Synchronization of time-delayed systems with chaotic modulation and cryptography
Introduction
Time delay often appears in many control systems (such as aircrafts, chemical or process control systems) either in the state, the control input or the measurements. The time-delayed systems are infinite dimensional in nature and in most of the cases the delay time is the source of instability. The synchronization phenomenon in time-delayed systems [1], [2], [3], [4] has aroused great interests in communications due to its complexity. It is known that chaos synchronization [5], [6] is extensively exploited in applications of cryptography [7], [8], [9], [10].
The secure transport of messages had been the concern from early civilizations. Since then, different methods have been developed to assure that only the sender and the receiver would be able to read a message, while it would be illegible or without significant meaning to a third party. Today, this practice continues with more fervor. Wireless, wired, and optical communication networks are able to transport unimaginable amounts of data and thus privacy of information and security of the network are of the utmost concern because a good part of the transported information may be very sensitive and confidential. Confidentiality of information has been particularly popularized with the explosive growth of the Internet, which has touched most peoples lives. Several encryption schemes with different secret keys have been developed time to time to protect the data and image files. As a result, the development of unbreakable cipher keys, cipher key distribution, identification of malicious actors, source authentication, physical-link signature identification, countermeasures, and so forth has been the major thrust of research efforts with regard to cyber-security.
The work in this paper is twofold. The first part contains a new process of chaos synchronization between two time-delayed system and its application in cryptographic encoding is discussed in the second part. The main motivation of this work is to produce a safe method of encryption for the digital messages with the help of synchronized time-delayed systems. Time-delayed systems are basically complex in nature for the delay parameter . In this paper, we consider a delayed system with chaotic modulation which makes the nature of the system more complicated. The main contribution is the new encryption scheme which is illustrated in details with two example for a text and picture messages.
This paper is arranged as follows. In Section 2 we discussed the nature of the time-delayed system with modulated chaotic delay time. In Section 3 we investigated the synchronization between those two modulated time-delayed systems. Section 4 is focussed on cryptography, a new method have been introduced to encrypt the digital data and an illustrated example is given for the text messages. Section 5 is about image file encryption using the same technique.
Section snippets
The system and chaotic modulation
The delay differential equation describing the system iswhere and are positive system parameters, is time delay. The system (1) is used as a prototype model to observe self-oscillations in the shipbuilding industry. The system is in chaotic state for the parameter values . In our previous communication [11] we replaced the time delay parameter with a periodic modulation of the form . Here, we choose the delay timeas a function
Synchronization between coupled systems
Consider the driving system asand a response system defined bywhere is defined in (2). The two systems are coupled with the variable delay time as well as the linear constant feedback coupling k. We choose k = 2.5. All other system parameters are into the corresponding chaotic state given by Fig. 1(a). Fig. 2 represents the time variation of the synchronization error between the driving system (3) and response system (4). From this figure
Digital cryptography using synchronized systems
In our previous correspondences [11], [15] we have shown a method of digital cryptography with the synchronized chaotic systems. In this section, we propose another new method of digital cryptography with a high security level.
It is easy to see from Fig. 2 that the driving system (3) and the response system (4) synchronizes for . We now want to utilize the driving and response systems for cryptographic encoding. The sender will use system (3) and the receiver will use system (4) to get
Image encryption
In this section, we want to encrypt an image file [17], [18] using the chaotic time-delayed system. A grey scale digital image usually represented as two-dimensional arrays with the pixel values. Our basic idea of encryption is to shuffle the position of the pixels randomly (by row-wise shuffling and column-wise shuffling) to disturb the correlation between them. Let the dimension of image be and position matrix of pixel value is where and . Here
Conclusion
In this paper we have studied the synchronization phenomenon between two identical time-delayed systems with chaotic modulated delay time. Numerical calculations well support the effectiveness of the proposed method. A new method is implemented for digital encryption which is based on theory of Lucas numbers and synchronized chaotic system. Two different examples are given for text messages and image encryption.There are several encryption schemes with synchronized chaotic systems. Our method
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