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

23-10 Industrial Communication Systems
Remark 23.1: One can replace the amplifier function in (23.11) by k′(t) = Kg(x 1 , x 2 , x 3 ), where g is a
predefined function. Since the function g is a function of the state variables of the chaotic system, the
sensitivities to the parameter mismatches will be different for different g.
The scrambled signal v R is given by
vR( t) = E( p( t)) = p( t) ⊕ k( t),
(23.13)
where
p(t) is the plaintext
k(t) is given in (23.12)
E(p(t)) is the ciphertext
⊕ denotes XOR operation
23.3.3 the Decrypter
Both Chua’s circuit and the impulsive controller in the decrypter are given by
⎧x̃̇ 1 = kα( x̃ 2 − x̃ 1 − f ( x̃
1)),
⎪
⎨x̃̇ 2 = k( x̃ 1 − x̃ 2 + x̃
3),
⎪
⎪x̃̇
2 = k( −βx̃
2 − γx̃
⎩
3),
t ≠ τ ; l = 1, 2, …,
(23.14)
l
and
−
−
⎡x̃
̃
1( τl)
⎤ ⎡x1( τl
) ⎤ ⎡Q( x1( τl)) − x̃
1( τl
) ⎤
⎢ ⎥ ⎢ ⎥ ⎢
⎥
⎢x̃
2( τl)
⎥ = ⎢ −
x̃
2( τl
) ⎥
−
− B ⎢Q( x2( τl)) − x̃
2( τl
) ⎥, l = 1, 2 , …,
(23.15)
⎢ ⎥ ⎢ ⎥ ⎢
⎥
⎢
⎣x
̃3( τl)
⎥ ⎢ − ⎥ ⎢
− ⎥
⎦ x̃
⎣⎢
3( τl
)
⎦⎥
⎣⎢
Q( x3(
τl)) − x̃
3( τl
)
⎦⎥
where
B is a 3 × 3 matrix to be designed to satisfy certain inequality for synchronization
Q(·) is a predefined quantizer
τ l − are the time instant immediately prior to the time instant τ l
Let T l = (τ l − τ l − 1 ), (τ 0 = 0, l = 1, 2, …), denote impulsive time intervals.
In the decrypter, the plaintext is recovered via
( )
̃ /
k( t) = ⎢K( x̃ ( t) + x̃ ( t) + x̃
( t)) ⎥
⎣
1 2 2 2 3 2 1 2 ⎦ + λ mod( 256 ),
(23.16)
̃p ( t) = Ẽ ( p( t)) = v ( t) ⊕ k̃
( t),
(23.17)
R
where
Ẽ (p(t)) is the recovered encrypted signal
k˜ (t) is recovered in the receiver circuit and should approximate k(t)
© 2011 by Taylor and Francis Group, LLC