EXPERIMENTAL VALIDATION OF PSEUDO TRUE RANDOM

NUMBER GENERATION AND SYNCHRONIZATION USING

NESTED LINEAR CHAOTIC MAPS BASED ON TMS320C6416

Q. Nasir, A. M. Abid and A. S. Elwakil

Electrical & Computer Engineering Department, University of Sharjah, Sharjah, U.A.E.

Keywords:

Pseudo-true random bit generators, Chaotic maps, Chaos synchronization, NIST statistical test suite.

Abstract:

A Pseudo True Random Binary Generator (PTRBG) based on a Nested Linear Chaotic Maps (NLCM) is

proposed. Implementing the synchronization of chaotic systems presents a challenge. The paper proposes an

implementation of a generation and synchronization method of PTRBG using NLCM and backward iteration

synchronization approach. A prototype has been developed through Texas Instruments TMS320C6416 DSP

development kit. Randomness tests of the generated bits of the PTRB is performed using the NIST statistical

test suite.

1 INTRODUCTION

Random and pseudo-random numbers are used in

many areas including test data generation, Monte-

Carlo simulation techniques, generation of spread-

ing sequences for spread spectrum communications,

and cryptography(Ahmed and Siyal, 2006). Pseudo-

random spreading sequences used in spread spec-

trum communications must be repeatable, while for

most simulations using random numbers repeatabil-

ity is not necessary. In cryptographic and secu-

rity applications depends on the randomness of the

source and the unpredictability of the used random

bits(Tang and Tang, 2005). Various encryption tech-

niques for secure transmission have been studied. The

approaches include time domain scrambling tech-

niques (Tang and Tang, 2005), and permutation and

depermutation of Fast Fourier Transform (FFT) co-

efﬁcients (Ahmed and Siyal, 2005). In recent years

many researchers have noticed a close relationship

between chaos and cryptography. Chaos appeared

to be another paradigm to protect data and seems to

be promising in the areas of security and cryptog-

raphy. Chaos based encryption techniques such as

in (Tang and Tang, 2005), (Drutarovsk and Galajda,

2006) are considered practical because they provide a

good combination of speed, high security, complex-

ity, reasonable computational overheads.

Chaotic circuits represent an efﬁcient alternative

to classical TRBG (Drutarovsk and Galajda, 2006).

Studies in nonlinear dynamics show that many of the

seemingly complex systems in nature are described

by relatively mathematical equations(Sprott, 2003).

Although chaotic systems appear to be highly irreg-

ular, they are also deterministic in the sense that it

is possible to reproduce them with certainty. These

promising features of chaotic systems attracted many

researchers to try chaos as a possible medium for

secure communication.The nonlinear phenomenon

of chaos poses a promising alternative for pseudo-

random number generation due to its unpredictable

behaviour.

The chaotic system generates unpredictable

pseudo random orbits which can be used to generate

TRNGs (True Random Number Generators). Many

different chaotic systems have been used to generate

TRNGs such as Logistic map (Sajeeth et al., 2001),

and its generalized version (Matthews, 1989), Cheby-

shev map, (Ahmed and Siyal, 2006) piecewise linear

chaotic maps (NLCM) (Masuda and Aihara, 1999)

and piecewise nonlinear chaotic maps (Tao et al.,

1999). Chaotic systems are characterized by a sensi-

tivity dependence on initial conditions, and with such

initial uncertainties, the system behaviour leads to

large uncertainty after some time.

A TRBG produce long sequences made of per-

fectly independent bits and when restarted, it never

reproduces a previously delivered sequence. To as-

sess the statistical properties and investigate the ran-

domness of the TRBGs, several test suites are avail-

able such as AIS 21, AIS 31, FIPS 140 and the

NIST statistical suite (NIST, 2001). TRBGs are usu-

145

Nasir Q., M. Abid A. and S. Elwakil A..

EXPERIMENTAL VALIDATION OF PSEUDO TRUE RANDOM NUMBER GENERATION AND SYNCHRONIZATION USING NESTED LINEAR

CHAOTIC MAPS BASED ON TMS320C6416.

DOI: 10.5220/0003310001450150

In Proceedings of the 1st International Conference on Pervasive and Embedded Computing and Communication Systems (PECCS-2011), pages

145-150

ISBN: 978-989-8425-48-5

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

ally difﬁcult to implement; chaos-based random num-

ber generators implemented in the software provide

a good method to produce pseudo-true random num-

bers. Chaotic systems or discrete chaotic maps are

random like and deterministic but unpredictable in the

long term. Consequently, the evaluation of chaotic

systems in the ﬁeld of cryptography has been exten-

sively studied for several decades. This is due to

the highly unpredictable and random-look nature of

chaotic signals. Chaotic signals are aperiodic and ex-

hibit sensitive dependence on initial values. Since

they are governed by one or more control parameters,

a small perturbation in these parameters can cause a

large change in the state of the system. These fea-

tures make chaos certainly suitable for applications

that need a high level of security.

Chaotic streams can be generated using a num-

ber of chaotic maps such as Logistic map, 2-D Henon

map, Chebyshev map, piecewise linear chaotic maps

(PWLCM), piecewise nonlinear chaotic map, etc (Li,

2003). With two chaotic systems communicating over

a noisy channel, the problem of synchronization be-

comes an essential part to study. In a communication

system that uses chaos for encryption; synchroniza-

tion is the ability of the receiver to recover the origi-

nal message transmitted. This can be achieved only if

the transmitter and receiver have the same copy of the

chaotic binary sequence. For this reason several syn-

chronization techniques exist in literature (Millerioux

and Mira, 2001)- (Lau, 2006).

In spite of the extreme sensitivity to initial con-

ditions of chaotic systems, synchronization can still

be achieved. In (Millerioux and Mira, 2001), the ob-

server synchronization method implemented depends

on continuously feeding the chaotic system at the re-

ceiver with the error (difference) between the origi-

nal chaotic sequence at the transmitter and the esti-

mated sequence at the receiver . The impulsive syn-

chronization technique in (Yong-Ai and Yi-Bei, 2002)

involves the use of small control impulses where the

chaotic maps must be asymptotically stable. Ac-

cording to (De Angeli et al., 1995), using the dead

beat synchronization method, the synchronization er-

ror reach zero in exactly two steps. Another syn-

chronization method was proposed in (Min et al.,

2006), where synchronization is achieved by mixing

discrete chaotic signals and using the output to drive

the chaotic systems at the transmitter and the receiver.

Synchronization using backward iterations and anal-

ysis of chaotic systems using symbolic dynamics was

introduced in (Cong et al., 1999). In this method a

number of backward iterations from a random initial

condition are sufﬁcient to reproduce an exact copy of

the chaotic signal produced at the transmitter. This

type of PTBGs can be easily included as part of Soft-

ware Deﬁned Radios (SDRs).

In this paper an implementation of software gen-

eration and synchronization of the chaotic binary se-

quences generated by NLCM is achieved by using the

backward iteration synchronization approach (Cong

et al., 1999), (Stojanovski and Kocarev, 1997). An

implementation test bed for a complete communica-

tion system based on a TMS320C6416 DSP develop-

ment kit is then described. Randomness of the pseudo

true random bits (PTRB) generated is assessed using

the NIST statistical test suite.

The rest of the paper is organized as follows. An

overview of the main blocks in the chaotic communi-

cation system proposed is given in Section II. Sec-

tion III describes the TRBGs studied. The chaotic

synchronization method is explained in Section IV.

The results of subjecting the bit sequences to NIST

randomness test suite is provided in Section V. Sec-

tion VI describes the TMS320C6416 DSP implemen-

tation. Finally the conclusions are drawn in Section

VII.

2 SYSTEM DESCRIPTION

The block diagram of the proposed chaotically en-

crypted communication system is shown in Figure

1. The analog speech waveform is sampled and

quantized using the PCM waveform coding process.

The binary speech (message) signal is then masked

(XORed) by the PTRBs generated using Nested Piece

Wise Linear Map (NPWLM).

Figure 1: Block diagram of the proposed chaotic voice com-

munication system.

As illustrated in Figure 2, the ﬁrst transmitted

(training) packet contains the synchronization bits

produced by the PTRBG in the transmitter. The rest

of the transmitted packets contain the encrypted data

PECCS 2011 - International Conference on Pervasive and Embedded Computing and Communication Systems

146

Figure 2: Illustration of the synchronization bits and en-

crypted binary stream.

(the message signal XORed with the PTRB.

At the receiver, once the demodulated signal is re-

ceived, demultixplexing is performed to extract the

ﬁrst packet and get the synchronization bits. The

backward iteration synchronization technique is ap-

plied using the training bits which was transmitted

without encryption to get an the initial condition of

the chaos generator used in the transmitter. Starting

the chaos generator at the receiver with the estimated

initial condition, an estimated copy of the binary se-

quence generated at the transmitter can be produced.

At this stage XORing the encrypted signal with the

receiver generated chaotic binary sequence to retrieve

the original message signal.

3 PSEUDO-TRUE RANDOM BIT

GENERATION

A TRNG should be able to produce inﬁnitely long se-

quences of independent equiprobable bits and when

restarted it should never reproduce a previously de-

livered sequence (nonrepeatable) as it uses differ-

ent initial conditions. The processes are also non-

deterministic (a given sequence of numbers cannot be

reproduced). Chaotic generators can be used to pro-

duce Pseudo TRBG (PTRBG) and therefore provide

an efﬁcient alternative to TRBGs. Generally speaking

ideal or TRBGs are extremely difﬁcult to implement

by software. However because of the irregular be-

havior of chaotic systems, chaotic generators can be

used to produce PTRBG and therefore provide an efﬁ-

cient alternative to TRBGs. Several PTRBG based on

chaotic systems exist in literature, Study in this paper

is focused on the nested PWLM (NPWLM).

Drutarovsky and Galajda (Drutarovsk and Gala-

jda, 2006) proposed a modiﬁed form of the one di-

mensional PWLM called Nested . The map is ex-

pressed as:

x (n + 1) =







2x(n) − 2 x > 1/2

2x(n) −1/2 < x < 1/2

2x(n) + 2 x < −1/2

(1)

Equation 2 can be split and rewritten as (Dru-

tarovsk and Galajda, 2006):

(n) =



x(n) − A

x(n) + A

x(n) > 0

x(n) < 0

(2)

x(n + 1) =



B(x

(n) − A)

B(x

(n) + A)

(n) > 0

(n) < 0

(3)

where parameters A and B are 1.3 and 2 respec-

tively. The domain of x(n) is between −1, and +1.

The bifurcation diagram (Abid et al., 2009), (Abid,

2009), (Abid et al., 2010) of the NPWLM shows that

when the control parameter B is 2, chaos is still gen-

erated. This splitting can be thought of as a form of

cascading or nesting one map into another, therefore

equations (3) and (4) are called NPWLM.The PTR-

BGs are generated using a threshold function called

the generating partition described in the following

section.

4 CHAOTIC SYNCHRONIZATION

In simple terms; synchronization for discrete time

systems can be thought of as the ability of the re-

ceiver to recover an identical copy of the chaotic se-

quence generated at the transmitter. The concept of

using symbolic dynamics to process chaotic signals

was studied in (Min et al., 2006), (Cong et al., 1999),

where the inﬁnite number of ﬁnite-length chaotic sig-

nals can be partitioned into a ﬁnite number of signals

sets. Suppose X is the set of all possible signals for

x(n) that the chaotic map can generate. Using the def-

initions of symbolic dynamics the set X can be parti-

tioned into M disjoint partitions E

of the phase space

S such that

i=1

= S and E

= Φ for i 6= j .

In order to obtain enough information, pairs of bits

are needed to be transmitted to the receiver instead of

single bits. The ﬁrst bit is taken from the ﬁrst map and

the second bit from the second map. Therefore two

generating functions b

(n) and b

(n) are needed for

the nested PWLM to generate the symbolic sequence

given by:

(n) =



0 x

(n) < 0

1 x

(n) ≥ 0

(n) =



0 x(n) < 0

1 x(n) ≥ 0

(4)

The two bits from b

(n) and b

(n) are used to

decide on the appropriate inverse chaotic map to be

used, therefore a total of 38 bits are needed to retrieve

the original initial condition.

Suppose that for an initial condition x(0) and

a length of N bits the symbolic sequence B =

(0)b

(0),b

(1)b

(1),...,b

(N − 1) b

(N − 1)]

is generated and transmitted, where N is the number

of bits required to reach an acceptable estimate of the

EXPERIMENTAL VALIDATION OF PSEUDO TRUE RANDOM NUMBER GENERATION AND

SYNCHRONIZATION USING NESTED LINEAR CHAOTIC MAPS BASED ON TMS320C6416

147

initial condition. The receiver use the following

equation to retrieve the initial condition.

x (0|N) = f

−1

(0)

−1

(1)

◦ f

−1

(1)

−1

(1)

◦···◦ f

−1

(N−1)

−1

(N−1)

(η)

(5)

where f

−1

b(0)

is deﬁned as the inverse mapping of f

at bit 0, η is a point chosen randomly from the domain

of f

−1

b(N−1)

(η).

When NPWLM is implemented , the following

function b(n) is used (what is called a generating

partition) to generate the symbolic sequence.

b(n) =



0 x(n) < 0

1 x(n) ≥ 0

partition E

(6)

Certainly for larger N, the estimate is more accurate.

Practically, the error between the original chaotic se-

quence at the transmitter and the estimated sequence

at the receiver must be reduced. Suppose the follow-

ing system is available at the transmitter x(n + 1) =

f [x(n)] where x(n) is the chaotic value, and at the

receiver the distorted value y(n = x(n) + v(n) is re-

ceived, where v(n) is the channel noise. We tar-

get by synchronization to obtain a close estimate

of x

from the available information y(n) such that

lim

n→∞

x(n) − x(n)

= 0 (Millerioux and Mira, 2001).

In this paper, experimental trials show that a min-

imum of 38 bits are needed to be sent to the receiver

to achieve synchronization. The N = 38 bits are suf-

ﬁcient to reproduce the same chaotic sequence at the

receiver. The number of bits required to synchronize

the receiver is less than that when using chipcon plat-

form which was N = 50. Fig. 2 illustrates the concept

of the synchronization technique where the transmit-

ter sends the ﬁrst 38 bits unscrambled to the receiver

to be used for synchronization and then the encrypted

binary signal. Starting from the 38

received bit and

depending whether it is a zero or one; the receiver

starts to iterate the appropriate inverse chaotic map

starting from a random initial point η. Theoretical

analysis for estimating the number of bits required for

synchronization will be carried out in future work.

5 RANDOMNESS TESTS

Statistical testing is employed to provides a mecha-

nism for making quantitative decisions that a genera-

tor produces numbers that appear to be true random.

The intent is to determine whether there is enough ev-

idence to ”reject” a conjecture or hypothesis about

the true randomness of the generated bits. Any ran-

dom bit generator proposed for use in a cryptographic

protocol must be subject to statistical tests. The sets

of tests available are: a Federal Information Process-

ing Standard (FIPS 140-1) statistical test which was

lately replaced with FIPS 140-2, the German Appli-

cation Notes and Interpretation of the Scheme (AIS

31) and the National Institute of Standards and Tech-

nology (NIST) test suite(NIST, 2001).

Many chaotic maps introduce biases in the binary

sequence. In many cases, before testing the sym-

bolic sequences generated by the chaotic maps, post-

processing of the produced sequences has to be per-

formed in order to reduce any biases in the produced

distribution. Therefore, the well-known Von Neu-

mann’s (VN) deskewing technique can be employed.

The technique consists of converting the bit pair 01

into output 0, 10 into output 1 and of discarding bit

pairs 00 and 11.

Randomness tests are used to analyze the distribu-

tion pattern of the generated data. There is an array

of statistical tests available to test the randomness of

random and pseudorandom number generators. Even

though these statistical tests do not provide deﬁnite

results, it is possible to interpret these results with

care and caution to determine the randomness of a

generator. The general rule of thumb is more tests

the better. The generator bit stream was subjected to

a plenty of statistical tests for randomness used by

The National Institute of Standard and Technology

(NIST; an agency of the U.S. Commerce Departments

Technology Administration (NIST, 2001). It is how-

ever important to note that the test suite is suitable for

identifying deviations of binary sequences from ran-

domness. However factors contributing to these de-

viations are numerous and it is possible to expect a

certain number of failures from a particular generator.

The binary sequences generated by the PWLM and

NPWLM map were passed to NIST statistical suite to

test their randomness. Each NIST statistical test as-

sesses a binary sequence to establish whether there is

signiﬁcant evidence to suggest that the null hypothe-

sis (H

) should be rejected in favor of the alternative

hypothesis. Here the null hypothesis H

is that the

sequence being tested is random, while the alterna-

tive hypothesis H

, is that the sequence being tested

is not random. Thus for each applied test a decision

is made to accept or reject the null hypothesis based

on statistical evidence. Each test statistic obtained for

each individual test is used to calculate a P-value that

indicates the strength of the evidence against the null

hypothesis. Thus for each test, the P-value is the prob-

ability that a perfect random number generator would

have produced a sequence that is less random than

the tested sequence, given the particular nonrandom-

ness being gauged by that particular test. In this work,

PECCS 2011 - International Conference on Pervasive and Embedded Computing and Communication Systems

148

the binary sequences generated by the PWLM and the

NPWLM were passed to NIST statistical suite to test

their randomness. They were subjected to 12 of the

16 tests of the suite. Each map was used to gener-

ate 1000 sequences each having a length of 1000000

bits. The Random Excursions, Random Excursions

Variant and Non-overlapping templates tests are not

applicable for these bit streams.

The DSP kit used for implementing the system

deﬁnitely with ﬁnite precision; therefore the chaotic

map implemented on a ﬁnite precision machine can

be called a ’pseudo’ chaotic map. Furthermore to de-

termine the effect of post processing, the bit streams

produced by the maps were ﬁrst tested without per-

forming any form of post processing, then the Von

Neumann’s deskewing technique and the XOR post

processing technique were independently applied. A

signiﬁcance level of 0.01 was chosen for the tests. For

each of the tests a P-value is calculated which is the

probability that a perfect RNG would have produced

a sequence less random than the sequence that was

tested, given the kind of non-randomness assessed by

the test. If a P-value for a test is 1, then the se-

quence appears to be perfectly random (Li, 2003).

Table 1 summarizes the results for single precision

PWLM and NPWLM. It is evident from Table 1 that

the NPWLM is a good choice to be used as a PTRNG.

The binary sequences generated by the NPWLM with

XOR applied as a post processing technique, pass the

all the tests except the FFT test.

Table 1: Summary of NIST results for TMS320C6416

based PWLM and NPWLM.

PWLCM NPWLM

Type of Post Processing None XOR VN None XOR VN

Frequency Pass Fail Pass Pass Pass Pass

Block Pass Fail Pass Fail Pass Pass

Cumulative Sums Pass Fail Pass Pass Pass Pass

Runs Fail Fail Fail Pass Pass Pass

Longest Runs Pass Fail Fail Pass Pass Fail

Rank Fail Fail Fail Pass Pass Pass

FFT Fail Fail Fail Fail Fail Fail

Overlapping Fail Fail Fail Pass Pass Fail

Universal Fail Fail Fail Fail Pass Pass

Apen Fail Fail Fail Pass Pass Fail

Serial Fail Fail Fail Pass Pass Fail

Linear Complexity Pass Pass Fail Pass Pass Pass

Figure 3: Implementation setup of the TMS320C6416 de-

velopment kit.

6 SYSTEM IMPLEMENTATION

The proposed system is implemented on ﬁxed-point

TMS320C6416 DSP manufactured by Texas Instru-

ments Corporation (TI) (TexasInstruments, 2011).

The generation and synchronization code is writ-

ten in C language and compiled using Code Com-

poser Studio Workbench Software. The program

starts by generating the ﬁrst 38 chaos bits using the

NPWLM and sending them without encryption. This

is followed by further chaotic generation and XORing

of the symbolic sequence with the digital data sam-

ples. Then the required data transmission of the en-

crypted packets is carried out. The receiver part of

the program receives the encrypted data and performs

synchronization bits by estimating the initial value us-

ing the backward iteration synchronization technique

to regenerate an exact copy of the chaotic symbolic

sequence generated at the transmitter. At this point

the encrypted data in the buffer can be decrypted to

retrieve the original message and play it with the kit

DAC.

Real-Time Data Exchange (RTDX) is used

to provide real time, continuous visibility into

the way TRBG software application operate in

TMS320C6416. RTDX allows transfer the random

bits generated in the DSP to a host PC for testing.

On the host platform, an RTDX host library oper-

ates in conjunction with Code Composer Studio. In

RTDX an output channel should be conﬁgured within

NPWLM code which resides on the DSP kit. The

generated data from NPWLM is written to the out-

put channel. This data is immediately recorded into

a C6416 DSP buffer deﬁned in the RTDX C6416 li-

brary. The data from this buffer is then sent to the

host PC through the JTAG interface. The RTDX host

library receives this data from the JTAG interface and

records it into either a memory buffer for testing pur-

poses.

EXPERIMENTAL VALIDATION OF PSEUDO TRUE RANDOM NUMBER GENERATION AND

SYNCHRONIZATION USING NESTED LINEAR CHAOTIC MAPS BASED ON TMS320C6416

149

7 CONCLUSIONS

Application of the backward iteration synchroniza-

tion method for linear chaotic nested maps was in-

troduced and implemented. Pseudo-true random bits

generated the PWLM, and NPWLM using double and

single precisions test bed (DSP chip) were tested. The

test results have shown that the bits generated by the

NPWLM with XOR applied as a post processing tech-

nique and using a single precision representation of

numbers, pass the maximum number (11 out of 12) of

tests for 1000 binary sequences each having a length

of 1000000 bits. The chaotic map was implemented

on TMS320C6416 DSP development kit. The ap-

plication of the backward iteration synchronization

method to nested maps required the estimate of the

initial condition from two concatenated maps as com-

pared to one for normal chaotic map. After extended

experimental tests, the results proved that a mini-

mum of 38 synchronization chaotic bits are needed

to achieve synchronization.

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