A Developed Discrete Fourier Transform based Cryptosystem

This paper provides a developed cryptosystem for data encryption based on using the Discrete Fourier Transform (DFT). The encryption side using the DFT method for performing the main security development that is considered as the main encryption process in the developed cryptosystem. The primary ciphered plaintext data must be prepared as a block of (N) data, and entered to the DFT part. The decryption side uses as an inverse DFT method, to get back the primary ciphered plaintext from the received cipher text. The proposed method applied to classical encryption method, and can be applied to a stream cipher method. The validity of the developed encryption process is illustrated through many numerical examples.


INTRODUCTION
The cipher system provides confidential information in such a way that it's meaning is unintelligible to unauthorized person. The historical development of cryptography can be divided into a number of stages. First stage is the handwritten systems such as a simple letter substitution that could be implemented using pencil and paper or simple mechanical machines. The second stage dates from the beginning of the twentieth century (the time the telegraph became truly established) the late 1950s. These systems normally used complex mechanical and electro-mechanical machines [1].
Diffie and Helman in 1976 [2] proposed a system for distributing the secret key to be used in conventional cryptosystem over public (insecure) communication channel. The disguised information about the key is sent over an insecure channel in such a way that only the authorized partner can deduce the key [1]. The process of applying a key to transfer back from the cipher text to the plain text is known as deciphering. The block diagram of cryptosystem is shown in Fig. (1). The set of all messages is called the "Message Space" and is denoted by {M}. The set {C} of all cryptograms is called "Cryptogram Space" and the set of all keys is denoted by {K}. In any practical situation M, C and K will be finite sets [3]. One crucial requirement of cipher system is that knowledge of the cryptogram, key and algorithm must enable the recipient to determine the message uniquely. Thus if c  t(m), then (m) must be uniquely determined by c and t. In other words, (t) must be reversible and (m) must be the image of c under t -1 . In [6], it was given a generalization of the DFT. If (s) be a periodic sequence whose elements lie in a finite field. It presented an algorithm which calculates the minimal polynomial of (s), assuming that a period of (s) is known [4].
The main objective of this paper is: " A proposing method for developed data enciphering and deciphering using the DFT with some well-known cipher systems and evaluating the new developed cryptosystem".
Section two introduces the DFT and its main properties. Section three provides the use of DFT in Cryptography, i.e., discussion of the proposed procedure for using of the DFT based developed cipher systems. Section four provides evaluation of the developed system through a given mathematical example. Section five provides the conclusions and future works. The DFT with additive ciphers or multiplicative ciphers [5] is developed for producing new cipher system that consists of same substitution cipher system in connection with DFT.
This means that the ciphered output of the classical substitution cryptosystem is passed to another mapping or transformation that is achieved by the DFT.

Example (1):
Encipher and decipher the message "Notation", using the proposed DFT based cipher system.

Evaluation of the Proposed Developed DFT based Cipher Systems
An example of evaluation the security of the proposed cipher system will be considered to illustrate the evaluation task.

The Proposed Monoalphebetic -DFT Cipher
Actually it is well known that this system has a limited security due to many factors, such as: -The limited number of alphabet size which results in a limited number of the used keys.
-The statistical behavior of the used language in preparing the input data of the cryptosystem.
These factors cause the possibility of ease breaking of ciphered output. Hence any proposition to enhance the security must overcome these factors. The following example illustrates this concept.

Example (2):
Use the Monoalphabetic to encipher the plaintext given below: "since its not to easy to hand write or type write bold face letters we should write a vector either with an arrow over it".
-Draw a histogram for the cipher letters before using the DFT and the coefficients resulting for the DFT, these graphs are shown in Fig. (3) and (4).
According to the coefficient histogram Fig. (4), it is difficult to recognize the statistics of the natural language, while in the ciphertext histogram Fig. (3), it is clear that it will be easy for cryptanalyst to find a way to break the ciphertext through the existence of the statistical behavior of the English language.

Conclusion and Future Work
The use of the DFT (in the developed cryptosystem) causes a failure of the frequency analysis of the statistical cryptanalysis, and the failure of the trying of all possible used enciphering keys. Hence the use of the DFT with the conventional cipher systems (monolaphabetic and polyalphabetic) will enhance the security of the resulted encryption messages. This will results in more complexity to the attack approaches (methods) used to cryptanalysis of these developed cryptosystems. For the future work, it will be suitable to try