IMPROVING SECURITY IN CHAOTIC SPREAD SPECTRUM

COMMUNICATION SYSTEMS WITH A NOVEL ‘BIT POWER

PARAMETER SPECTRUM’ MEASURE

Branislav Jovic

Department of Electrical and Computer Engineering, University of Auckland, 38 Princes street, Auckland, New Zealand

Charles Unsworth

Department of Engineering Science, Univesity of Auckland, 70 Symonds street, Auckland, New Zealand

Keywords: Security, Spread Spectrum, Communications, Chaos, PC Synchronization, Bit Power Parameter Spectrum.

Abstract: Due to the broadband nature and the high sensitivity to parameter and initial conditions in chaotic signals,

chaotic spread spectrum (SS) communication systems have been regarded as highly secure. However, it is

often easier to decrypt chaotic parameter modulation (CPM) based SS systems than was originally thought.

In this paper, a single user CPM based chaotic communication system implementing Pecora-Carroll (PC)

synchronization is described. Following this, the CPM based communication system, employing the chaotic

carrier generated by the Burger’s map is proposed. To highlight the security aspect a new measure called

‘Bit Power Parameter Spectrum’ (BPPS) is introduced. The BPPS is then used to identify parameters that

provide high secure and insecure regions for the chaotic map. Furthermore, it is demonstrated how a binary

message can be decrypted easily if the parameters of the map exist in the insecure region of the BPPS and

how security is optimised if the parameters exist in the secure region of the BPPS. The results are contrasted

with those of the standard Lorenz CPM based system. The BPPS measure shows that the Lorenz CPM

based system is easily decrypted for nearly all parameter values thus rendering the carrier insecure.

1 INTRODUCTION

In 1990 Pecora and Carroll (PC) discovered that

chaotic systems can be synchronized. Along with the

broadband nature and the high sensitivity of chaotic

systems to parameter and initial condition

perturbations, chaotic synchronization allowed

researchers to design SS chaotic communication

systems. These systems were primarily designed

with the aim of the increased security over the

existing SS systems. However, as will be discussed

shortly, chaotic communications are often insecure.

A PC synchronization scheme can be viewed as

a master-slave synchronization system (Jovic et al.,

2006a). The master system provides at least one of

its chaotic outputs to the slave system. The slave

system uses the given master output (driving signal),

to elegantly synchronize itself to the master system,

regardless of its initial conditions. The master-slave

system can also be viewed as the transmitter-

receiver communication system. Since the

introduction of the PC synchronization method a

number of communication schemes based on this

method have been proposed, (Wu and Chua, 1994;

Oppenheim et al., 1992; Cuomo and Oppenheim,

1993; Jovic et al., 2006a). These include such

methods as the chaotic masking (CS) (Oppenheim et

al., 1992), the chaotic parameter modulation (CPM)

(Cuomo and Oppenheim, 1993) and the initial

condition modulation (ICM) (Jovic et al., 2006a).

Other chaotic SS communication systems, such as

those based on DS-CDMA synchronization also

exist and have been studied in (Jovic et al. 2007a).

In contrast to PC synchronization where the

master-slave system either synchronizes or does not,

it is also possible to design controllers which enforce

synchronization. Such design techniques have been

investigated for both chaotic flows (Jovic and

Unsworth, 2007b) and chaotic maps (Millerioux and

Mira, 1998, 2001; Yan, 2005). In a number of cases

it has been shown that these techniques can be

273

Jovic B. and Unsworth C. (2007).

IMPROVING SECURITY IN CHAOTIC SPREAD SPECTRUM COMMUNICATION SYSTEMS WITH A NOVEL ‘BIT POWER PARAMETER SPECTRUM’

MEASURE.

In Proceedings of the Second International Conference on Security and Cryptography, pages 273-280

DOI: 10.5220/0002125302730280

Copyright

c

SciTePress

applied to chaotic communications (Millerioux,

1998; Nan, 2000). In (Millerioux and Mira, 1998,

2001) synchronization of piecewise linear chaotic

maps in a master-slave configuration is investigated.

In particular, finite time synchronization is

considered and the conditions for it discussed. It is

shown that finite time synchronization requires the

eigenvalues of the error system matrix to be equal to

zero. The significance of the results in relation to

secure chaotic communications is also discussed.

A similar method to that of the master-slave map

synchronization of (Millerioux and Mira, 2001) is

proposed here. In our method the general approach

to master-slave synchronization of chaotic maps is

presented and the requirements for synchronization

outlined. It is shown that the synchronization is

achieved by keeping the eigenvalues of the error

system matrix within the unit circle in the z domain.

Furthermore, the method of implementing the

synchronized master-slave system within a CPM

based secure SS chaotic system is demonstrated.

With the development of secure communication

techniques based on the concept of chaotic

synchronization, eavesdropping techniques have also

been developed in parallel, highlighting the lack of

security in many of the proposed systems. The

eavesdropping techniques include those based on the

prediction attacks (Short, 1994), short-time zero-

crossing rate (STZCR) attacks (Yang, 1995),

generalized synchronization attacks (

Álvarez et al.,

2004c), return map attacks (Pérez and Cerdeira,

1995), spectral analysis attacks (

Álvarez et al.,

2004b), and parameter estimation attacks (Álvarez et

al., 2004a), among other.

In perhaps the broadest of terms the chaotic

communication eavesdropping techniques, in the

literature today, can be divided into those which

directly extract the transmitted message without the

knowledge of the dynamics of the transmitter (Short,

1994; Yang, 1995;

Álvarez et al., 2004b), and those

which make certain assumptions about the dynamics

of the transmitter before attempting the extraction of

the message (

Álvarez et al., 2004a, 2004c).

In this paper, it is demonstrated how one can

decrypt a binary message from a CPM based SS

communication system with no prior knowledge of

the dynamics of the transmitter. The message

extraction technique is based on the average power

of the received signal which for a secure system

must be equal for both bits 0 and 1. The carrier

powers of bits 0 and 1 must be equal, or very nearly

equal, to each other to eliminate the possibility of

recognising the message from these (

Álvarez et al.,

2004c). It is shown that in terms of the bit power

security, the Burgers’ map CPM system can be

optimized and outperforms the Lorenz CPM system.

2 SS CHAOTIC

SYNCHRONIZATION BASED

COMMUNICATION SYSTEMS

In this section, the chaotic communication system

with the receiver based on the PC chaotic

synchronization, namely chaotic parameter

modulation (CPM) is briefly described.

A block

diagram of a SS chaotic communication system

based on the CPM concept is shown in Figure 1. A

requirement for the CPM scheme is for the master-

slave system to synchronize for a given driving

signal (Jovic and Unsworth, 2007b).

Slave system

n

Transmitter Channel Receiver

∧

m

∧

x

r

x

x

Master system

m

Detector

Figure 1: A block diagram of the chaotic communication

system based on the parameter modulation concept.

In Figure 1, the message signal m varies between

the two particular values, depending on whether a

binary 0 or a binary 1 is to be transmitted. The

message is incorporated into a certain modulating

parameter of the master system causing it to change

its value with the change in the message. The

parameters of the slave system are fixed at all time.

When the master-slave parameters are identical

synchronization occurs. This forces the

synchronization error to zero, indicating that bit 0

has been transmitted. Alternatively, with the master-

slave parameter mismatch the system does not

synchronize, indicating that bit 1 has been

transmitted. Assuming that the additive white

Gaussian noise (AWGN) component n is near zero,

and that sufficient amount of time has passed for

r

x

and

∧

x

to synchronize, the transmitted message m is

recovered in the form of

∧

m

. The choice of the

modulating parameter of the master chaotic system

must be chosen with care to ensure chaotic

properties of the system at all time. This ensures the

increased security within the communication system.

SECRYPT 2007 - International Conference on Security and Cryptography

274

3 GENERAL APPROACH TO

THE DESIGN OF THE

SS CHAOTIC

SYNCHRONIZATION BASED

COMMUNICATION SYSTEMS

In this section, the general approach to the design of

the synchronized chaotic maps is proposed and

applied to the design of the CPM based SS

communication systems.

3.1 Synchronization of Chaotic Maps

In (Pecora and Carroll, 1990), the chaotic

synchronization concept with a single signal of the

master system supplied to the slave system is

considered. The general result of this is that the

master-slave system either synchronizes or does not

(Pecora and Carroll, 1990; Jovic et al, 2006a). In this

subsection, the design of nonlinear controllers for

the chaotic map master-slave systems is proposed. In

particular, the method is demonstrated on the two

dimensional Burgers’ chaotic map. These controllers

then ensure the synchronization among the master-

slave systems. The design of the nonlinear control

laws is via the following theorem:

Theorem 1:

Suppose:

nnnnn

eUeAe +=

+1

,

∀

0≥n

,

1)()( <=+ BeigUAeig

nn

.

Then:

0→

n

e

, as

∞→n

,

∀

n

Re ∈

0

.

The theorem states that the equilibrium 0, of the

error system

1+n

e , is globally asymptotically stable

if and only if all eigenvalues of

nn

UAB +=

have

magnitude less than one.

Special case: If the matrix B is a function of n,

then the condition that

nn

BB −

+1

remains

bounded must also be satisfied.

In the above theorem the brackets | | denote the

magnitude of the eigenvalues of a matrix, and the

brackets || || denote the Euclidian norm. In the

following sections theorem 1 is used for the purpose

of synchronizing one, two and three dimensional

master-slave chaotic maps.

3.2 SS Communication System based

on the Synchronization of Burgers’

Map Master-Slave Chaotic System

In this subsection, the master-slave synchronization

of the Burgers’ map master-slave system is

considered and the CPM based SS communication

system proposed.

The Burgers’ map (Whitehead and MacDonald,

1984) is given by equation 1:

nnnn

nnn

YXbYY

YaXX

+=

−=

+

+

1

2

1

(1)

With the parameters a = 0.75 and b = 1.75 the

system is chaotic.

The design procedure of the synchronizing

nonlinear control laws of the Burgers’ map CPM

based SS chaotic communication system of Figure 2

is now explained. Let the error be defined by

equation 2:

n

n

n

XXe −=

∧

1

(2a)

n

n

n

YYe −=

∧

2

(2b)

In order to demonstrate the design of the controller

of Figure 2 assume no noise in the system. It follows

then that:

nnr

YY

=

. The difference error, (the error

system), can then be represented by equation 3:

n

nn

nn

n

n

n

n

n

n

n

n

n

n

n

n

n

uYXYXbYYb

YYe

uYYaXXa

XXe

2

1

1

1

2

1

2

2

1

1

1

1

+−+−=

−=

++−−=

−=

∧∧∧

+

+

∧

+

∧∧

+

+

∧

+

(3)

Equation 3 can also be represented by equation 5,

keeping in mind the identities of equation 4:

n

n

n

nnn

nn

n

n

n

n

n

n

eXeYYXYX

eYeYYY

21

22

2

2

∧∧∧

∧∧

+=−

−−=+−

(4)

n

n

nn

n

n

n

n

n

nnn

uXbeeYe

uYYeaee

221

1

2

121

1

1

)(

)(

+++=

+−−+=

∧

+

∧

+

(5)

With theorem 1 in mind matrix equation 6 is

formed:

nnnnn

eUeAe +

=

+1

(6)

IMPROVING SECURITY IN CHAOTIC SPREAD SPECTRUM COMMUNICATION SYSTEMS WITH A NOVEL

‘BIT POWER PARAMETER SPECTRUM’ MEASURE

275

Transmitter

(Master system)

nnnn

n

nn

YXbYY

Y

XmaX

+=

−

+

=

+

+

1

2

1

)19.0(

0

Y

0

0

YY ≠

∧

Receiver

(Slave system)

n

nnnn

n

nnn

uYXYbY

uYXaX

2

1

1

2

1

++=

+−=

∧∧∧

+

∧

∧∧

+

∧

n

m

n

Y

nr

Y

n

Y

∧

∫

dt

2

)(

∧

m

0

X

0

0

XX ≠

∧

nyn

ee =

2

Controller:

n

n

n

n

nr

n

n

eXbu

eYYu

22

21

)(

)(

∧

∧

+−=

+=

n

u

2

n

u

1

Figure 2: The Burgers’ map SS chaotic communication system based on the parameter modulation concept.

where:

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

=

n

n

n

nn

nn

n

nn

nn

n

e

e

e

uu

uu

U

aa

aa

A

2

1

2221

1211

2221

1211

,,

.

Therefore:

n

nn

nn

n

nn

nn

n

e

uu

uu

e

aa

aa

e

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎦

⎤

⎢

⎣

⎡

=

+

2221

1211

2221

1211

1

(7)

Modifying equation 5 to fit the matrix form of

equation 7, equation 8 is obtained:

n

n

n

nn

n

n

n

n

n

n

e

uu

uu

e

XbY

YYa

e

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

−−

=

∧

∧

+

iviii

iii

1

(8)

where:

nnnnn

eueuu

2ii1i1

+=

,

n

n

nnn

eueuu

2iv1iii2

+=

.

Therefore:

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+++

+−−+

=

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

−−

=+=

∧

∧

∧

∧

n

n

n

n

n

n

n

n

n

n

nn

n

n

n

n

nn

uXbuY

uYYua

uu

uu

XbY

YYa

UAB

iviii

iii

iviii

iii

(9)

Following theorem 1 the control laws can be

chosen in the following manner:

)(,0,,0

iviiiiii

n

n

n

n

n

nn

XbuuYYuu

∧∧

+−==+==

(10)

With the control laws of equation 10, the matrix

B of equation 9 takes the form of equation 11:

⎥

⎦

⎤

⎢

⎣

⎡

=

0

0

n

n

Y

a

B

(11)

It is then readily verifiable that the eigenvalues

of matrix

n

B of equation 11 are equal to 0 and a.

Furthermore, the theorem 1 above requires matrix

B

to be constant. As the matrix

B is a function of n, it

must also be ensured that

nn

BB −

+1

remains

bounded to guarantee global asymptotic stability

which is the requirement for synchronization. The

fact that

nn

BB −

+1

remains bounded is

demonstrated by equation 12:

0

0

0

0

0

0

0

0

0

1

)1(

)1(

→

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

=

+

−+

+

−++

n

i

i

n

in

in

nninin

Ya

a

Y

a

Y

a

Y

a

eBBB KK

∀

|a| < 1, and as ∞→i . (12)

Therefore, as stated in equation 12, in order for

the master and slave systems of Figure 2 to

synchronize, the parameter

a must be kept within the

unit circle in z domain.

The control laws

n

u

1

and

n

u

2

are therefore

given by equations 13 and 14, and incorporated into

Figure 2.

SECRYPT 2007 - International Conference on Security and Cryptography

276

n

n

n

nnnnn

eYYeueuu

22ii1i1

)( +=+=

∧

(13)

n

n

n

n

nnn

eXbeueuu

22iv1iii2

)(

∧

+−=+=

(14)

The important feature of the master-slave system of

Figure 2 is that it only requires the master signal

n

Y

to synchronize the master and slave systems. This

fact is of particular importance for communications

as only one signal needs to be transmitted thus

reducing the required bandwidth (Jovic et al.,

2006a,b).

In Figure 2, the master system parameter set of

015.0=a

and

75.1=b

has been chosen to represent

a bit 0. The master system parameter set of

205.0

=

a

and

75.1=b

has been chosen to represent a bit 1.

The reasoning behind such choice of parameters is

clarified in the next section. Note that the message

m

of Figure 2 takes on the values of 0 and 1 depending

on the polarity of a bit transmitted. The slave system

parameters are set for all time at

015.0=a

and

75.1=b

, so that synchronization at the receiver side

signals a bit 0 and de-synchronization signals a bit 1.

Both parameter sets,

015.0

=

a

and

75.1=b

, and

205.0=a

and

75.1=b

generate chaotic behaviour

within the system (Whitehead and MacDonald,

1984).

The transmitted signal

n

Y is shown in Figure 3

when the series of 10 bits is transmitted, that is,

when

m = [0 0 1 0 1 1 0 1 0 1]. Figure 3 also shows

the corresponding squared synchronization error,

2

ny

e

, under noiseless conditions.

0 500 1000 1500 2000 2500 3000 3500 4000

-1.5

-1

-0.5

0

0.5

1

1.5

Yn

Transmitted signal

n

0 500 1000 1500 2000 2500 3000 3500 4000

0

0.01

0.02

0.03

0.04

0.05

0.06

ey

2

n

n

Error squared

Transient

Bit 1

Bit 0

10

Figure 3: The transmitted signal

n

Y

and the squared

synchronization error

2

ny

e

.

The received bits are detected by squaring and

integrating the error

ny

e

. The output of the integrator

is then compared to the predetermined threshold and

the decision is made whether a bit 0 or a bit 1 was

sent. Note that the spreading factor of 400 has been

used to represent one bit. By definition, the

spreading factor denotes the number of discrete

points (chips) contained within one information bit.

It is the ratio of a bit period to a chip period (Jovic

and Unsworth, 2007b). A transient period of 10

chips has been allowed for the case of Figure 3.

During the transient period there is no data

transmission taking place.

4 BIT POWER SECURITY

ISSUES OF THE SS CHAOTIC

COMMUNICATION SYSTEMS

In this section, highly secure and insecure regions of

the Burgers’ map CPM based SS communication

system are identified using a new measure called the

‘Bit Power Parameter Spectrum’ (BPPS). The

analysis is also performed on a standard Lorenz

CPM based SS system. It is shown that an

eavesdropper can readily decode the message,

without any assumptions about the system, if the

system is operated outside the secure regions of the

BPPS.

4.1 Security Evaluation of the Burgers’

Map CPM based SS Chaotic

Communication System

For any secure chaotic communication system it is

imperative that the power of the chaotic carriers

representing bits 0 and 1 be approximately equal to

avoid the possibility of decoding information by a

third party simply based on the average powers of

the chaotic carriers (Álvarez et al., 2004c). In order

to perform the security analysis on the Burger’s map

communication system of Figure 2, and thus explain

the choice of the modulating parameters, the average

power of the chaotic carriers representing bits 0 and

1 is now analysed. To do so the average power of a

number of bits (1024) is first calculated and the

mean of those powers and the corresponding

standard deviation found. A number of points are

then obtained for a number of different sets of

chaotic parameters and the average power graph,

with the error bars, versus the varied parameter,

plotted. A pseudo random binary sequence (PRBS)

generator has been used to model the transmitted

bits. The plots have been produced with the concept

of security in mind. If the average power of the

IMPROVING SECURITY IN CHAOTIC SPREAD SPECTRUM COMMUNICATION SYSTEMS WITH A NOVEL

‘BIT POWER PARAMETER SPECTRUM’ MEASURE

277

chaotic carriers of bits 0 and 1 are different during

the same transmission, with the confidence intervals

which do not overlap, then the security of the system

based on those carriers is jeopardized.

Figure 4 shows the BPPS of the chaotic carriers

representing the bits transmitted. For the bits 1 the

parameter

b is always kept constant at 1.75 with the

parameter a varied in steps of 0.01 from a = 0 to a =

0.75. For the case of Figure 4 bits 0 are represented

by the parameter values:

a = 0.6 and b = 1.75 at all

times. Bits 1 must then be represented by some other

parameter values in order to achieve successful

communication. From Figure 4 it can be observed

that the average power of the chaotic carriers is

approximately the same, (and the deviation of this

power), when the parameter

a is kept in the region: 0

< a < 0.22, whereas it differs drastically outside of

this region. Therefore, choosing the parameter sets

for bits 0 and 1 anywhere outside this region would

jeopardize the security of the system. Thus,

choosing the parameter values:

a = 0.6 and b = 1.75

to represent bits 0 is not suitable for the security

reasons. In order to remedy this let the parameter

values representing bits 0 be:

a = 0.015 and b = 1.75.

In this case Figure 5 is obtained. From Figure 5 it is

observed that the carrier powers of the bits 0 and 1

have approximately equal values thus offering

increased security over the choice of parameters of

Figure 4.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Bit powe

r

Parameter a representing bits 1

Bits 0

Bits 1

Figure 4: The BPPS within the Burgers’ CPM based SS

chaotic communication system when the bits 0 are

represented by the parameter set: a = 0.6, b = 1.75.

Based on the findings of Figure 5, it is now

shown that choosing the parameter set: a = 0.015

and b = 1.75, to represent bits 0, and the parameter

set,

a = 0.205 and b = 1.75, to represent bits 1,

produces the best performance in terms of the bit

error rate (BER). In Figure 6, the BER vs. the bit

energy to noise power spectral density ratio (E

b/No)

curves have been plotted. Figure 6 demonstrates the

progressive improvement represented by the BER

curves with the parameter

a varied in the secure

region of Figure 5 from

a = 0.0625 up to a = 0.205

in steps of 0.0475. The parameter b has been set to

1.75 for both bits 0 and 1. The parameter a

representing bit 0 has been set to:

a = 0.015. Note

that the best BER performance is achieved by

choosing the parameter sets, representing bits 0 and

1, to be as far apart as possible from each other

within the secure region of Figure 5. Also note

further improvement in the BER curve, marked by

the open circles, as one exits the secure region.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Bit power

Parameter a representing bits 1

Bits 0

Bits 1

Secure region

Figure 5: The BPPS within the Burgers’ CPM based SS

chaotic communication system when the bits 0 are

represented by the parameter set: a = 0.015, b = 1.75.

30 35 40 45 50 55 60 65

10

-4

10

-3

10

-2

10

-1

10

0

BER

Eb/No (dB)

Bits 1 par. a = 0.0625

Bits 1 par. a = 0.11

Bits 1 par. a = 0.1575

Bits 1 par. a = 0.205

Bits 1 par. a = 0.26

Figure 6: The secure region BER curves of the SS chaotic

communication system based on the parameter modulation

of the Burgers’ chaotic map with the progressively

increasing bits 1 parameter a.

Figure 7 illustrates the effect on security caused by

choosing inappropriate parameter sets which

produce chaotic carriers of different power. In case

of Figure 7 bits 0 have been represented by the

parameter set of

a = 0.6 and b = 1.75, while bits 1

have been represented by the parameter set of

a =

0.205 and b = 1.75. The average power of the

SECRYPT 2007 - International Conference on Security and Cryptography

278

transmitted signal of Figure 7 has been evaluated

using the sliding window of 400 chips in length (the

spreading factor of a single bit). The sliding

window is then shifted one chip in time and the

average power evaluated again. This process is

repeated until the end of the transmitted signal. It

can be observed from Figure 7 that the average

power of the chaotic carriers of the transmitted bits

oscillates periodically with the change of the binary

message. In contrast to Figure 7, Figure 8 illustrates

the effect on security caused by choosing the

appropriate parameter sets which produce chaotic

carriers of approximately equal power. In case of

Figure 8 bits 0 have been represented by the

parameter set of

a = 0.015 and b = 1.75, while bits 1

have been represented by the parameter set of a =

0.205 and b = 1.75.

0 500 1000 1500 2000 2500 3000 3500 4000

-0.5

0

0.5

1

1.5

Bits

Binary message

0 500 1000 1500 2000 2500 3000 3500 4000

-2

0

2

Yn

Transmitted signal

0 500 1000 1500 2000 2500 3000 3500 4000

0.2

0.3

0.4

0.5

Power

Time

Average Power

Figure 7: The binary message, the transmitted signal

n

Y

and the average power of the transmitted signal. Bits 0

parameter set: a = 0.6 and b = 1.75. Bits 1 parameter set:

a = 0.205 and b = 1.75.

0 500 1000 1500 2000 2500 3000 3500 4000

-0.5

0

0.5

1

1.5

Bits

Binary message

0 500 1000 1500 2000 2500 3000 3500 4000

-2

0

2

Yn

Transmitted signal

0 500 1000 1500 2000 2500 3000 3500 4000

0.2

0.3

0.4

0.5

Power

Time

Average Power

Figure 8: The binary message, the transmitted signal

n

Y

and the average power of the transmitted signal. Bits 0

parameter set: a = 0.015 and b = 1.75. Bits 1 parameter

set: a = 0.205 and b = 1.75.

4.2 Security Evaluation of the Lorenz

CPM based SS Chaotic

Communication System

In (Cuomo and Oppenheim, 1993), the Lorenz CPM

based SS chaotic communication system has been

presented. In this scheme the binary message is used

to alter the parameter

b of the master (transmitter)

between 4 and 4.4 depending on whether a bit 0 or

bit 1 is to be transmitted. However, at the slave

(receiver) side the parameter

b is fixed at 4 for all

time. Thus, the synchronization either occurs or does

not, depending on the state of the parameter

b at the

transmitter (master) side. The other Lorenz

parameters, namely

σ and r, are fixed at 16 and 45.6,

respectively. A BPPS as that of Figures 4 and 5 is

plotted in Figure 9 but for the Lorenz CPM based

chaotic communication system of (Cuomo and

Oppenheim, 1993). In this case the parameter

b of

the bits 1 is varied from 0.1 to 10 in steps of 0.1 with

the other parameters being fixed at the constant

values specified above. From Figure 9 one can see

that there are no secure regions where one can

operate the system as the power of the bits 1

increases, almost linearly, with the parameter

b.

Therefore, to minimise the impact on the security,

the parameters

b representing bits 0 and 1, must be

kept as close to each other as possible.

0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bit power

Parameter b representing bits 1

Bits 0 power

Bits 1 power

Figure 9: The BPPS within the Lorenz CPM based SS

chaotic communication system. The close up is shown in

the upper left hand corner.

5 CONCLUSIONS

In this paper, a method of synchronizing chaotic

maps and its implementation within a CPM based

SS chaotic communication system has been

proposed. The security of the proposed, as well as of

the existing SS chaotic communication systems, has

3 3.5 4 4.5 5

1.2

1.4

1.6

1.8

IMPROVING SECURITY IN CHAOTIC SPREAD SPECTRUM COMMUNICATION SYSTEMS WITH A NOVEL

‘BIT POWER PARAMETER SPECTRUM’ MEASURE

279

then been evaluated in terms of the average power of

the chaotic carriers of the bits transmitted. In order

to do so, a novel analysis technique, termed the ‘Bit

Power Parameter Spectrum’ (BPPS), has been

proposed. Without any assumptions about the

system architecture or its characteristics, the BPPS

has been used to show that the CPM based SS

systems are not as secure as often thought.

The design of the nonlinear control laws for the

synchronization of the chaotic map master-slave

systems has been proposed and demonstrated on the

two dimensional Burgers’ map master-slave system.

Following this, the method of implementing the

synchronized master-slave system within a CPM

based secure SS communication system has been

demonstrated on the two dimensional Burgers’ map.

The nonlinear control laws were designed in such a

way to force the synchronization among the master

and slave systems using only one signal of the

master system. This is of particular importance for

communications as only one signal needs to be

transmitted thus reducing the required bandwidth.

Finally, the Lorenz CPM based SS chaotic

communication system has been presented. The

security of the proposed and the existing CPM SS

chaotic communication systems has been evaluated

in terms of the average power of the chaotic carriers

of the bits transmitted using the newly proposed

technique of BPPS. It has then been shown that due

to the largest BPPS overlap region, the Burgers’ map

CPM based SS chaotic communication system can

be optimized and is thus more secure than the

Lorenz CPM based SS system. As the BPPS relies

on the evaluation of average power, the security

optimization is thus achieved by assuming that an

eavesdropper has no knowledge of the system

architecture or its dynamics. Furthermore, it has

been shown that the BER performance of the

Burgers’ map CPM based SS chaotic

communication system can also be optimized. The

optimization is achieved by choosing the parameter

sets, representing bits 0 and 1, to be as far apart as

possible within the secure operating region of the

BPPS.

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