(IJCNS) International Journal of Computer and Network Security, 38 Vol. 2, No. 8, 2010
A New Approach to Measure Quality of Image Encryption Alireza Jolfaei1 and Abdolrasoul Mirghadri2 1
2
Faculty and Research Center of Communication and Information Technology, IHU, Tehran, Iran
[email protected]
Faculty and Research Center of Communication and Information Technology, IHU, Tehran, Iran
[email protected]
Abstract: Image encryption techniques are applied widely in the digital world today to assure information security. Although more and more encryption algorithms appear, they lack a method to evaluate the encryption quality. Visual inspection is not enough on judging the quality of encrypted images. So, we propose three classes of measurements based on the pixel’s position changing, value changing and both value-position changing. In order to evaluate the efficiency of methods, measurements were applied on three different chaotic image encryption algorithms based on the baker’s map, Arnold cat map and standard map. Experimental results indicate the performance of the measurement techniques in terms of producing results that are consistent with the judgment by visual inspection. Keywords: encryption quality, chaotic image encryption, baker’s map, Arnold cat map, standard map
1. Introduction Nowadays, along with the development of digital technologies and telecommunication networks, there is a substantial increase in the demand for private and secure movement of highly confidential imagery data over public channels. The concern for protection of information is increasing at an alarming rate. It is important to protect the confidentiality of imagery data from unauthorized access. Security breaches may affect user’s privacy and reputation. So, data encryption is widely used to confirm security in open networks such as the internet. Digital image is a massive two-dimensional data. The smallest unit of an image is a pixel. In a digital image, each pixel represents a different level of color intensity. According to the capacity of human visual perception in distinguishing different levels of intensity, the entire range of intensity is divided into 256 levels. Thus, the level of intensity in each pixel has a value between 0 and 255. This range is demonstrated by a byte (8 bits). Therefore, each pixel is equal to one byte. For example, a gray scale image with size of 256×256 pixels is approximately 65 KB. So, an image with a small size has a large data volume. However, due to large data size and real time requirement, it is not reasonable to use conventional encryption methods. Thus, a major recent trend is to minimize the computational requirements for secure multimedia distribution. During last two decades, chaotic dynamical systems have attracted the attention of cryptographers due to their definable and pseudo-random behavior. In consequence of increased interest in this field, a large number of chaos based image encryption schemes have been proposed [1, 2, 3].
Designing good image encryption schemes has become a focal research topic since the early 1990s. So far a number of image encryption quality measures have been proposed [4, 5, 6]. However, Most of the previous studies on image encryption were based on visual inspection to judge the effectiveness of the encryption techniques. Unfortunately, there are no classified measures to justify and compare the effectiveness of proposed schemes. However, in [7], Elkamchouchi and Makar presented quantitative measures of the encryption quality based on maximum deviation and correlation. Afterwards, they proposed an improved version of maximum deviation measure and named it as irregular deviation measurement. In this paper, we present new classified tests for encryption quality measurement and implement these tests on three common encryption schemes based on baker’s map, Arnold cat map and standard map and compare the results. This paper is organized as follows. In the next section three image encryption schemes based on chaotic maps are briefly overviewed. In Section 3, the new classified measures of encryption quality are introduced. Experimental results for presented encryption schemes are reported in section 4. Finally, some conclusions are given in Section 5.
2. Chaotic Image Encryption Algorithm The increasing interests in utilizing chaotic dynamics in various cryptographic applications have ignited tremendous demands for chaos generators with complex dynamics but simple designs. The mixing property of chaotic maps is of particular interests for cryptographic designs. Due to the differences in formulations, the nature of the generated chaotic maps may not be the same and hence their characteristics are different. Among chaotic maps, 2D baker’s map, Arnold cat map and standard map attract much attention. These prevalent maps are described as follows. 2.1 Baker’s Map The baker’s map, invented by Eberhard Hopf in 1937, is an intuitively accessible, two-dimensional chaos-generating discrete dynamical system [8]. This is a simple example of a map similar to a horseshoe, although it is a discontinuous map [9]. Consider the map F for the half-open square [0,1) × [0,1) onto itself where
F (x , y ) = (σ (x ), g (a, x , y ))
(1)
(IJCNS) International Journal of Computer and Network Security, Vol. 2, No. 8, 2010
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σ ( x) = 2x mod 1 , 0 ≤ σ ( x) < 1 1 ⎧1 0≤x < ⎪⎪ 2 ay 2 g (a, x , y ) = ⎨ 1 1 ⎪ (ay + 1) ≤ x 0 is the control parameter. In order to map image pixels to another in a bijective manner, the discretized version of standard map is required. In [14], Fridrich stated the criterion for continuous map discretization. So, the discretized standard map is N N N , Y =y , K =k attained by substituting X = x , 2π 2π 2π which maps from [0, 2π)×[0, 2π) to N×N. The discretized map is as follows
⎧x n +1 = (x n + y n ) mod N , ⎪ 2π x n +1 ⎨ ⎪ y n +1 = ( y n + k sin( N )) mod N . ⎩ (a)
(6)
(8)
This map reduces the computational complexity by operating in integer domain. So, it is more suitable for realtime data encryption.
3. Measurement of Encryption Quality (b) Figure 1. Baker’s map: (a) geometrical nature of the baker’s map, (b) area contraction by the map F.
2.2 Arnold Cat Map The Arnold cat map is a discrete system that stretches and folds its trajectories in phase space. Vladimir Arnold discovered the ACM in the 1960s and he used the image of a cat while working on it [11]. Assume that the dimension of the original grey scale image is N×N. Arnold cat map is described as follows:
p ⎤ ⎡x n ⎤ ⎡x n +1 ⎤ ⎡x n ⎤ ⎡1 ⎢ ⎥ = A ⎢ ⎥ mod N = ⎢ ⎥ ⎢ ⎥ mod N , (5) ⎣q pq + 1⎦ ⎣ y n ⎦ ⎣ y n +1 ⎦ ⎣y n ⎦ where p and q are positive integers and det (A) = 1, which makes the map area-preserving. The (xn+1, yn+1) is the new position of the original pixel position (xn, yn) when Arnold cat map is performed once. The period T of the Arnold cat
Image encryption quality measures are figures of merit used for the evaluation of image encryption techniques. We classify these measures into three categories: methods based on the pixel’s position changing, methods based on the pixel’s value changing and methods based on both pixel’s value and position changing. We present these measures as follows. 3.1 Measurement Based on the Position Changing Here, we propose a method to justify the confusion property of a chaotic map. That is to test the average distance change (ADC) among indices of closed pixels in plain-image and indices of relocated pixels in cipher-image. If an H×W image is permuted by chaotic map, then for the four neighbor pixels in the plain-image {(i–1, j), (i+1, j), (i, j–1), (i, j+1): (i = 1, 2,…, H–2), (j = 1, 2 ,…, W–2)}, the average distance change is defined as A DC (i , j ) =
1 [ D ((i ′ − 1, j ′), (i − 1, j )) + D ((i ′ + 1, j ′), (i + 1, j )) (9) 4 + D ((i ′, j ′ − 1), (i , j − 1)) + D ((i ′, j ′ + 1), (i , j + 1))],
(IJCNS) International Journal of Computer and Network Security, 40 Vol. 2, No. 8, 2010
D ((i ′, j ′), (i , j )) = (i ′ − i )2 + ( j ′ − j )2 , (10)
where (i', j') is the location of the pixel permuted from the one (i, j). Thus, the average distance change in the whole image is ADC =
1 (H − 2)(W − 2)
H −2 W −2
∑ ∑ ADC (i , j ). (11) i =1
j =1
Seen from Eq. (11), the average distance change is always bigger than 0, unless the permuted image is the same as the original one. The bigger ADC, the more confused the original image. The ADC is in relation with iteration time. 3.2 Measurement Based on the Value Changing Plain-image pixels values change after image encryption as compared to their original values before encryption. Such change may be irregular. This means that the higher the change in pixels values, the more effective will be the image encryption and hence the encryption quality. So the encryption quality may be expressed in terms of the total changes in pixels values between the plain-image and the cipher-image. Ahmed et al. proposed a measure for encryption quality that is expressed as the deviation between the original and encrypted image [4]. This method is determined as follows: Let P, C denote the original image (plain-image) and the encrypted image (cipher-image) respectively, each of size W×H pixels with L grey levels. P (x , y ),C (x , y ) ∈ {0,..., L − 1} are the grey levels of the images P, C at position (x, y), 0
