CHAPTER 12 CRYPTOGRAPHY OF A GRAY LEVEL IMAGE USING A MODIFIED HILL CIPHER

Transcription

1 177 CHAPTER 12 CRYPTOGRAPHY OF A GRAY LEVEL IMAGE USING A MODIFIED HILL CIPHER

2 Introduction The study of cryptography of gray level images [110, 112, 118] by using block ciphers has gained considerable impetus in the recent years. The transformation of an image from its original form to some other form, such that it cannot be deciphered what it is, is really an interesting one. In a recent investigation [125, 128], we have developed two large block ciphers by modifying the Hill cipher [5]. In these ciphers, the key is of size 512 bits and the plaintext is of size 2048 bits. In one of the chapters [112], the plaintext matrix is multiplied by the key on one side and by its modular arithmetic inverse on the other side. From the cryptanalysis and the avalanche effect, we have noticed that the cipher is a strong one and it cannot be broken by any cryptanalytic attack. In the present chapter, our objective is to develop a block cipher, and to use it for the cryptography of a gray level image. Here, we have taken a key containing 64 decimal numbers (as it was in [125]), and generated a key matrix of size 32 x 32 by extending the key in a special manner (discussed later), and applied it in the cryptography of a gray level image. In Section 2, we have developed a procedure for the cryptography of a gray level image. In Section 3, we have used an example and illustrated the process. Finally, in Section 4, we have drawn conclusions from the analysis.

3 Development of a Procedure for the Cryptography of a Gray level image Consider a gray level image whose gray level values can be represented in the form of a matrix given by P = [Pij], i = 1 to n, j = 1 to n. (2.1) Here, each Pij lies between Let us choose a key k let it be represented in the form of a matrix given by K = [Kij], i = 1 to n, j = 1 to n, (2.2) where each Kij is in the interval [0, 255]. Let C = [Cij], i = 1 to n, j = 1 to n (2.3) be a matrix, obtained on encryption.

4 180 The process of encryption and the process of decryption, which are quite suitable, for the problem on hand, are given in Fig. 1. Here, Mix ( ) is a function used for mixing thoroughly the decimal numbers (on converting them into binary bits) arising in the process of encryption at each stage of iteration. IMix ( ) is a function which represents the reverse process of Mix ( ). For a detailed discussion of these functions, and the algorithms involved in the processes of encryption and decryption, we refer to [125].

5 Illustration of the cryptography of the image Let us choose a key Q consisting of 64 numbers. This can be written in the form of a matrix given by Q = (3.1) The length of the secret key (which is to be transmitted) is 512 bits. On using this key, we can generate a new key E in the form Q R E = (3.2) S U where U = Q T, in which T denotes the transpose of a matrix, and R and S are obtained from Q and U as follows. On interchanging the 1 st row and the 8 th row of Q, the 2 nd row and the 7 th row of Q, etc., we get R. Similarly, we obtain S from U. Thus, we have E = (3.3)

6 182 The size of this matrix is 16 x 16. This can be further extended to a matrix L of size 32 x 32 where E F L = (3.4) G H where H = E T, in which T denotes the transpose of a matrix, and F and G are obtained from E and H as follows. On interchanging the 1 st row and the 16 th row of E, the 2 nd row and the 15 th row of E, etc., we get F. Similarly, we obtain G from H. Thus, we have L. Here, F = (3.5) G = (3.6)

7 H = (3.7) Now we obtain the key matrix K given by E F K = (3.8) G J where J is obtained from H by rotating, circularly, two rows in the downward direction J = (3.9)

8 184 The afore mentioned operations are performed for 1) enhancing the size of the key matrix to 32 x 32, and 2) obtaining the modular arithmetic inverse of K, in a trial and error manner. The modular arithmetic inverse of K is obtained as W X K 1 = (3.10) Y Z W = (3.11) X = (3.12)

9 Y = (3.13) Z = (3.14) From (3.8) and (3.10), we can readily find that K K 1 mod 256 = K 1 K mod 256 = I. (3.15) Let us consider the image of a hand, which is given below. Fig. 2. Image of a Hand

10 186 This image can be represented in the form of a binary matrix P given by P = (3.16) where 1 denotes black and 0 denotes white.

11 187 On adopting the iterative procedure given in Fig. 1, we get the encrypted image C C = (3.17) On using (3.8), (3.10), (3.17), and the procedure for decryption (See Fig. 1.(b)), we get back the original binary image P, given by (3.16).

12 188 From the matrix C, on connecting each 1 with its neighbouring 1, we get an image which is in a zigzag manner (See Fig. 3). Fig. 3. Encrypted image of the hand It is interesting to note that, the original image and the encrypted image differ totally, and the former one, exhibits all the features very clearly, while the later one does not reveal anything. 6.4 Computations and Conclusions In this analysis, we have made use of a modified Hill cipher for encrypting a binary image. Here we have illustrated the procedure by considering a pair of examples: (1) the image of a hand, and (2) the image of upper half of a person. Here, we have noticed that, the encrypted image is totally different from the original image, and the security of the image is completely enhanced, as no feature of the original image can be traced out in any way from the encrypted image. This analysis can be extended for the images of signatures and thumb impressions.