Message Embedded Cipher Using 2-D Chaotic Map

International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
13
MESSAGE EMBEDDED CIPHER
USING 2-D CHAOTIC MAP
Mina Mishra1
and Dr. V.H. Mankar2
1
Ph. D. Scholar, Department of Electronics & Telecommunication, Nagpur University,
Nagpur, Maharashtra, India
minamishraetc@gmail.com
2
Senior Faculty, Department of Electronics Engineering, Government Polytechnic,
Nagpur, Maharashtra, India
vhmankar@gmail.com
ABSTRACT
This paper constructs two encryption methods using 2-D chaotic maps, Duffings and Arnold’s cat maps
respectively. Both of the methods are designed using message embedded scheme and are analyzed for
their validity, for plaintext sensitivity, key sensitivity, known plaintext and brute-force attacks. Due to the
less key space generally many chaotic cryptosystem developed are found to be weak against Brute force
attack which is an essential issue to be solved. For this issue, concept of identifiability proved to be a
necessary condition to be fulfilled by the designed chaotic cipher to resist brute force attack, which is a
basic attack. As 2-D chaotic maps provide more key space than 1-D maps thus they are considered to be
more suitable. This work is accompanied with analysis results obtained from these developed cipher.
Moreover, identifiable keys are searched for different input texts at various key values.
The methods are found to have good key sensitivity and possess identifiable keys thus concluding that
they can resist linear attacks and brute-force attacks.
KEYWORDS
Message embedded scheme, Arnolds Cat map, Duffings map, Identifiability.
1. INTRODUCTION
For last several years many efforts have been made to use chaotic systems for enhancing some
features of communications systems. Chaotic signals are highly unpredictable and random-like
nature, which is the most attractive feature of deterministic chaotic systems that may lead to
novel (engineering) applications. Some of the common features between chaos and
cryptography [1] [2] are being sensitivity to variables and parameters changes. An important
difference between chaos and cryptography lies on the fact that systems used in chaos are
defined only on real numbers, while cryptography deals with systems defined on finite number
of integers. Nevertheless, we believe that the two disciplines can benefit from each other. Thus,
for example, as it is shown in this paper, new encryption algorithms can be derived from chaotic
systems. On the other hand, chaos theory may also benefit from cryptography: new quantities
and techniques for chaos analysis may be developed from cryptography.
During the past two decades, there has been tremendous interest worldwide in the possibility of
using chaos in communication systems [3][4]. Many different chaos-based decryption
algorithms have been proposed up to date.
The aim of this paper is to construct and crypt analyze two of the stream symmetric chaotic
ciphers[5] constructed using one of the latest chaotic scheme known as message-embedded

14
scheme [6] [7]. Both of the developed methods use 2-D chaotic maps, Duffings and Arnolds Cat
map. Parameters of the respective chaotic maps act as secret key in the ciphers due to which
complexity and key space is increased compared to 1-D chaotic map. Both of the ciphers are
analyzed for key space, avalanche effect and strength against Brute-force and Known-plaintext
attack.
A cryptanalytic method based on the identifiability concept, solves the problem of less key
space in chaotic ciphers. It is possible to test about the cipher strength against Brute-force attack
using it. Both of the mentioned ciphers are concluded to provide security against the Brute-force
attack[8]. Identifiability concept fulfils the necessary condition but not sufficient as the
developed cryptosystems must be tested for sensitivity and other statistical tests to result in a
robust cipher. Thus both the ciphers are tested for sensitivity and it is concluded that some of the
keys selected from domain of key space[9] of the ciphers seem to have good key sensitivity and
resist known plaintext attack for the available first two characters of plaintext [10] [11].
This paper is organized into five sections as follows. Section II, presents the background and in
section III algorithm for encryption used in developing ciphers is provided. Then in section IV,
analysis result in tabulated form and discussions are presented. Section V, discusses about the
conclusions derived.
2. BACKGROUND
Message-Embedded Scheme: According to this scheme at the transmitter side, the plain text is
encrypted by an encryption rule which uses non-linear function and the state generated by the
chaotic system in the transmitter. The scrambled output signal is used further to drive the
chaotic system such that the chaotic dynamics is changed continuously in a very complex way.
Then another state variable of the chaotic system in the transmitter is transmitted through the
channel.
At the receiver side, the reconstruction of the plaintext is done by decrypting the input by using
the reverse of encryption method.
This method can be illustrated along with diagram as shown in fig 1.
Chaotic
function
Non-linear
function
Non-linear
function
Chaotic
function
P
Y
C
C
X X Y
P
(a)Transmitter end (encrypt or) (b) Receiver end (decrypt or)
Figure 1. Message-embedded chaotic cryptosystem
P: plaintext; C: cipher text; X: state of chaotic function;
Z &Y: Intermediate encrypted plaintext;
In the present work modified form of maps has been used that is generated and its randomness
can be clearly seen in fig 2(a) and (b) and fig 3(a) and (b) respectively.

15
Arnold’s Cat Map: Arnolds Cat map is a 2-D discrete-time dynamical system, which takes a
point(x, y) in the plane and maps it to a new point using equations:
[ ];
),
(
)
(
2
mod
)
1
(
)
1
( N
k
y
k
x
a
k
x +
−
=
+
[ ];
),
(
)
1
(
)
(
mod
)
1
( N
k
y
b
k
x
k
y −
+
=
+
a, b and N are parameters on which the map depends. At a=0.3, b=0.345, map exhibits chaotic
nature.
Figure 2. (a) Plot of Arnolds Cat Map at x(0) = 0.5, y(0) = 0.06, a = - 3.5, b = 0.9, n=1000.
Figure 2. (b) Plot of Arnolds Cat Map at x(0) = 0.5, y(0) = 0.06, a = - 3.5, b = 0.9, n=5000.

16
Duffings map: Duffings map is a 2-D discrete-time dynamical system, which takes a point
(x, y) in the plane and maps it to a new point using equations:
);
(
)
1
( k
y
k
x =
+
3
)
(
)
(
)
(
)
1
( k
y
k
ay
k
bx
k
y −
+
−
=
+
a and b are parameters on which the map depends. At a = 2.75, b = 0.2, map exhibits chaotic
nature.
Figure 3. (a) Plot of Duffings Map at x(0) = - 0.04, y(0) = 0.2, a= 2.75, b=0.1, n=1000.
Figure 3. (b) Plot of Duffings Map at x(0) = - 0.04, y(0) = 0.2, a= 2.75, b=0.1, n=5000.

17
Non-Linear Function: Modular function is used as non-linear function in the construction of
ciphers.
The Mod numeric function returns the remainder when the dividend is divided by the divisor.
The result is negative only if the dividend is negative. Both the numbers must be integers. The
function returns an integer. If any number is NULL, the result is NULL. For example:
Mod (5, 3) returns 2.
Mod (-5, 3) returns -2.
Cryptanalysis: Cryptanalysis is the study of attacks against cryptographic schemes to disclose
its possible weakness. During crypt analyzing [12] a ciphering algorithm, the general
assumption made is that the cryptanalyst knows exactly the design and working of the
cryptosystem under study, i.e., he/she knows everything about the cryptosystem except the
secret key. It is possible to differentiate between different levels of attacks on cryptosystems.
They are briefly explained as follows:
1. Cipher text-only attack: The attacker possesses a string of cipher text.
2. Known plain text: The attacker possesses some portion of plain text and the corresponding
cipher text.
3. Chosen plain text: The attacker has obtained temporary access to the encryption machinery.
Hence he/she can choose a plain text string, p, and construct the corresponding cipher text
string.
4. Chosen cipher text: The attacker has obtained temporary access to the decryption
machinery. Hence he/she can choose a cipher text string, c, and construct the corresponding
plain text string.
5. Brute Force Attack: A brute force attack is the method of breaking a cipher by trying every
possible key. The brute force attack is the most expensive one, owing to the exhaustive search.
In addition to the five general attacks described above, there are some other specialized attacks,
like, differential and linear attacks.
Differential cryptanalysis is a kind of chosen-plaintext attack aimed at finding the secret key in
a cipher. It analyzes the effect of particular differences in chosen plaintext pairs on the
differences of the resultant cipher text pairs. These differences can be used to assign
probabilities to the possible keys and to locate the most probable key.
Linear cryptanalysis is a type of known-plaintext attack, whose purpose is to construct a linear
approximate expression of the cipher under study. It is a method of finding a linear
approximation expression or linear path between plaintext and cipher text bits and then extends
it to the entire algorithm and finally reaches a linear approximate expression without
intermediate value.
Security Analysis: Various cryptanalytic procedures are developed to test the validity of newly
constructed ciphers and they are as follows:
(a) Key Space Analysis: Key space belongs to the chaotic region of the system in case of
chaotic ciphers. The total key space is a product of all the parameters involved. Once the key

18
has been defined and key space has been properly characterized, the good key is chosen
randomly from the large key domain.
(b) Identifiability Test method: From the crypto graphical point of view, the size of the key
space should not be smaller than 2100 to provide a high level security so that it can resist all
kind of Brute force attack. A fundamental issue of all kinds of cryptosystem is the key. No
matter how strong and how well designed the encryption algorithm might be, if the key is
poorly chosen or the key space is too small, the cryptosystem will be easily broken.
Unfortunately, chaotic cryptosystem has a small key space region and it is non-linear because
all the keys are not equally strong. The keys should be chosen from the chaotic region. To solve
the problem of small key space and weakness against brute force attack, identifiability concept
is quite advantageous.
A cryptanalytic procedure, known as output equality based on the identifiability concept, is
carried out on the developed ciphers. It is found that in chaotic ciphers, there exists a unique
solution for a particular input for certain domain of values of parameters. The response of any
system to a particular input is the solution of that particular system and it contains all the
information about the parameters of system. In the discussed ciphers, system parameters are
acting as a secret key. This type of analysis is also known as parametric analysis.
The output equality method is explained as follows:
“For the same inputs and initial condition, transmitter system is parameterized at different
values of parameter taken from the existing domain of parameter space, if the output response of
the system obtained after some value of iteration, parameterized at a particular value coincides
with the output response of the same system parameterized at some other value of parameter
within the domain for the same number of iteration, then both the parameters are said to be
equal and identifiable. The system is said to possess unique solution at that particular value of
parameter and the system is said to be structurally identifiable.”
If parameter of the transmitter is identifiable, it is more difficult for the eavesdropper to find it
by a brute force attack. Consequently, this parameter can play the role of the secret key against
brute force attack. If parameter is not identifiable, the eavesdropper has a higher favorable
chance to find it by a brute force attack and thus, the parameter vector cannot play the role of
the secret key against brute force attack.
(c) Plaintext sensitivity Test Method: The percentage of change in bits of cipher text obtained
after encryption of plaintext, which is derived by changing single bit from the original plaintext
from the bits of cipher text obtained after encryption of original plaintext. With the change in
single bit of plaintext, there, must be ideally 50% change in bits of cipher text to resist
differential cryptanalysis (chosen-plaintext attack) and statistical analysis.
(d) Key sensitivity Test Method: The percentage of change in bits of cipher text obtained after
encryption of plaintext using key, which is flipped by single bit from the original key, from bits
of cipher text obtained after encryption of plaintext using original key, which requires ideally
50% change in cipher text bits to resist Linear and statistical attacks.
(e) Known plaintext attack Method: For observing this attack on developed cryptosystem it is
assumed that the opponent knows everything about the algorithm, he/she has the corresponding
cipher text of plaintext and some portion of plaintext. With this much information, the opponent
tries to find out the secret key.

19
3. ALGORITHM FOR THE DEVELOPED CIPHERS
Encryption Algorithm:
Step-1: Read plaintext and key vector.
Step-2: Convert plaintext into its ASCII values.
Step-3: Each value of ASCII values are transformed using following steps:
(a) The chaotic map is iterated for a number of times to output a random state.
(b) ASCII value is mixed with non-Linear function and the output state of chaotic system
obtained after a fixed value of iteration.
(c) Again chaotic system is iterated for a fixed number of times and an output state is obtained.
(d) The response obtained in (b) is mixed with the output state obtained in (d) and output values
are obtained as cipher text.
Step-4: Convert cipher text into characters.
Step-5: Read the cipher text.
Decryption algorithm is reverse of encryption process and the original information is retained
using the same secret key using which encryption is being done and it is kept secret between
authenticated sender and receiver only.
4. RESULTS AND ANALYSIS
The simulated result data produced after analyzing both the ciphers using above discussed
(section II) cryptanalytic procedures is summarized with the help of table 1 and 2. Twenty
different values of keys are chosen from key space of respective ciphers and are analyzed for its
security. From both the observation tables, it can be seen that plaintext sensitivity of Arnold’s
cat cipher ranges from 0.5 to 2.5 % and Duffings cipher ranges from 0.5 to 2 %, which is not
sufficient. Key sensitivity for each of the cipher ranges from 0 to 36 % and from 0 to 51 %,
respectively. Thus key sensitivity property of some keys from both the ciphers shows
satisfactory values. Both ciphers are robust against known plaintext attack for the available first
two characters of plaintext. Key space of ciphers shows lesser range than compared to the
required limit i.e. 2100 to resist Brute-force attack but identifiable keys conclude that the
developed ciphers can resist Brute-force attack.
a. Arnold’s Cat cipher: Key space is from [-5 0.4] to [-0.9 1.5] = 5 x 1016

20
Table 1. Analysis Table for Arnold’s Cat
Sl.
No
.
Plain text Key
value
Ciphertext Plaintext
sensitivity
(in %)
Key
sensitivity
(in %)
Domain for
key
With
increment
= 0.0001
Identif
iability
of key
for
iteratio
n value
=2 or 3
Robustnes
s against
known
plaintext
attack for
p=[p1 p2].
Wheth
er key
can act
as
secret
key
against
Brute
Force
attack
?
1. What is
your
name?
[0.003
4
0.0013
]
k¢³¬¿k´¾kÄºÀ½k¹¬¸° 1.9737 24.3421 [0
0]to[0.005
0.004]
I R YES
2. I am going
to market.
[-
4.4977
0.2034
]
kk¬¸k²º´¹²k¿ºk¸¬½¶°¿y 1.7045 21.5909 [-4.5
0.2]to[-
4.495
0.204]
I R YES
3. My
college
name is
s.s.c.e.t.
[-
2.9982
-
0.6981
]
kÄk®º··°²°k¹¬¸°k´¾k¾
y¾y®y°y¿y
1.2500 25.4167 [-3 -0.7]to[-
2.995 -
0.696]
I R YES
4. Hello!how
are you?
[-2.992
-0.694]
i®µµ¸j±¸Àiª»®iÂ¸¾ 1.3158 0 [-2.995 -
0.696]to[-
2.99 -0.692]
NI R NO
5. Sita is
singing
very well.
[-2.99
-0.692]
h±¼©h±»h»±¶¯ ±¶¯ h
¾ºÁh¿´´v
0.4630 36.1111 [-2.99 -
0.692]to[-
2.985 -
0.688]
NI R NO
6. Ram
scored 98
marks in
Maths.
[-2.985
-0.688]
lµÀ-
lµ¿l¿µº³µº³lÂ±¾ÅlÃ±¸
¸z
0.4630 24.0741 [-2.985 -
0.688] to[-
2.98 -
0.684]
NI R NO
7. Jaycee
publication
.
[-
0.9957
0.0101
]
lÅ¯ ±±l¼Á®¸µ¯ -
Àµ»ºzU
0.5952 23.8095 [-1 0.01]to[-
0.995
0.014]
I R YES
8. Thank
you,sir.
[-0.99
0.018]
k³¬¹¶kÄºÀw¾´½y 2.5000 26.6667 [-0.995
0.014]to[-
0.99 0.018]
NI R NO
9. The match
was very
exciting.
[-
0.9853
0.0213
]
k³°k¸¬¿®³kÂ¬¾kÁ°½
Äk°Ã®´¿´¹²y
1.2931 0 [-0.99
0.018]to[-
0.985
0.022]
I R YES
10. I will be
leaving at
9p.m.
[0.203
5
1.4009
]
llÃµ¸¸l®±l¸±Âµº³l-
Àl¼z¹z
0.4630 28.2407 [0.2
1.4]to[0.205
1.404]
I R YES
11. How are
you?
[ -5
0.4]
ºÂk¬½°kÄºÀ 3.1250 31.25 [-5.0 0.4] to
[-4.995
0.4031]
NI R NO
12. Meet me
after 5p.m.
[-4
0.5]
¯ ¯ ¾j·¯ j«°¾¯ ¼j ºx·
x
2.6316 38.1579 [-4.0 0.5 ]
to [-3.995
0.5031 ]
NI NR NO
13. I have a
gift for
you.
[-3
0.6]
h°©¾-h©h¯ ±®¼h®
·ºhÁ·½v
1.1364 24.4318 [-3. 00 0.6 ]
to [-2.995
0.6031 ]
I NR YES
14. We will go
for walk.
[-2
0.7]
l£±lÃµ¸¸l³»l²»¾lÃ-¸·z 0.5952 31.5476 [ -2 0.7] to
[ -1.995
0.0.7031]
NI NR NO
15. Study
different
papers.
[-1
0.8]
¿À¯ Äk¯ ´±±°½°¹¿k»
¬»°½¾y
0.5435 31.2500 [- 1 0.8] to
[-0.995
0.8031]
I R YES
16. How to do
analysis?
[- 0.9
0.9]
l »ÃlÀ»l°»l-º-¸Å¿µ¿ 0.6250 27.5000 [-0.9 0.9] to
[-0.895
0.9031]
I R YES

21
17. Hai!
Where are
you going?
[- 4.99
1.0]
l -µml£´±¾±l-¾±lÅ»
Ál³»µº³
0.4808 20.6731 [- 4.99 1.0]
to [-4.98
1.01]
NI NR NO
18. Dolly, are
you
coming
with me?
[- 3.78
1.1]
j ¹¶¶Ãvj«¼¯ jÃ¹¿j-¹·³¸
±jÁ³¾²j·¯
0.4032 0.0 [-3.78 1.1 ]
to [-3.77
1.1 ]
NI R NO
19. Children
are playing
in park.
[ -2.2
1.2]
j ²³¶®¼¯ ¸j«¼¯ jº¶«Ã³
¸±j³¸jº«¼µxj
0.4032 24.5968 [ -2.2 1.2] to
[-2.2 1.3 ]
NI NR NO
20. I shall go
to cinema.
[- 2.5
1.4]
h h»°©´´h¯ ·h¼·h«±¶-
µ©v
0.5682 19.8864 [- 2.5 1.4] to
[ -2.5 1.5]
NI NR NO
NI – Non – Identifiable; I – Identifiable; R- Robust; p[p1 p2…p n]- First ‘n’ characters of
available plaintext string. ]; NR – Not robust;
b. Duffings cipher: Key space is from [1.8 -0.59] to [2.9 0.2] = 9 x 1014
Table 2. Analysis Table for Duffings.
Sl.
No
.
Plaintext Key
value
Ciphertext Plaintext
sensitivity
(in %)
Key
sensitivity
(in %)
Domain
for key
With
increment
= 0.0001
Identifi
ability
of key
for
iteratio
n value
=2 or 3
Robustnes
s against
known
plaintext
attack for
p=[p1 p2].
Wheth
er key
can act
as
secret
key
against
Brute
Force
attack
?
1. What is
your
name?
[1.803 -
0.584]
k¢³¬¿k´¾kÄºÀ½k¹¬¸° 1.9737 28.9474 [1.8 -
0.59]to[1.
805 -
0.586]
NI R NO
2. I am going
to market.
[1.8995
0.0068]
!J!bn!hpjoh!up!nbslfu/ 1.2987 0 [1.895
0.006]to[1
.9 0.01]
I R YES
3. My
college
name is
s.s.c.e.t.
[1.9015
0.0100]
!Nz!dpmmfhf!obnf!jt!t
/t/d/f/u/
0.9524 0 [1.897
0.008]to[1
.902
0.012]
I R YES
4. Hello!how
are you?
[1.8127
-0.5804]
!Ifmmp"ipx!bsf!zpv@ 1.5038 21.0526 [1.81 -
0.582]to[1
.815 -
0.578]
I R YES
5. Sita is
singing
very well.
[1.7547
0.1521]
!Tjub!jt!tjohjoh!wfsz!x
fmm/
1.0582 0 [1.75
0.15]to[1.
755 0.154]
I R YES
6. Ram
scored 98
marks in
Maths.
[1.4045
0.0020]
!Sbn!tdpsfe!:9!nbslt!jo
!Nbuit/
0.9524 0 [1.4
2.15]to[1.
405 2.154]
I R YES
7. Jaycee
publication
.
[1.7544
0.1540]
!Kbzdff!qvcmjdbujpo/ 1.4286 0 [1.75
0.15]to[1.
755 0.154]
I R YES
8. Thank
you,sir.
[1.6247
0.0937]
!Uibol!zpv-tjs/ 1.9048 0 [1.62
0.09]to[1.
625 0.094]
I R YES
9. The match
was very
exciting.
[2.802 -
0.094]
"Vjg"ocvej"ycu"xgt{"
gzekvkpi0
0.4926 51.2931 [2.8 -
0.1]to[2.8
05 -0.096]
NI R NO
10. I will be
leaving at
9p.m.
[2.0045
0.1509]
!J!xjmm!cf!mfbwjoh!b
u!:q/n/
1.0582 0 [2
0.15]to[2.
005 0.154]
I R YES
11. How are
you?
[1.81 -
0.44]
Ipx!bsf!zpv@ 2.3810 0 [1.81 -
0.44] to
[1.81 -
0.43]
NI R NO

22
12. Meet me
after 5p.m.
[1.82 -
0.57]
!Nffu!nf!bgufs!6q/n/ 1.4286 48.7500 [1.82 -
0.57] to
[1.83 -
0.57]
NI R NO
13. I have a
gift for
you.
[1.85 -
0.47]
J!ibwf!b!hjgu!gps!zpv/ 0.6494 50 [ 1.85 -
0.47] to
[1.85 -
0.46]
NI NR NO
14. We will go
for walk.
[ 1.89 -
0.3]
"Yg"yknn"iq"hqt"ycn
m0
0.6803 0 [ 1.89 -
0.3] to
[1.9 -0.3]
NI NR NO
15. Study
different
papers.
[1.9 0.2] !Tuvez!ejggfsfou!qbqf
st/
1.1905 0 [1.9 0.2]
to [1.7
0.2]
NI R NO
16. How to do
analysis?
[2.01
0.11]
!Ipx!up!ep!bobmztjt@ 1.4286 0 [2.01 0.11]
to [2.02
0.11]
NI R NO
17. Hai!
Where are
you going?
[ 2.05
0.14]
!Ibj"!Xifsf!bsf!zpv!hpj
oh@
1.0989 0 [2.03 0.14
] to [ 2.05
0.14]
NI NR NO
18. Dolly, are
you
coming
with me?
[ 2.3
0.15]
"Fqnn{."ctg"{qw"eqok
pi"ykvj"ogA
0.4608 0 [ 2. 2 0.15]
to [ 2.3
0.15]
NI NR NO
19. Children
are playing
in park.
[2.5
0.18]
!Dijmesfo!bsf!qmbzjo
h!jo!qbsl/!
0.9217 0 [2.5 0.18]
to [ 2.6
0.18]
I NR YES
20. I shall go
to cinema.
[2.6
0.19]
J!tibmm!hp!up!djofnb/ 1.2987 0 [2.5 0.19]
to [2.6
0.19]
I NR YES
It is seen from the observation table that each of the key possesses individual virtues to resist the
respective attacks.
5. CONCLUSIONS
This paper presents two encryption methods using Duffings and Arnold’s cat maps designed
using message embedded scheme and are analyzed for their validity, for plaintext sensitivity,
key sensitivity, known plaintext and brute-force attacks. As 2-D chaotic maps provide more key
space than 1-D maps thus they are considered to be more suitable. Less key space of the
developed ciphers concludes that they cannot resist brute force attack which is an essential issue
to be solved. For this issue, concept of identifiability proved to be a necessary condition to be
fulfilled by the designed chaotic cipher to resist brute force attack.
The methods are found to have good key sensitivity and possess identifiable keys thus
concluding that they can resist linear attacks and brute-force attacks. A comparison table no.3
shows that both ciphers are found to resist Brute-force attack as they consist of identifiable keys.
Key sensitivity property is also good for some of the keys selected from domain of key space.
Ciphers are determined to resist known plaintext attack for available first two characters of
plaintext. If available characters are not the starting characters of plaintext then ciphers shows
robustness against the attack for available any number of plaintext characters.
Table 3: Comparison between the two ciphers
Name of
Cipher
key Space
Range of plaintext
sensitivity
Range of key
sensitivity
Identifiable
key
Robust
against
known
plaintext
attack
Whether key
space > 2100
Duffing’s 9 x 1014
0.5 to 2 % 0 to 51 % Yes Yes No
Arnold’s
Cat map
5 x 1016
0.5 to 2.5 % 0 to 36 % Yes Yes No

23
ACKNOWLEDGEMENTS
The authors would like to thank the anonymous reviewers for their valuable suggestions and the
proposed references.
REFERENCES
[1] G. Jakimoski and L. Kocarev, “Chaos and Cryptography: Block Encryption Ciphers Based on
Chaotic Maps,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and
Applications, vol. 48, no. 2, pp. 163–169, 2001.
[2] F. Anstett, G. Millerioux, and G. Bloch, “Global adaptive synchronization based upon
polytopic observers,” in Proc. IEEE Int. Symp. Circuits Syst., Vancouver, BC, Canada, May
2004.
[3] T. Yang, “A survey of chaotic secure communication systems” Int. J. Comput. Cogn., vol. 2,
no. 2, 2004.
[4] L. P. de Oliveira and M. Sobottka, “Cryptography with chaotic mixing” Chaos, Solitons and
Fractals, vol. 35, pp. 466–471, 2008.
[5] G. Alvarez and S. Li, “Some Basic Cryptographic Requirements for Chaos-based
Cryptosystems” Int. J. Bifurc. Chaos, 2006.
[6] G. Millérioux, A. Hernandez, and J. Amigó, “Conventional cryptography and message-
embedding” in Proc. 2005 Int. Symp. Nonlinear Theory and its Applications (NOLTA 2005),
Bruges, Belgium, Oct. 18–21, 2005.
[7] F. Anstett, G. Millerioux, and G. Bloch, “Message-embedded cryptosystems: Cryptanalysis
and identifiability,” in Proc. 44th IEEE Conf. Decision and Control, Sevilla, Spain, Dec. 12–
15, 2005.
[8] Ruming Yin, Jian Yuan, Qiuhua Yang, Xiuming Shan, Xiqin Wang, “Linear Cryptanalysis
for a Chaos-based Stream Cipher” World Academy of Science, Engineering and Technology
60, 2009.
[9] N. Masuda and K. Aihara, “Cryptosystems with Discretized chaotic maps”, IEEE Trans.
Circuits and Syst. I, vol. 49, pp. 28–40, Jan. 2002.
[10] Floriane Anstett, Gilles Millerioux, and Gérard Bloch, “Chaotic Cryptosystems:
Cryptanalysis and Identifiability”, IEEE transactions on circuits and systems—I, Vol. 53, No.
12, December 2006.
[11] G. Alvarez, F. Montoya, M. Romera, and G. Pastor, “Cryptanalysis of a chaotic encryption
system,” Physics Letters A, vol. 276, pp. 191–196, 2000.
[12] T. Beth, D. E. Lazic, and A. Mathias, “Cryptanalysis of Cryptosystems Based on Remote
Chaos Replication”, New York: Springer-Verlag, 1994.
Authors
Mina Mishra, is pursuing Ph.D. (Engg) from Nagpur University, Maharashtra, India.
She received M.E. degree specialization in communication in the year 2010. Her
research area covers chaotic systems, chaotic cryptology, network security and secure
communication.
Vijay H. Mankar received M. Tech. degree in Electronics Engineering from VNIT,
Nagpur University, India in 1995 and Ph.D. (Engg) from Jadavpur University,
Kolkata, India in 2009 respectively. He has more than 16 years of teaching
experience and presently working as a Lecturer (Selection Grade) in Government
Polytechnic, Nagpur (MS), India. He has published more than 30 research papers in
international conference and journals. His field of interest includes digital image
processing, data hiding and watermarking.

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