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CONTENTS
1. Introduction
2. Chaos - What is it?
3. Types of systems
4. Chaos - Early conception
5. Chaos control
6. History of Cryptology
7. Chaotic encryption – An Overview
8. An example of chaotic wave generation
9. Chaotic Encryption – Ultimate in security
10. Robustness in synchronization
11. Chaotic synchronization
12. Open loop chaotic synchronization
13. The Set-Up
14. Drawbacks
15. Merits
16. 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.
ABCDEFGHIJKLMNOPQRSTUVWXYZ
CDEFGHIJKLMNOPQRSTUVWXYZAB
As this was not much of a success newer ideas came up.
The next step towards this direction was the use of
scattered relationship between alphabets for coding.
Scattered relationship between alphabets
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.
As we can see no information is evident on the
chaotic waveform and it looks absolutely random.
Xnew = Xold * a * (Xold-1)
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(AA’) 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.