A New Algorithm of Image Encryption Based on 3D Arnold Cat

3D Arnold cat map can be applied in image encryption, and it has more security and better effect. However, its period is fixed. The original image will be returned to itself if iterating some times. On the basis of 3D Arnold cat map, it presented an algorithm of image encryption which separates the original image to many same blocks and no period. Simulation analysis shows that the encryption algorithm has characters of strong keys, better effect and fast.


Introduction
Image encryption is very important method in many fields such as military, medical system and communication [1]. The classic algorithms of image encryption [2,3,4], including Arnold cat transform, baker's transformation, Hilbert transformation and Zigzag transformation and so on, in which the Arnold cat transformation is most widely used, scrambling effect is relatively best, but the key of Arnold cat is smaller, and visual effects is also less than ideal [5]. At the same time, these image encryption algorithms applying mathematics transformation are carried out from position scrambling, the image pixel values have not changed, that is, no change the gray histogram, so the attacker can analyze through statistical means. From the substance of image scrambling, this paper proposes an algorithm of image encryption based on 3D Arnold cat map, combined with logistic chaotic map to image encrypt. The encryption system can be applied in medical image processing and transformation. The experiments show that the feasibility and effectiveness of the algorithm.

The 3D Arnold Cat Map and Logistic Map
Arnold Cat Map. Arnold cat transformation is a classical encryption algorithm. 3D Arnold cat map is shown as Eq.1.
Where a, b, c, d and e are positive integers, x F and y F is the original pixel positions while x F ' and y F ' is scrambled pixel positions. z F is a temp parameter and z F ' is scrambled pixel value. Chaotic Map. The technology of image encryption that based on chaos is a code encryption technology that having developed in recent years. It looks upon the original image as the binary data stream that according to some encoded mode, then encrypts the image by using chaotic signal. The reason that Chaos is fit to image encryption is closely related to some of its dynamics characteristics. The chaotic signal has natural concealment, high sensibility to initial condition and to tiny perturbation motion, all those make the chaotic signal has an ability of long time unforeseeable. The security of this encryption system depends on the degree of approximation between signal and random numbers that produced by secret key stream generator (be chaotic). The secret key stream is getting higher security as it approaching random numbers, whereas it is easily to be broken through. Logistic map is an example among nonlinear equation which can be applied on the experiment mathematic studies triumphantly. Although it is simple, it can embody all the nature of nonlinearity phenomenon. Its function is shown as Eq.2.
. If µ =4 then the system is in chaotic state, and the sequence that the system produces now has the characteristics of randomness, erotic, and the sensibility sensibility to original value. And the range of it is (0,1). All these characteristics can provide a very good maintenance for the image encrypt operation.

Image Encryption Algorithm based on 3D Arnold Cat
On the basis of 3D Arnold cat map, an improved algorithm is defined as Eq.3.
). (4) Initial value of a, b, c, d, e and K to iterate K times to get position scrambled image.

Simulation Experiment and Analysis
The original gray image of 256×256 is shown as Fig.1 (a). When the parameters of B=16, K=1, x 0 =0.1368, a=2, b=4, c=6, d=5and e=2, the scrambled image is shown as Fig.1 (b). When B=16, K=100, x 0 =0.9876, a=2, b=4, c=3, d=23 and e=201 is shown as Fig.1 (c). When K=200, B=32, x 0 =0.9456, a=11, b=14, c=23, d=3 and e=101 is shown as Fig.1 (C). It can get better effect no matter the parameters. Therefore, the key space of the presented encryption algorithm is unlimited. It is impossible for unauthorized users to decode the encrypted image by means of an exhaustive searching for the possible choices in the security key space. As a result, the image is protected by a high level of security. In cryptanalysis, the chosen-plaintext attack is an attack model in which the attacker can choose a number of plaintexts and then get their corresponding cipher texts. In this manner, the attacker can choose any useful information as plaintext in order to deduce the security keys of encryption algorithms, or to reconstruct the original plaintexts from the unknown cipher texts. If the image pixel values are not changed by the encryption process, the chosen-plaintext attack can break the encrypted image without knowing the encryption algorithm or its security keys. The presented 3D Cat Map based image encryption algorithm changes image pixel values while changing the locations of all image pixels. This ensures that the encrypted image data is not useful in the case of a chosen-plaintext attack. As a result, the presented algorithm is able to withstand chosen-plaintext attacks. Ability of resisting statistic attack. Studies have indicated that there exists inverse ratio between the stand or fall of an image scrambling effect and the correlative degree of adjacent pixel points, the more correlativity the worse displacement effect ,on the contrary the correlativity is getting less then the scrambling effect is better. Testing the correlativity of horizon (vertical) adjacent pixel points in the scrambling image, the method is as follows: Take the image pixel with its horizon (vertical) direction next pixel form an adjacent pixel couple, and randomly sampling 100 couples like this, then making use of Eq.4, Eq5 and Eq.6 to calculate the related coefficients of horizon (vertical) adjacent pixel points separately. Among the formula: X is grey value of the former pixel point; Y is grey value of the latter pixel point; cov (x , y) is the covariance of x, y.
Among the formula: r xy is the related coefficients. Carrying out the experiment analysis on the adjacent pixel points of the primitive image Fig.1 (a) and the encrypted image Fig.1 (b), the final outcome is expressed as Table 1.  Fig.2 it can be seen that the distribution of primitive image pixel gray values is concentrated on some values, while the pixel gray values after the encryption are scattering in the entire pixel value space. Accordingly it indicates that this encryption method has very good characteristics of gray evenly distribution. Thereby, it can fight against certain degree of statistic analysis attack.

Summary
On the basis of 3D Arnold cat map, it presented an algorithm of image encryption which separates the original image to many same blocks and no period. Simulation analysis shows that the encryption algorithm has characters of strong keys, better effect and fast.