CONTENTS

 1.     Introduction2.     Chaos - What is it?3.     Types of systems4.     Chaos - Early conception5.     Chaos control6.     History of Cryptology7.     Chaotic encryption – An Overview8.     An example of chaotic wave generation9.     Chaotic Encryption – Ultimate in security10. Robustness in synchronization11. Chaotic synchronization12. Open loop chaotic synchronization13. The Set-Up14. Drawbacks15. Merits16. Conclusion 

INTRODUCTION

The idea of encrypting data can be dated back to the caesarian period where he used keys to encrypt alphabets one by one. The demand for better and foolproof encrypting technique led to immense research on encryption and now we have a large number of techniques of encryption of which chaotic encryption has proved to be the most promising one.

 

CHAOS – What is it?

Chaos refers to a type of complex dynamical behavior that possess some special features such as being extremely sensitive to small variations in initial conditions.It is said 20th century would be known for three things,quantum mechanics, relativity theorem, and the third and most recent chaos. 

 

Types of systems

All systems can be basically divided into three types: 1.Deterministic systems These are systems for which for a given set of conditions the result can be predicted and the output does not vary much with change in initial conditions.  2. Stochastic systems

These systems, which are not as reliable as deterministic systems. Their output can be predicted only for a certain range of values.  3. Chaotic systems

Chaotic systems are the most unpredictable of the three systems. Moreover they are very sensitive to initial conditions and a small change in initial conditions can bring about a great change in its output. 

CHAOS - EARLY CONCEPTION – The journey from being a nuisance to the ultimate tool

 

Due to its inherent instability chaos was considered to be neither controllable nor predictable and hence useless but ironically recent researches have proved that chaos can be harnessed for beneficial purposes.

CHAOS CONTROL

Chaos control refers to the situation where chaotic dynamics is weakened or eliminated by appropriate controls; while anti-control of chaos means that chaos is created, maintained, or enhanced when it is healthy and useful. Both control and anti-control of chaos can be accomplished via some conventional and nonconventional methods such as microscopic parameter perturbation, bifurcation monitoring, entropy reduction, state pinning, phase delay, and various feedback and adaptive controls.

It has been shown that the sensitivity of chaotic systems to small perturbations can be used to direct system trajectories to a desired target quickly with very low and ideally minimum control energy

Chaos may be used to enhance the artificial intelligence of neural networks, as well as increase coding- decoding efficiency in signal and image communications.

HISTORY OF CRYPTOLOGY – Why the need for chaotic encryption ever came up?

The first ideas of encryption came up in the caesarian period. The idea was to use keys to displace alphabets as shown. 

ABCDEFGHIJKLMNOPQRSTUVWXYZCDEFGHIJKLMNOPQRSTUVWXYZAB

 

As this was not much of a success newer ideas came up.

Scattered relationship between alphabets

The next step towards this direction was the use of scattered relationship between alphabets for coding.

CHAOTIC Encryption – An overview.

The basic idea of chaotic encryption is to modulate a chaotic wave with a massage so that the message remains in the transmitted wave remains invisible in both time and frequency domains.

AN EXAMPLE OF CHAOTIC WAVEFORM GENERATION

 

To get an idea about chaos and how a chaotic waveform would look like let us consider an example of a chaotic wave generated. In the given example the output wave follows the formula  

Xnew = Xold * a * (Xold-1) 

Where Xnew stands for the new state and Xold for the just previous state. Here as we can see every value depends on the previous state and hence going back the system on a whole depends on the initial conditions.

Xnew = Xold * a * (Xold-1)

As we can see no information is evident on the chaotic waveform and it looks absolutely random.

CHAOTIC ENCRYPTION –

ULTIMATE IN SECURITY

Chaotic encryption is both technically simple and inexpensive to embed on a microchip -- two factors that make it attractive to any company looking for a low-cost way to protect data communications. It would also include the cellular phone industry, since all cellular phones contain microprocessors and since the encryption systems the industry now has in place have failed to stem an explosion in theft of service.  

For successful decryption of a chaotic encrypted data three elements should match at the transmitter and receiver end.  

1. Original conditions:- Original condition is a value, chosen

by the chaotic encryption system's proprietary protocol that gets plugged into a formula called a map.

 2. Map :-

The map is composed of parameters, and is shared by both the sender of a message, who uses it to scramble the underlying data, and by the receiver, who uses it as a descrambler.

 3. Parameters:-

The various parameters at the receiver and transmitter end should match.

 

Up to 10,000 maps can be hardwired into single microchip.Any of the three parameters is kept changing frequently.

ROBUSTNESS OF SYNCHRONISATION

This is most crucial in chaotic synchronization. Good quality of synchronization does not guarantee good retrieval of message signal due to the sensitivity of the synchronized trace to any perturbation, including the perturbations caused by the intrinsic noise of the transmitter and that of the receiver .If some perturbation temporally desynchronizes the synchronized transmitter and receiver for a period of time, the message signal within this period cannot be recovered.

Robust synchronization will not occur until the value of coupling strength index K is not above 0.15, but an increase of K thereafter does not necessarily result in an increase in robustness of synchronization.

It has been found that optimum value of coupling strength for robust synchronization is = 0.4 

CHAOTIC SYNCHRONISATION

As has been mentioned earlier the most demanding task in the process of chaotic encryption is to synchronize the transmitter and the receiver. Two ways to achieve this synchronization would be 1.  Use two- transmission channel system :-

Here we have two separate

channels, one to carry the encrypted signal and the other to carry the chaotic carrier wave for synchronization at the receiver side.  2. Single transmission channel :-

Here the carrier chaotic waveform itself acts as driving signal for synchronizing the receiver with the transmitter. 

OPEN-LOOP CHAOTIC

SYNCHRONISATION

Both the above-mentioned techniques of synchronization make use of complex closed loops, and the former needs filters to make recovered message signal recognizable while the latter is limited to transmit a digital signal. Hence we came up with the idea of open loop chaotic synchronization. Here no optical feedback path exists.

The transmitting laser is operated to chaos under proper injection conditions. Since the message signal is injected into transmitter

The transmitter is driven from one chaotic state to another. Hence this makes the process of decryption by an eavesdropper totally impossible.

THE SET-UP

Here we consider a set-up for open loop chaotic synchroniosation of injection locked semiconductors. The given set-up can carry a message regardless of wheather it is analog or digital , by amplitude modulation frequency modulation for the input Ei(t)  

In the given set up for an input of Ei(t) we get an output given by  E’I(t) = ( Kr/ Kt) * S(t) – Kr * A’(t))

= (Kr/ K) * (Ei(t) + K(A-A’))

(Kr/K) * Ei(t) ( K is kept low) 

 where Kr is total measurement of coupling coefficient of receiver and transmission coefficient of BS3 Kt is total measurement of coupling coefficient of transmitter and transmission coefficient of BS2  (A-A’) is the synchronization error. 

In order to keep the signal buried under the chaotic signal we need to keep the message signal amplitude sufficiently low.  The output Ei(t) is recovered without any auxiliary optical or electronic filter provided the condition Ei(t) >> K(A-A’) is satisfied.

K is taken small.

Hence we see amplitude is affected by demand for privacy than quality.

DRAWBACKS

No system goes without any drawbacks however perfection is taken care of. Hence chaotic synchronization also has a few drawbacks to mention. They are mainly

  .. Sensitivity to initial conditions :-

It has been mentioned repeatedly the chaotic synchronization is extremely sensitive to its initial conditions and any small perturbation could lead to disastrous effects. 

  Phase sensitivity :- This is again another drawback we face in

chaotic encryption. By saying the system is phase sensitive we mean that for the signal K*Ei(t-t’) the corresponding chaotic wave should be Kt * A(t-t’). 

A successful circuit implementation in a chaotic environment is generally difficult, due to the extreme sensitivity of chaos to parameter variations and noise perturbations, and the nonrobustness of chaos to the structural stability, within the physical devices 

MERITS

The merits we have by adopting chaotic encryption are many. As has been mentioned throughout the text chaotic encryption is accredited to be 1.   

Ultimate in encryption. 2.    Cost effective  3.    Utilizes chaos, which was considered a nuisance. 

CONCLUSION

Chaotic encryption is the technology of tomorrow.

It is one solution to a cost-effective and foolproof encryption technique.