Vol.29 No.3/4 JOURNAL OF ELECTRONICS (CHINA) July 2012

CONSTRUCTION OF THE ENCRYPTION MATRIX BASED ON CHEBYSHEV CHAOTIC NEURAL NETWORKS1

Zou Ajin Wu Wei* Li Renfa** Li Yongjiang (Information College, Guangdong Ocean University, Zhanjiang 524088, China)

*(Biomedical Engineering Department, Wenzhou Medical University, Wenzhou 325000, China) **(Department of Computer and Communication, Hunan University, Changsha 410012, China)

Abstract The paper proposes a novel algorithm to get the encryption matrix. Firstly, a chaotic se-quence generated by Chebyshev chaotic neural networks is converted into a series of low-order integer matrices from which available encryption matrices are selected. Then, a higher order encryption matrix relating real world application is constructed by means of tensor production method based on selected encryption matrices. The results show that the proposed algorithm can produce a “one-time pad ci-pher” encryption matrix with high security; and the encryption results have good chaos and auto-correlation with the natural frequency of the plaintext being hidden and homogenized.

Key words Neural network; Encryption matrix; Chaos; Tensor production

CLC index TP309

DOI 10.1007/s11767-012-0796-9

I. Introduction With the rapid development of the communi-

cations, a great deal of concern has been raised in data security. The data encryption technology is an important method of data security in reality. In 1929, Hill proposed a polygram substitution ciphers (also known as matrix transformation password) method which employs encryption matrix to en-crypt plaintext, and then decrypts the cipher text with an inversed encryption matrix[1,2]. However, the encryption matrix and its inverse matrix require integer elements and being sufficient large in di-mension[3]; that makes the matrix difficult to con-struct. In recent years, the matrix tensor theory has attracted a lot of attention and has been widely applied in digital image processing and secure communication[4–7]. Some researchers have proposed a method to construct high order encryption matrix by calculating tensor product of second-order re-versible matrix[1,7]. The drawback of the method is that the constructed encryption matrix can only be

1 Manuscript received date: October 19, 2011; revised date:

May 13, 2012. Supported by the National Natural Science Foundation of China (No. 61173036). Communication author: Li Yongjiang, born in 1967, male, Ph.D., Associate Professor. College of Information, Guangdong Ocean University, Zhanjiang 524088, China. Email: LyjRiver@21cn.com.

in even order (2n order), so it is hard to meet the needs of practical problems.

In order to overcome the disadvantages men-tioned above, this paper proposes a novel algorithm of Construction the Encryption Matrix Based on Chebyshev Chaotic Neural Networks (CEMBCCNN). Firstly, we use the Logistic chaos sequence to construct a Chebyshev Chaotic Neural Network (CCNN)[8,9], to get security chaos sequence; then the sequence is further used to construct some lower order chaos encryption matrices. According to actual problem and performing tensor product calculation, a higher order chaos encryption matrix is further obtained. An example of digital image encryption is given to evaluate the proposed algo-rithm; and the results are compared with ones found in the published literatures[10–15].

II. Related Mathematic Theory The matrix tensor product, also called the

Kronecker product, has definitions and theorems below[16]. Definition 1 Define ,A B as matrix, and =A ( ) ,ij m na × then

ZOU et al. Construction of the Encryption Matrix Based on Chebyshev Chaotic Neural Networks 249

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a a

a a a

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

⊗ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

B B B

B BA B

B B B

B (1)

is Kronecker product of ;A or is the tensor product of A and .B Definition 2 Define the tensor products of n matrices ( )1,2, ,i i n=A as

1

n

ii=

= ⊗A A (2)

Theorem 1 If A is a -orderm invertible matrix and B is a -ordern invertible matrix, then ⊗A B is also an invertible matrix, and

1 1 1( )− − −⊗ = ⊗A B A B (3)

Theorem 2 If A is an -orderm matrix and B is an -ordern matrix, then

n m⊗ =A B A B (4)

Encrypting data through encryption and de-cryption matrices, the elements of the matrices must be integers. The matrix with integer elements can be constructed according to Ref. [3]. Theorem 3 If all elements of n n×A are integers and det( ) 1,=A then all elements of 1−A are also the integers.

From Theorems 2 and 3, we can conclude that Corollary 1: If each matrix of ( )1,2, ,i i n=A is the encryption matrix, then 1

ni i== ⊗B A is also the

encryption matrix.

III. CCNN Model and Encryption Matrix Structure 1. CCNN model

Based on the logistic chaos sequence 1kx + = (1 ),k kx xμ= − where (3.569945 , 4],μ ∈ we con-

struct a single-input and single-output, three-level CCNN recursion model according to Refs. [8,9]. The model is shown in Fig. 1. The weight between the input-level and the hidden-level is constant and equals 1. The weight iw between the hidden-level and the output-level will be determined after the model being trained. The active function of the hidden-level neurons is a group of Chebyshev or-thogonal polynomials that can be calculated ac-

cording to a recurrence formula. The network described in Fig. 1 includes

Fig. 1 CCNN model

Input-level o x= The input of hidden neuron net ; 1,2,i o i= =

, .n The outputs of hidden-level neuron io = (net ),i iT where (net )i iT is a group of Chebyshev

orthogonal polynomials which can be deduced ac-cording to Eq. (5)

1 2

2 1

( ) 1, ( )

( ) 2 ( ) ( ),

1,2, , 2

j j j

T x T x x

T x xT x T x

j n

+ +

⎧ = =⎪⎪⎪⎨⎪ = −⎪⎪⎩= − (5)

Output-level 1 1 ( )n ni ii i i iy w o wT x= == =∑ ∑

Suppose the training samples are ( , ),t tx d t = 1,2, , ,s where s is the sample number, tx is the CCNN input, td is the ideal output of Logistic chaos sequence. We use the BackPropagation (BP) learning algorithm to train CCNN model with

Error , 1,2, ,t t te d x t s= − = Training target 2

11/2 st tJ e== ∑

Weight correlation formula

(net ), 1,2, ,

( 1) ( ) ( ), 1,2,

j t j jj

j j j

Jw eT j n

w

w k w k w k k

η η⎧ ∂⎪⎪Δ =− = =⎪⎪ ∂⎨⎪⎪ + = +Δ =⎪⎪⎩

(6)

where 0 1η< < is the learning rate, and k is the number of times of study.

2. The algorithm of encryption matrix production

After CCNN is constructed, the encryption matrix is generated according to the algorithm below: Step 1 Give 1x as input of CCNN and get se-quence 1 2 ,qx x x x= where q is the length of se-quence ;x let 1,i = define the mold being m and the encryption matrix being ;n n×A Step 2 Pickup subsequence 21 1i i i n

x x x x′+ + −

= from ,x and transform all elements of x ′ into in-

250 JOURNAL OF ELECTRONICS (CHINA), Vol.29 No.3/4, July 2012

tegers between 0 ~ m by calculating y = mod(1000 , ),x m′ 1;i i← + If 2,i q n≤ − go to Step 3, otherwise, end the process; Step 3 Transform y into an n n× matrix ,A i.e.

reshape( , , ),y n n=A if det( ) 1,=A output A and turn to Step 2, otherwise, go to Step 2.

The above algorithm may mostly produce q − 2n number of n n× encryption matrices, which

can be recorded as 2( 1,2, , ; ),i i t t q n= ≤ −A then we may produce a higher order encryption matrix by the Eq. (2).

IV. Simulation Examples For the standard Logistic chaos sequence 1kx + 4 (1 ),k kx x= − let the chaos initial value 0 0.2,x =

the number of sample 100,s = and the training rate 0.1.η = After 566k = times of training at a 1 3 1× × Chebyshev neural network, the training error reaches 139.8897 10 ,J −= × indicating that CCNN has been constructed successfully. Example 1 Let 510q = and arbitrarily set 1x = 0.1 at CCNNM illustrated in Fig. 1, we get se-quence 1 2 .qx x x x=

Let 4n = and 16,m = by the algorithm in Subsection II.2, we get below seven 4 4× low-order encryption matrices

1 2

3 4

5

1 1 1 0 6 9 8 11

5 7 5 15 6 6 8 15, ,

2 0 5 5 4 9 7 11

1 0 3 6 15 14 12 2

2 5 0 5 2 9 6 4

2 3 0 6 7 10 6 14, ,

3 7 0 8 11 11 6 3

4 0 1 3 13 10 5 4

3 5 10 7

1 4 12 12

15 15 14 12

1 1 1 2

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡

=

A A

A A

A 6

7

8 9 7 12

11 6 0 14, ,

12 4 1 10

5 9 7 11

5 4 0 5

7 4 1 3

8 8 4 15

9 7 1 9

⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

A (7)

Example 2 If only change 1x from 0.1 to 0.1000000000000001 and keep the other parame-ters fixed 5( 10 , 4,q n= = and 16),m = then we can get seven different encryption matrices

1 2

3 4

5

6 1 12 15 2 7 10 13

10 4 8 7 2 0 11 11, ,

6 2 5 4 1 0 5 5

15 3 9 1 0 2 8 9

13 2 3 6 1 8 13 6

4 3 6 6 0 4 5 4, ,

13 3 4 7 10 13 8 9

14 12 2 9 7 7 13 0

0 4 1 2

1 14 2 5

4 7 14 7

1 1

′ ′

′ ′

′

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

=

A A

A A

A 6

7

0 12 5 6

10 15 1 5, ,

2 11 15 8

1 10 9 7 3 10 3

2 15 9 5

15 2 8 6

1 10 4 5

15 8 13 6

′

′

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

A (8)

Example 3 If let 51 0.1, 10 , 5,x q n= = = m =

10, similarly we get

5 6 7 1 8

0 2 3 7 8

4 8 6 2 7

3 5 1 8 1

2 3 4 2 6

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(9)

and so on five 5 5× encryption matrices (abbrevi-ated) .

Below we illustrate how to construct a higher order encryption matrix 256 256×A through the ma-trix tensor product, to encrypt and decrypt a standard image lena.bmp provided by Matlab (the picture size is 256 256× pixels and the gray level equals 0 ~ 255). The original image lena.bmp is transformed into an integer matrix 256 256.×B The matrix A is further calculated tensor production

41 (i i== ⊗A A A is also a encryption matrix ac-

cording to Corollary 1) to obtain the encryption

ZOU et al. Construction of the Encryption Matrix Based on Chebyshev Chaotic Neural Networks 251

results mod( , 256).=C AB According to Theorem 1, the decryption results should be 1=mod( ,′ −B A C 256).

The encryption and decryption results are il-lustrated in Fig. 2 in which Fig. 2(a) is primitive image; Fig. 2(b) is the encryption result; Fig. 2(c) is the autocorrelation function of encryption result and Fig. 2(d) is decrypted result.

Fig. 2 Digital image encryption based on encryption matrix

V. Analysis and Comparison 1. Security analyses

According to the results of simulation Ex-ample 1 and Example 2, the encryption matrix is completely different when the random initial value 1x varied from 0.1 to 0.1000000000000001, which illustrate that a small change to the key will produce different effect of encryption and the key sensitivity is high. Besides the key space of Logistic chaos is 1610 , if taking the other pa-rameters ( , , )q m n into account, the key space is more huge by mathematic combinatorial so as the exhaustive attack can not be feasible.

Based on chaos’s sensitivity, giving a dif-ferent initial chaos value 1x into CCNN will generate different chaos sequence 1 2 qx x x x=

and so on for a different encryption matrix ,A so that a “one-time pad cipher” encryption is realizable. According to the Shannon informa-tion theory, this kind of encryption matrix is theoretically un-decryptable.

Fig. 2(c) indicates that the encryption result is very good in chaos characteristic and in autocor-relation, which satisfies the requirement of cryp-tology.

The above analyses show that the encryption matrix and the encryption result have very high security.

2. Correlation analysis of neighboring pixels

In order to evaluate the correlation of neighboring pixels in original image and in ciphered image, we select neighboring pixels from both im-ages (level, vertical, or opposite angle), and cal-culate correlations according to Eq. (10)[10].

cov( , )

( ) ( )xy

x yr

D x D y= (10)

where

( )( ) ( )( )

( )( )

1

2

1 1

1cov( , )

1 1( ) , ( )

N

i i i ii

N N

i i ii i

x y x E x y E yN

E x x D x x E xN N

=

= =

⎧⎪⎪ = − −⎪⎪⎪⎨⎪⎪ = = −⎪⎪⎪⎩

∑

∑ ∑

(11)

where x and y represent grey value of two ad-jacent pixels in image. Fig. 3 and Fig. 4 illus-trate the correlations of horizontally adjacent pixels of original image and ciphered image, respectively.

Fig. 3 Correlations of horizontally adjacent pixels in the original image

Tab. 1 gives the correlation coefficients of horizontal, vertical, and the diagonal adjacent pixels of original image and ciphered image. It is obvious, according to Tab. 1, that the correlation coefficient approaches 1 in the original image

252 JOURNAL OF ELECTRONICS (CHINA), Vol.29 No.3/4, July 2012

indicating the adjacent pixels are highly correlated; and, the correlation coefficient of ciphered image approaches 0 , indicating the adjacent pixels are almost non-correlated; that proves that the original image’s statistical nature has been well proliferated into the stochastic ciphered image.

Tab. 1 Correlation coefficients of two adjacent pixels in two images

Direction Original image Ciphered image

Horizontal 0.9682 0.0005

Vertical 0.9451 0.0024

Diagonal 0.9489 0.0765

Fig. 4 Correlations of horizontally adjacent pixels in the ciphered image

If let 41 ,i i

′ ′== ⊗A A then the correlation coeffi-

cients of horizontal, vertical, and the diagonal adjacent pixels of ciphered image are

0.0010,0.0044,− and 0.0651. So the average of the experiments (A and )′A are 0.00025, 0.0034,− and 0.0708.

The correlation results of image Lena by dif-ferent encryption methods are given in Tab. 2. According to the comparison results indicated in Tab. 2, the proposed algorithm gets better corre-lation results than ones presented by Refs. [10–15] at horizontal and vertical direction but is a little bit poor in the diagonal direction. Generally speaking, the proposed algorithm gives better encryption results.

3. The performance comparison on encryption matrices

Ref. [7] discusses the construction of an en-cryption matrix based on the tensor product, and

gives the method for constructing a higher order matrix from second-order invertible matrices by tensor product, but does not explain how to get the second-order invertible matrix. The proposed al-gorithm can automatically produce arbitrary low- order matrix iA and control the elements ija size of iA by modifying the value of mold m (i.e. 0 );ija m≤ < and can conveniently construct a higher order encryption matrix A which meets the requirement of real world application, by tensor product of various low-order matrix .iA The per-formance comparison of encryption matrices con-structed by two methods is shown in Tab. 3.

Tab. 2 The comparison of correlation coefficients generated by different encryption algorithms

Algorithm Horizontal Vertical Diagonal

CEMBCCNN –0.00025 0.0034 0.0708

Ref. [10] –0.0142 –0.0074 –0.0183

Ref. [11] –0.01589 –0.06538 –0.03231

Ref. [12] 0.0500 0.0400 0.0200

Ref. [13] –0.0148 –0.0083 –0.0076

Ref. [14] 0.0028 0.0029 0.0046

Ref. [15] 0.0064 0.0084 0.0107

Tab. 3 The performance comparison of encryption matrices constructed by two algorithms

Algorithm Order of iA,ija parameter

of iA Order of A

Chaos of A

Ref. [7] 2 − Even ×

CEMBCCNNAny positive

integer Automatically

generated Any posi-

tive integer √

4. The analysis of computational efficiency on encryption algorithms

The encryption algorithm involving -dimen-n sional matrix multiplication is mod( ,256),=C AB then its time complexity is 3( ).nΟ If n is small, the algorithm is feasible; If n is large, it is also feasible by block encryption.

VI. Conclusions How to quickly and effectively construct an

invertible matrix as an encryption key and extract relative inverse matrix as a decryption key is the core for realizing secure communication through encryption matrix. Based on the Chebyshev chaotic neural network, this paper proposes a novel

ZOU et al. Construction of the Encryption Matrix Based on Chebyshev Chaotic Neural Networks 253

algorithm to construct the encryption matrix, and gives an encryption example. Both the theoretical analysis and the simulation results indicate that this algorithm can automatically produce arbitrary order encryption matrix, and the encryption results have good chaos autocorrect characteristics, sat-isfying the cryptology requirement.

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