A New Image Permutation Approach Using Combinational Chaotic Maps

INTRODUCTION

With the fast developments of computer network technology, more and more images are transmitted over the Internet. Sensitive image, such as military image, must be protected while transmitted over public channels. How to protect the security of images becomes a serious problem. The encryption is an important tool to protect security of images from unauthorized access. Some traditional encryption technologies such as Data Encryption Algorithm etc. can be used to protect the security of images. But for some intrinsic features of image, such as bulk data capacity and high correlation among adjacent pixels, it is known that conventional encryption technologies have obvious limitations on practical image encryption (Socek et al., 2007; Mazloom and Eftekhari-Moghadam, 2009).

Image permutation is a method or a process for protecting image from undesirable attacks by converting it into a new image not recognizable by its attackers. Permutation can against statistical cryptanalysis. At the same time, it can charge the correlation among adjacent pixels. Image permutation usually is taken as a very important part of image encryption. Some new technologies are used in image permutation; act as SCAN patterns (Maniccam and Bourbkis, 2004), chaotic map (Feng et al., 2006) etc. SCAN patterns generate very large number of scanning paths or space filling curves. The image encryption is performed by SCAN-based permutation of pixels and a substitution rule which together form an iterated product cipher. The SCAN patterns require the size of plain image must be even.

Chaos based image encryption provides a useful supplement to conventional methods (Kocarev, 2001; Zheng and Gao, 2011; Wang et al., 2011). Chaos has many characteristics which can be connected with the “confusion” and “diffusion” property in encryption, such as sensitive dependence on initial conditions and parameters, broadband power spectrum, randomness in the time domain, ergodicity, low-dimensional etc. In fact the idea of using chaos for encryption can be trace back to the classical Shannon’s paper (Shannon, 1949) in which the basic stretch-and-fold mechanism of chaos was proposed which could be used for encryption.

In particular, many chaotic image encryption schemes are based on permutation. There are some typical chaotic maps suitable for image permutation such as the Standard map, the Logistic map, the Cat map, the Baker map, the Tent map, the chaotic matrix etc. Among them, the Baker map, the Cat map and the Standard map (Patidar et al., 2011) attracted much attention. Fridrich (1998) obtained a symmetric image encryption scheme. It is shown that the permutations induced by the Baker map behave as typical random permutations. The Baker map can be used in image permutation. But the version B of the Baker map doesn’t have simple formula and the key are limited by size of image. Two symmetric image encryption schemes based on three-dimensional chaotic maps are proposed (Mao et al., 2004; Chen et al., 2004). They employ chaotic maps to shuffle the positions of image pixels which are the image permutation. In order to strengthen security of image encryption, a large chaotic permutation matrix was designed to achieve the high performance of pseudorandom permutation in by Yoon and Kim (2010).

A new chaotic map, the line map was proposed by Feng et al. (2006). An image encryption scheme based on the two maps of the line map was developed. But there are two serious problems in the study: It has many week keys and duplicate keys and the encryption time may disclosure the key. It seriously affects the security of the encryption algorithm.

In order to avoid the above-mentioned problems, the study proposed a new image permutation approach using combinational chaotic maps. There are four chaotic maps different from such two maps. A new image permutation approach based on the four maps is developed which use a six decimal numbers as the key. The simulation results validate the permutation approach.

THE KEY GENERATION

On many occasions, people are required using six decimal numbers as a password, such as Internet, banks, games. Here the key in image permutation also is six decimal numbers. Suppose the key is k₁k₂k₃k₄k₅k₆, here 0 = k_i = 9 (i = 1, 2, 3, 4, 5, 6). The process of the key generation is as following:

Fig. 1:	The process of the key generation

Table 1:	The relationship between the number in the key and the order of four maps

The process of the key generation can be seen in Fig. 1. It can use key₂ to encrypt image. Image permutation can be achieved by chaotic maps. Here suppose there are four chaotic maps in the study: map A, map B, map C and map D. Each number in the key means the order of four maps. It can see in Table 1. A simple example is given here. Assuming the key is “1821”. From Table 1 the number “1” means firstly map B, then map A, map C, map D. By analogy it can see the process of permutation is firstly map B, then map A, map C, map D, map D, then map A, map B, map C, map B, then map C, map A, map D, map B, then map A, map C, map D.

INTRODUCTION OF THE CHAOTIC MAPS

Suppose that a square image consists of NxN pixels with L gray levels. The key chaotic map utilizes an important characteristic of images, which is, each pixel in image can be inserted into the corresponding two adjacent pixels.

The new chaotic map is realized by processing image stretch-and-fold. Firstly, a square image was divided into two isosceles triangles along the diagonal, utilizing the difference of the pixel numbers of two adjacent columns of the triangles, each pixel in a column was inserted to the next adjacent column. Then, the plain image could be stretched to a line. Finally, the line was folded over to a new square image whose size was the same as the plain image.

Fig. 2 (a-b):	Principle of the chaotic map, (a) The left map, (b) The right map

Fig. 3:	The process of the left map with a 4x4 image

The map shuffles the positions of image pixels. All the process is invertible which is seen in Fig. 2. The chaotic map is divided into two forms: The left map and the right map.

In order to explain the chaotic map more clearly, an example is given here. The process of the left map permutation is shown in Fig. 3. The image with 4x4 pixels, that is N = 4. In upper triangle, pixel (3, 3) can be inserted before pixel (2, 2). Pixel (2, 3) can be inserted between pixels (2, 2) and (1, 2). Pixel (1, 3) can be inserted between pixels (1, 2) and (0, 2) and so on. Then, the pixels join to a part of line: (3,3), (2,2), (2,3), (1,2), (1,3), (0,2)…. in lower triangle, pixel (3,0) can be inserted before (3,1). Pixel (2,0) can be inserted between pixel (3,1) and (2,1) and so on. The pixels join to another part of line. Then, it connects two parts to a line. Lastly it is from a line to a square image different from the originally one. The right map is symmetric with the left map.

Supposing the dimension of a square image is NxN, where N is an integer. A (i, j) is the matrix of a square image, in which each element corresponds to a gray-level value of the pixel (i, j); L(i), i = 0, …, N-1, j = 0, …, N-1. N is a one dimensional vector mapped from A.

(1)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(2)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(3)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(4)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(5)

where, A’ is the matrix of the mirror image of a square image A:

(6)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(7)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(8)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(9)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

Method one is shown in Fig. 4a. The map from line L to image B is described with the following formula:

(10)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

Fig. 4 (a-b):	The process of folding algorithms

Method two is shown in Fig. 4b. The map from line L to image B is described with the following formula:

(11)

while j≥i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

(12)

while j = i, N-j is odd number, i = 0, …, N-1, j = 0, …, N-1.

A NEW IMAGE PERMUTATION APPROACH

A new image permutation approach is carried out based on the four maps. The plain image and cipher image are shown in Fig. 5. It has 256x256 pixels with 256 grey levels. The plain image is encrypted using the maps by the key₁ “1111” and key₂ “182142157171”. It can be seen that the plain image has been encrypted. The plain image and the cipher image are equal for every pixel; the decrypted image is recovered completely. It shows the image encryption using the chaotic map has no message loss. The time of encryption by key₁ is 0.1319 sec and the time of encryption by key₂ is 0.8478 sec; the time of decryption by key₁ is 0.0929 sec and the time of encryption by key₂ is 0.7844 sec. (the CPU of PC is Intel’s core i5 2.5 GHz, the ram is 4G and the operating system is Windows 7).

Correlation of two adjacent pixels in a cipher image, cov (x, y) = E (x-E(x)) (y-E(y)):

where, x and y are gray-scale values of two adjacent pixels in the image.

Fig. 5 (a-c):	(a) Plain image and Cipher image of (b) key1 and (c) key2

Fig. 6 (a-c):	Correlations of adjacent pixels (a) Plain image and Cipher image of (b) key1 and (c) key2

Table 2:	Correlation coefficients of two adjacent pixels

Figure 6 shows the correlations of two horizon-tally adjacent pixels in plain image and cipher image (key₁ and key₂): The correlation coefficients are 0.9442, 0.0004 and 0.0007.

Fig. 7 (a-b):	The sensitivity of keys (a) Decrypted cipher image of key1 and (b) Decrypted cipher image of key2

Table 3:	Self-correlation of images

Similar results for diagonal and vertical directions were obtained and are shown in Table 2: