Abstract: Image encryption is essential to assure information security. Universal modular transformation is a widely applied encryption method, but it has some disadvantages such as a relatively small quantity of keys and insecurity. A uniform block encryption method is proposed in this study to improve the image entropy of encryption and an image is encrypted with the universal modular transform combined with a chaotic mapping. Experimental results show that this algorithm can provide good encryption quality and reach high safety. Moreover, the proposed algorithm is robust to shear attack and noise attack.
INTRODUCTION
With the development of computer and communication, the security of the multimedia digital image has become the focus of Internet. Universal modular transformations such as Fibonacci transformation (Zou and Liu, 2007), Arnold Cat transformation (Huang and Xiao, 2009; Chen, 2006) and affine modular transformation (Zou et al., 2000) are the most widely used methods nowadays for their simplicity. However, such algorithms have to be iterated several times to achieve good results and they have too small key quantities to resist exhaustive attack. A new approach which leads to uniform scrambling is proposed to increase the entropy of encrypted image in this study. Furthermore, a uniform block encryption algorithm combined with a class of chaotic mapping (Kwok and Tang, 2007; Fan and Jiang, 2004; Huang and Feng, 2007; Ren et al., 2008) based on the universal modular transformation is proposed to increase the key quantities. Finally, experimental results are provided to verify the effectiveness of the proposed method.
UNIVERSAL MODULAR TRANSFORMATION
Suppose an image consisting of NxN pixels, where (x, y) and (x', y') are the coordinate of a pixel in the original image and the scrambled image, respectively. The universal modular transformation is given as:
| (1) |
where, mod (N) means the modular, which is to ensure that the pixels of the image still fall on the original region after transformed.
When the matrix determinant
UNIFORM BLOCK ENCRYPTION
The essence of image encryption or image scrambling is to reduce the correlation of pixel positions and the correlation of pixel values until they are irrelevant. From the perspective of information theory, digital image should be seen as information source composed of the non-overlapping blocks with arbitrary shapes and numbers and the pixels of image can be seen as messages of the information source. Image scrambling process can be considered as the process of enhance uncertainty, also as the process of increasing the amount of image information. The correlation of the natural image pixels is deemed to be the largest in respective block and the uncertainty is as well. If some original image pixels in block σ1 still distribute in the same block σ2 of the scrambled image, even if the block σ1 and σ2 have different situation, the certainty of these pixels are still large and the corresponding information content generated by scrambling is less. Therefore, the study defines the concept of image position entropy by Eq. 2:
| (2) |
where, B is the total number of image blocks, Hk(P) is the entropy of the kth block and denotes the average information capacity in this block. H(S) is the image position entropy.
| (3) |
where, P(i,j) is the probability of pixel which coordinates (i,j) in original image appears at the kth block in the scrambled image. By the property of discrete information source entropy, we known that Hk(P) will reach the maximum when the probability P(i,j) is equal. At the same time H(S) will get its maximum value. So, the perfect state of image scrambling is the random pixel in respective block of the scrambled image has the equal probability of coming from the random situation in original image. It also means that average information content will get the maximum when the probability that pixels in the same block of original image distribute into different blocks is equal. Temporality the effect of image scrambling is the best. The definition of uniform scrambling proposed is as follows:
| • | All the pixels in the same block of original image distribute into all the blocks of scrambled image and the every block has one pixel at least, without regarding to the order of the pixels appearance, accordingly all the pixels in the same block of scrambled image come from different blocks of original image, this condition is described uniform scrambling |
Figure 1 shows that all the pixels in the first block of the original image are distributed into all the blocks of the scrambled image. Thus, the ideal block numbers is N for an original image of size NxN. But the number maybe not the integer for images of different size, so we can take the two numbers which has the approximate value to N.
| Fig. 1: | Image uniform encryption |
UNIFORM BLOCK ENCRYPTION ALGORITHM BASED ON UNIVERSAL MODULAR TRANSFORMATION
The basic principle of the algorithm: In order to apply the definition of uniform scrambling to Universal modular transformation, we need to improve the transformation because the classical methods are maps for the whole image, while according to the proposed method pixels should mapped from blocks to the whole image. The Universal modular transformation in block (i, j) is described by Eq. 4:
| (4) |
where, N is the size of the image, B is the total number of original image blocks just means there are BxB blocks in the image and aij, bij, cij, dij, eij, fij is the parameters of each block. i, j∈(1,B) and i, j are positive integers.
Uniform block encryption algorithm based on universal modular transformation is given by Eq. 5:
| (5) |
where, X and Y are the coordinates of original image pixels, X' and Y' are the coordinates of whole scrambled image pixels, X'' and Y'' are the coordinates of image pixels which are taken uniform block encryption on whole scrambled image m, k are the times of scrambling.
The equation means that we first scramble the image in the whole for several times and then encrypt image k times by uniform block encryption algorithm. If m = 0 that means encrypt image k times by uniform block encryption algorithm directly.
The study proposed a scheme to realize uniform block encryption algorithm by affine modular encryption transformation. To simplify the arithmetic and make the transformation area-preserving, the parameters are constrained. The formula is described by Eq. 6:
| (6) |
The affine modular transformation of digital image keeps the cycles of Arnold Cat transformation while it has more parameters to choice, so it can increase the secrecy of image. The equation of uniform block encryption algorithm based on affine modular transformation is described by Eq. 7:
| (7) |
where ao and bo are parameters of image scrambling in the whole, ai, bi, ei and fi are parameters after image taken in uniform block encryption, i∈(1,B2) and i is positive integers. (k1, k2) is the coordinate of partitioned matrix for image block, n1, n2 are the number of iterative process.
The proposed method can take ao, bo, ai, bi, ei and fi as cipher keys and the key space is large because parameters of different blocks are different. It is impractical remembering all the parameters so we use chaotic map to deal with that.
Improved logistic map: The equation of logistic map is described with Eq. 8:
| (8) |
where, 0<μ≤4, xn∈(0,1).
The logistic map is chaotic map when 3.5699456<μ≤4, we often make μ = 4. For Logistic map does not satisfy uniform distribution, a better random system is given as:
| (9) |
where, yn is also the chaotic sequence and it satisfies uniform distribution. Simulation shows that the system response is extremely sensitive to the initial values, so we can amplify each value of yn and get round-off to generate the parameters of different blocks.
The Implementation of the algorithm
Encryption scheme:
| • | Step 1: Provide four initial values x1(0), x2(0), x3(0), x4(0) to generate chaotic sequence {x1(k)},{x2(k)}, {x3(k)}, {x4(k)}; then improving the four sequences to {y1(k)},{y2(k)}, {y3(k)}, {y4(k)} by Eq. 9. Get the sequence values which are in the place after the second and third of decimal point to decimal integers. Array those decimal integers to integer sequences {a(k)}, {b(k)}, {e(k)}, {f(k)} by nature order and assign these values to the parameters ai, bi, ei, fi. in different blocks |
| • | Step 2: Perform affine modular transformation n1 times for original image F(X,Y) with a0 = a(0), b0 = b(0), e0 = e(0), f0 = f(0). Then the whole scrambled image F'(X',Y') is obtained |
| • | Step 3: Divide the image F'(X',Y') into BxB blocks, generate the last scrambled image by taking Eq. 7 n2 times with the parameters {a(k)}, {b(k)}, {e(k)}, {f(k)} |
Decryption scheme:
| • | Step 1: Perform the same operation as step 1. in the encryption scheme. Then, Calculate the lowest common multiple T2 for different cycle with ai, bi , let n22=T2-n2 |
| • | Step 2: Split the image F''(X'',Y'') into N2/B2 blocks, generate image F'(X',Y') by taking Eq. 7 n22 times with the parameters {a(k)}, {b(k)}, {e(k)}, {f(k)} |
| • | Step 3: Take affine modular transformation n1 times for image F'(X',Y') with parameters a0 = a(0), b0 = b(0), e0 = e(0), f0 = f(0). After that the decrypted image F(X,Y) is generated |
RESULTS AND DISCUSSION
To demonstrate the effectiveness of the proposed method, experimental results are given in this section. For the classical Lena image with 256x256 pixels as in Fig. 2, the initial values for the chaotic mapping is set as x1(0) = 0.1, x2(0) = 0.2, x3(0) = x4(0) = 0.3 and the image is divided into 16x16 blocks.
Encryption experiments and some analysis: In this part, the universal modular encryption algorithm without uniform block scrambling is compared with the proposed method under the condition of the same number of transformation times.
| Fig. 2: | Original image |
| Fig. 3: | Comparing the effect of encryption |
| Fig. 4: | Decrypting image adopting little difference key. (a) x1(0)
= 0.218, (b) x1(0) = 0.218+10-12 |
The experimental results are given in Fig. 3, where, Fig. 3a-d are the output results by the Universal modular encryption algorithm with 1, 4, 64 and 192 times of transformations, respectively. And Fig. 3e-h are the corresponding experimental results with the uniform block scrambling encryption algorithm.
From the above comparison, it can be seen that the Universal modular encryption algorithm without uniform block scrambling has the periodic characteristics and the encryption results may not be good with the first several times of iteration. Also, the encryption quality fluctuates with increased iteration times. With the uniform block scrambling method, however, good encryption result can be obtained even without any iteration and thus may be better than the Universal modular encryption method.
Key sensitivity experiments and some analysis: Suppose an image is encrypted with the key x1(0) = 0.218, x2(0) = 0.25, x3(0) = 0.7, x4(0) = 0.95 and it is decrypted with x2(0), x3(0), x4(0) fixed to the right value. After that, the decryption results with x1(0) = 0.218 and x1(0) =0.218000000001 are shown in Fig. 4a and b, respectively.
From this experiment results, it is shown that the original image can not be obtained even with a bias of 10-12 in one of the key components. Thus, the proposed encryption method in this paper can have a very large key space and ensure the security of the encryption from exhaustive attacking with the hardware condition of c today.
Anti-shearing attack experiments: In this experiment, an image is respectively encrypted with the universal modular encryption algorithm with and without uniform block scrambling at first. Then, the shearing attack is performed on the encryption results with the shearing area being about 10~20%. Image restoration is performed with the attacked encryption results at last as can be seen from Fig. 5a-d and 6a-d, respectively.
From the above experimental results, it can be seen that if the universal modular encryption algorithm is applied with limited iteration times, the contour of the restored image is smeared when suffering from the shear-attack. With the uniform block scrambling encryption algorithm, the contour can be better restored and the information loss can be less because the shear-attacked part of the image can be distributed to the whole image and thus makes it more robust to shear-attack.
Anti-noise experiments: Zero-mean Gaussian noise with the variance of 0.5 is added to images encrypted by the affine transform scrambling encryption algorithm and universal modular encryption algorithm, respectively. The decryption results are shown in Fig. 7a-d and 8a-d.
From the decryption results, it can be seen that when the encryption result is polluted by the Gaussian noise, the contour of the restored image can be better reserved by the uniform block scrambling encryption algorithm.
| Fig. 5: | The cutting effect and decrypting by affine transform (a)
encrypting one time and cutting, (b) decrypting with a, (c) encrypting 64
times and cutting and (d) decrypting with c |
| Fig. 6: | The cutting effect and decrypting by uniform algorithm. (a) encrypting one time and cutting, (b) decrypting with a, (c) encrypting 64 times and cutting and (d) decrypting with c |
| Fig. 7: | The noising effect and decryption by affine transform. (a)
encrypting one time and adding noise, (b) decrypting with a, (c) encrypting
64 times and adding noise and (d) decrypting with c |
| Fig. 8: | The noising effect and decryption by universal algorithm.
(a) encrypting one time and adding noise, (b) decrypting with a, (c) encrypting
64 times and adding noise and (d) decrypting with c |
In present experiments, independent runs of present proposed method, we choose the classical Lena image with 256x256 pixels as original image and compare performance of our proposed method with the affine modular transform methods through three experiments. Figure 3 shows that the affine modular encryption algorithm without uniform block scrambling is compared with the proposed method under the condition that we scramble original image 1, 4, 64 and 192 times. Figure 5 and 6 show the universal modular encryption algorithm is applied with limited iteration times, the contour of the restored image is smeared when suffering from the shear-attack. Figure 7 and 8 show that Zero-mean Gaussian noise with the variance of 0.5 is added to images encrypted by the affine transform scrambling encryption algorithm and uniform modular encryption algorithm.
In addition, Fig. 4 show the proposed encryption method in this study can have a very large key space and ensure the security of the encryption from exhaustive attacking with the hardware condition of today. As a whole, our approach is superior to the existing universal modular encryption algorithm methods. The traditional universal modular transformation algorithms have to be iterated several times to achieve good encryption results and they have too small key quantities to resist exhaustive attack. This method can obtain good encryption result even without any iteration and it has the advantage such as large quantity of keys, robust to cutting attack and noise attack. Also, it has feasibility and validity.
CONCLUSIONS
From essential analyzing to image scrambling, a concept of uniform block scrambling is proposed from enhancing encryption image entropy and makes use of affine modular transform, combines to chaos theory, provides a image block uniform encryption algorithm. A number of experiments analysis proved that the proposed algorithm has the advantage such as large quantity of keys, robust to cutting attack and noise attack. Also, it has feasibility and validity.
ACKNOWLEDGMENT
This study was supported by the National Natural Science Foundation of China (60572030) and the Chun Hui Program of Chinese Education Department (Z2007-1-15014).