A Modified Feistel Cipher Involving Modular Arithmetic Addition and Modular Arithmetic Inverse of a Key Matrix

In this investigation, we have modified the Feistel cipher by taking the plaintext in the form of a pair of square matrices. Here we have introduced the operation multiplication with the key matrices and the modular arithmetic addition in encryption. The modular arithmetic inverse of the key matrix is introduced in decryption. The cryptanalysis carried out in this paper clearly indicate that this cipher cannot be broken by the brute force attack and the known plaintext attack.


INTRODUCTION
In a recent investigation [1], we have developed a block cipher by modifying the Feistel cipher.In this, we have taken the plaintext (P) in the form of a pair of matrices P 0 and Q 0 , and introduced a key matrix (K) as a multiplicant of Q 0 on both its sides.In this analysis the relations governing the encryption and the decryption are given by P i = ( K Q i-1 K ) mod N, (1.1) Q i = P i -1 P i , and Q i-1 = ( K -1 P i K -1 ) mod N, ( Here, multiplication of the key matrix, mod operation and XOR are the fundamental operations in the development of the cipher.The modular arithmetic inverse of the key plays a vital role in the process of the decryption.Here N is a positive integer, chosen appropriately, and n denotes the number of iterations employed in the development of the cipher.
In the present paper, our objective is to develop a block cipher by replacing the XOR operation in the preceding analysis by modular arithmetic addition.The iteration process that will be used in this cipher is expected to offer a strong modification to the plaintext before it becomes finally the cipher text.Now, we present the plan of the paper.We introduce the development of the cipher, and present the flowcharts and the algorithms, required in this analysis, in section 2. In section 3, we deal with an illustration of the cipher and discuss the avalanche effect, then in section 4 we study the cryptanalysis of the cipher.Finally, in section 5, we mention the computations carried out in this analysis and draw conclusions.

II. DEVELOPMENT OF THE CIPHER
Let us now consider a plaintext P. On using the EBCIDIC code, the plaintext can be written in the form of a matrix which has m rows and 2m columns.This is split into a pair of square matrices P 0 and Q 0 , wherein both the matrices are of size m.
The basic equations governing the encryption and the decryption, in the present investigation, assume the form The flowcharts depicting the encryption and the decryption processes of the cipher are presented in Figures 1and 2.
Here it may be noted that the symbol || is used for placing one matrix adjacent to the other.The corresponding algorithms can be written in the form as shown below.

III. ILLUSTRATION OF THE CIPHER
Let us now consider the plain text given below.
Dear Janaki, I have received your letter.You sent me a wonderful cryptography program and a letter along with that.I started working six years back on web security.Really I am finding it very difficult to hit upon an interesting problem.As the complexity of network security is growing in all directions.I am slowly losing my hope; I am thinking now whether I would really contribute in a significant manner and get a Ph.D. Please come and join me, so that, we shall lead a comfortable life. (3.1) Consider the first 128 characters of the above plaintext.This is given by Janaki, I have received your letter.You sent me a wonderful cryptography program and a letter along with that.I started w (3.2) On using the EBCIDIC code we get Read K, P, n The process of Encryption The process of Decryption On comparing (3.7) and (3.9), after converting them into their binary form, we find that the two cipher texts under consideration differ by 518 bits out of 1024 bits.From the above analysis, we conclude that the cipher is expected to be a strong one.

IV. CRYPTANALYSIS
The different approaches existing for cryptanalysis in the literature are 1.Cipher text only attack( Brute Force Attack ) 2. Known plaintext attack 3. Chosen plaintext attack 4. Chosen cipher text attack In this analysis, the key is a square matrix of size m.
Thus the size of the key space = (2) 8m2 If we assume that the time required for encryption is 10 -7 seconds then the time required for the computation with all the keys in the key space [1] (2.4m 2 -15) = 3.12 x 10

x 10
As this time is very large, the cipher under consideration cannot be broken by the brute force attack.Now let us examine the known plaintext attack.In the case of this attack, we know as many plaintext cipher text pairs as we require.In the light of this fact, we have P 0 , Q 0 and P n , Q n in as many instances as we want.Keeping the quotations governing the encryption in view ( see algorithm for encryption ), we can write the following equations connecting the plaintext and the cipher text at different stages of the iteration process.
In view of the above system of equations we can write the entities at the n th stage of the iteration as follows: Here it is to be noted that, the initial plaintext can be obtained by concatenating P 0 and Q 0 .The cipher text which we get at the end of the iteration by concatenating P n and Q n .
Though we have as many relations as we want between the cipher text and the plain text, the key matrix K cannot be determined as the equations (4.2) are nonlinear and involving mod N. In the light of the above discussion, we conclude that this cipher cannot be broken by the known plaintext attack.
In the literature of the cryptography [2], it is well known that a cipher must be designed such that it withstands at least the first two attacks.As the relations given in (4.2) are very complex, it is not possible either to choose a plaintext or to choose a cipher text to attack the cipher.In the light of the afore mentioned facts, we conclude that this cipher is a strong one and it cannot be broken by any means.

V. COMPUTATIONS AND CONCLUSIONS
In this paper, we have developed a block cipher by modifying the Feistel cipher, in this modification, the plaintext is taken in the form of a matrix of size mx(2m).The iteration process is carried out by dividing this matrix into two equal halves wherein each one is of size mxm.In this analysis, we have used multiplication of a portion of the plaintext (Q 0 ) with a key matrix on both the sides of Q 0 .Here we have made use of modular arithmetic addition as a primary operation in the cipher.The modular arithmetic inverse of the key is used in the decryption process.
Programs are written for encryption and decryption in C language.The entire plaintext given in (3.1) is divided into 4 blocks.Wherein each block contains 128 characters.We have appended the portion of the last block with 15 characters so that it becomes a full block.On carrying out encryption (by using the key and the algorithm for encryption), we get the ciphertext corresponding to the complete plaintext (3.1).This cipher has acquired a lot of strength in view of the multiplication with key matrix, the modular arithmetic addition and mod operation.From the cryptanalysis, it is worth noticing that the cipher is a strong one.
www.ijacsa.thesai.orgWhen m = 8 , the time required for the entire computation can be obtained as 138.6 Now let us study the avalanche effect.To this end, we change 4 th row, 2 nd column element from 116 to 117 in (3.3).