, 1) (4, 1) (1, 1)
rations of a 4�4 image.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548542
4.5.1. Setup phase
I is the secret image to be shared.
1.
I will be considered as the initial configuration of a 2-D CAwith the same size, say, C0¼ I¼{px,y,1rx,yrR}.
2. Consequently, the 8 Moore neighborhood matrixes corre-sponding to C0 can be obtained according to the boundaryconditions
NW0 ¼ fpx�1,y�1,1rx,yrRg, N0 ¼ fpx,y�1,1rx,yrRg,
NE0 ¼ fpxþ1,y�1,1rx,yrRg, W0 ¼ fpx�1,y,1rx,yrRg,
E0 ¼ fpxþ1,y,1rx,yrRg, SW0 ¼ fpx�1,yþ1,1rx,yrRg,
S0 ¼ fpx,yþ1,1rx,yrRg, SE0 ¼ fpxþ1,yþ1,1rx,yrRg
4.5.2. Local transition function given by formula (1)
Due to the security condition of (k, n)-threshold schemes(krn), i.e., any k or more than k parts can recover the secretdata; any k�1 or fewer than k parts cannot compute the secretdata. The f given by formula (1) has the least variables amongformula (1)–(4). In our scheme, we select formula (1) as f, whereone variable is the central configuration C0 and the other twovariables NC0
1 ,NC02 are two neighborhood configurations chosen
out of the 8 Moore neighborhood configurations. ½NC01 ,NC0
2 � pair isradial symmetry or rotational symmetry.
4.5.3. Sharing phase
1.
Fig. 5. The evolvement process of 2-D CA with formula (1).
Choose two radial symmetry or rotational symmetry neigh-borhood configurations out of the 8 Moore neighborhoodconfigurations, NC0
1 , NC02 AfNW0, N0, NE0, W0, E0, SW0, S0,
SE0g. Starting from NC01 ,NC0
2 ,C0, and iterating t(k�R/2oto(kþ1)�R/2,kAZ) times for periodic boundary condition ort(k�Roto(kþ1)�R,kAZ) times for reflecting boundarycondition with f given by formula (1), the t-th centralconfiguration matrix Ct is computed as follows:
C1 ¼NC01 � NC0
2 � C0
C2 ¼NC11 � NC1
2 � C1
. . .. . .
Ct ¼NCt�11 � NCt�1
2 � Ct�1
2.
The 8 neighborhood configurations corresponding to thecentral configuration Ct: NWt, Nt, NEt, Wt, Et, SWt, St, SEt arealso subsequently obtained according to the boundary con-ditions, at the same time are NCt1 and NCt2. The evolvement
process is display in Fig. 5.
3. The shares to be distributed among the n participants,P1, y, Pn, are computed as follows:3.1 If n¼3, the three shares are: shr1¼Ct, shr2¼NCt
1,shr3¼NCt
2.3.2 If no3, the two shares are: shr1¼Ct, shr2¼NCt
1 � NCt2, or,
shr1¼NCt1, shr2¼Ct � NCt
2, or, shr1¼NCt2, shr2¼Ct � NCt
1.3.3 If l�2þ3onr(lþ1)�2þ3,(0r l), divide NCt
1,NCt2|fflfflfflfflffl{zfflfflfflfflffl}
2
,
NCt�11 ,NCt�1
2|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}2
,. . .. . ., NCt�l1 ,NCt�l
2|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2
, NCt�l�11 ,NCt�l�1
2 ,Ct�l�1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}3
into
n non-overlapped groups. In the i-th (1r irn) group,apply Boolean XOR operation to the neighborhood con-figurations, and the i-th share shri is computed. For
example, if only Ct in the i-th group, shri¼Ct; if Ct, NCt1,
NCt2 in the i-th group, shri¼Ct � NCt
1 � NCt2; if NCt�1
1 ,NCt1,
NCt�22 ,NCt�l
2 in the i-th group, shri¼NCt�11 � NCt
1�
NCt�22 � NCt�l
2 , and so on.
4.5.4. The recovery phase
To recover the secret image, all the n shares are needed.
1.
Compute the initial central configuration: ~C0¼ shr1 � shr2�� � �� shrn�1 � shrn
2.
The 8 neighborhood configurations corresponding to ~C0: ~N ~W0,
~N0, ~N ~E
0, ~W
0, ~E
0, ~S ~W
0, ~S
0, ~S ~E
0are subsequently obtained
according to the boundary conditions, at the same time are
~N ~C0
1 and ~N ~C0
2.0 0 0
3.
Starting from ~C , ~N ~C 1, ~N ~C 2, and iterating m¼R/2� (kþ1)�t�1times for periodic boundary condition or m¼R� (kþ1)�t�1times for reflecting boundary condition with f given byformula (1), the m-th central configuration matrix ~Cmis
obtained, which is the secret image I:
~C1¼ ~N ~C
0
1 �~N ~C
0
2 �~C
0
~C2¼ ~N ~C
1
1 �~N ~C
1
2 �~C
1
. . .. . .
~Cm¼ I
Two points should be noted:
1.
If I is a grayscale or color image, the � is bit-wise XORoperation. For example:pix1¼210 and pix2¼169 are two pixels of a grayscale image,the bit-wise XOR operation of the two pixel as follow:pix1�pix2¼(210)10�(169)10¼(11010010)2�(10101001)2¼(01111011)2¼(123)10
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548 543
2.
In the case of a color image, the same procedure is appliedindependently to each individual layer, and in the sharingphase the resultant 3 layers are combined to form the n imageshares.Fig. 6. The secret ima
Fig. 7. Shares and the recovered im
5. Simulation results
In this section, we design several experiments by computersimulations to verify the feasibility and applicability of the
ges to be shared.
age of the three (3, 3) schemes.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548544
proposed sharing scheme. The secret is respectively binary,grayscale and color image with 256�256 pixels shown in Fig. 6.
Fig. 7 displays three (3, 3) schemes, and Fig. 8 displays one (6,6) scheme. The (3, 3) scheme for secret binary image in Fig. 7 (a):t¼110, periodic boundary, the neighborhood configurations cho-sen are C0, NW0, SE0; The (3, 3) scheme for secret grayscale imagein Fig. 7(b): t¼158, reflection boundary, the neighborhood con-figurations chosen are C0, W0, E0; the (3, 3) scheme for secret colorimage in Fig. 7 (c): t¼212, reflection boundary, the neighborhoodconfigurations chosen are C0, N0, S0; the (6, 6) scheme in Fig. 8:t¼93, periodic boundary, the neighborhood configurations cho-sen are C0, NE0, SW0. In the 4 cases, we have obtained shareswhich have the same size as the original one, i.e. 256�256 pixels,and the recovered image is exactly the same as the original one.
It is worth noting that during the interim time marks; theimage is greatly distorted, making the shares impossible to berecognized. Take the grayscale image for example only, 3 shareimages of different iterate time are shown in Fig. 9. The experi-ment condition is as the same as that in Fig. 7 (b), except for theiterating time, Fig. 9. (a) t¼80, (b) t¼182, (c) t¼247. The PeakSignal to Noise Ratio (PSNR) is also calculated.
6. Analysis
In this section we demonstrate two objectives. The first is toprove that our proposed scheme satisfies four general criteria of
~C0¼ shr1 � shr2 � . . .� shrn�1 � shrn
¼NCt1 � NCt
2 � . . .� NCt�l�11 � NCt�l�1
2 � Ct�l�1
¼NCt1 � NCt
2 � . . .� NCt�l�11 � NCt�l�1
2 � NCt�l�21 � NCt�l�2
2 � Ct�l�2
¼NCt1 � NCt
2 � . . .� NCt�l�11 � NCt�l�1
2 � NCt�l�21 � NCt�l�2
2 � . . .� NC11 � NC1
2 � C1
¼NCt1 � NCt
2 � . . .� NCt�l�11 � NCt�l�1
2 � . . .� NC11 � NC1
2 � NC01 � NC0
2 � C0
|{z}C0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
C1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ct�l|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ct|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ctþ 1
So, ~C0¼ Ctþ1, ~N ~C
0
1 ¼NCtþ11 , ~N ~C
0
2 ¼NCtþ12 , continue evolve m¼R/2�t�1 (or m¼R�t�1) times, the process can be expressed as follows:
~Cm¼ ~N ~C
m�1
1 � ~N ~Cm�1
2 � ~Cm�1
¼ ~N ~Cm�1
1 � ~N ~Cm�1
2 � ~N ~Cm�2
1 � ~N ~Cm�2
2 � ~Cm�2
¼ ~N ~Cm�1
1 � ~N ~Cm�1
2 � ~N ~Cm�2
1 � ~N ~Cm�2
2 � . . .� ~N ~C1
1 �~N ~C
1
2 �~C
1
¼ ~N ~Cm�1
1 � ~N ~Cm�1
2 � ~N ~Cm�2
1 � ~N ~Cm�2
2 � . . .� ~N ~C1
1 �~N ~C
1
2 �~N ~C
0
1 �~N ~C
0
2 �~C
0
|{z}Ctþ 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ctþ 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ctþ 3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
CR=2�1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}CR=2
secret image sharing: security, accuracy, computational complex-ity and shadow size, also called pixel expansion. The second is todemonstrate that the proposed scheme can resist substitutionattacks and statistical attacks.
6.1. Security analysis
A (k, n)-threshold scheme (krn) is said perfect if theknowledge of any k�1 or fewer shares provides no information
about the original secret ([13,14]). This is also the securitycondition.
We set out to prove that our share satisfies the securitycriterion; that is, preventing a share from leaking any informationabout the original secret image. From Fig. 7(a)–(c) and Fig. 8, wecan see that each share is a noise-like image; therefore, thesecurity of the proposed scheme is guaranteed when subjectedto the human visual system. In the experiment of the (6, 6)scheme in Fig. 8, assuming that sh2 is unknown, the image isrecovered from the other 5 shares. Fig. 10 displays the recoveredimage. In this case, no information about the original image isobtained. Through the following proposition, we prove that theshadow mages created by our algorithm above indeed satisfy thesecurity condition.
Proposition. Consider the 2-D CA with f given by formula (1), if any
one shares is unknown, and then it is impossible to recover the secret
image.
Proof. In the sharing phase, without loss of generality, the n
(l�2þ3onr(lþ1)�2þ3,(0r l)) shares computed from NCt1,
NCt2, NCt�1
1 , NCt�12 ,. . ., NCt�l�1
1 , NCt�l�12 , Ct�l�1. In the recovery
phase, firstly compute the initial central configuration ~C0, accord-
ing to the evolution of the 2-D CA with local transition function(1), the process can be expressed as follows:
So, ~Cm¼ CR=2 ¼ I or ~C
m¼ CR ¼ I.
Now, without loss of generality, we can assume that the
unknown share is the i-th (1r irn�1) shri ¼CNt�l�11 , then no
information about the configuration Ct� l can be obtained. Conse-
quently, no information about the configurations Ct�lþ1, Ct�lþ2,
. . ., Ct , Ctþ1 can be obtained, i.e., no information about ~C0
can be
obtained. Sequentially, no information about the configurations
Ctþ2, Ctþ3,. . ., CR2�2, C
R2�1 can be obtained. Eventually, no infor-
mation about the configuration CR/2 or CR can be obtained. Note
Fig. 10. The recovered image when shr2 is unknown.
Fig. 8. Shares and the recovered image of the (6, 6) scheme.
Fig. 9. Share images of different iterate time: (a) psnr¼8.0330, (b) psnr¼7.9635, and (c) psnr¼7.9813.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548 545
that a similar result holds if the number of unknown shares is
greater than one.
As a consequence, for the secret sharing scheme proposed it is
impossible to recover the secret starting from n-1 (or less) shares
and it is perfect. &
6.2. Computational complexity
The proposed scheme employs linear 2-D CA, and the compu-tations carried out during all phases consists of computingneighborhood configurations according to boundary conditions,computing evolutions of the CA or shares with Boolean XOR orbitwise XOR operations, which is not computationally intensive.The total computation time of the scheme is proportional to theiteration cycle which is related with the size of the secret image.Therefore, the scheme can be considered efficient.
6.3. Accuracy and shares size
A (k, n)-threshold scheme (krn) is called ideal if the size ofevery share is equal to the size of the shared secret ([13,14]). Themerit achieved by the scheme is that the recovered imagecoincides exactly with the secret image, i.e. there is no loss ofresolution or contrast. The results of the experiments in section 5are visualized evidences to this merit, as well as the well knowncalculate character of Boolean XOR operation is a guarantee intheory. Moreover, as the size of every share is equal to the size ofthe secret image (the shares computed from neighborhood con-figurations of the same 2-D CA), the proposed scheme is ideal andindeed satisfies the precision condition.
6.4. Substitution attacks
We have studied the robustness of the scheme against theattacks in which a share is modified. Let us consider that theattacker has access to n�1 shares, but no information aboutthe remainder one. If he wants to recover the original image, theonly possibility is to generate at random a new share in order tocomplete the set of n necessary shares to evolve the CA. In theexperiment of the (6, 6) scheme in Fig. 8, we generate at random anew share and substitute this new share for sh2, and thenreconstruct the image with the random sh2 and the other5 original shares. Fig. 11 displays the recovered image.
Let us consider a more favorable case for the attacker. Supposethat he knows n�1 shares and partial pixels of the remaindershare are unknown. Then, he can generate at random those pixelsand try to recover the original image. In the experiment of the(3, 3) scheme in Fig. 7(b), we generate at random1/16 pixels ofshr2, the recovered images are shown in Fig. 12, its psnr¼8.8244.
In the two cases, the attacks have no success since noinformation about the original image is obtained.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548546
6.5. Statistical properties
We have performed statistical analysis in order to prove theconfusion and diffusion properties of the proposed scheme. Theseproperties allow the scheme to strongly resist statistical attacks.This analysis is performed by a test on the histograms and by thecorrelations of adjacent pixels of the original image and its shares.For the sake of simplicity, we consider the (3, 3) scheme for secretgrayscale image (Fig. 7 (b)) only.
The histograms of the original image and of the shares areshown in Fig. 13. It is immediate to observe that the histograms of
Fig. 12. The original shr2, the tampered shr2
Fig. 13. Histograms of the orig
Fig. 11. shr2 generated at random and the corresponding recovered image.
the shares are fairly uniform and they are significantly differentfrom the histogram of the original image.
In order to test the correlation between two adjacent pixels inthe four images of the previous example, we have randomlyselected 1000 pairs of two horizontally adjacent pixels, 1000 pairsof two vertically adjacent pixels, and 1000 pairs of two diagonallyadjacent pixels, for the original image as well as for its shares. Ineach case, we have computed the correlation coefficient of eachpair and the results obtained are shown in Table 1.
As it is shown in Table 1, the correlation coefficients are farapart. For example, in the original image, the correlation coeffi-cient for two horizontally adjacent pixels is 0.9567, which is verynear to 1, as it was expected. Nevertheless, in the three shares,these coefficients are 0.0017, 0.0031, and 0.0011, respectively, i.e.these are very close to 0. Similar results for vertical and diagonal
and the corresponding recovered image.
inal image and its shares.
Table 1Correlation coefficients of two adjacent pixels in the original image and in its
shares.
Original image Share 1 Share 2 Share 3
Horizontal 0.9567 0.0017 0.0031 0.0011
Vertical 0.9802 0.1431 0.1419 0.1431
Diagonal 0.9530 0.0450 0.0920 0.0909
Fig. 14. Correlation of vertically adjacent pixels.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548 547
pixels were obtained (see Table 1). This guarantees the confusionand diffusion of the pixels.
Finally, Fig. 14 presents the correlation distributions of twovertically adjacent pixels in the original image and in its shares.The correlation distributions of two diagonally and horizontallyadjacent pixels have the same appearance. As it was expected, theadjacent pixels in the original image are distributed along diagonaldue to the fact that the gray levels of those pixels have a very nearvalue. Nevertheless, the distribution of the three classes of adjacentpixels for the shares seems to be random. The results allow us tostate that it strongly resists the statistical attacks.
The proposed scheme is identical in terms of confusion anddiffusion properties with the scheme of [11] which usingreversible MCA.
7. Conclusions and future work
In this paper we have investigated the application of neighbor-hood configurations of 2-D CA and an attribute of the BooleanXOR operation, in order to design a new sharing scheme for secretimage. Our study is the first one that exploits the neighborhoodconfigurations of a general linear 2-D CA in secret image sharing.
A new (n, n)-threshold scheme for sharing secret binary, grayscaleand color images is presented. We have proved that the scheme isideal and perfect and also shown that it is secure againstsubstitution attacks and resists the most important statisticalattacks due to the fact that it exhibits good statistical properties.Moreover, simulation results verified the proposed scheme isfeasible and applicable.
Since the research of secret image sharing based on neighbor-hood configurations of 2-D CA is still in its infancy, the presentmethod features the following problems. First of all, the proposedmethod can be applied only if the resolution of the images is apower of 2. This weakness can be easily overcome by modifyingthe images using (adding) extra pixels as frame in order toaccomplish the correct dimension. Moreover, as to shares of afew iterating times, their noise-like effect is not ideal. Thisproblem can be solved initially by avoiding using these specialiterating times to compute shares. Furthermore, in ourscheme 3 cells (neighborhood configurations) must be chosen,in addition to the central cell (configuration), the remainder twocells (neighborhood configurations) must be radial symmetry orrotational symmetry. This selectional restriction is resulted in bythe calculate rule of Boolean XOR operation. The potential weak-nesses of this method may be the focus of future work.
J. Jin, Z.-h. Wu / Optics & Laser Technology 44 (2012) 538–548548
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