A Neural Network Approach for Visual Cryptography - Tai-Wen Yue's ...
A Neural Network Approach for Visual Cryptography Tai-Wen
Yueand
Suchen Chiang
Department of Computer Science and Engineering, Tatung University 40 Chungshan North Road, Section 3, Taipei 104, Taiwan [email protected] snd suchentlcomm.cs e.ttu.edu.tw
Abstract Visual cryptography finds many applications in cryptographic field such as key management, message concealment, authorization, autbentication, identification, and entertainment. In the paper, we propose a novel approach for visual cryptography using neural networks (NNs). To perform encrypting, the input to the NN is a set of graylevel images, and the output is a set of binary images (shares) that fulfills the desirable access scheme. This approach is considerably different from the traditional one, and can be applied to cope with very complex access schemes.
1 Introduction Visual cryptography is a cryptographic scheme to achieve visual secret sharing [5, 6]. Given a visually recognizable target image, the (n, k) encrypting scheme (or access scheme) is to cryptologically separate the image into a set of n shares, called shadow images which are recorded in transparencies, such that the pictorial meaning stored in the original image is recognizable if and only if a viewer is able to acquire at least k shares and stack them together. One can also define more complex encrypting schemes using the so-called access structures [1, 2]. Figure l(a) shows the code book devised by Naor and Shamir to fulfill the (2, 2). Figure l(b) shows an example that uses such a encrypting strategy. Apparently, complicated code books should be required to account for complex access schemes. Besides? using such a table lookup strategy, the target image must be a binary one, and the size of the shadow images w1ll be enlarged. In this paper, we propose a neural network (NN) approach for visual cryptography. The NN model to conduct this research is the so-called quantum neural networks (Q’tron NNs; for short) [7, 9, 10]. For encrypting, the target image input to the NN is a gray image. After the NN settles down, a set of binary shadow images, each of them has the same size as the target, will be resulted. By stacking enough number of the shadow images, the superposed version will be a halftone image to mimic the target image. This implies that decrypting is performed by eyes, i.e., done without the help of machines. Because Q ‘tron NN’s are intrinsically auto-invertible, the NN designed for encrypting can automatically be used for decrypting. That is, by reversing the operating modes of Q’trons, the NN then serves to reconstruct the target image using the set of shadow images as input. This, hence, allows us to perform decrypting with the help of machines, although this is not absolutely needed. We propose a Q’tron NN paradigm to achieve the aforementioned goals. In the article, a Q’tron NN for the (2, 2) access scheme will be built to enlighten the main concept. However, with skillful modification of the energy function and applying the persistent noise-injection mechanism [7, 10] peculiarly for a Q’tron NN, we can generalize the paradigm to cope with complex access schemes. Although the generalized version is not discussed in the paper, some experimental results corresponding to various access schemes will be demonstrated to manifest the effectiveness of the approach. The content of the paper is organized as follows: In Section 2, we’ll give a brief introduction on the Q’tron NN model. Section 3 reviews the strategy of using a Q’tron NN for image halftoning and restoration [8]. Section 4 discusses the strategy to build a Q’tron NN for the (2,2) visual cryptography. Section 5 gives experimental results. Finally, we will draw some conclusions in section 6.
2 Q’tron
NN
Model
In this model of NN, the basic processing elements are called Q’tron’s. The number of output-level of each Q’tron can be greater than two. Specifically, let pi represent the ith Q’tron in a Q’tron NN, where the output of Ki, denoted as Qi, takes up its value in a finite integer set {O, 1,..., qi – 1} with qi (z 2) being the number of output-level. In addition, Q~ is also weighted by a specific positive value a~ called active weight, which stands for the unit excitation strength of that Q’tron. The term a~Q~ is thus referred to as the active value of vi, which represents the total excitation strength of that Q’tron. In a Q’tron NN, for a pair of connected Q’trons LLi and M, there is only one connect ion strength, i.e. Tij = Tji. In Q’tron NN model, each Q’tron in the NN is allowed to be noise-injected. Thus, noise-injected
stimulus gi for the Q’tron pi is defined as: n fii = ‘Hi +Ni
= ~Tij(ajQj)
+Ii +Ni,
(1)
j=l where Hi denotes the noise-free net stimulus of pi, which apparently ~Tij(a~Q~),
and eXteWWLl StimUIUSI
k equal to the sum of internal stimuli, namely,
The term Ni denotes the additive noise that is fed into pi, and n denotes
j=l
0-7695-0619-4/00/$10.00 (C) 2000 IEEE
Pixel
Probability
❑
P=O.5 p = 0.5
■
P:O.5 -0.5
~
Shares #l
#2
❑ ❑ ❑
Supeqmsition of the twos hares
❑ ❑ ■ ■
m m m
S1
S2
S1+S2
(a)
0) Figure 1: (a) Code book for (2, 2); (b) a (2,2) example.
the number of Q’trons in the NN. A Q’tron NN is said to run in simple mode if it is noise-free, i.e., Ni = O for all i; otherwise, it is said to run in full mode. At each time step only one Q’tron is selected for level transition subject to the following rule:
Qi(t + 1) = Qi(t) + AQ~(t), with AQ,(t)
= {
+1 –1 o
whenever fii (t) > ~ lTiiai \and Q~(t) < q~– 1; whenever fii(t) < –~ lTiiail and Qi(t) > O; otherwise;
(2)
where we have assumed that the Z?hQ’tron is selected at time t+ 1. From the model description for a Q’tron NN, one can easily seen that if we let each Q’tron, say ~i have qa = 2, ai = 1 and Tai = O, and let the NN run in simple mode, then the Q’tron NN model is reduced to the original Hopfield model [3, 4]. The system energy 8 embedded in a Q’tron NN, called Liapunov energy, is defined by the following form:
Where n is total number of Q’trons in the NN, and K can be any suitable constant. It was shown that, in simple mode, the energy & defined above will monotonically decrease with time. Therefore, if a problem can be mapped into one which minimizes the function & given in the above form, then the corresponding NN, hopefully, will autonomously solve the problem after & reaches a global/local minimum. However, there may be enormous number of local minima which, in fact, represent unsatisfactory solutions. In the Q ‘tron NN approach, Ni serves to control the solution quality to a problem [9]. In brief, given the desirable solution quality, the allowable range of noise strength injected into each Q’t ron can be systematically determined. By persist ently injecting such strengthbounded noises into Q’trons, the NN eventually will settle down on a state corresponding to a satisfactory solution provided such a solution does exist. To solve visual cryptography described by a complex access structures, the so-constructed Q’tron NNs, however, are almost useless if the NN is to run in simple mode. Interestingly, one can avoid injecting noise to the Q’t ron NN for the (2,2) access scheme if its energy function is ‘magically’ constructed (to be discussed shortly). In addition, to make each Q’tron to be able to function either as an input or as an output node, a Q’tron can either be operated in clamp-mode, i.e., its output-level is clamped fixed at a particular level, or in free mode, i.e., its output-level is allowed to be updated according to the level transition rule specified in Eq. 2.
3 Image Halftoning Halftoning is a process to convert gray (continuous-tone) images into binary (halftone) images. Upon display, it is hoped that, by blurring the eyes, the halftone image will appear similar to the original gray image. In the sequel, we’ll use an integer value between O and 255 to represent a pixel value in a gray image, where O represents a pure white and 255 a darkest black, and use O and 1 to represent an uninked (white, or more precisely, transparent) pixel and an inked (black) pixel in a halftone image, respectively. Consider two M x N images, say, G and H which represent a graytone and a halftone image, respectively. We’ll use two M x N Q’tron planes, say Plane-G and Plane-H to represent images G and H, respectively, as shown in Figure 2 (a).. With such a representation scheme, we’ll use Q: ~ {O, 1, ..., 255} and Q~ E {O, 1} to represent the ijih and klth pixels in Plane-G and Plane-H, G– — a G=landq$=q ‘– — a ‘=255 and G = 256 for Q’tron in Plane-G, and akl respectively. This implies aij
qfi = qH = 2 for Q’tron in Plane-H. The Energy
Function
for Image
Halftoning
and Restoration
Given a gray image, image halftoning can be done by constructing a binary image that, in average sense, preserves
0-7695-0619-4/00/$10.00 (C) 2000 IEEE
Pkine-G (Gray Image)
‘Z’s~ Plan-Ii (Halftone image )
(b)
(a) Figure 2:
❑ 00 ❑ mo ❑ 00
j-cl
Reconstructed
Halftone
Original
(a) Q’tron NN for image halftoning and restoration; (b) a simulation reult. Table 1. Error squares for stackingrule
❑ nnoclnn ❑ tlnclncln
12nnnn ❑ 0000 ❑ nmln ❑ 0000 ❑ 0000
❑ ❑
clcmcmnu
❑ nnmclun ❑ clnclclcln ❑ ncinoun ❑ 000000
)’4
f’=3
S1 %hp 0000 00
0
on In In
0.25
szh~ S1 00 on lB ❑ lmon
lBOD
1~
0.25
lEOEIOCI
1
l~lm
1~
0.25
lmlnocl
4
2.2s 1
Figure 3: The r-neighborhood structures for r=l, 2, and 3.
the luminance everywhere in a small area on the original image. We define energy function ~1 for that as follows: 2 El=;x
~ Z55Q;::;:fi { (k,l)efvr(i,j) ——
~ Q$ 2=; (k,l)ENr(2,j) }{
aHQfi – aGQfl ~ x (k,l)
Yueand
Suchen Chiang
Department of Computer Science and Engineering, Tatung University 40 Chungshan North Road, Section 3, Taipei 104, Taiwan [email protected] snd suchentlcomm.cs e.ttu.edu.tw
Abstract Visual cryptography finds many applications in cryptographic field such as key management, message concealment, authorization, autbentication, identification, and entertainment. In the paper, we propose a novel approach for visual cryptography using neural networks (NNs). To perform encrypting, the input to the NN is a set of graylevel images, and the output is a set of binary images (shares) that fulfills the desirable access scheme. This approach is considerably different from the traditional one, and can be applied to cope with very complex access schemes.
1 Introduction Visual cryptography is a cryptographic scheme to achieve visual secret sharing [5, 6]. Given a visually recognizable target image, the (n, k) encrypting scheme (or access scheme) is to cryptologically separate the image into a set of n shares, called shadow images which are recorded in transparencies, such that the pictorial meaning stored in the original image is recognizable if and only if a viewer is able to acquire at least k shares and stack them together. One can also define more complex encrypting schemes using the so-called access structures [1, 2]. Figure l(a) shows the code book devised by Naor and Shamir to fulfill the (2, 2). Figure l(b) shows an example that uses such a encrypting strategy. Apparently, complicated code books should be required to account for complex access schemes. Besides? using such a table lookup strategy, the target image must be a binary one, and the size of the shadow images w1ll be enlarged. In this paper, we propose a neural network (NN) approach for visual cryptography. The NN model to conduct this research is the so-called quantum neural networks (Q’tron NNs; for short) [7, 9, 10]. For encrypting, the target image input to the NN is a gray image. After the NN settles down, a set of binary shadow images, each of them has the same size as the target, will be resulted. By stacking enough number of the shadow images, the superposed version will be a halftone image to mimic the target image. This implies that decrypting is performed by eyes, i.e., done without the help of machines. Because Q ‘tron NN’s are intrinsically auto-invertible, the NN designed for encrypting can automatically be used for decrypting. That is, by reversing the operating modes of Q’trons, the NN then serves to reconstruct the target image using the set of shadow images as input. This, hence, allows us to perform decrypting with the help of machines, although this is not absolutely needed. We propose a Q’tron NN paradigm to achieve the aforementioned goals. In the article, a Q’tron NN for the (2, 2) access scheme will be built to enlighten the main concept. However, with skillful modification of the energy function and applying the persistent noise-injection mechanism [7, 10] peculiarly for a Q’tron NN, we can generalize the paradigm to cope with complex access schemes. Although the generalized version is not discussed in the paper, some experimental results corresponding to various access schemes will be demonstrated to manifest the effectiveness of the approach. The content of the paper is organized as follows: In Section 2, we’ll give a brief introduction on the Q’tron NN model. Section 3 reviews the strategy of using a Q’tron NN for image halftoning and restoration [8]. Section 4 discusses the strategy to build a Q’tron NN for the (2,2) visual cryptography. Section 5 gives experimental results. Finally, we will draw some conclusions in section 6.
2 Q’tron
NN
Model
In this model of NN, the basic processing elements are called Q’tron’s. The number of output-level of each Q’tron can be greater than two. Specifically, let pi represent the ith Q’tron in a Q’tron NN, where the output of Ki, denoted as Qi, takes up its value in a finite integer set {O, 1,..., qi – 1} with qi (z 2) being the number of output-level. In addition, Q~ is also weighted by a specific positive value a~ called active weight, which stands for the unit excitation strength of that Q’tron. The term a~Q~ is thus referred to as the active value of vi, which represents the total excitation strength of that Q’tron. In a Q’tron NN, for a pair of connected Q’trons LLi and M, there is only one connect ion strength, i.e. Tij = Tji. In Q’tron NN model, each Q’tron in the NN is allowed to be noise-injected. Thus, noise-injected
stimulus gi for the Q’tron pi is defined as: n fii = ‘Hi +Ni
= ~Tij(ajQj)
+Ii +Ni,
(1)
j=l where Hi denotes the noise-free net stimulus of pi, which apparently ~Tij(a~Q~),
and eXteWWLl StimUIUSI
k equal to the sum of internal stimuli, namely,
The term Ni denotes the additive noise that is fed into pi, and n denotes
j=l
0-7695-0619-4/00/$10.00 (C) 2000 IEEE
Pixel
Probability
❑
P=O.5 p = 0.5
■
P:O.5 -0.5
~
Shares #l
#2
❑ ❑ ❑
Supeqmsition of the twos hares
❑ ❑ ■ ■
m m m
S1
S2
S1+S2
(a)
0) Figure 1: (a) Code book for (2, 2); (b) a (2,2) example.
the number of Q’trons in the NN. A Q’tron NN is said to run in simple mode if it is noise-free, i.e., Ni = O for all i; otherwise, it is said to run in full mode. At each time step only one Q’tron is selected for level transition subject to the following rule:
Qi(t + 1) = Qi(t) + AQ~(t), with AQ,(t)
= {
+1 –1 o
whenever fii (t) > ~ lTiiai \and Q~(t) < q~– 1; whenever fii(t) < –~ lTiiail and Qi(t) > O; otherwise;
(2)
where we have assumed that the Z?hQ’tron is selected at time t+ 1. From the model description for a Q’tron NN, one can easily seen that if we let each Q’tron, say ~i have qa = 2, ai = 1 and Tai = O, and let the NN run in simple mode, then the Q’tron NN model is reduced to the original Hopfield model [3, 4]. The system energy 8 embedded in a Q’tron NN, called Liapunov energy, is defined by the following form:
Where n is total number of Q’trons in the NN, and K can be any suitable constant. It was shown that, in simple mode, the energy & defined above will monotonically decrease with time. Therefore, if a problem can be mapped into one which minimizes the function & given in the above form, then the corresponding NN, hopefully, will autonomously solve the problem after & reaches a global/local minimum. However, there may be enormous number of local minima which, in fact, represent unsatisfactory solutions. In the Q ‘tron NN approach, Ni serves to control the solution quality to a problem [9]. In brief, given the desirable solution quality, the allowable range of noise strength injected into each Q’t ron can be systematically determined. By persist ently injecting such strengthbounded noises into Q’trons, the NN eventually will settle down on a state corresponding to a satisfactory solution provided such a solution does exist. To solve visual cryptography described by a complex access structures, the so-constructed Q’tron NNs, however, are almost useless if the NN is to run in simple mode. Interestingly, one can avoid injecting noise to the Q’t ron NN for the (2,2) access scheme if its energy function is ‘magically’ constructed (to be discussed shortly). In addition, to make each Q’tron to be able to function either as an input or as an output node, a Q’tron can either be operated in clamp-mode, i.e., its output-level is clamped fixed at a particular level, or in free mode, i.e., its output-level is allowed to be updated according to the level transition rule specified in Eq. 2.
3 Image Halftoning Halftoning is a process to convert gray (continuous-tone) images into binary (halftone) images. Upon display, it is hoped that, by blurring the eyes, the halftone image will appear similar to the original gray image. In the sequel, we’ll use an integer value between O and 255 to represent a pixel value in a gray image, where O represents a pure white and 255 a darkest black, and use O and 1 to represent an uninked (white, or more precisely, transparent) pixel and an inked (black) pixel in a halftone image, respectively. Consider two M x N images, say, G and H which represent a graytone and a halftone image, respectively. We’ll use two M x N Q’tron planes, say Plane-G and Plane-H to represent images G and H, respectively, as shown in Figure 2 (a).. With such a representation scheme, we’ll use Q: ~ {O, 1, ..., 255} and Q~ E {O, 1} to represent the ijih and klth pixels in Plane-G and Plane-H, G– — a G=landq$=q ‘– — a ‘=255 and G = 256 for Q’tron in Plane-G, and akl respectively. This implies aij
qfi = qH = 2 for Q’tron in Plane-H. The Energy
Function
for Image
Halftoning
and Restoration
Given a gray image, image halftoning can be done by constructing a binary image that, in average sense, preserves
0-7695-0619-4/00/$10.00 (C) 2000 IEEE
Pkine-G (Gray Image)
‘Z’s~ Plan-Ii (Halftone image )
(b)
(a) Figure 2:
❑ 00 ❑ mo ❑ 00
j-cl
Reconstructed
Halftone
Original
(a) Q’tron NN for image halftoning and restoration; (b) a simulation reult. Table 1. Error squares for stackingrule
❑ nnoclnn ❑ tlnclncln
12nnnn ❑ 0000 ❑ nmln ❑ 0000 ❑ 0000
❑ ❑
clcmcmnu
❑ nnmclun ❑ clnclclcln ❑ ncinoun ❑ 000000
)’4
f’=3
S1 %hp 0000 00
0
on In In
0.25
szh~ S1 00 on lB ❑ lmon
lBOD
1~
0.25
lEOEIOCI
1
l~lm
1~
0.25
lmlnocl
4
2.2s 1
Figure 3: The r-neighborhood structures for r=l, 2, and 3.
the luminance everywhere in a small area on the original image. We define energy function ~1 for that as follows: 2 El=;x
~ Z55Q;::;:fi { (k,l)efvr(i,j) ——
~ Q$ 2=; (k,l)ENr(2,j) }{
aHQfi – aGQfl ~ x (k,l)
