Modified Hill Cipher with Interlacing and Iteration

: In this research, we have developed a block cipher by taking a large key matrix of size nxn and a plaintext matrix containing n rows and two columns. In this, the plaintext column vectors, operated by the key matrix are thoroughly interlaced at each stage of the iteration. As a typical example, we have taken the key in the form an 8 · 8 matrix and the plaintext in the form of an 8 · 2 matrix. Here the key is of the size 384 binary bits and the plaintext is of size 112 binary bits. The cryptanalysis carried out in this research clearly indicates that the cipher cannot be broken by any cryptanalytic attack.


INTRODUCTION
The classical Hill cipher [1] is a typical block cipher which depends mainly on the modular arithmetic inverse of a key matrix. In this, the encryption and the decryption are governed by the relations C = KP mod 26 (1) and P = K −1 C mod 26 (2) where, K is the key matrix, P the plaintext, C the ciphertext and K −1 is the modular arithmetic inverse of K. It is well known that, though this cipher is very strong against brute force attack, it can be broken by the known plaintext attack, as we have direct relations, given by (1) and (2), for the cipher.
In a pioneering research, Sastry and Janaki [2] have obtained the modular arithmetic inverse of a matrix in a systematic manner and have pointed out that the Hill cipher cannot broken by the known plaintext attack, when the elements of the plaintext are transposed in an effective manner.
In the present research, our objective is to modify the Hill cipher by introducing a key matrix, which is significantly large in size (as the strength of a cipher increases with the length of the key) and by considering a plaintext vector which undergoes transposition, repeatedly, on account of interlacing. Here our interest is to modify the Hill cipher such that it cannot be broken by any cryptanalytic attack.

DEVELOPMENT OF THE CIPHER
Consider a plaintext consisting of 2n characters. By using the ASCII code, the corresponding plaintext matrix can be written in the form P = [P ij ] where i = 1 to n, j = 1 to 2.
Let K = [K ij ], i = 1 to n and j = 1 to n, be the key matrix. Let us suppose that C = [C ij ], i = 1 to n and j = 1 to 2, be the corresponding ciphertext. Then, C can be obtained by using the relation C = KP mod 128 (3) After obtaining the modular arithmetic inverse of K, denoted by K -1 , from (3), we get The Eq. 3 and 4 describe the process of encryption and the process of decryption.
We now introduce the concept of interlacing. Let us represent the decimal numbers in the two columns of P in terms of their binary bits. As each number lies between zero and 127, we get only seven binary bits corresponding to each number. Thus we have where, T is the transpose of the vector.
Here, b ij are the binary bits corresponding to the numbers in the first column and d ij are those of the second column. In order to illustrate the process of interlacing, let us consider a simple case where n = 8. Here we have Now, let us mix b ij and d ij , i = 1 to 8, j = 1 to 7, the binary bits of the first and second columns of the plaintext and write them in terms of a pair of matrices as shown below.
In these matrices, each b ij lies adjacent to its corresponding d ij .
Now we obtain the decimal numbers corresponding to the binary bits of each row, in the above two matrices and reconstruct the modified plaintext matrix [P ij ], i = 1 to 8 and j = 1 to 2. Similarly we can obtain [P ij ], for i = 1 to n, j = 1 to 2, in general.In the process of decryption, we carryout the reverse process of the above interlacing, which is hereafter called as decomposition.
The cipher involving interlacing and iteration is shown in the schematic diagram given in Fig.1 read n,N,K,P for I = 1 to n { for j = 1 to 7 { b ij = temp(l); d ij = temp(l+1); l = l + 2 ; } } 6. convert binary bits to decimal numbers; }

ILLUSTRATION OF THE CIPHER
Consider the following plaintext: The World Bank has given an assistance of 100 billion dollars for the community development in our country. Let us have progress in all directions.
Let us focus our attention on the first sixteen characters of the above plaintext. This is given by The World Bank h.
By using the ASCII code, the matrix corresponding to the above plaintext can be constructed, in a row wise manner, as Here, as we have sixteen numbers, the size of the plaintext block is 112 binary bits.
Let us consider a key matrix given by Here, each element in the key matrix is less than 64. In view of this fact, as each number can be represented in terms of 6 binary bits, the size of the key matrix is 6x64 i.e., 384 binary bits.
On using the algorithm for encryption we get This is the same as (9) The above steps clearly indicate the encryption and the decryption processes underlying in the cipher.

CRYPTANALYSIS
In the case of the classical Hill cipher, it is well known that the cipher can be broken by applying the known plaintext attack. In the present cipher, we have introduced interlacing and iteration. On account of these two concepts, the binary bits arising due to the interaction between the key and the plaintext, are undergoing a thorough diffusion and confusion. Thus the cipher is expected to be a very strong one.
In what follows, let us discuss the ciphertext only attack, the known plaintext attack and the chosen plaintext/ciphertext attacks.
In the case of the ciphertext only attack, the ciphertext is known to us. In this, the key matrix is of size nxn. As each element of the matrix can be represented in terms of binary bits, the size of the key space is, in general, 2 7n² . However, in the present analysis, each element of the key matrix is taken to be less than 64. Thus it can be represented in six binary bits. Therefore the key space is of size 2 6n² . From these facts, when n is greater than or equal to four, we readily conclude that the cipher cannot be broken by the brute force attack.
Let us now consider the known plaintext attack. In this case, we know as many plaintext and ciphertext pairs as we require. Though we know as many Ps and the corresponding Cs as we want, we do not have a direct relation between them as the P is under going transposition at every stage of the iteration. Thus we cannot construct an equation of the form X = KY mod 128.
(as we could do in the case of the Hill cipher [1] ) and determine K by obtaining the modular arithmetic inverse of Y. Hence, the cipher cannot be broken by the known plaintext attack.
Further, we notice that any special choice of the plaintext vector or the ciphertext vector will not help the attacker as the plaintext interacting with the key, is interlaced at every stage of the encryption process.
From the above discussion, we find that the strength of the cipher is enhanced enormously by the interlacing and the iteration introduced into the cipher.