on the fractional order extended kalman filter and its application to chaotic cryptography in noisy...
TRANSCRIPT
On the fractional-order extended Kalman filter and its application to
chaotic cryptography in noisy environment
By Hoda Sadeghian, Hassan Salarieh, Aria Alasty, Ali Meghdari
Presentation by Mostafa Shokrian Zeini
Chaotic Systems
Chaotic systems characteristics
Very complex and nonlinear
behaviour
Dependence of the trajectories
to the initial conditions
Impossible long-time prediction
1990, Pecora & Carroll: synchronization of chaotic systems
Chaos synchronization: the slave/response system should track the master/drive system trajectories.
A synchronization system consists of a transmitter module, a channel for communication and a receiver module.
Chaotic Systems
-Application:
*secure communication
Chaotic Cryptography
BUT
• Most of the works in chaotic communication have modeled the chaotic systems in deterministic form.
IN THIS
CASE
• in real world applications due to random uncertainties such as stochastic forces on physical systems and noisy measurements caused by environmental uncertainties, a stochastic chaotic behavior is produced instead of a deterministic one.
the deterministic differential equation of a system must be substituted by a stochastic one.
Chaotic Communication
As to increase the complexity of the transmitter, one may use a fractional-order stochastic chaotic system as transmitter.
A chaotic fractional-order dynamical equation produces a complex behavior which makes the masked signal more encrypted and consequently hard to decipher.
The complexity of fractional-order systems is due to dealing with integration and derivation of non-integer orders.
Fractional-order Stochastic Chaotic Systems
Kalman Filter for Linear Fractional-order Stochastic Systems
Lemma
Then the Kalman filter can be obtained via following steps
Kalman Filter for Linear Fractional-order Stochastic Systems
Kalman Filter for Linear Fractional-order Stochastic Systems:
Design & Calculations
Kalman Filter for Linear Fractional-order Stochastic Systems:
Design & Calculations
Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems
Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems:
Design & Calculations
Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems:
Design & Calculations
Except , is not a Wiener process and thus the Kalman filter presented here is dealing with non-Wiener process.
The above theorems can be easily verified in the case of to be exactly equal to the classical Kalman and extended Kalman filter.
FKF Design Remarks
Cryptography and Synchronization
Cryptography and Synchronization
Cryptography and Synchronization
Using fractional-order Chen system for chaotic fractional-
order cryptography
Fractional-order
Chen system
the output of
noisy channel
Using fractional-order Chen system for chaotic fractional-
order cryptography
where
Using fractional-order Chen system for chaotic fractional-
order cryptography
The first synchronized state of the fractional-order stochastic Chen chaotic system and its estimated error
Using fractional-order Chen system for chaotic fractional-
order cryptography
The second synchronized state of the fractional-order stochastic Chen chaotic system and its estimated error
Using fractional-order Chen system for chaotic fractional-
order cryptography
The third synchronized state of the fractional-order stochastic Chen chaotic system and its estimated error
The
embedded
message signal
The output
for encrypti
on
Using fractional-order Chen system for chaotic fractional-
order cryptography
where
Using fractional-order Chen system for chaotic fractional-
order cryptography
Encryption/Decryption of signal via fractional-order synchronization in the fractional-order stochastic Chen chaotic system and its estimated
error