Image Encryption Using Advanced Hill Cipher Algorithm

Image Encryption Using Advanced Hill Cipher Algorithm

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  ACEEE International Journalon Signal and Image Processing Vol 1, No. 1, Jan 2010 Image Encryption Using Advanced Hill Cipher Algorithm Bibhudendra Acharya1, Saroj Kumar Panigrahy2, Sarat Kumar Patra3, and Ganapati Panda3 1 Department of E & TC, NIT Raipur, Chhattisgarh-492010, India bibhudendra@gmail.com 2 Department of CSE, NIT Rourkela, Orissa-769008, India skp.nitrkl@gmail.com 3 Department of ECE, NIT Rourkela, Orissa-769008, India {skpatra, gpanda}@nitrkl.ac.in Abstract—The Hill cipher algorithm is one of the symmetric engineering [1]. key algorithms that have several advantages in data Substitution cipher is one of the basic components of encryption. But, the inverse of the key matrix used for classical ciphers. A substitution cipher is a method of encrypting the plaintext does not always exist. Then if the encryption by which units of plaintext are substituted key matrix is not invertible, then encrypted text cannot be decrypted. In the Involutory matrix generation method the with ciphertext according to a regular system; the units key matrix used for the encryption is itself invertible. So, at may be single letters (the most common), pairs of letters, the time of decryption we need not to find the inverse of the triplets of letters, mixtures of the above, and so forth. The key matrix. The objective of this paper is to encrypt an receiver deciphers the text by performing an inverse image using a technique different from the conventional Hill substitution [8]. The units of the plaintext are retained in Cipher. In this paper a novel advanced Hill (AdvHill) the same sequence as in the ciphertext, but the units encryption technique has been proposed which uses an themselves are altered. There are a number of different involutory key matrix. The scheme is a fast encryption types of substitution cipher. If the cipher operates on scheme which overcomes problems of encrypting the images single letters, it is termed a simple substitution cipher; a with homogeneous background. A comparative study of the proposed encryption scheme and the existing scheme is cipher that operates on larger groups of letters is termed made. The output encrypted images reveal that the polygraphic. A monoalphabetic cipher uses fixed proposed technique is quite reliable and robust. substitution over the entire message, whereas a polyalphabetic cipher uses a number of substitutions at Index Terms—Encryption, Decryption, Hill Cipher, Image different times in the message— such as with Encryption, Advanced Hill Cipher. homophones, where a unit from the plaintext is mapped to one of several possibilities in the ciphertext. Hill I. INTRODUCTION cipher is a type of monoalphabetic polygraphic substitution cipher. Hill cipher is a block cipher that has Owing to the advance in network technology, several advantages such as disguising letter frequencies information security is an increasingly important of the plaintext, its simplicity because of using matrix problem. Popular application of multimedia technology multiplication and inversion for enciphering and and increasingly transmission ability of network deciphering, its high speed, and high throughput [5,7]. gradually leads us to acquire information directly and In this paper, we have proposed an advanced Hill clearly through images. Cryptography, the science of (AdvHill) cipher algorithm which uses an Involutory key encryption, plays a central role in mobile phone matrix for encryption [1]. The objective of this paper is to communications, pay-TV, e-commerce, sending private overcome the drawback of using a random key matrix in emails, transmitting financial information, security of Hill cipher algorithm for encryption, where we may not ATM cards, computer passwords, and touches on many be able to decrypt the encrypted message, if the key aspects of our daily lives. Cryptography is the art or matrix is not invertible. Also the computational science encompassing the principles and methods of complexity can be reduced by avoiding the process of transforming an intelligible message (plaintext) into one finding inverse of the matrix at the time of decryption, as that is unintelligible (ciphertext) and then retransforming we use Involutory key matrix for encryption. Using this that message back to its original form. In modern times, key matrix we encrypted gray scale as well as color cryptography is considered to be a branch of both images. Our algorithm works well for all types of gray mathematics and computer science, and is affiliated scale as well as color images except for the images with closely with information theory, computer security, and background of same gray level or same color. The organization of the paper is as follows. Following This research work was carried out at the Department of ECE, NIT the introduction, the basic concept of Hill Cipher is Rourkela, Orissa-769008, India. outlined in section II. Section III discusses about the Corresponding author: bibhudendra@gmail.com modular arithmetic. In section IV, a method of generating 37 © 2010 ACEEE DOI: 01.ijsip.01.01.08
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  ACEEE International Journalon Signal and Image Processing Vol 1, No. 1, Jan 2010 Involutory key matrix is explained. Section V presents III. MODULAR ARITHMETIC the proposed method of image encryption using advanced The arithmetic operation presented here are addition, Hill Cipher (AdvHill) algorithm. Experimental results are subtraction, unary operation, multiplication and division discussed in section VI. Finally, section VII describes the [9]. Based on this, the Involutory matrix for Hill cipher concluding remarks. algorithm is generated. The congruence modulo operator has the following properties: II. HILL CIPHER It is developed by the mathematician Lester Hill. The 1. a ≡ b mod p if n (a − b ) core of Hill cipher is matrix manipulations. For encryption, algorithm takes m successive plaintext 2. (a mod p ) = (b mod p ) ⇒ a ≡ b mod p letters and instead of that substitutes m cipher letters. In Hill cipher, each character is assigned a numerical value like a = 0, b = 1, ... , z = 25 [5, 9]. The substitution of 3. a ≡ b mod p ⇒ b ≡ a mod p ciphertext letters in the place of plaintext letters leads to m linear equation. For m = 3 , the system can be 4. a ≡ b mod p and b ≡ a mod p ⇒ a ≡ c mod p Let Z p = [0, 1,..., p − a ] the set of residues modulo p . described as follows: C1 = ( K 11 P1 + K 12 P2 + K 13 P3 ) mod 26 If modular arithmetic is performed within this set Z p , C 2 = ( K 21 P1 + K 22 P2 + K 23 P3 ) mod 26 … (1) the following equations present the arithmetic operations: C1 = ( K 31 P1 + K 32 P2 + K 33 P3 ) mod 26 Addition: This case can be expressed in terms of column vectors (a + b ) mod p = [(a mod p )+ (b mod p )] mod p and matrices: Negation: ⎛ C1 ⎞ ⎡ K11 K12 K13 ⎤⎛ P ⎞ ⎜ ⎟ ⎢ ⎜ 1⎟ ⎜ C2 ⎟ = ⎢ K 21 K 22 K 23 ⎥⎜ P2 ⎟ ⎥ … (2) − a mod p = p − ( a mod p ) ⎜ C ⎟ ⎢K ⎝ 3 ⎠ ⎣ 31 K 32 K 33 ⎥⎜ P3 ⎟ ⎦⎝ ⎠ Subtraction: or simply we can write as C = KP , where C and P are column vectors of length 3, representing the plaintext (a − b ) mod p = [(a mod p ) − (b mod p )] mod p and ciphertext respectively, and K is a 3 × 3 matrix, which is the encryption key. All operations are performed Multiplication: mod 26 here. Decryption requires using the inverse of (a ∗ b ) mod p = [(a mod p ) ∗ (b mod p )] mod p the matrix K . The inverse matrix K −1 of a matrix K is defined by the equation KK -1 = K -1 K = I , where I is Division: the Identity matrix. But the inverse of the matrix does not always exist, and when it does, it satisfies the preceding (a / b ) mod p = c when a = (b ∗ c ) mod p equation. K −1 is applied to the ciphertext, and then the plaintext is recovered [2,6]. In general term we can write The following exhibits the properties of modular as follows: arithmetic. For encryption: Commutative Law: C = E k ( P) = K p … (3) (ω + x ) mod p = ( x + ω ) mod p (ω ∗ x ) mod p = (x ∗ω ) mod p For decryption: Associative law: P = D k (C ) = K C = K K p = P -1 -1 … (4) [(ω + x ) + y ] mod p = [ω + (x + y )] mod p Distribution Law: If the block length is m, there are 26 different mm [ω ∗ (x + y )] mod p = [{(ω ∗ x )mod p}∗ {(ω ∗ y )mod p}]mod p letters blocks possible, each of them can be regarded as a letter in a 26m -letter alphabet. Hill's method amounts to a Identities: monoalphabetic substitution on this alphabet [5]. (0 + a ) mod p = a mod p and (1∗ a ) mod p = a mod p 38 © 2010 ACEEE DOI: 01.ijsip.01.01.08
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  ACEEE International Journalon Signal and Image Processing Vol 1, No. 1, Jan 2010 ⎡10 2⎤ Let A22 = ⎢ ⎥, Inverses: ⎣ 3 4⎦ For each x ∈ Z p , ∃y such that ⎡ 3 11⎤ (x + y ) mod p = 0 then y = − x then, A11 = ⎢ ⎥. ⎣10 9 ⎦ For each x ∈ Z p ∃y such that (x ∗ y ) mod p = 1 If k is selected as 2, IV. GENERATION OF INVOLUTORY KEY MATRIX ⎡9 4 ⎤ then, A12 = k ( I − A11 ) = ⎢ ⎥, The proposed AdvHill algorithm uses an involutory ⎣6 10⎦ key matrix for encryption technique. The various proposed methods can be found in literature [1]. One of ⎡2 12⎤ the methods is explained below. and A21 = ⎢ ⎥ A is called a involutory matrix if A = A −1 . The ⎣5 5 ⎦ analysis presented here for generation of involutory key matrix is valid for matrix of +ve integers that are the ⎡ 3 11 9 4 ⎤ ⎢10 9 6 10⎥ residues of modulo arithmetic of a number. This So, A = ⎢ ⎥ will be the involutory matrix. algorithm can generate involutory matrices of order ⎢ 2 12 10 2 ⎥ n × n where n is even. ⎢ ⎥ ⎣5 5 3 4⎦ ⎡ a11 a12 ... ... a1n ⎤ V. IMAGE ENCRYPTION USING ADVHILL TECHNIQUE ⎢a 21 a22 ... ... a2 n ⎥ ⎥ Let A= ⎢ ... ... As we note that Hill cipher can be adopted to encrypt ⎢ ... ... ... ⎥ be an n × n involutory grayscale and color images, proposed AdvHill algorithm ⎢ ... ... ... ... ... ⎥ can also be used for grayscale and color images. For ⎢ ⎥ ⎣ an1 an 2 ... ... ann ⎦ grayscale images, the modulus will be 256 (the number ⎡A A12 ⎤ of levels is considered as the number of alphabets). In the matrix partitioned to A = ⎢ 11 A22 ⎥ , where n is even case of color images, first decompose the color image ⎣ A21 ⎦ into (R-G-B) components. Second, encrypt each n n component (R-G-B) separately by the algorithm. Finally, and A11 , A12, A21 & A22 are matrices of order × concatenate the encrypted components together to get the 2 2 each. encrypted color image [10]. The algorithm is given below So, A12 A21 = I − A11 = (I − A11 )(I + A11 ) 2 and the block diagram for the encryption process is shown in Figure 1. If A12 is one of the factors of I − A11 then A21 is the 2 Algorithm AdvHill: other. Solving the 2nd matrix equation results A11 + A22 = 0 . Step1. A involutory key matrix of dimensions m × m Then form the matrix. is constructed. Step2. The plain image is divided into m × m Algorithm: symmetric blocks. n n 1. Select any arbitrary × matrix A22 . Step3. The ith pixels of each block are brought 2 2 together to form a temporary block. 2. Obtain A11 = − A22 . a. Hill cipher technique is applied onto the 3. Take A12 = k (I − A11 ) or k (I + A11 ) where k is a temporary block. scalar constant. b. The resultant matrix is transposed and Hill cipher is again applied to the this matrix. 4. Then, A21 = 1 (I + A11 ) or 1 (I − A11 ) . Step4. The final matrix obtained is placed in the ith k k block of the encrypted image. 5. Form the matrix completely. Example: (For Modulo 13) 39 © 2010 ACEEE DOI: 01.ijsip.01.01.08
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  ACEEE International Journalon Signal and Image Processing Vol 1, No. 1, Jan 2010 Figure. 1. The block diagram for proposed AdvHill algorithm. VI. EXPERIMENTAL RESULTS We have taken different images and encrypted them using original Hill and our proposed AdvHill algorithm and the results are shown below in Figure 2 and 3. It is clearly noticeable from the Figure 2(e, g), that original Hill Cipher can’t encrypt the images properly if the image consists of large area covered with same colour or gray level [8]. But our proposed algorithm works for any images with different gray scale as well as colour images. In Figure 3, it is found that our proposed AdvHill algorithm can able to encrypt the image properly as compared to original Hill Cipher algorithm. Figure. 2. Original images (a, c, e, g, i) and corresponding encrypted images (b, d, f, h, j) by original Hill Cipher Algorithm 40 © 2010 ACEEE DOI: 01.ijsip.01.01.08
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  ACEEE International Journalon Signal and Image Processing Vol 1, No. 1, Jan 2010 REFERENCES [1] Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, Saroj Kumar Panigrahy. 2007. Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm, International Journal of Security, Vol 1, Issue 1, 2007, pp. 14-21. [2] Imai H., Hanaoka G., Shikata J., Otsuka A., Nascimento A.C. 2002. Cyptography with Information Theoretic Security. Information Theory Workshop, 2002, Proceedings of the IEEE, 20-25 Oct 2002. [3] Lerma, M.A., 2005. Modular Arithmetic. http://www.math.northwestern.edu/~mlerma/problem_solv ing/results/modular_arith.pdf. [4] Li, S., Zheng, X., 2002. On the Security of an Image Encryption Method. ICIP2002. http://www.hooklee.com/Papers/ICIP2002.pdf. [5] Menezes, A. J., P.C. Van Oorschot, S.A. Van Stone. 1996. Handbook of Applied Cryptography. CRC press. [6] Overbey, J., Traves, W., Wojdylo, J., 2005. On the keyspace of the Hill cipher. Cryptologia, 29(l):59-72. [7] Petersen, K., 2000. Notes on Number Theory and Cryptography. http://www.math.unc.edu/Faculty/petersen/Coding/cr2.pdf. [8] Saeednia, S., 2000. How to make the Hill cipher secure. Cryptologia, 24(4):353-360. [9] Stallings, W. Cryptography and Network Security.2005. 4th edition, Prentice Hall. Figure. 3. Original images (a,d) and corresponding encrypted images (b,e) by original Hill Cipher Algorithm and (c,f) by our proposed AdvHill [10] ISMAIL I.A., AMIN Mohammed, DIAB Hossam, How to algorithm repair the Hill cipher, Journal of J Zhejiang Univ SCIENCE A, vol. 7(12), pp. 2022-2030, 2006. VII. CONCLUSION [11] Y. Rangel-Romero, R. Vega-García, A. Menchaca- This paper suggests efficient method of encryption of image. Proposed AdvHill algorithm is more secure to Méndez, D. Acoltzi-Cervantes, L. Martínez-Ramos, M. brute force attacks as compared to original Hill cipher Mecate-Zambrano, F. Montalvo-Lezama, J. Barrón- Vidales, N. Cortez-Duarte, F. Rodríguez-Henríquez, algorithm. A Brute Force Attack requires 2 7 +8*(n / 2) 2 Comments on How to repair the Hill cipher, Journal of J number of key generations; where n is the order of key matrix. AdvHill is a fast encryption technique which can Zhejiang Univ SCIENCE A, pp. 1-4, 2007. provide satisfactory results against the normal hill cipher technique. The proposed scheme is resistant against known plaintext attacks. 41 © 2010 ACEEE DOI: 01.ijsip.01.01.08