On the fractional order extended kalman filter and its application to chaotic cryptography in noisy...
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Transcript of On the fractional order extended kalman filter and its application to chaotic cryptography in noisy...
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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
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Chaotic Systems
Chaotic systems characteristics
Very complex and nonlinear
behaviour
Dependence of the trajectories
to the initial conditions
Impossible long-time prediction
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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
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-Application:
*secure communication
Chaotic Cryptography
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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
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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
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Kalman Filter for Linear Fractional-order Stochastic Systems
Lemma
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Then the Kalman filter can be obtained via following steps
Kalman Filter for Linear Fractional-order Stochastic Systems
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Kalman Filter for Linear Fractional-order Stochastic Systems:
Design & Calculations
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Kalman Filter for Linear Fractional-order Stochastic Systems:
Design & Calculations
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Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems
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Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems:
Design & Calculations
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Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems:
Design & Calculations
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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
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Cryptography and Synchronization
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Cryptography and Synchronization
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Cryptography and Synchronization
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Using fractional-order Chen system for chaotic fractional-
order cryptography
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Fractional-order
Chen system
the output of
noisy channel
Using fractional-order Chen system for chaotic fractional-
order cryptography
where
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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
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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
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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
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The
embedded
message signal
The output
for encrypti
on
Using fractional-order Chen system for chaotic fractional-
order cryptography
where
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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