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On the fractional-order extended
Kalman filter and its application
to chaotic cryptography in noisy
environment
By Hoda Sa...

Chaotic Systems
Chaotic systems
characteristics
Very complex and
nonlinear
behaviour
Dependence of the
trajectories to t...

1990, Pecora & Carroll: synchronization of chaotic
systems
Chaos synchronization: the slave/response system
should track...

-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...

As to increase the complexity of the transmitter, one may
use a fractional-order stochastic chaotic system as
transmitte...

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

d. Updating equation
c. Kalman gain equation
b. Innovation equation
a. Output prediction equation
Kalman Filter for Line...

and
where
e. State prediction equation
Kalman Filter for Linear Fractional-order
Stochastic Systems:
Design & Calculatio...

Then the Kalman filter can be obtained via following steps
Extended Kalman Filter for Non-linear
Fractional-order Stocha...

c. Kalman gain equation
b. Innovation equation
where
a. Output prediction equation
Extended Kalman Filter for Non-linear...

where
where the initial condition assumed to be at time
e. State prediction equation
d. Updating equation
Extended Kalma...

Except 𝛼 = 1, 𝜔 𝑘 is not a Wiener process and thus the
Kalman filter presented here is dealing with non-Wiener
process.
...

while the output of the communication channel would be
The output/measurement of this system can be assumed as
The Caput...

To decrypt the message in the receiver, assuming the
synchronization via nonlinear filtering has been done
The noise of ...

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-orde...

Using fractional-order Chen system
for chaotic fractional-order
cryptography
The first synchronized state of the fractio...

Using fractional-order Chen system
for chaotic fractional-order
cryptography
The second synchronized state of the fracti...

Using fractional-order Chen system
for chaotic fractional-order
cryptography
The third synchronized state of the fractio...

The
embedded
message
signal
The output
for
encryption
Using fractional-order Chen system
for chaotic fractional-order
cr...

Using fractional-order Chen system
for chaotic fractional-order
cryptography
Encryption/Decryption of signal via fractio...
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Reference:
H. Sadeghian, H. Salarieh, A. Alasty, A. Meghdari. "On the Fractional-Order Extended Kalman Filter and its Application to Chaotic Cryptography in Noisy Environment" Applied Mathematical Modelling, 2014, 38, pp. 961-973.

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On the fractional order extended kalman filter and its application to chaotic cryptography in noisy environment

  1. 1. 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
  2. 2.  Chaotic Systems Chaotic systems characteristics Very complex and nonlinear behaviour Dependence of the trajectories to the initial conditions Impossible long- time prediction
  3. 3.  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
  4. 4.  -Application: *secure communication Chaotic Cryptography
  5. 5.  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
  6. 6.  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
  7. 7.  Kalman Filter for Linear Fractional-order Stochastic Systems Lemma
  8. 8.  Then the Kalman filter can be obtained via following steps Kalman Filter for Linear Fractional-order Stochastic Systems
  9. 9.  d. Updating equation c. Kalman gain equation b. Innovation equation a. Output prediction equation Kalman Filter for Linear Fractional-order Stochastic Systems: Design & Calculations
  10. 10.  and where e. State prediction equation Kalman Filter for Linear Fractional-order Stochastic Systems: Design & Calculations
  11. 11.  Then the Kalman filter can be obtained via following steps Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems
  12. 12.  c. Kalman gain equation b. Innovation equation where a. Output prediction equation Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems: Design & Calculations
  13. 13.  where where the initial condition assumed to be at time e. State prediction equation d. Updating equation Extended Kalman Filter for Non-linear Fractional-order Stochastic Systems: Design & Calculations
  14. 14.  Except 𝛼 = 1, 𝜔 𝑘 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 𝛼 = 1 to be exactly equal to the classical Kalman and extended Kalman filter. FKF Design Remarks
  15. 15.  while the output of the communication channel would be The output/measurement of this system can be assumed as The Caputo derivative of fractional order is defined as Let assume the transmitter module as a fractional-order stochastic chaotic system be Cryptography and Synchronization
  16. 16.  To decrypt the message in the receiver, assuming the synchronization via nonlinear filtering has been done The noise of communication channel would change this output in the form of Let now assume the transmitter module has got another output with embedded message that should be encrypted. Cryptography and Synchronization
  17. 17.  Cryptography and Synchronization
  18. 18.  Using fractional-order Chen system for chaotic fractional-order cryptography
  19. 19.  Fractional- order Chen system the output of noisy channel Using fractional-order Chen system for chaotic fractional-order cryptography where
  20. 20.  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
  21. 21.  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
  22. 22.  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
  23. 23.  The embedded message signal The output for encryption Using fractional-order Chen system for chaotic fractional-order cryptography where
  24. 24.  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

Reference: H. Sadeghian, H. Salarieh, A. Alasty, A. Meghdari. "On the Fractional-Order Extended Kalman Filter and its Application to Chaotic Cryptography in Noisy Environment" Applied Mathematical Modelling, 2014, 38, pp. 961-973.

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