A chaotic sequence for chaos shift keying (CSK) that provides auto- and cross-correlation properties (that are similar to those of random white noise) is used for spread spectrum systems. Due to its bifurcation behavior (depending on the initial condition), the number of chaotic sequences that can be generated by a single formula is not restricted and will not repeat itself. These characteristics provide an increase in system capacity and security performance. The paper presents a study of two different commonly used chaotic logistic maps and a modified chaotic logistic map for CSK spread spectrum system. The newly modified logistic map provides similar bits error rate (BER) performance to the best logistic map. Yet, it also provides an additional chaotic parameter for the control of its dynamic property, hence increasing the system security and capacity

Hussain, Z

Lau, Y

A chaotic sequence for chaos shift keying (CSK) that pro-vides auto- and cross-correlation properties (that are simi-lar to those of random white noise) is used for spread spec-trum systems. Due to its bifurcation behavior (depending on the initial condition), the number of chaotic sequences that can be generated by a single formula is not restricted and will not repeat itself. These characteristics provide an increase in system capacity and security performance. The paper presents a study of two different commonly used chaotic logistic maps and a modified chaotic logistic map for CSK spread spectrum system. The newly modified logis-tic map provides similar bits error rate (BER) performance to the best logistic map. Yet, it also provides an additional chaotic parameter for the control of its dynamic property, hence increasing the system security and capacity. 1

Yuu-seng Lau

Zahir M. Hussain

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A new approach in chaos shift keying for secure communication

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A New Approach in Chaos Shift Keying for Secure Communication
Yuu-Seng Lau and Zahir M. Hussain
School of Electrical and Computer Engineering, RMIT University
124 La Trobe Street, Melbourne, Victoria 3000, Australia
Emails: s9701549@student.rmit.edu.au, zmhussain@ieee.org
Abstract
A chaotic sequence for chaos shift keying (CSK) that pro-
vides auto- and cross-correlation properties (that are simi-
lar to those of random white noise) is used for spread spec-
trum systems. Due to its bifurcation behavior (depending
on the initial condition), the number of chaotic sequences
that can be generated by a single formula is not restricted
and will not repeat itself. These characteristics provide
an increase in system capacity and security performance.
The paper presents a study of two different commonly used
chaotic logistic maps and a modified chaotic logistic map
for CSK spread spectrum system. The newly modified logis-
tic map provides similar bits error rate (BER) performance
to the best logistic map. Yet, it also provides an additional
chaotic parameter for the control of its dynamic property,
hence increasing the system security and capacity.
1. Introduction
The principle of spread spectrum (SS) systems is to
spread the original information over a broad bandwidth of
frequencies. These systems rely on the spreading sequence
to provide a white noise like auto- and cross correlation
property. Conventionally, a pseudorandom or a pseudo-
noise (PN) sequence generator is used for SS systems, but
it suffers from the periodicity problem. Because the gener-
ation of the pseudorandom sequence, which is a fixed num-
ber of states where the state-machine run through each state
in a deterministic manner. The periodicity behavior of pseu-
dorandom sequence compromises the overall system secu-
rity, and reduces the system capacity as well.
Due to intensive research in modelling non-linear dy-
namical systems using deterministic linear methods, which
has made it possible to use chaos theory in this arena. The
introduction of these non-linear chaos theories has offered
several new applications and performance enhancements
to exiting communication systems. A chaotic generator is
an unlimited states state-machine, therefore it can produce
non-linear and non-repeating sequences. It is very hard
Figure 1. Modulator and demodulator block
diagram for chaos shift keying
to predict chaotic patterns and sequences even when the
chaotic function is known to the interceptor. This is because
the chaotic function provides a bifurcation behavior, hence
a different estimation of the initial condition will lead to a
very different chaotic sequence. As a result, This non-linear
and non-periodic behavior of the chaotic generator offers
potential advantage over conventional pseudo-noise based
system in terms of security, synchronization and system ca-
pacity of a SS communication. Since synchronization was
solved by Pecora and Carroll in 1990 [1], there have been
increasing numbers of proposed schemes that utilize chaos
theory in SS communication for both analog and digital sys-
tems. Such schemes include but are not limited to chaotic
masking, chaos modulation, chaos shift keying (CSK), and
chaotic CDMA sequence.
In this paper, we propose a modified logistic chaotic map
for CSK spread spectrum communication and compare it
with two different logistic chaotic maps.
2. Chaos Shift Keying (CSK)
The simplest one-bit information chaos shift keying
(CSK) modulation scheme for SS systems uses a pair of
chaotic sequences (g1 and g2 ) with different bit energies to
transmit the binary information For the data bit(αl = +1)
during the lth bit period, g1 sequence is radiated from the
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transmitter , and for the (αl = −1), g2 sequence is trans-
mitted. The number of chaotic symbol transmitted for one
data bit is dependent on the spreading factor (2β). The out-
put of the CSK transmitter is sk = αlgv,k , v decides which
chaos sequence to be send.
The chaotic sequence for CSK g1 and g2 can be gen-
erated in three different ways. First method: it uses two
different chaotic generators. Second method: generating
the two sequences using different initial conditions of the
same chaotic generator. Third method: the two sequences
are generated by the same chaotic generator and same ini-
tial condition but multiplied by two different constants. For
simplicity, we used the last method to generate two chaotic
sequences g2 = −g1, as shown in Fig. 1. The demodulation
can be performed by a correlator at the receiver, as shown
in Fig. 1.
The performance of the CSK system in an AWGN envi-
ronment can be derived following the method used in [2].
For a correlator type receiver, the correlator output for the
lth bit yl is given by yl =
∑2βl
k=kk rkgk, kk = 2β(l− 1)+1
rk = sk + ηk is the received signal in an AGWN environ-
ment. ηk is an additive Gaussian white noise.
yl = αl
2βl∑
k=2β(l−1)+1
g2k +
2βl∑
k=2β(l−1)+1
ηkgk (1)
The first term is the required signal and second term is
noise. If we consider a sum of a large number of random
variables in the system, we can assume the system to follow
a normal distribution (Central Limit Theorem). The proba-
bility of error is
BERCSK = Prob(αl = 1)× Prob(yl ≤ 0 | αl = 1)
+ Prob(αl = −1)× Prob(yl > 1 | αl = −1)
=
1
4
[erfc( E[yl | (αl = +1)]√
2var[yl | (αl = +1)]
)
+erfc( −E[yl | (αl = −1)]√
2var[yl | (αl = −1)]
)] (2)
From [2], the variance of yl | (αl = +1) equals the
variance of yl | (αl = −1) and the E[yl | (αl = +1)] =
−E[yl | (αl = −1)]. Hence, equation (2) can be simplified
to
BERCSK =
1
4
[erfc( E[yl | (αl = +1)]√
2var[yl | (αl = +1)]
)
+erfc( E[yl | (αl = +1)]√
2var[yl | (αl = +1)]
)]
=
1
2
[erfc( E[yl | (αl = +1)]√
2var[yl | (αl = +1)]
)] (3)
where erfc(.) is the complementary error function defined
as erfc(ψ) ≡ 2√
π
∫∞
ψ
e−λ
2
dλ.
For AWGN environment, we know that the mean
E[ηk] = 0. As a result, the mean of yl | (αl = +1) is
E[yl | (αl = +1)] =
2βl∑
k=2β(l−1)+1
E[g2k]
+
2βl∑
k=2β(l−1)+1
E[ηk]E[gk]
= 2βPs (4)
The average power of the chaotic signal is Ps = E[g2k].
The variance of yl | (αl = +1) is
var[yl | αl = +1] = 2cov[
2βl∑
k=2β(l−1)+1
g2k,
2βl∑
k=2β(l−1)+1
ηkgk]
+ var[
2βl∑
k=2β(l−1)+1
ηkgk]
+ var[
2βl∑
k=2β(l−1)+1
g2k] (5)
From [2] we know that
2cov[
2βl∑
k=2β(l−1)+1
g2k,
2βl∑
k=2β(l−1)+1
ηkgk] = 0
var[
2βl∑
k=2β(l−1)+1
ηkgk] = βNoPs
var[
2βl∑
k=2β(l−1)+1
g2k] = 2βΛ
where Λ = var[g2k]. Hence
var[yl | αl = +1] = 2βΛ + βNoPs. (6)
Substituting (4) and (6) into (3), the BER for the CSK
can be found as follows
BERCSK =
1
2
erfc( 2βPs√
4βΛ + 2βNoPs
)
=
1
2
erfc( 1√
Λ
βP 2s
+ NoEb
) (7)
where Eb = 2βPs. From equation (7), the BER can be
improved by reducing the variance of g2k (= Λ) , increasing
the spreading factor (2β), or increasing the signal power Ps
(the E[g2k]).
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3. Chaos Sequence
For simplicity, one dimension chaotic logistic map is
studied in this section. To obtain the BER performance for
each logistic map, the signal power Ps and the variance Λ
are either calculated from their probability density function
(pdf) or numerically.
3.1. Logistic Map 1
One of the simplest chaotic logistic maps used for gen-
eration chaotic sequences is given by [2, 3]
gn+1 = 1− 2g2n (8)
which has the invariant probability density function [2, 4]
ρ(g) =
{
1
π
√
1−g2 , if | g |< 1
0 , otherwise
(9)
Since ρ(g) is an even function, the mean value E[gk] is
E[gk] =
∫ ∞
−∞
gρ(g)dg =
∫ 1
−1
gρ(g)dg = 0 (10)
and the auto-covariance of {xk} is equal to zero [2]
cov[gi, gk] = E[gj , gk]− E[gj ]E[gk] = E[gjgk] = 0.
(11)
Hence Ps and Λ can be manually calculated
Ps = E[g2k] =
∫ ∞
−∞
g2ρ(g)dg =
∫ 1
−1
g2
1
π
√
1− g2 dg =
1
2
(12)
Λ = var[g2k] = E[g4k]− E2[g2k] =
∫ 1
−1
g4ρ(g)dg − 1
4
=
1
8
(13)
The BER performances for this logistic map are shown in
Figs. 2 - 5.
3.2. Logistic Map 2
Another dynamic system capable of exhibiting chaos
used in spreading spectrum is proposed in [5]
gn+1 = agn(1− gn) (14)
where a is the bifurcation parameter or control, considered
to be in the interval 3.57 < a ≤ 4 for the system to be a
non-period chaos system.
Table 1. Statistics of M - Logistic Map 2
a E[g2k] or Ps Λ
4 12
1
8
3.97 920 0.111
3.95 0.410 110
3.90 0.392 0.095
Since logistic map 2 provides a bifurcation parameter,
it increases the system security. The attacker needs to
know both the initial condition and the bifurcation param-
eter value a before it can demodulate the transmitted infor-
mation. The drawback of this system is the output of the
above sequence lies in the interval 0 ≤ gn ≤ 1, which
will decrease the Ps and lead to a lower BER performance.
The probability density function for this system is not pro-
vided. Numerical simulation is used to obtain Ps = 38 and
Λ = 0.1333 when a = 4. The BER performance for this
logistic map is shown in Figs. 2 - 5 for different spreading
factors. A modified version of this logistic is proposed in
next section which is shown to have a higher value of Ps
and a lower value for Λ.
3.3. Modified Logistic Map 2 (M-Logistic)
Modified logistic map 2 is a scaled and shifted version
of the logistic map 2.
gn+1 = agn(1− gn)
jn+1 = 2(gn+1 − 0.5) (15)
where jn is the output chaotic sequence for CSK and a is
the bifurcation parameter or control, considered to be in the
interval 3.57 < a ≤ 4. Again, the probability density func-
tion for this system is not provided. Numerical simulation
is used to obtain Ps and Λ for different values of a.
From Table 1, the BER can be calculated using equation
(7). The BER performances for this logistic map are shown
in Figs. 2- 5.
Figs. 2 - 5 show that logistic map 1 and M-logistic map 2
with a = 4 provide same BER. However, M-logistic map 2
can enhance the security due to its extra bifurcation param-
eter a, although it still provides a BER performance close
to logistic map 1.
Similar to the initial condition case, the parameter a for
logistic map 2 and M-logistic map 2 need to be precise. A
slight difference in these parameters will generate totaly dif-
ferent chaotic sequences. Hence, this will provide an extra
parameter for security enhancement. This extra bifurcation
parameter a not only acts to enhance security but it also can
be used as a multi-user and a synchronization parameter.
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−8 −6 −4 −2 0 2 4 6 8 10
10−4
10−3
10−2
10−1
100
Theoretical BER performance for CSK with SF = 10
Eb/No in dB
BE
R
Logistic Map 1 & M−Logistic Map 2, a = 4
Logistic Map 2 
M−Logistic Map 2, a = 3.97
M−Logistic Map 2, a = 3.95
M−Logistic Map 2, a = 3.9
Figure 2. Theoretical BERperformance for dif-
ferent CSK logistic maps with SF = 10
−8 −6 −4 −2 0 2 4 6 8 10
10−5
10−4
10−3
10−2
10−1
100
Theoretical BER performance for CSK with SF = 40
Eb/No in dB
BE
R
Logistic Map 1 & M−Logistic Map 2, a = 4
Logistic Map 2 
M−Logistic Map 2, a = 3.97
M−Logistic Map 2, a = 3.95
M−Logistic Map 2, a = 3.9
Figure 3. Theoretical BERperformance for dif-
ferent CSK logistic maps with SF = 40
In general, an increase in the spreading factor will pro-
vide better BER performance. However, when the spread-
ing factor is at a higher value range (larger than 100), the
BER performance curves of the three logistic maps ap-
proach each other.
4. Conclusion
We proposed a modified logistic chaotic map for chaos-
shift-keying (CSK) spread spectrum communication and
compared it with two different logistic chaotic maps. It
is shown that the proposed scheme provides BER perfor-
mance close to the optimum performance. In addition, it
provides an extra parameter for better security performance.
This parameter can also be used for other purposes such as
synchronization in a multi-user environments.
References
[1] L. Pecora, and T. Carroll, “Synchronization in chaotic
systems”, Phys. Rev. Lett, vol. 64, 1990, pp.821-824.
−8 −6 −4 −2 0 2 4 6 8 10
10−5
10−4
10−3
10−2
10−1
100
Theoretical BER performance for CSK with SF = 100
Eb/No in dB
BE
R
Logistic Map 1 & M−Logistic Map 2, a = 4
Logistic Map 2 
M−Logistic Map 2, a = 3.97
M−Logistic Map 2, a = 3.95
M−Logistic Map 2, a = 3.9
Figure 4. Theoretical BERperformance for dif-
ferent CSK logistic maps with SF = 100
−8 −6 −4 −2 0 2 4 6 8 10
10−6
10−5
10−4
10−3
10−2
10−1
100
Theoretical BER performance for CSK with SF = 300
Eb/No in dB
BE
R
Logistic Map 1 & M−Logistic Map 2, a = 4
Logistic Map 2 
M−Logistic Map 2, a = 3.97
M−Logistic Map 2, a = 3.95
M−Logistic Map 2, a = 3.9
Figure 5. Theoretical BERperformance for dif-
ferent CSK logistic maps with SF = 300
[2] F. C. M. Lau, C. K. Tse, M. Ye, and S. F. Hau, “Coexistence
of chaos-based and conventional digital communication
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[3] S. S. Rao, and S. P. Howard, “Correlation performance
of chaotic signals in spread specturm systems”, in Proc.
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[4] T. Kohda, and A. Tsuneda, “Even- and odd-correlation
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[5] G. Heidari-Bateni, and C. D. McGillem, “A chaotic direct-
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A new approach in chaos shift keying for secure communication

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