INTERNATIONAL JOURNAL FOR ADVANCE RESEARCH IN ENGINEERING AND TECHNOLOGY WINGS TO YOUR THOUGHTS.. Design of a Cryptosystem Using Two-Level Hill Cipher

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1 Design of a Cryptosystem Using Two-Level Hill Sarla Dewangan 1, Mrs. Shikha Pandey 2, Mohammad Imroze Khan 3 1 M-Tech scholar, 2 Assistant Professor, 3 Assistant Professor 1,2 Rungta College of Engineering & Technology, Bhilai (C.G.), National Institute of Technology, Raipur (C.G.), sarla.dewangan@gmail.com, 2 shikhamtech2008@gmail.com, 3 imroze786@gmail.com Abstract: Pioneered in the field of polygraph substitution cipher, operation of Hill is based upon the linear algebraic equation by working on more than three symbols simultaneously. Its high speed of processing is advantageous and so it has inherent property of opposing the analysis based upon frequency letter. The advantage of faster processing time is due to the mode of operation, in which matrices are multiplied linearly. But due to its linearity, it s very difficult to decrypt any encrypted message as it consists of all matrices of a given dimension that are invertible over Z m. This is the same reason why it is considered to be difficult for decrypting image of same shade over a large space, which makes it useless to withhold any specific aspect out of an image, which might be pointing out any impression over it. The main aim of this paper is to use a technique for image encryption, which is different from the conventional one. Keywords: Cryptosystem, Encryption, Decryption, Hill 1. INTRODUCTION With the advancement of network technology, there is a crucial problem regarding the security of the information. Increasing network transmission ability and popular multimedia technology application gradually leads us to retrieve information clearly and directly from images. Therefore, the security of data is very complicated and imperative issue now [3]. Cryptography, which is the science of encryption, plays an important role in communication like mobile phones, pay-tv, e-commerce, private transformation, financial information transmission, ATM cards security, computer passwords etc are those examples which touches on many things in our daily lives [2, 9]. Science is the study of principles, rules and laws, whereas art is the study of methods to do something. Hence, we consider Cryptography as the art and science, because it includes the principles and methods of transforming an intelligible message (plaintext) into one that is unintelligible ( text) and then retransforming the message into its original form [1, 2, 6]. Now-a-days, Cryptography is also compared with both mathematics and computer science and it has also the affiliation to information theory, computer security and engineering [2]. Substitution cipher is one of the most basic components of the classical cipher. It is an encryption method, which substitutes the units of plaintext with cipher text according to particular system. The units may be in the form of single letters, pairs of letters, triplets of letters, mixture of the above and so forth. By performing the inverse substitution, the receiver can decipher the text [5, 8, and 10]. The units of plaintext are retained in the similar sequence as in the cipher text, but the units themselves are changed. There are different types of substitution cipher. The cipher is called as simple substitution ciphers, when the cipher works on single letters. The cipher is said to be polygraphic, when it works on group of letters. can be classified in terms of mono alphabetic and poly alphabetic. A fixed substitution is used by the mono alphabetic cipher over the entire message, whereas a number of substitutions are used by the poly alphabetic cipher at different times in the message. Hill cipher is one of the mono alphabetic polygraphic substitution cipher. It is a block cipher which has many advantages like distinguishing letter frequencies of the plaintext, its simplicity because of the use of the multiplication of matrices, and inversion for enciphering and deciphering, its high speed and high throughput [1, 3, and 6]. In our paper, we encrypt gray scale as well as color images using the pseudorandom key matrix. 2. PSEUDO-INVERTIBLE KEY MATRIX The pseudo-inverse of a m x n matrix A is a matrix that generalizes to arbitrary matrices the notion of inverse of a square, invertible matrix. The pseudo-inverse can be expressed from the singular value decomposition (SVD) of A, as follows. Page 37

2 Let the SVD of A be A = U S VT Where U, V are both orthogonal matrices and S is a diagonal matrix containing the (positive) singular values of A on its diagonal. Then the pseudo-inverse of A is the n x m matrix defined as A = V S UT Here, A has the same dimension as the transpose of A. The pseudo-inverse matrix has the following properties: AA = (AA )* A A = (A A)* AA A = A A AA = A 3. HILL CIPHER Hill cipher is an application of linear algebra to cryptology. It was developed by the mathematician Lester Hill. The Hill cipher algorithm takes m successive plaintext letters and substitutes them by m cipher text letters. The substitution is determined by m linear equations in which each character is assigned a numerical value (a=0, b=1 z = 25) [4, 8, 10]. Let m be a positive integer, the idea is to have m linear combinations of the m alphabetic characters in one plain text element and produce corresponding m characters in one cipher text element. Then, a m m matrix A as shown below in equation (1) is used as a key of the system such that A is invertible modulo 26 [1, 2]. Let a ij be the entry of A. A = a 11 a 12 a 1m a 21 a 22 a 2m a m1 a m2 a mm (1) For the plain text block, the numerical equivalents of m letters are given by Equation (2) as follows. x = (x 1, x 2 x m ) (2) and a key matrix A, the corresponding cipher text block are given by Equation (3) as follows. y = (y 1, y 2,.,y m ) (3) Thus the Encryption will be computed as follows in Equation (4) (y 1, y 2,., y m )) = (x 1, x 2,., x m )A (mod 26) (4) The cipher text is obtained from the plain text by means of a linear transformation. Thus the Decryption will be computed as follows in Equation (5) (x1, x2,.,xm) = (y1, y2,.,ym) A 1 (mod26) (5) Where A -1 is given by Equation (6) A 1 = a 11 a 12 a 1m a 21 a 22 a 2m a m1 a m2 a mm Since the block length is m, there are 26 different m letters blocks possible, each of which can be regarded as a letter in a 26 letter combination. Hill s method amounts to a mono-alphabetic substitution on this alphabet. 1 (6) A11 is a 1 x 1 matrix and is given by the Equation (7) A11 = a 11, (7) A12 is a 1 x (n-1) matrix and is given by the Equation (8) A12 = a 12 a 13 a 1n (8) A21 is a (n-1) x 1 matrix and is given by the Equation (9) A21 = a 21 a 31 a n1, (9) A22 is a (n-1) x (n-1) matrix and is given by the Equation (10) A22 = a 22 a 23 a 2n a 32 a 33 a 3n a n2 a n3 a nn (10) So, A12 A21 = I - A112 = I - a112 (11) And A12 (a11i + A22) = 0. [1,7] (12) 4. TWO-LEVEL HILL CIPHER As we know that both grayscale and color images can be encrypted using Hill cipher technique [6], they can also be encrypted using our proposed Two-Level Hill algorithm, the modulus will be 256(the no. of levels is considered as the no. of alphabets). On working with color images on encryption side, firstly the color image is decomposed into R-G-B components. Secondly, each component is encrypted by the algorithm individually. At last, the encrypted components are combined together to obtain the encrypted color image. In a Two Level Hill, Hill- technique is applied twice onto the blocks. Once, it is applied onto the Page 38

3 temporary block. Then, the resultant matrix is transposed b) the resultant matrix is transposed and Hill and Hill cipher is again applied to this matrix. The cipher is again applied to this matrix. algorithms are given below and the block diagram for the The final matrix obtained is placed in the ith encryption and decryption processes is shown in Figure 1 block of encrypted image. and Figure 2 respectively. Finally encrypted image of length L is obtained. 4.1 Sender Side: At the Sender s side, we implement encryption process. 4.2 Receiver Side: At the Receiver s side, we implement decryption process. Original image (P) Construct i th temporary block from i th pixel of each block Rgb2Gray (P1) Divide P1 in m x m symmetric Blocks (provide padding) Pseudo invertible Key Matrix image of order m x m (P2) Inverse Hill of Length L Place Resultant Image of Length L in i th Block to form m x m matrix Two Level Hill Inverse Transpose Inverse Hill Two Level Hill Hill Transpose Hill Restore Temporary Block to form a m x m matrix by replacing i th pixel of each block Pseudo invertible key matrix image of order m x m (P) Place Resultant Image in i th block of Separate Pseudo invertible Key to obtain m x m Symmetric Blocks Find Image Resolution Original Image of Length L Figure 1: Sender s End Flowchart Encryption: Generate a Pseudorandom key image of order m x m (P2). Convert original image P in RGB2Gray (P1). Divide P1 into m x m symmetric blocks. The i th pixels of each block are brought together to form a temporary block. a) Hill cipher technique is applied onto the temporary block. Figure 2: Receiver s End Flowchart Decryption: Encrypted image of length L is placed such that it forms an m x m matrix. Inverse Hill cipher is applied to m x m matrix. The resultant matrix is reverse transposed and inverse Hill cipher is again applied to resultant matrix to form a temporary block. Temporary block is restored to form a m x m symmetric blocks by replacing i th pixel in each block Page 39

4 Separate pseudorandom key (P2) from m x m symmetric blocks to restore original image. 5. RESULT Hill cipher is a block cipher that has several advantages such as disguising letter frequencies of the plaintext, its simplicity because of using matrix multiplication and inversion for enciphering and deciphering, its high speed, and high throughput. However, Hill cipher succumbs to a known plaintext attack and can be easily broken with such attacks [2,3,]. Although the algorithm presented in this paper aims at image encryption, it is not just limited to this area and can be widely applied in other information security fields such as video encryption. (d) (a) (b) (e) Figure 3: Encryption Using Two-Level Hill (a) Original Image, (b) Histogram of Original Image, (c) Key Image, (d) Histogram of Key Image, (e) We have taken one image as original image, another as the key image, which are shown in Figure 3(a) and 3(c) respectively. The histograms of both the images are shown in Figure 3(b) and 3(d) respectively. They are encrypted them using the Two-level Hill cipher algorithm. The result is shown in Figure 3(e) as the encrypted image. Figure 4 shows the result of decryption process. Here, we have decrypted the image shown in Figure 3(a). (c) (a) Page 40

5 (b) Figure 4: Decryption Using Two-Level Hill (a) Decrypted Image, (b) Histogram of Decrypted Image 6. CONCLUSIONS The competent method of encryption of an illustration is presented in this paper. Two-Level Hill is a fast encryption technique which can provide satisfactory results against the normal hill cipher technique. It can easily resist to the infamous Known Plain Text Attack (KPTA) for secure transmission of graphic encoded information while taking over various limitation in the aspect of efficiency and security of various encryption technologies of previous generation. Hence this hybrid system proves to be a very sound technique for transferring messages from sender to the receiver, achieving confidentiality as well as message authentication. REFERENCES [1] Bibhudendra Acharya, Saroj Kumar Panigrahy, Sarat Kumar Patra, and Ganapati Panda. Image Encryption Using Advanced Hill Algorithm, International Journal of Recent Trends in Engineering, Issue. 1, Vol. 1, May [2] Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, Saroj Kumar Panigrahy. Novel Methods of Generating Self-Invertible Matrix for Hill Algorithm, International Journal of Security, Vol 1, Issue 1, 2007, pp [3] S.K.Muttoo, Deepika Aggarwal, Bhavya Ahuja. ASecure Image Encryption Algorithm Based on Hill System, Buletin Teknik Elektro dan Informatika (Bulletin of Electrical Engineering and Informatics) Vol.1, No.1, March 2012, pp. 51~60 ISSN: [4] Rushdi A. Hamamreh, Mousa Farajallah, Design of a Robust Cryptosystem Algorithm for Non-Invertible Matrices Based on Hill, International Journal of Computer Science and Network Security, [5] Bibhudendra Acharya, Mohammad Imroze Khan, S K Patra, G Panda. Implementation of Hybrid Cryptosystem Using Non-Invertible Matrices Based on Hill and RSA Algorithm [6] Saroj Kumar Panigrahy, Bibhudendra Acharya, Debasish Jena Image Encryption Using Self-Invertible Key Matrix of Hill Algorithm 1st International Conference on Advances in Computing, Chikhli, India, February 2008 [7] Bibhudendra Acharya, Debasish Jena, Sarat Kumar Patra, and Ganapati Panda. Invertible, Involutory and Permutation Matrix Generation Methods for Hill System, International Conference on Advanced Computer Control, [8] W. Stallings, Cryptography and Network Security, 4 th edition, Prentice Hall, 2005 [9] Blakley G.R., Twenty years of cryptography in the open literature, Security and Privacy 1999, Proceedings of the IEEE Symposium, 9-12 May 1999 [10] A. J. Menezes, P.C. Van Oorschot, S.A. Van Stone, Handbook of Applied Cryptography,CRC press, 1996 Page 41