Research on 4-dimensional Systems without Equilibria with Application

TELKOMNIKA, Vol.16, No.2, April 2018, pp. 811~826
ISSN: 1693-6930, accredited A by DIKTI, Decree No: 58/DIKTI/Kep/2013
DOI: 10.12928/TELKOMNIKA.v16i2.8507  811
Received January 4, 2018; Revised January 21, 2018; Accepted February 16, 2018
Research on 4-dimensional Systems without Equilibria
with Application
Ruibin Hao*
1
, Lequan Min
2
, Hongyan Zang
3
Schools of Mathematics and Physics, University of Science and Technology Beijing, Beijing China, 100083
*Corresponding author, e-mail: haoruibin0001@163.com, minlequan@sina.com, zhylixiang@126.com
Abstract
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
Keywords: chaotic map, pseudorandom number generator, randomness test, avalanche encryption
scheme
Copyright © 2018 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Chaos is a kind of complex dynamic behaviors generated from determined nonlinear
systems. Chaotic behaviors are extremely sensitive to initial conditions, difficult to predict in a
long-term [1-3]. Chaos synchronization (CS) is of essential importance for many physical,
biological and engineering systems. Pecora and Carroll’s poineer work on GS communication
[4] has made the research on GS to developed rapidly [5-11]. The apparant random behaviors
of chaotic systems makes them to provide new tools for cryptography and other fields [12-25].
In cryptographic terms, the strict key avalanche criterion means that when any bit of the
key change, each binary bit of the ciphertext should have a change with the probability of one
half [26,27]. In 2013, a d-bit segment stream encryption scheme with avalanche effect (SESAE)
has been presented [28]. The feature of the SESAE is to make each bit of the decrypted
plaintext changed to 1 with probability of (2
d
-1)/2
d
if using an ideal d-bit PRNG [28]. Following
[28], some 2
16
-word PRNGs have been designed [19,29,30], which provide a new tool in
cryptography.
Dynamic chaotic systems without equilibria have generally complex dynamic behaviors
[31], and more suitable to design PRNGs because there are corresponding chaos criterion
theorems on them. In a recent paper [19], we have firstly intorduded a kind of disrete chaotic
system without equilibria (DCSE), used a DCSE to design a PRNG applying to SESAE.
Conseqently, studing new theorems on DCSE, PRNGs and their applications to SESAE
is important both for theoretical researchs and pactical applications. This paper firstly set up two
new propositions for determining 4-dimensional DCSE. Our propositions extend the results
obtained in [19]. And then introduces such a DCSE. Thirdly construct an DCSE-based
generalized CS (GCS) system, and simulate the complex dynamics of the system. Fourthly
designs a DCSE-GCS-based PRNG. and uses the NIST FIPS 140-2 test suite [32] to test the
randomness of the GCS PRNG, the RC4 algorithm and the ZUC algorithm [33]. Finally, using
the GCS PRNG and the SESAE [28] encrypts an RGB image with numerical analysis.

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TELKOMNIKA Vol. 16, No. 2, April 2018 : 811 – 826
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2. Definition and the GCS Theorem
Definition 1: (Similar to [34,35 ]).Consider two systems,
( 1) ( ( ))k F k X X (1)
( 1) ( ( ), ( ))k G k k Y Y X (2)
where
T
1( ) ( ( ), , ( ))nk x k x kX L (3)
1
T
( ) ( ( ), , ) ,( )mk y k y k m n Y L (4)
1
T
( ( )) ( ( ( )), , )( ( ) )nF k f k f kX X XL (5)
1
T
( ( ), ( )) ( ( ( ), ( )), , ( ( ), ( ))) .mG k k g k k g k kY X Y X Y XL (6)
If there exists a transformation
: n m
H ¡ ¡ (7)
1
T
( ( )) ( ( ( )), , )( ( ) )mH k h k h kX X XL (8)
and a subset n m
X YB B B   ¡ ¡ such that all trajectories of (1) and (2) with initial conditions
in B satisfy lim ( ( )) ( ) 0,
k
H k k

 X Y‖ ‖ then the two systems (1) and(2) are said to be in GS with
respect to the transformation ( ( ))H kX . System (1) is called the driving system, while system (2)
is the driven system. In particular, if the two systems are chaotic, then the GS is called a
generalized chaos synchronization (GCS).
In order to construct a new discrete chaotic system with the GCS property, the following
theorem is needed.
Theorem 1: [11] Let , , , ( )m FX Y X X and ( , )G Y X be defined by (3)-(6), and
1
T
( ( ), , ( ))m mx k x kX L
Suppose that
1 2
T
( ) ( , , , )m mH y y yX L (9)
is an invertible transformation. If the two systems (1) and (2) are in GCS via the transformation
( )mHY X , then the function ( , )G Y X given in (2) will have the following form:
( , ) ( ( )) ( , )m mG H F q Y X X X Y (10)
Where
1 2
T
( ) ( ( ), ( ), , ( ))m mF f f fX X X XL
and the function
1 2
T
( , ) ( ( , ), ( , ), , ( , ))m m m m mq q q qX Y X Y X Y X YL
guarantees that the zero solution of the following error equation is asymptotically stable:

TELKOMNIKA ISSN: 1693-6930 
Research on 4-dimensional Systems without Equilibria with Application (Ruibin Hao)
813
( 1) ( ( 1)) ( 1) ( , )m mk H k k q     e X Y X Y (11)
3. Two Propositions on Chaotic System Without Equilibria
Consider a general parametric form of four-dimensional discrete system:
Form A:
1 1 1 1 2 1
2 2 1 2 3 4 2 3 4
3 3 3 1 2 5 6
4 4 4 1 2
( 1) ( ) ( ( ), ( ), )
( 1) ( ( ), ( ), ( ), ( ), , , )
( 1) ( ) ( ( ), ( ), , )
( 1) ( ) ( ( ), ( ))
x k x k f x k x k
x k f x k x k x k x k
x k x k f x k x k
x k x k f x k x k

  
 
  
  

  
   
(12)
Where , 1,2, ,6.n
ia i ¡ L Now we give the following
Proposition 1: If the following conditions hold, then system (12) has no equilibrium.
i 1 1 2 1( , , ) 0f x x a  if and only if 1 2x x
ii 3 1 2 5 6( ( ), ( ), , ) 0f x k x k a a  if and only if 2
1 2 6x x a
iii 4 6 6( , ) 0f a a 
Proof. Firstly, solve for the equilibrium of the first equation of system (12). Condition (i) gives
1 1 1 1 2 1( ) ( ) ( ( ), ( ), )x k x k f x k x k a 
1 2( ) ( )x k x k (13)
Substituting (13) into the third equation of system (12) and letting 3 3( 1) ( )x k x k  gives
3 1 2 5 60 ( ( ), ( ), , )f x k x k a a (14)
and
1 2 6( ) ( )x k x k a  (15)
Then substituting (15) into the fourth equation of system(12) and letting 4 4( 1) ( )x k x k  gives
6 60 ( , ) 0f a a  (16)
This contradiction shows that system (12) has no equilibria. This completes the proof.
Form B:
1 2 3 4 1 2 3 4
1
1 1 2 3 4
1 2 3 4
2
2 1 2 3 4
1 1
3 3
3 1 2 3 4
( ( ), ( ), ( ), ( )) ( ( ), ( ), ( ), ( ))
( 1)
( ( ), ( ), ( ), ( ))
( ( ), ( ), ( ), ( ))
( 1)
( ( ), ( ), ( ), ( )
sin( ( ))
( 1) ( )
( ( ), ( ), ( ), ( )
g x k x k x k x k e x k x k x k x k
x k
f x k x k x k x k
g x k x k x k x k
x k
f x k x k x k x k
x k
x k x k
f x k x k x k x k


 
 
  
2 2
4 4
3 1 2 3 4
)
sin( ( ))
( 1) ( )
( ( ), ( ), ( ), ( ))
x k
x k x k
f x k x k x k x k











   


(17)
where 1 2, 0   . Now we give the following
Proposition 2: If the following conditions hold, then system (17) has no equilibrium.

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i 1 2 3 4| g( ( ), ( ), ( ), ( )) | <x k x k x k x k M
ii 1 2 3 40 | ( ( ), ( ), ( ), ( )) |e x k x k x k x k N 
iii 1 2 3 4( ( ), ( ), ( ), ( )) 0, 1,2,3,4.i if x k x k x k x k i  
iv
1 1
| |
M N
 

 ,
2 2
| |
M
 

Proof. Solving the equilibrium point is to solve the following Equations:
1 2 3 4 1 2 3 4
1
1 1 2 3 4
1 2 3 4
2
2 1 2 3 4
1 1
3 1 2 3 4
( ( ), ( ), ( ), ( )) ( ( ), ( ), ( ), ( ))
( ) (18 1)
( ( ), ( ), ( ), ( ))
( ( ), ( ), ( ), ( ))
( ) (18 2)
( ( ), ( ), ( ), ( ))
sin( ( ))
0= (1
( ( ), ( ), ( ), ( ))
g x k x k x k x k e x k x k x k x k
x k
f x k x k x k x k
g x k x k x k x k
x k
f x k x k x k x k
x k
f x k x k x k x k


 
 
2 2
4 1 2 3 4
8 3)
sin( ( ))
0= (18 4)
( ( ), ( ), ( ), ( ))
x k
f x k x k x k x k








 


 


(18)
Then 1 1( ( )) 0sin x k  , because of (18-3) and conditions (iii). Those imply
1 1( ) , 0, 1, 2x k m m     L
then
1
1
( ) , 0, 1, 2
m
x k m


    L
And we can know 1
1
| ( ) |
M N
x k


 form the proposition 2
that is
1 1
| | , 0, 1, 2,
m M N
m

 

    L (19)
then 0m  because of (19) and condition (iv), so 1( ) 0x k  and similarly 2 ( ) 0x k  .
then
3 4 3 4
3 4
(0,0, ( ), ( )) (0,0, ( ), ( )) 0
(0,0, ( ), ( )) 0
g x k x k e x k x k
g x k x k
 


then
3 40 (0,0, ( ), ( )) 0e x k x k 
This contradiction shows that system (17) has no equilibria. This completes the proof.
Table 1 shows fourteen systems which satisfy propositions 1 and 2, respectively. The
corresponding Lyapunov exponents and initial conditions are listed in the table. The largest
Lyapunov exponents of all systems are positive. Therefore they are chaotic systems. The
chaotic orbits of the state variables 1 ( )x k , 2 ( )x k , 3 ( )x k and 4 ( )x k of the systems are shown in
Figure 1 and Figure 2. It can be observed that, although the same initial conditions are used, the
chaotic systems have different dynamical characteristics.

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Figure 1. Chaotic orbits of the variables of the form a listed in Table 1

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Table 1. 4-dimensional Discrete Chaotic Systems without Equilibrium

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Figure 2. Chaotic orbits of the variables of the form b listed in Table 1
4. A chaotic System-Based GCS Theorem
Firstly, we construct a 4-dimensional polynomial system
1 1 2 1
2 2 1 3 4
3 3 1 2
4 4 2
( 1) ( ) 0.01( ( ) ( ))
( 1) 1.0025 ( ) 0.001 ( ) ( ) 0.001 ( )
( 1) ( ) 0.001( ( ) ( ) 25)
( 1) ( ) 0.1sin( ( )).
x k x k x k x k
x k x k x k x k x k
x k x k x k x k
x k x k x k
   
    
 
   
   
X (20)
From Proposition 1, system (20) has no equilibria. Calculated Lyapunov exponents of this
system are 0.00104, 0, 0.00089, 0.00761  . Therefore, it is chaotic. System (20) is used as the
driving system of our GCS system.

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Second construct an invertible matrix:
4 2 8 7
5 8 3 1
7 0 2 3
4 3 1 7
A
 
 
 
 
 
 
(21)
with the transformation 4 4
:H ¡ ¡ defined as follows:
1 2 3 4( ) ( ( ), ( ), ( ), ( ))H A h h h hX X X X X X@ (22)
Let
1
( , ) ( )
8
q  X Y AX Y (23)
where ( , )q X Y is used to ensure the error Equation (11) be asymptotically stable.
Using Theorem 1, we can select a driven system as following form:
1
2
3
4
( 1)
( 1)
( 1) [ ( ( ))] ( ( ), ( )).
( 1)
( 1)
y k
y k
k F k q k k
y k
y k
 
 
    
 
 
 
Y A X X Y (24)
Therefore system (20) and (24) are in GCS with respect to transformation (22). Now choose
(25) and (26) as initial conditions:
T
(0) (0.2,0.1,0.75, 2) X (25)
(0) (0)Y AX (26)
The numerical simulated chaotic orbits of state variables 1 2 3 4, , ,x x x x and 1 2 3 4, , ,y y y y for
the first 50000 iterations are shown in Figure 3 and Figure 4, respectively. The evolution of state
variables: 1 ( )k x k , 2 ( )k x k , 3 ( )k x k , 4 ( )k x k and 1 ( )k y k , 2 ( )k y k , 3 ( )k y k , 4 ( )k y k are
shown in Figure 5 and Figure 6. It can be observed that the dynamic behaviors of the chaotic
system demonstrate chaotic attractor. Moreover, as the theory predicts, with respect to
transformation ( )H A k X and ( )kY are showed in generalized synchronization in Figure 7.

819
Figure 3. Chaotic trajectories of variables: Figure 4. Chaotic trajectories of variables:
(a) 1 2 3( ) ( ) ( )x k x k x k  ,(b) 1 2 4( ) ( ) ( )x k x k x k 
(c) 1 3 4( ) ( ) ( )x k x k x k  ,(d) 2 3 4( ) ( ) ( )x k x k x k 
(a) 1 2 3( ) ( ) ( )y k y k y k  ,(b)
1 2 4( ) ( ) ( )y k y k y k 
(c) 1 3 4( ) ( ) ( )y k y k y k  ,(d)
2 3 4( ) ( ) ( )y k y k y k 
Figure 5. The evolution of state variables: Figure 6. The evolution of state variables:
(a) 1( )k x k (b) 2 ( )k x k
(c) 3 ( )k x k (d) 4 ( )k x k
(a) 1 ( )k y k (b) 2 ( )k y k
(c) 3 ( )k y k (d) 4 ( )k y k
Figure 7. The state vectors and are in generalized synchronization with respect to the: (a),
(b), (c), (d)

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5. Chaotic Pseudorandom Number Generator and Pseudorandomness tests
5.1. Pseudorandom Number Generator
Denote
{ ( ) | 1,2,3,4},
{ ( ) | 1,2,3,4}
i i
i i
x k i
y k i
 

 
X
Y
(27)
where ix s , iy s are defined by system (20) and (24).
Now introduce a transformation 16
1 : {0,1, ,2 1}T  ¡ L , which transforms the chaotic streams of
systems(27) into key streams. Let 15
10L  , 3 1S X Y  .Then, the chaotic PRNG 1T is defined by
16
1( ) mod(round(( ( min( )) / (max( ) min( ))),2 )T L  S S S S S (28)
The seeds of the chaotic PRNG are the initial conditions of the GCS system, which can
be chosen via random number generator. Therefore, the output key streams of the chaotic
PRNG can be obtained via the transformation (28) on the chaotic streams of the GCS systems
(20) and (24).
5.2. Pseudorandomness Test
The FIPS 140-2 test consists of four sub-tests: Monobit Test, Poker Test, Runs Test and
Long Runs Test. Each test needs a single stream of 20,000 one and zero bits from the
keystream generator. Any failure in the first three tests means that the corresponding quantity of
the sequences falls out the required intervals listed in the second column of Table 2. The Long
Runs test is passed if there are no runs of length 26 or more.
Two previous papers [36,37] have pointed out that the required intervals of Monobit test
and Porker test correspond significant 4
10 
 for the normal cumulative distribution and the 2

distribution, respectively; however the required intervals of the Runs tests correspond
approximately the significant 7
1.6 10 
  for the normal cumulative distribution. If one selects
the significant 4
10 
 of all tests, the corresponding accepted intervals are those as ones listed
in the third column of Table 2 [36-37]. Then, we denote the accepted intervals by G FIPS 140-2
test criterion.
Table 2. The Required Intervals of FIPS 140-2 Monobit Test, Porker Tests, Runs Test. Here,
MT, PT, and LT Represent the Monobit Test, the Porker Test and the Long Runs Test, k
Represents the Length of the Run of a Tested Sequence. 2
 DT Represents 2
 Distribution
Test Item
FIPS 140-2
required intervals
4
10 

Accepted Intervals
Golomb’s
Postulates
MT 9,725~10,275 9,725_10,275 10000
PT 2.16~46.17 2.16_46.17 2
 DT
LT < 26 < 26 ----
k Run Test Run Test Run Test
1 2,315~2,685 2,362~2,638 2,500
2 1,114~1,386 1,153~1,347 1,250
3 527~723 556~694 625
4 240~384 264~361 313
5 103~209 122~191 156
6+ 103~209 122~191 156
According to Golomb's three postulates on the randomness, the ideal pseudorandom
sequences should satisfy [38], the ideal values of the first three tests should be listed in the
fourth column of Table 2. Finally, FIPS 140-2 test suite is used to test the randomness
performance. One needs to change the keystreams with values 16
{0,1, ,2 1}L to binary
keystreams via the following transformation:

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16
:{0,1, ,2 1} {0,1}T  % L
which is defined by
22 21T T T% o (29)
16
{0,1, ,2 1}N
 y L
21 ( ) 2 ( )T dec biny y
Let 2 ( )dec binz Y . Then
22 ( ) (:)T z z
where 2dec bin and (:)z are both Matlab commands.
The FIPS 140-2 test is used to check 1,000 keystreams randomly generated,
respectively by the chaotic PRNG with perturbed randomly initial conditions (25) and (26) and
the matrix (21) in the range 16
| | [10 ,1]
ò . All sequences pass the FIPS 140-2 test and 16
sequences fail to pass the G FIPS 140-2 test. The test results are listed in the third column of
Table 3, which the results are described by mean values  standard deviation (Mean  SD).
Table 3. The Confident Intervals of FIPS 140-2 Tested Values of 1,000 key Streams Generated
by the CHAOTIC PRNG, the RC4 and ZUC PRNG. Here, SD Represents the Standard
Deviation
Test
item
bits
PRNG RC4 ZUC
Mean  SD Mean  SD Mean  SD
MT
0 9999.0  69.813 9999.7  70.092 9998.4  71.843
1 10000.9  69.813 10000  70.092 1002  71.843
PT - 15.175  5.568 14.87  5.433 15.043  5.549
LT
0 13.577  1.841 13.6  1.8214 13.605  1.841
1 13.642  1.930 13.604  1.884 13.595  1.931
1
0 2500.3  45.770 2500.9  45.568 2501.9  45.735
1 2498.7  46.972 2501.4  46.398 2502.7  45.121
2
0 1249.4  31.030 1250.5  31.372 1252.1  32.606
1 1250.9  31.613 1249  31.048 1249.5  32.221
3
0 624.42  23.051 624.95  22.964 624.09  22.648
1 625.58  22.912 625.65  22.93 624.64  23.455
4
0 312.75  16.662 311.71  16.548 312.56  16.748
1 313.73  16.245 312.17  16.822 312.72  16.506
5
0 156.36  11.758 156.41  12.069 155.65  12.097
1 155.99  12.096 156.6  11.958 156.66  12.369
6+
0 156.13  11.811 156.15  11.792 155.75  11.719
1 156.53  11.551 155.79  11.979 155.82  11.497
The Rivest Cipher 4 (RC4) has been widely used in popular protocols such as Secure
Sockets since it's designed in 1987. The RC4 algorithm as PRNG can be designed via the
Matlab commands which is shown in Figure 8.

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Figure 8. The Matlab commands of RC4 algorithm to design PRNG
Here, “randint(1, 2^L, [0 2^L-1])” generates a vector of uniformly distributed random integers
{0,1, ,2 1}L
L of dimension 2L
; “mod” means modulus after division; “zeros(1, N)” is a zero raw
vector of dimension .N Consequently, the RC4 algorithm based L -bit segment PRNG is
designed. Next, using FIPS 140-2 test to test the 1,000 keystreams randomly generated by RC4
PRNG. Results show that 1 and 12 sequences fail to pass the FIPS 140-2 test and G FIPS 140-
2 test, respectively. The statistic test results are listed in the forth column of Table 3.
Furthermore, ZUC is a stream cipher that forms the heart of the third generation
partnership project (3GPP) confidentiality algorithm 128-EEA3 and the 3GPP integrity algorithm
128-EIA3. Then, using FIPS 140-2 test to test the 1,000 keystreams randomly generated by the
ZUC algorithm [33]. It demonstrates that all of the sequences pass the FIPS 140-2 test, and 21
sequences fail to pass the G FIPS 140-2 test. The test results are listed in the fifth column of
Table 3. Finally, compare all test results shown in Table 3. It can be observed, the statistical
properties of the pseudorandomness of the sequences generated via the three PRNGs don't
have significant differences.
5.3. Key Space
The key parameters set of the proposed CHAOTIC PRNG includes the initial condition
(0)X , (0)Y and the matrix ,( ).i jA  It can be proved that if the perturbation matrix ,( )i j 
satisfies ,| | 1.0035i j  , the matrix A   is still invertible. Therefore the chaotic PRNG have
4+4+16 key
parameters denoted by
1 2 24{ , , , }s k k kK L (30)
The perturbed keys have the forms
1 2 24( ) [ , , , ]s s     K K L (31)
The Matlab platform uses double precision decimal computations. That means each
computed decimal number has 16 bits' accuracy. Therefore, one can select
16
10 | | 1, 1, ,24,i i
   L that is, 1 2 160.i a a a  L , where [0,1, ,9]ia  L . Therefore, the 24 keys
have a key space which is larger than 24 16 1275
10 2 .
 Now, compare the difference between the
key stream S with 20000 codes length generated by the key set (30) with the key streams pS
generated by the perturbed key set (31), respectively.
The comparison results are shown in the third column of Table 4, where SV denotes the
statistic values, DC denotes the different codes, and CC denotes the correlation coefficients.
Observe that the average percent of different codes is 50.0136%, which is very closed to the
ideal value 50%. And the average of the correlation coefficients is 0.00583440, also very closed
to the ideal value of 0.

823
Table 4. The Statistic Data Describes the Percentages of the Codes of the Key Stream
Variations between S and pS as well as S and mS
Item SV pS mS
DC
Min 48.5900% 48.7299%
Mean 50.0136% 49.9855%
Max 51.1000% 51.0700%
CC
Min 00000871 0.00001005
Mean 0.00583440 0.00554509
Max 0.02823380 0.02533419
Next, compare the same key stream S with the 1000 streams mS generated by the
Matlab function randi([0 1], 1, 20000). The comparison results are shown in the fourth column of
Table 4. Observe that the average percentage of different codes is 49.9855% and the average
of the correlation coefficients is 0.00554509. The results suggest that the key stream S has no
significant correlations with the perturbed key streams pS and the streams mS . In summary, the
effective key space of the CHAOTIC PRNG is 10
24x16
(larger than 1275
2 ), which is larger than the
key space 10
24x15
obtained in [19].
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
Figure 9 (a) Original Image (b) Decrypted Image via the key Streams P .Ten Decrypted Images
via Key Streams Generated with Slighted Perturbed Initial Conditions and the Matrix within the
Range 15 10
[10 ,10 ] 
: (c) 1,1I (d) 3,1I (e) 4,1I (f) 4,2I ,(g) 5,1I ,(h) 23,1I (i) 23,2I (j) 24,1I ,(k) 24,2I and (l) 25,1I
6. Simulations on SESAE
Consider the avalanche effect of the CHAOTIC PRNG, which is used to encrypt an RGB
image "tower" with 250  140 pixels. The simulation is implemented via the Matlab R2016a
platform. The SESAE experiments on CHAOTIC PRNG are described as follows: Prosedures
(1)-(4) are the same as those given in [19]. The receiver randomly disturbs the initial conditions

 ISSN: 1693-6930
824
(25) and (26), and also the matrix (21), for 1000 times in the range 15 10
| | [10 ,10 ] 
ò , then obtain
disturbed key streams:
, 1,2, ,1000.iP i  L (32)
The receiver uses 1 2{ , , , }i NP p p p L to decrypt the ciphertext and obtains a decrypted plaintext:
1
( , ), 1,2, ,1000.i iM E C P i
  L (33)
After changing iM s
to RGB images, one can find that all images become almost pure white-
colored ones. There are  840000 0,1 codes in each decrypted image. Among the decrypted
images,the minimum of 0 s
is 2 and the maximum of 0 s
is 24.
Denote ,i jI the j th image that has i zero codes. The first five images with minimum zero
codes and the last five images with maximum zero codes are shown in Figure 9(c)-(l).
Therefore, the percentages of the numbers of “1” codes in the 1000 decrypted images are within
the range[0.999970, 0.999998], which are near to the ideal value 16 16
(2 1) / 2 0.999985  ,
and are similar to those given in [19].
Table 5 lists some statistical data of the norms between the original key stream 0S and
the key stream ,i jS used in the above ten decrypted images, respectively. The results suggest
that there are no significant correlations between the norms and the corresponding decrypted
image, and similar to those obtained in [19].
Table 5. Differences between the Original Keystream 0S and the keysteams ,j iS , Measured by
Norm 0 ,|| ||j iS S
10
0 ,|| || 10j iS S 
 
1,1S 3,1S 4,1S 4,2S 5,1S
0S 3.151 2.836 3.138 2.344 2.723
23,1S 23,2S 24,1S 24,2S 25,1S
0S 3.092 2.828 2.847 2.950 2.817
Remark: To resist attacks, one may consider implementing an “one-time-pad” scheme into
CHAOTIC PRNG: Let X be a set in the seed space (initial conditions) of the CHAOTIC PRNG,
and assume that Alice and Bob share a one-to-one map :f X X . Before each
communication, Alice randomly selects an element xX and sends it to Bob. Then, they both
use ( )f x as the seed for one-time encryption.
In summary, the simulation shows that using the CHAOTIC PRNG and SESAE to encrypt
RGB images is able to generate encrypted images with significant avalanche effects.
7. Concluding Remarks
The main results of this paper are summarized as follows:
a. This paper proposes two propositions on 4-dimensional discrete systems without equilibra,
which extend the results obtained by [19].
b. A 4-dimensional discrete chaotic system is proposed. Using the system and the GS
theorem designs an 8-dimensional GCS system.
c. Using the 8-dimensional GCS system constructs a chaotic PRNG. The key space of our
PRNG is 10
24x16
(larger than 1275
2 ) is larger than the key space 10
24x15
obtained in [19] and
the key space 2
128
obtained by the ZUC algorithm.

825
d. Using the FIPS 140-2 test critrions tests the keystreams generated via the CHAOTIC
PRNG, the RC4 algorithm and the ZUC algorithm. The results show that the randomness of
the sequences generated via the chaotic PRNG and others are similar.
e. Numerical simulations show that the CHAOTIC PRNG is able to generate significant
avalanche effects, and the percentages of the “1” code in the decrypted texts for different
keystreams are larger than 0.999970, which is very closed to the idea value of
16 16
(2 1) / 2 0.999985  and similar to those given in [19]. Therefore, it verifies the proposed
chaotic PRNG is a qualified candidate for SESAE.
In summary, the proposed chaotic PRNG is a promising candidate for practical
applications. Further comparison with different state-of-the-art PRNG schemes in terms of
computational complexity, storage requirement, communication cost, etc., it will be carried out in
future research along the same lines.
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