In this paper, we propose on image encryption method which uses NC(Nonlinear Cycle) and 2D CAT(TwoDimensional Cellular Automata Transform) in sequence to encrypt medical images. In terms of the methodology, we use NC to generate a pseudo noise sequence equal to the size of the original image. We then conduct an XOR operation of the generated sequence with the original image to conduct level 1 NC encryption. Then we set the proper Gateway Values to generate the 2D CAT basis functions. We multiply the generated basis functions by the altered NC encryption image to conduct the 2nd level 2D CAT encryption. Finally, we verify that the proposed method is efficient and extremely safe by conducting an analysis of the key spatial and sensitivity analysis of pixels.
1. INTRODUCTION
The rapid progress in information and communications technologies has facilitated the accessing of various information and this has in turn improved the quality of life of individuals. Especially notable is the fact that the progress of information and communications technology has allowed digital imagery contents to be circulated beyond the limitations of time and space, providing us with useful multimedia information. Recently, a ubiquitous environment has become the focus of attention so as to maximize the utilization of such digital imagery contents. However, a ubiquitous environment raises many serious issues in terms of information security. Important information can be stolen by theirs who many use it for fraudulent purposes, this along with other accompanying issues such as hacking or the breach of privacy and copyrights are becoming serious social issues
[1
,
2]
. With the protection of information becoming such a high social priority, many studies have proposed methods of image encryption in order to prevent information leaks and to protect information
[3

11]
. Ateniese proposed using Visual Cryptography to conduct encryption. Scharinger proposed encryption using the Kolmogorov flow map, Wong a method based on the chaotic standard map, and Chen proposed an image encryption method utilizing the 3D chaotic cat maps. Zhang also proposed a method of image encryption using chaotic maps, Pareek applied the of chaotic logistic map and Zhou applied the use of discretized chaotic maps
[4

10]
. The aforementioned method of Visual Cryptography breaks down the original image into pixel units to conduct encryption and therefore has the problem of being unable to conduct no loss recovery of the image
[4]
. Also, encryption methods using Maps are accompanied by problems such as the complexity of the stage of generating the Map
[7]
, the inability to conduct no loss recovery
[9]
, and the low level of encryption
[5
,
6
,
8

10]
. In this paper, we propose a new medical image encryption method using NC(Nonlinear Cycle) and 2D CAT(TwoDimensional Cellular Automata Transform) to overcome the problems of existing methods such as the complexity of the method, the problem with recovery, and the low level of encryption. In terms of the methodology, we first generate PN(pseudo noise) sequences in accordance with the size of the original image, and apply an XOR operation on said sequences and the original image to convert the image into a level 1 NC encrypted image. We then set gateway values to generate 2D CAT basis functions. The generated basis functions are multiplied to the NC converted image to conduct level 2 CAT encryption. Finally, experiments and stability analysis are conducted to compare the methods proposed herein with other papers in order to verify that the proposed method offers high levels of encryption.
2. PROPOSAL METHOD
This paper proposes an medical image encryption method using levels of NC and 2D CAT. The proposed method is conducted by first using NC to generate PN sequences and then using this to create a basis image. The XOR operation is applied to the generated basis image and the original image to create a level 1 NC encrypted image. Then, 2D CAT gateway values are used to convert the level 1 encrypted image into the final encrypted image. The flowchart of the proposed encryption process is depicted in
Fig. 1
.
Flowchart of proposed encryption method.
 2.1 NC
NC is a Nonlinear shift register which has the structure of a LFSR(Linear Feedback Shift Register) with an added NOT operator, and the input bit of which is nonlinear compared to its former state. This method can be used for encryption since it will have a long cycle if the feedback function is efficiently selected. By using NC we can conduct XOR and NOT operations of binary plaintext and binary secret keys to each bit to generate the ciphertext, and in terms of decryption we can generate a stream cipher system that applies XOR and NOT operations to each bit of the secret key to each bit of the ciphertext to acquire the plaintext. As this method only applies XOR and NOT operators to each bit, there is no expansion of errors and the method is faster and simpler compared to block cipher algorithms. Also, NC has stronger defenses against known plaintext attacks compared to LFSR, as well as an enhanced cycle
[11]
.
The structure of the NC proposed by this paper is shown in
Fig. 2
. The NC structure is composed of 8 bit and Nonlinear feedback circuit XOR and NOT operators. The Nonlinear feedback function
f
is given in equation (1) and
F
is the complemented vector and represents the NOT operator.
Proposed NC structure.
The generation of the NC basis image utilizes equation (1). The generated NC basis image is shown in
Fig. 3
.
NC basis image.
 2.2 2D CAT
The basic form of CAT is the 1D CA(OneDimensional Cellular Automata) in which all cells are arranged in a linear manner and form a 3neighborhood structure
[11]
. Equation (2) is a state transition function where
f
is a local transition function which has combination logic and an arrangement state of 2
^{3}
different neighborhoods. CA has 2
^{23}
= 256 state transition function.
2D basis functions generate 2D basis functions
A_{ijkl}
in 2D CA space
a
≡
a_{ijt}
(
i
,
j
,
t
= 0,1,2,...,
N
1). The 2D basis functions are created using the 1D basis functions
A_{ik}
as shown in equation (3).
The generation of 2D basis functions equations use the gateway values laid out in
table 1
. Gateway values are values used to generate 2D CAT basis functions, and are generated using the Wolfram Rule, the Number of Cells per Neighborhood, the Initial Configuration, the Boundary Configuration, and Basis Function Type etc.
Gateway Values
The state transition function equation of cells generated by the gateway values in
table 1
are as in equation(4).
In equation (4), when
r
= 1 and
t
+ 1, the condition is 0 ≤
W_{j}
≤ 2.
α_{j}
is composed of combinations of neighborhood cell states. This is the 1D 3neighborhood structure. Therefore, m=3 and
W
_{23}
=
W
_{8}
. Here, the states of the cells are defined as
a
_{0k}
,
a
_{1k}
,
a
_{2k}
where
t
(
t
=
k
) is time.
a_{ik}
refers to the state of the cell number
i
when
t
=
k
. The 2D CAT basis functions generated by the gateway values of
table 1
are shown in
Fig. 4
.
2D A_{ijkl} dualcoefficient basis functions.
A
_{00kl}
is the block at the extreme upper left corner. The top row represents 0 ≤
j
≺ 8;
i
= 0. The left column is
j
= 0;0 ≤
i
≺ 8. A
_{ij00}
is the upper left corner of each block. The white rectangular dots represent 1(addition) while the block dots are 1(subtraction). When 2D image space consists of
n
×
n
cells and
f
is the function defined in the spatial domain
i
,
j
, the 2D CAT equation of
f_{ij}
is as in equation (5)
[12
,
13]
.
c_{kl}
is the 2D CAT coefficient. Equation (6) is used to encrypt the image.
The steps to getting the 2D CAT basis functions are shown in
Fig. 5
.
2D basis functions generation process.
3. EXPERIMENT AND EVALUATION
To evaluate the performance of this encryption method, 8 bit gray level medical images were used. The NC and 2D CAT proposed by this paper were each applied to the original image and the ensuing changes were studied. In order to study the various changes of these images, 50 medical images were used in the experimentation and some are shown in
Fig. 6
.
Experimental medical images.
In terms of the time required to conduct encryption and decryption, the proposed method was faster and simpler than the algorithms using maps
[6

10]
as encryption is conducted by bit stream units. For the purposes of this paper, experimentation was conducted using Matlab on a computer with an Intel(R) Core(TM) i52400 CPU @3.10GHZ, 2G of memory, and running on Windows 7. As a result, the average encryption and decryption time was 0.5~1 seconds. The evaluation proposed in this chapter comparatively analyzed the three methods of using NC, 2D CAT, and using both NC and 2D CAT in a 2 level method. The histogram and PSNR(Peak Signal to Noise Ratio) values were used as the standards of evaluation.
In equation (7), 255 is the maximum value of pixels indicating 8 bit noise or brightness, and is expressed as
dB
. Also,
MSE
is the error square average, with the
i
and
j
values representing the width and height of the image, respectively, and calculates the variance of the same location of two image data of the same size. The image subjected to experimentation and the histogram are shown in
Fig. 7
.
Original medical image and its Histogram.
First is the stream encryption method using NC a cipher system that has periodic characteristics. In terms of the methodology, periodic sequences are generated to generate a basis image as shown in
Fig. 3
. The generated basis image and original image are subjected to a XOR operation to acquire the NC applied encrypted image.
Fig. 8
shows the results of XOR operating the NC basis image with the original image.
Encrypted medical image and Histogram by NC.
Next, the encryption method using 2D CAT generates the 2D CAT basis functions shown in
Fig. 4
and multiplies it to the original image to encrypt the image.
Fig. 9
shows the results of applying 2D CAT to the original image.
Encrypted medical image and Histogram by 2D CAT.
Finally, the image acquired by applying NC and 2D CAT in sequence to the original image is shown in
Fig. 10
.
Encrypted medical image and Histogram by NC and 2D CAT.
To comparatively analyze the performance of each method, the histogram and PSNR values were used to study how the pixels of the images were distributed. If we take a look at the histogram first we see that the methods using NC and 2D CAT individually have irregular cycles and high pixel densities(refer to
Fig. 8
,
Fig. 9
). However, in the method applying NC and 2D CAT in sequence, a relatively more even histogram develops compared to the previously mentioned methods and the pixel density is also low, signifying that the level of encryption is high(refer to
Fig. 10
). Also, the PSNR value, which measures the distortion of images, was quite low at 23
dB
with the method applying both methods in sequence, signifying that the image distortation was great, as seen in
Fig. 10
. Also, by applying this method to 50 images with irregular pixel cycles, we found that the PSNR value was even. When the PSNR<35
dB
, we can visually confirm the distortion of the image.
In conclusion we confirmed through the PSNR values and histogram that the encrypted message had even pixels with no recognizable correlation with the original image when using NC and 2D CAT together in sequence(refer to
Fig. 11
). Therefore, using the NC and 2D CAT methods in sequence leads to higher levels of encryption and a stronger defense against external attacks. When decrypting, we use equation (5) to conduct inverse CAT, as the 2D CAT basis functions
A_{ijkl}
are orthogonal. Then, we apply the XOR operator to the NC converted image to recover the original image without any loss.
Original medical images, NC converted images, Encrypted images by NC and 2D CAT, Histogram.
4. STABILITY ANALYSIS
 4.1 Key spatial analysis
The conditions laid out by this paper are 8cell, 2state, and 5neighborhoods. Therefore, 2D CA is defined in equation (8).
Here, T is time, K is number of cell state, m is cell number, M and N refer to the space of cell. Therefore 2D CA generates
N
^{2}
_{T}
=
K
^{km+3(N+M)+2T}
= 2
^{96}
(2
^{25+3(8+8)+2×8}
) keys. This is a greatly improved result compared to those of image encryption methods using normal CA keys. Also, NC is a stream encryption method with 1D periodic sequences, and has 2
^{8+8}
(2
^{W+H}
) different sequences. Here, W and H refer to the width and height of the image. Therefore the encryption method proposed by this paper is capable of generating 2
^{16+96 }
= 2
^{112}
keys, thus offering sufficient protection.
 4.2 Key sensitivity analysis
x_{i}
and
y_{i}
in equations (9) refer to the value of the neighborhood pixels and
N
refer to the numbers of total pixels.
The result in proposed method and Pareek
[9]
of each key sensitivity is shown in
table 2
. In this paper, signifies a higher alteration rate in key sensitivity compared to Pareek
[9]
.
Sensitivity analysis for the cipher to key and proposed image
Sensitivity analysis for the cipher to key and proposed image
5. CONCLUSION
This paper proposes a method of applying NC and 2D CAT in sequence to conduct medical image encryption. In terms of the methodology, we first generate PN(pseudo noise) sequences in accordance with the size of the original image to create a basis image. Then we apply an XOR operation on said sequences and the original image to convert the image into a level 1 NC encrypted image. We then multiply the 2D CAT basis functions to the NC converted image to acquire the final encrypted result. By applying two levels of encryption as such, we can enhance the level of protection. The paper also analyzed key space and other points of stability to comparatively analyze the performance of the proposed method with that of other methods. As a result of the comparative analysis, we concluded that the proposed method resulted in higher levels of encryptions than existing methods. This study is significant in that it provides a basis for creating various encryption algorithms, as altering the structure of NC is relatively simple, and different algorithms can be created by simply changing the initial values of the 2D CAT basis functions. In future studies, analyzing the properties of 2D CAT basis functions to apply the method in various fields and studies developing high efficiency image encryption methods should be actively conducted.
BIO
TaeHee Nam
He received the Ph.D. degree from the department of electronics engineering, Pukyong National University, Republic of Korea in 2010. Since march 1993, he has been a professor, department of biomedical engineering, Dongju College, Republic of Korea. His current research interests are in medical image processing and analysis.
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