Encryption Approach for Images based on Householder Reflector Scheme and Extended Hill Cipher Techniques

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1 Encryption Approach for Images based on Householder Reflector Scheme and Extended Hill Cipher Techniques Karima Djebaili 1 Lamine Melkemi 2 Department of Computer Science, University of Batna, Batna, Algeria 1 Department of Mathematics, University of Batna, Batna, Algeria 2 karima.djebaili@univ-batna.dz 1 lamine.melkemi@univ-batna.dz 2 Abstract In this paper we are concerned with the problem of protecting the transmission of digital images over insecure channel. Such an encryption image system is judged efficient and secure if statistical and differential tests reveal satisfactory results and the system should design to resist cryptanalysis attacks including brute force and known/chosen plaintext attacks. On the other hand the encryption and decryption process require reasonable time and hardware space. In this context we propose a symmetric key cryptosystem using matrix transformation and Householder reflector, which achieve the requirements related to its security and efficiency. We performed a series of tests and comparisons to confirm the efficiency of this method. Therefore, we may say that the proposed scheme has a high security and high capability to resist statistical, differential and known/chosen plaintext attacks, also this method eliminates the computational complexity involved in finding inverse of the key while decryption. Keywords: Image encryption; matrix transformation; Householder reflector; statistical tests; differential tests. I. ITRODUCTIO In our time, the security of information becomes obvious, especially with the increasing of communication networks. Several sensitive domains required the secure exchange of numerical images such as medical records, the diplomatic service and government in general [1]. Cryptography was used as a tool to protect sensitive information against unauthorized access [2]. There are many good algorithms for textual data but cannot be applied to numerical images (such as RSA, AES and DES) [3], for the reason that these algorithms do not take into consideration the structural and statistical properties of the images. In most cases, the values of the adjacent pixels of an image are powerfully correlated [4], so to encrypt it the encryption process should decrease this correlation. Many algorithms have been proposed to tackle the image security problem, such as the Hill cipher [5]. Hill cipher can be easily broken with a known/chosen plaintext attacks, also the need to the generation of a large invertible matrices and calculate its inverse is another obstacle against the use of this cipher in practice. To overcome these problems Krishna and Madhuravani [6], proposed a method which reduces the problem of computing inverse matrices, but it is still vulnerable to known/plaintext attacks. Another Hill modification is proposed in [7], is based on the permutation and multiplication using an initial vector, however the proposed algorithm is vulnerable to known/chosen plaintext attacks. The authors in [8], suggested the use of self-invertible matrices in a modified Hill encryption to eliminate the complexity of calculate the inverse of the key which resists chosen plaintext attacks but this last cannot encrypt an image that contain large areas of a single color. Another survey in image encryption approach using a combination of permutation technique introduced by [9]. Here permutation process was very complicated. A different permutation based image encryption technique [10], is based on random pixel permutation with the motivation to maintain the quality of the image, in this there is a high chance of error in key generation process. Another category of image encryption is based on using the chaotic systems due to the close relationship between chaos and cryptography [11]. In [12] a new method based on Fractional Wavelet Packet Transform (FWPT) is proposed, it has a drawback of limited key space and limited perceptual quality. The key space is the number of all the keys that can be used for the encryption. Several techniques have been proposed using one dimensional chaotic system [13], but these systems, present a weak of security due to the linear function in logistic map. On the other hand, various cryptanalysis have exposed some inherent drawbacks of chaotic cryptosystems [14], we enumerate the slow performance, small key space and the need of sensitive materials, which make it difficult to promote the chaotic digital encryption into practical service. In this paper a simple and secure algorithm for image encryption is proposed in order to overcome this disadvantages using: 1

2 1. Householder reflector, to eliminates the computational complexity involved in finding inverse of the key while decryption. 2. Shared a privet-key vector (short key) instead of a matrix. 3. Using two method which add the confusion and definition properties 4. Matrix transformation method which allowed a good encryption and a large key space. II. PRICIPLE TOOLS In this section we give the tools of our model. 1. Matrix Transformation Let as denote the set of all n n matrices as M, left and right action operations in the set M are denoted by and respectively, can be expressed in the following way: In the matrix equation G = A, the array in A is flipped row- wise down for w i steps where w is a privet-key vector and G M. In the matrix equation H = A the column in A is flipped column-wise left to right for w i steps where w is a privet-key vector and H M. 2. The Householder reflector Suppose A is a square matrix. The inverse matrix A -1 of the same size, satisfy the following: Definition 1 The matrix A is invertible if there exists a matrix A -1 such that: A -1 A=I and AA -1 =I. (1) Definition 2 The matrix A is called involutary matrix if: AA=I. (2) The main problems we wish to solve in this paper are: 1. Find a good way to write an involutary matrix in order to eliminate the complexity of finding the inverse of the matrix during decryption process. 2. Facility the transmission and the save of the secret key (matrix) by transmit a vector instead of the matrix. Householder reflector is orthogonal reflector across the plane orthogonal to a unit normal vector z can be expressed in matrix form as: Definition 3 Let z Z n with z t z = 1. Then the n n matrix: H=I-2zz t. (3) is called a Householder reflector. 2.1 Properties of Householder reflector a) H is orthogonal, i.e. HH t = I. b) H is symmetric, i.e. H = H t. c) H is involutary, i.e. HH = I. Where H t is the transpose matrix. Proof a) The symmetry is from: H t = (I 2zz t ) t = I t (2zz t ) t = I 2(z t ) t (z) t = I 2zz t = H. b) The orthogonality is from: HH t = (I 2zz t )(I 2zz t ) = I 4zz t + 4zz t zz t = I 4zz t + 4z(z t z)z t = I 4zz t + 4zz t = I. c) The involutary is same as orthogonality, due to symmetry, since H = H t. 3. Diffie-Hellman key agreement protocols The challenge in symmetric key encryption is that how a party A can transmit a secret key throw insecure channel? There are many protocols in litterateur, provided a solution to the key distribution problem. Diffie-Hellman key agreement provided the first practical solution to the key distribution problem [15], allowing two parties, to establish a shared secret by exchanging messages over an open channel. III. The proposed Algorithm The following notation will be used throughout this algorithm: 1. A denotes the image of pixels. 2. x(i, j) denote the indexed set of elements in any matrix X. 3. w denotes the privet-key vector. 4. H denotes the Householder reflector. 5. n denotes the number of rows or columns in A. 6. C denotes the encryption image. 7. is the exclusive-or (XOR) operation defined as follows:0 0 = 0, 0 1 = 1, 1 0 = 1, 1 1 = 0. The new algorithm which consists of two phases is presented as follows: A. The Encryption Process The detail of encryption process is described below and the diagram of this process is presented in Fig.1. Step.1 let assume that w = (w 1,w 2,w 3,...,w n ) Z n where, n 2 i=1 w i mod n = 1. (4) using definition 3 we can generate H as follows: H=I-2 ww t. (5) Step.2 To add the confusion properties to the encryption process, the matrix B is calculate from the matrix A as follows: b(i,j)=a(j,i), for i,j =1,2,3, n. (6) Step.3 Using matrix transformation we can calculate the matrix R as follows: R= wi B wi. (7) 2

3 Original image DH protocol Cipher image DH protocol Confusion phase Householder Reflector Diffusion phase Householder Reflector Matrix transform Hill encryption Hill encryption Matrix transform Diffusion phase Confusion phase Cipher image Original image Fig. 1. The diagram of encryption process Fig. 2. The diagram of decryption process Step.4 In this step the Hill encryption can be represented by the matrix multiplication: Q=R*H. (8) Step.5 To add the diffusion properties to the encryption process, the matrix C is calculate as follows: c(i,j)=q(j,i) ((w(i)+h(i,j) mod n), for i=1,2,3, n and j=1,2. (9) c(i,j)=q(j,i) ((c(i,j-1)+c(i,j-2) mod n), for i=1,2,3, n and j= 3,4,5 (10) B. Decryption process The decryption process is similar to that of the encryption process illustrated above, but with the reverse operational steps. The diagram of this process is presented in Fig.2. Step.1 Using the privet- key vector w, Calculate H uses definition 3. Step.2 Recover matrix Q from Eq. 9 and Eq.10 as follows: q(i,j)=c(j,i) ((w(i)+h(i,j) mod n), for i=1,2,3, n and j=1,2 (11) q(i,j)=c(j,i) ((q(i,j-1)+q(i,j-2) mod n), for i=1,2,3, n and j= 3,4,5 (12) Step.3 Due to the proprieties of Householder reflector, recover the matrix R operation can be written similarly to Eq. 8 (without need the calculation of matrix inverse): R=Q*H. (13) Step.4 inversing matrix transformation we can calculate the matrix B as follows: B= wi R wi. (14) Step.5 The matrix A is calculated as follows: a(i,j)=b(j,i), for i,j =1,2,3, n. (15) IV. SECURITY AALYSIS AD TEST RESULTS Some experimental results are given in this section to demonstrate the efficiency of our algorithm. Experimental analysis of the new algorithm presented in this paper has been done with two gray-scale images and two color images with the size , for color images same technique implemented for each color band (red, green and blue).the original images and the corresponding encrypted images are shown in Fig Statistical Analysis To prove the robustness of the proposed model, we have performed statistical analysis by calculating the histograms, the entropy and the correlations for the original and encrypted images. We have found that the values are good Visual and Histogram Analysis If the attacker analyzes the histogram of an encrypted image by using some statistical analysis to get some useful information of the original image he should fail, if we ensure that encrypted and original images do not have any statistical similarities. The histograms of original and encrypted images are shown in Fig.4. The histograms of the encrypted images have uniform distribution which is significantly different from original images and have no statistical similarity in appearance. Therefore, the proposed algorithm does not provide any clue for statistical attacks. 3

4 Fig. 3. (a,b,c,d) Original images. (e,f,g,h) Corresponding encrypted images. Fig. 4. (a,b,c,d) Histogram of Original images.(e,f,g,h) Corresponding histogram of encrypted images Information Entropy Analysis Entropy shows the level of uncertainty in any communication system. The entropy, H(x) of any data can be calculated as: 2n 1 1 H m = i=0 p(m) i log 2. (16) p(m i ) Where p(x i ) represents the probability of symbol x i. When data is encrypted for a source that generates 2 n symbols with equal probability, its entropy should be close to n bits ideally. In the case where the entropy is less than n bits, then there exists a certain degree of predictability. For the encrypted image with 256 symbols the entropy value should be close to 8. The entropy value of the encrypted image using our model is , so the encryption system is secure against the entropy attack Correlation Coefficient Analysis Any image encryption system is said to be good, if encryption algorithm hides all attributes of an original image, and encrypted image is totally random and highly uncorrelated. The correlation coefficient between two images mathematically can be written as: c = ( 1 1 i=1 x i x y i y i=1 x i x 2 )( 1 y = 1 x = 1 i=1 y i y 2 ). (17) i=1 y i (18) i=1 x i (19) where x and y are intensity values of two adjacent pixels in the image and is the number of adjacent pixels selected from the image to calculate the correlation. Fig.5 shows the correlation distribution of two vertical, horizontal and diagonally adjacent pixels in the original and encrypted image. 4.2 Key Space/Sensitivity Analysis To prevent some attacks like brute force, the key spaces should be large. So an encryption schemes considered secure if its key space is large enough. In our algorithm the key space K can be equal to K = bytes, this is very large because an attacker needs to try all possible keys. So our model is free from the brute force attack. 4

5 In another hand, the ideal image encryption procedure should be sensitive with the secret key.it means that the change of a single bit in the secret key should produce a completely different encrypted image. Fig.6 shows the sensitivity of our encryption system using a gray-scale image (the same results if we use a color image). 4.3 Differential Analysis In general, a suitable property for an encrypted image is being delicate to the little changes in original image (e.g., changing only one pixel). Attacker can make a little modify in the input image to notice changes in the result. By this technique, the significant relationship between original and encrypted image can be found. To analyze the impact of one pixel modify on the whole encrypted image by the proposed algorithm, two common measures are used: PCR (umber of Pixels Change Rate) between two images with the same size is defined as follows: PCR = i,j D(i,j ) 100%. (20) W H UACI (Unified Average Changing Intensity) between two images with the same size is defined as follows: UACI= 1 H W C i i,j C 2 (i,j ) i,j 100% (21) 255 C 1 and C 2 : two ciphered images, whose corresponding original images have only one-pixel difference. C 1 and C 2 have the same size. C 1 (i,j) and C 2 (i, j): gray-scale values of the pixels at grid (i, j). D(i, j): determined by C 1 (i, j) and C 2 (i, j), if C 1 (i, j) = C 2 (i; j), then, D(i, j) = 1;otherwise, D(i, j) = 0. W and H: columns and rows of the image. The results in Table 1 illustrate that a small modify in the original image will results in a significant difference in the encrypted image. Therefore, the proposed schema has a high ability to resist differential attack. TABLE 1. DIFFERETIAL AALYSIS BETWEE ORIGIAL IMAGE AD ECRYPTED IMAGE PCR UACI Cryptanalysis The encryption process is judged efficient if is effectively free from the cryptanalysis vulnerability, where cryptanalysis is the science of recovering original image (or text) without access to the private key (also called codecracking or code-breaking). The following analyses are used to demonstrate the weakness of some famous attacks in image encryption to break our model. Known/Chosen plaintext attack in this attack, the attacker has a prior knowledge of the encryption process as well as Fig. 5. (a,c,e)correlations of two horizontal, vertical and diagonal adjacent pixels in the original image. (b,d,f)correlations of two horizontal vertical and diagonal adjacent pixels in the encrypted image an original and encrypted image pair, the objective is to find the private key. Let us assume that the attacker tries to decrypt an image encrypted using the algorithm proposed in this paper, by randomly picking a key from the key space, and compares the resulting decrypted image to the original image. The probability of finding the correct key would be approximately 1/K where K is the key space; this is very low probability for finding the correct key. Another technique allowed the attacker to break down an image encryption is the use of particular images (ex: the all zero bloc image) to get some helpful information about the secret image. Fig.7 demonstrated the efficacy of our model where the encryption of all zero bloc images is totally a deferent image. V. COMPARATIVE AALYSIS Table 2 gives a comparison of our algorithm with the classical Hill and some existing algorithms using these comparison factors: a) PCR. b) UACI. c) eed inversed key matrix. d) Vulnerable if there are all zero blocks. e) Key sensitivity. TABLE 2. SCHEMES COMPARAISO AALYSIS USIG DIFFERET a) b) c) d) e) Hill Yes Yes Krishna Yes o Panigrahy o Yes Our algorithm o o

6 Fig. 6. Key sensitivity analysis. (a) Original Lena image, (b) Encrypted image using the original key, (c) Encrypted image using the key modified last bit, (d) Difference between (b) and (c),(e) Decrypted Lena using the original key, (f) Decrypted Lena using the modified key (a) Fig. 7. (a) zero image. (b) the encryption image of zero image VI. COCLUSIO The proposed algorithm improved image security based on matrix transformation and Householder reflector. At the time of decryption the receiver do the same process of encryption so the computational complexity of finding inverse of the decryption key can be eliminated. Histogram analysis shows that histograms of encrypted images are uniformly distributed, so the algorithm is secure from frequency analysis attack. Entropy analysis shows that the algorithm has entropy that closes to ideal, so the algorithm is secure from leakage of information. The proposed algorithm has expected performance; the lowest correlation and a large key space. To quantify the difference between encrypted image and corresponding original image, two measures were used: PCR and UACI, the results show that a small change in the original image or in the (b) encryption key will result in a significant difference in the encrypted image. There are some effective encryption process cannot encrypt images that contain large areas of a single color (like image (d) in Fig.3 ), also can be easily broken with a known/chosen plaintext attacks, the new algorithm is used to encrypt image that overcomes these disadvantages. Also the method presented in this work can be used for medical images in modern healthcare. References [1] UM Bhatti, Cryptography and network security, in 11 th Research Seminar Series Workshop, [2] Christof Paar and Jan Pelzl, Understanding cryptography:a textbook for students and practitioners, Springer, [3] M. Van Droogenbroeck and R. Benedett, Techniques for a Selective Encryption of Uncompressed and Compressed Images, in Proceedings of Advanced Conceptsfor Intelligent Vision Systems (ACIVS) 2002, Ghent, Belgium, Sept [4] S. P. ana'vati and K. P. Prasanta, Wavelets: Applications to Image Compression-I, Joined of the Scientific and Engineering Computing. Vol. 9, o.3: 2004, PP [5] Ryan Doyle, Hills cipher: Linear algebra in cryptography, [6] AV Krishna and K Madhuravani, A modified hill cipher using randomized approach, International Journal of Computer etwork and Information Security (IJCIS), vol. 4, no. 5, pp. 56, [7] Saroj Kumar Panigrahy, Bibhudendra Acharya, and DebasishJena, Image encryption using self-invertible key matrix of hill cipher algorithm, [8] A Mitra, Y V Subba Rao, SRM Prasanna, et al., A new image encryption approach using combinational permutation techniques, International Journal of Computer Science, vol. 1, no. 2, pp , [9] M.aliBani, younes1+, and A.jantan2++, an image encryption Approach using a combination of permutation technique, [10] 15. S. P. Indrakanti, P. S. Avadhani, permutation based image encryption technique, [11] Rasul Enayatifar, Image encryption via logistic map function and heap tree, Int. J. Phys. Sci, vol. 6, no. 2, pp. 221, [12] Xu Shu-Jiang, Wang Ying-Long, Wang Ji-Zhi, and Tian Min, A novel image encryption scheme based on chaotic maps, in Signal Processing, ICSP th International Conference on. IEEE, 2008, pp [13] Zhang Han, Wang Xiu Feng, Li Zhao Hui, Liu Da Hai, and Lin You Chou, A new image encryption algorithm based on chaos system, in Robotics, Intelligent Systems and Signal Processing, Proceedings IEEE Conference for. IEEE, 2008, pp [14] Hongjuan Liu, Zhiliang Zhu, Huiyan Jiang, and Beilei Wang, A novel image encryption algorithm based on improved 3d chaotic cat map, in Young Computer Scientists, ICYCS The 9th International Conference for. IEEE, 2008, pp [15] Alfred J Menezes, Paul C Van Oorschot, and Scott A Vanstone, Handbook of applied cryptography, CRC press,