wilamowski-b-m-irwin-j-d-industrial-communication-systems-2011

Secure Communication Using Chaos Synchronization 23-9
impulsive controller. The impulsive differential system is essentially a Chua’s circuit, which is the same
as that used in the encrypter. The synchronization of the encrypter and the decrypter can be achieved
by using an impulsive controller with piecewise constant impulsive intervals. The key signal k(t) is a
combination of all three state variables of the chaotic systems. The ciphertext is obtained from an XOR
operation on the plaintext and the key sequence bit by bit. The decryption is the same as the encryption,
including the XOR operation of the transmitted scrambled signal with the key signal k˜(t). When the
chaotic systems in the decrypter and the encrypter are synchronized, the decrypter can find the same
key signal sequence k˜ (t) as that in the encrypter k(t).
23.3.2 the Encrypter
The dimensionless state equations of Chua’s circuit are given as
⎧ẋ
1 = kα( x2 − x1 − f ( x1)),
⎪
⎨ẋ
2 = k( x1 − x2 + x3),
⎪
⎪⎩
ẋ
3 = k( −βx2 − γx3),
(23.9)
where α, β, and γ are constants, k ∈ [−1, 1] and f(x) is the nonlinear characteristic of Chua’s diode in
Chua’s circuit given by
{ }
1
f ( x) = m1x + ( m0 − m1)
x + 1 − x −1 , (23.10)
2
with m 0 and m 1 being two negative constants. Since the signals are transmitted through a digital
channel, the synchronization pulses should be first quantized by a predefined quantizer Q(·), which
depends on the amplification factor K used in (23.11). Since chaos is very sensitive to initial condition,
the quantization error should be less than certain values to ensure that the encrypter and the decrypter
can be synchronized [HLYS00].
To provide the desired key sequence, the concept of a magnifying glass is introduced, which is composed
of an amplifier and an observer. They are given in details as follows.
The amplifier:
/
k′ ( t) = K( x ( t) + x ( t) + x ( t)) .
1 2 2 2 3 2 1 2 (23.11)
The observer:
( )
k( t) = ⎢⎣ k′
( t) ⎥ ⎦ + λ mod( 256 ),
(23.12)
where
K is a large number chosen to influence the sensitivity of the system
λ is an arbitrary integer
a is the integer truncation of a
© 2011 by Taylor and Francis Group, LLC