In this application the logistic map of the chaotic noise MOS generator shows chaotic behaviour. The variable i?? in this application has the same function as the variable we called k in our work, it is known as the control parameter. Values for x are taken between 0 and 1, and values for i?? are taken between 0 and 4. In the same way that our bifurcation diagram only converged to one steady state for values of k below 2, for values of below 3 the values for x(n+1) converge to one steady state also.
Similarly, as in our work when k is increased for values above 2, when is increased to values over 3, there are now 2 steady states. Furthermore, as is increased between the values of 3 and 4, the number of steady states reached also increases from 4 to 8 and then to 16. This is reflected in our own bifurcation diagram, as the value of k is increased between 2 and 2. 567. The number of steady states is increased also, in the same way. The bifurcation diagram shows the system behaviour; where one steady state turns to two steady states, the line is shown to split into two on the diagram, as shown in fig 51.
Likewise, 4 steady states emerge, then 8 and then 16 steady states the lines are shown to split further until the system develops into chaos. In this system, as i?? reaches values greater than 3. 5699456, there are no longer any steady states reached. The values for x(n) never converge and reach steady states. This is because the system is now in chaos, which is shown on the diagram as the shaded region. This relates to our work as the logistic equation we used reaches chaos, at which point the value of k is 2.567.
Through the course of time, the world has advanced through many ages, be it the middle ages or the ongoing digital age. With each age humanity has made great strives at improving themselves and their environment, various innovative ideas have been put forward and developed but as the case is with most, with each new innovation comes a whole new set of barriers. “Encryption is the conversion of data into a form, called a cipher text, that cannot be easily understood by unauthorized people (searchsecurity.com (c) 2000 contributors Robert Bauchle, Fred Hazen, John Lund, Gabe Oakley, and Frank Rundatz)”.
In recent years, with the advancement of technology, the world has become a global village where information can be sent half way across the world in a matter of seconds as a result the demand/flow of information in the form of data has risen exponentially and being able to secure your flow of data has become a necessity, it is this desideratum that brought about encryption and its uses/ methods.
Take the military for example, in war time situations communication is ‘sine que non’ for the military, not only do they need to be able to communicate with each other, they also need to be able stop the enemy from intercepting their communications. Apart from its military uses, encryption is also essential in private video conferencing, cable TV, storing personal data online and many other applications. The high demand for encryption has lead to numerous advancements in encryption techniques as well as their algorithms, one of them being chaos based encryption techniques.
Chaos based encryption techniques are praised for their good practical use, this is because of the high speed, complexity security and more they provide, an example of this can be seen in the Image encryption using chaotic logistic map article by N. K. Pareek, Vinod Patidar, K. K. Sud:- “Digital images have certain characteristics such as: redundancy of data, strong correlation among adjacent pixels, being less sensitive as compared to the text data i. e. a tiny change in the attribute of any pixel of the image does not drastically degrade the quality of the image and bulk capacity of data etc.((c) Image encryption using chaotic logistic map article by N. K. Pareek, Vinod Patidar, K. K. Sud)”.
Various methods of chaos based encryption techniques, each with their own pro’s and con’s ranging from its intended use. For real time image encryption, ciphersi which take less time and yet still provide the necessary security are preferred to those which take considerably long amounts of time and produce high levels of security, this is because the latter would bee of little use in real time proceses and applications as well as the fact that as the strength of the encryption rises so does its cost.
In the case of real time imaging, chaos based encryption have been seen as a best fit, various chaos based image encryption schemes have been developed, each one aimed at being able to provide a quick ad yet still highly secured means of image encryption, with a novel image encryption scheme based on the logistic map and cheat image being an example.