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

Secure Communication Using Chaos Synchronization 23-11
It can be known from (23.16) and (23.17) that the original signal can be recovered only when
two identical chaotic circuits in both the encrypter and the decrypter are synchronized. Similarly, an
intruder can know the original message only when he knows the parameters and the structure of the
circuits and the synchronization impulses.
Remark 23.2: The magnifying glass is used to transform the chaotic state variables that act
as key sequence before XOR with the plaintext. Assuming that there is a small mismatch that
results in perturbations Δx i (t) = σ i (i = 1, 2, 3), then the signal getting through the amplifier gives
⎛
1/
2
⎢ 3
⎛
⎞ ⎥
k( t) = ⎢K ( xi( t) + i) mod( )
⎝
⎜
i
⎠
⎟ ⎥
⎣⎢
⎦⎥ +
⎞
2
⎝
⎜ ∑ σ λ 256 . Since the parameter K is a large number, it can
= 1
⎠
⎟
be seen that the signal is enlarged many times, which implies that the parameters mismatch can be
enlarged. Thus, even a minor mismatch in the parameters will produce a large decryption error, resulting
in a decryption key sequence that is not the same as the encryption key signal. So, one cannot
recover the plaintext signal. The security of the chaotic communication system is thus improved.
Before transmitting the ciphertext through a digital channel, it is necessary to packetize the ciphertext.
To simplify the operation, a packetization algorithm is designed, in which the packet length is
fixed. Since the lengths of the impulsive intervals are piecewise constant, it is difficult to find the synchronization
impulse. The security of the system can then be improved. To maintain the simplicity of
the presented secure system, the length of the packets is the same as the length of the first impulsive
interval. The length of other impulsive interval is determined by the length of the first impulsive interval
and the parameters of the Chua’s circuits.
23.3.4 Synchronization Time Estimation
It is noticeable that the time-varying impulsive intervals do not affect the synchronization of two identical
chaotic circuits embedded in the encrypter and the decrypter, and the time required for synchronization
can be estimated. A scheme in [LLWS03] is presented to achieve this objective.
From the estimation, it is remarked that the value of K will influence the synchronization stable time
of two identical chaotic systems and the desired precision of the system. A larger K will require longer
synchronization time. On the other hand, the security of the system is higher with a larger K. A trade-off
should be obtained in practice.
23.3.5 Implementation
Text transmission is taken to illustrate the performance of the proposed chaotic secure communication
system. In order to make the presented encryption system widely used in the computer, 256 characters ASCII
code is used, which includes 128 standard ASCII codes and other 128 that are known as extended ASCII.
Thus, the ciphertext in the simulation has not only the standard characters but also some local symbols and
marks. When the ciphertext can be decrypted by the correct key, the message is exactly recovered. Otherwise,
the recovered message is spread in the extended ASCII code.
Choose the parameters of Chua’s circuit as k = 1, α = 9.35159085, β = 14.790313805, γ = 0.016072965,
m 0 = −1.138411196, and m 1 = −0.723451121. For synchronization to be ensured, matrix B is selected as
B = diag(−1.05, −1, −1) and the lengths of impulsive time intervals are chosen as T 2h−1 = 5 × 10 −3 .s and
T 2h = 2 × 10 −3 .s where h = 1, 2, …. Then starting with τ 0 = 0, the impulsive instants can be determined.
The initial condition is given by [x 1 (0); x 2 (0); x 3 (0)] = [−2.12; −0.05; 0.8] and [x˜1(0); x˜ 2(0); x˜3(0)] = [−0.2;
−0.02; 0.1], respectively. As the initial error is e(0) = [−1.92; −0.03; 0.7], so the encrypter and decrypter
are initially not synchronized. Choose the magnifying glass as K = 1 × 10 3 , λ = 12, and the quantization
step is q = 5 × 10 −8 .
© 2011 by Taylor and Francis Group, LLC