TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 82
2
9
~ 823
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.48
53
8229
Re
cei
v
ed
No
vem
ber 1
9
, 2013; Re
vi
sed
Octob
e
r 18, 2
014; Accepte
d
No
vem
ber
6, 2014
A New Image E
n
cryption Algorithm Based on
Two-
dimensional Coupled Chaotic Map
Li Tu*
1
, Li
y
u
an Jia
2
, Chi Zhang
3
, Saiqiu Guo
4
Schoo
l of Information Sci
enc
e and En
ge
erin
g,Hun
an Cit
y U
n
iversit
y
,
Yi
yan
g
, Hun
a
n
4130
00, Ch
ina
,
086-07
37
635
312
8
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: tulip1
9
0
3
@1
63.com
1
, jsjcar
ol@1
26.com
2
A
b
st
r
a
ct
In this pap
er, a kind of two-di
me
nsi
ona
l c
oup
led c
haoti
c
transcend
ent
al map (T
CCT
M) w
a
s
prop
osed. F
i
rs
tly, by usi
n
g
the T
CCT
M c
h
aotic s
equ
enc
es w
e
re
gen
er
ated,the
n
the
chaotic
seq
u
e
n
ce
s
w
e
re mo
difie
d
to gener
ate ch
aotic key
strea
m
that is more
suitab
le for
i
m
age e
n
cryptio
n
.
In the process of
encrypti
o
n
，
a
n
ori
g
i
nal
col
o
r i
m
a
g
e
w
a
s dec
o
m
pos
ed
into
three
i
m
a
ges
of re
d
,
gree
n a
n
d
blu
e
compo
nents, a
nd encry
pted the
m
in a differ
ent w
a
y
respe
c
tively. T
he experi
m
e
n
tal res
u
lts de
mo
nstra
t
e
that the extre
m
ely se
nsitive
to
the key, th
e e
n
crypt
ed
i
m
a
g
e
has
ran
d
o
m
-l
ike d
i
stributi
on
beh
avior
of gr
e
y
valu
es, the
ad
jace
nt pix
e
ls
h
a
ve
z
e
ro c
o
-c
orrelati
on
pro
p
e
rties. F
u
rther
mor
e
, the
al
g
o
rith
m s
how
s
the
adva
n
tag
e
s of large key sp
ace
and hi
gh sp
ee
d of encryptio
n
.
Ke
y
w
ords
:
tw
o-dime
nsi
o
n
a
l cou
p
l
ed ch
aotic transc
e
n
denta
l
eq
uati
o
n; positi
on scr
am
bli
ng, se
nsi
t
ivity,
Imag
e encry
pti
o
n
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Due
to the
ch
ara
c
teri
stics
of ea
sy-u
nde
rst
andi
ng and
attractive pre
s
entatio
n,
mu
ltimedia
conte
n
ts
su
ch as imag
e
and a
udio, h
a
ve bee
n wi
dely tran
smit
ted in Intern
et and m
obil
e
comm
uni
cati
ons.
peo
ple
can o
b
tain, u
s
e or p
r
o
c
e
s
s
digital ima
g
e
s
more freque
ntly. Since di
gital
media
su
ch
a
s
ima
ge, a
udi
o, and
video
are
ea
sy
to p
r
ocess,
copy
and tran
sfer,
the eme
r
g
e
n
c
e
of powerful to
ols raises
a serie
s
of p
r
obl
ems. It
ha
s b
e
com
e
e
s
s
e
n
t
ial t
o
se
cur
e
inf
o
rmat
io
n f
r
om
leakage
s. Ma
ny peopl
eha
s done
research
of this
area an
d o
b
tai
ned m
any a
c
hievement
s [
1
].
Some
cla
ssi
c en
cryption
te
chni
que
s
su
ch a
s
opti
c
al t
r
ansfo
rm
s a
n
d
ch
aotic ma
p
s
h
a
ve b
e
co
me
a vital rol
e
i
n
protectin
g
image
s d
ue
to the
in
crea
sing
re
qui
re
ment for ima
ge
stora
ge
a
nd
transmiss
ion [2-7].
Cha
o
s i
s
a p
a
rticul
arly int
e
re
sting n
on-l
i
near
effect. Cha
o
s the
o
ry
has b
een
establish
e
d
sin
c
e 197
0s by
many different re
sea
r
ch
area
s, such
as
physi
cs,
m
a
thematics, e
ngine
erin
g,
a
nd
biology, etc [
8
]. Becau
s
e
of the ch
ara
c
ters of
n
o
n
-
p
e
riodi
city, no
n-conv
e
r
ge
nce, ergo
dicity,and
high
sensitivity to initial conditions, whi
c
h i
s
related to cryptos
y
s
t
em,
c
h
aos
is
us
ed for
cryptolo
gy. Several ap
pro
a
ch
es a
r
e
seen in the
literatu
r
e that applie
s to co
nce
p
ts from
the
cha
o
t
i
c sy
st
e
m
s.
In recent yea
r
s, a va
riety of cha
o
s-ba
s
ed imag
e cryptosyste
m
s h
a
ve been
stu
d
ied. In
[9], a hype
rchaotic en
cryp
tion sch
e
me
i
s
p
r
e
s
ent
e
d
. The dra
w
b
a
cks
such
a
s
small key spa
c
e
and wea
k
se
curity of low-dimen
s
ion
a
l maps, hi
g
h
-d
imensi
onal
chaotic
syste
m
s we
re u
s
e
d
in
crypto
system
s. To meet the requi
rem
e
n
t
s of moder
n appli
c
ation
s
with high lev
e
ls of se
cu
rity, a
kind of
two
-
dimen
s
ion
a
l cou
p
led ch
a
o
tic
tra
n
sce
n
dental
m
ap (TCCTM)
i
s
prop
osed
in
this
pape
r, And it wa
s used in i
m
age en
cryption
2. Chaos Mo
del
2.1. Transc
enden
t
al
Equation
Functio
n
2.1
is a
tra
n
scen
dental
equ
ation, Feig
enb
a
u
m ha
s studi
ed its bifurcat
ion a
n
d
cha
o
tic chara
c
teri
stics, and
made its corresp
ondi
ng fig
u
re.
1,2,3,...n
k
),
asin(
π
s
x
k
1
k
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8229 – 82
37
8230
Here p
a
ra
me
ter a i
s
a n
on-n
egative
real n
u
mbe
r
,
from a
n
y initial value,
1
,
0
k
x
,
selected the i
n
itial values of x
0
=0.1234
a
nd a
=
3, Fig
u
re 1 is the
sc
a
tter plot of a
transce
nde
ntal
equatio
n. Figure 1 sho
w
s t
hat:
(1) When
pa
rameter
319
.
0
,
0
.
0
a
, no
matter
what
i
n
itial value
s
we
ch
oo
se, t
he final
result will be
close to 0;
(2) Wh
en
p
a
ram
e
ter
732
.
0
,
319
.
0
a
, the final result
will be
close to A non-zero
numbe
r,Thi
s
is a stabl
e sin
g
le value ;
(3) Whe
n
pa
ramete
r
856
.
0
,
732
.
0
a
, the
function
curv
e gets into two bran
che
s
,
the
iterative value x falls between two fixed
values, a sol
u
tion of perio
d 2;
(4) Whe
n
parameter
1
,
856
.
0
a
,it is a chaoti
c
map
p
ing;
(5) When
pa
ramete
r
1
a
, the iterative
re
sults may fal
l
in any
sub
-
interval of th
e
interval (-a,a) rando
mly, and it may be repe
ated.
This i
s
the e
r
godi
city of chao
s. With the
increa
sing of
para
m
eter a,
the map
app
e
a
rs bl
an
k win
dows pe
riodi
cally.
Figure 1. Bifurcatio
n and Bl
ank
wind
ow f
o
r
transce
nde
ntal equatio
n (a
=0:3,x
1
=
0
.1234)
Figure 2. Bifurcatio
n for the
improved
transce
nde
ntal equatio
n a
=
0.2, b=0.21,
x
1
=0.12, y
1
=0
.31, r
1
= r
2
=0:2
2.2.
Impro
v
ed T
w
o
-
dimensional Tran
scend
e
ntal Equation
A one-dim
e
nsio
nal equ
ation can g
enerate
ch
a
o
tic se
quen
ce thro
ugh
iterative
cal
c
ulatio
n, b
u
t
its key
spa
c
e i
s
ge
nerall
y
small, and i
t
s
se
cu
rity is
not high. Fo
r
this proble
m
,
we
prop
osed
a
n
improve
d
two
-
dime
nsi
onal cou
p
led ch
ao
tic tran
scen
d
ental ma
p, its math
ematical
expre
ssi
on is:
n
n
2
n
1
n
n
n
1
n
1
n
x
y
r
)
bsin(
π
s
3
y
x
y
r
)
asin(
π
s
3
x
(2)
Whe
r
e
)
12
,
12
(
,
),
2
,
0
(
,
),
1
,
0
(
,
2
1
y
x
r
r
b
a
.Took the initial values of
31
.
0
,
12
.
0
1
1
y
x
Th
e
bifurcation of the improve
d
transce
nde
ntal equatio
n is sho
w
n in Fig
u
re 2
：
(1) When
p
a
ram
e
ter
3
.
0
,
0
.
0
a
, the chaoti
c
mappin
g
con
v
erge
s to
a
non
ze
ro
numbe
r,it is called a fixed point, and it is a stabl
e sin
g
le value;
(2)
W
hen
pa
ramete
r
6
.
0
,
3
.
0
a
, the
functio
n
cu
rve gets in
to
two branches
,it is a
state of perio
d 2;
(3) Whe
n
pa
ramete
r
8
.
0
6
.
0
a
, the
chaoti
c
map
p
ing ap
pea
r
cha
o
tic state
mainly,
and it appe
ars blan
k wi
ndo
ws too;
(4) Wh
en
p
a
r
a
meter
8
.
0
a
, the cha
o
tic ma
pp
ing ge
nerate
s
a
stable
si
ngle value. It
doe
sn't
hav
e cha
o
s cha
r
a
c
t
e
rist
ic
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A New Im
age
Encryption A
l
gorithm
Base
d on Two
-
dim
ensi
onal
Cou
p
led Chaoti
c
Map (Li Tu
)
8231
3. Encr
y
p
tion
Algorith
m
and Decry
p
tion Schem
e
3.1. Encr
y
p
tion
Algorith
m
The en
cryptio
n
step
s are a
s
follows:
(1) Read
a
size
of 25
6*2
56 pixel
s
col
our
imag
e, calcul
ated it
s
red,
gre
en, a
nd bl
u
e
comp
one
nts,
saved it
s val
ue in th
ree t
w
o-dim
e
n
s
i
ona
l arrays
re
sp
ectively, then
conve
r
ted th
em
to 3 length of 256*25
6 one-dime
nsio
nal se
quen
ce
. Throu
gh i
t
erative cal
c
ulation from
the
improve
d
ch
aotic eq
uatio
n, it generat
ed two one
-dimen
sion
al arrays, they are na
med a
rray
B(Formed fro
m
x series) a
nd array I (Forme
d from
y serie
s
), their length are 2
56*25
6. In order
to incre
a
se t
he difficulty of the
ciph
ertext, took th
e first, the
si
xth and
the
fifth digit of t
he
element
s in
a
rray B afte
r th
e de
cimal
poi
nt to form a t
h
ree
-
di
git nu
mber, h
ad it o
n
256
rem
a
in
der
operation, an
d we got seq
uen
ce L1;
(2) First
we
encrypted
th
e ima
ge
of the
red
comp
onent, h
a
d
its valu
e o
n
a
rray
L1
remai
nde
r op
eration, conv
erted it to a 2-dime
nsi
onal
seq
uen
ce;
(3) T
hen
we
encrypted the imag
e of the gr
ee
n compon
ent,bui
lt a two-dim
ensi
ona
l
matrix M1, its column
len
g
th is 2, a
n
d
its line le
ngt
h is 6
553
6(2
56* 25
6). Put
the elem
ent
s of
array I on the
first ro
w of the matrix P, elem
ent
s of g
r
een
co
mpon
ent on the
se
con
d
line, the
s
e
numbe
rs 1,2,3...256*256
o
n
the se
co
nd
line, the two-
dimen
s
ion
a
l matrix p is al
so the d
e
cryp
tion
matrix. Then
sorte
d
the e
l
ements in th
e array I, th
at sort the first line of ma
trix P, took the
se
con
d
line
of so
rted
ma
trix P1, we
got a
one
-di
m
ensi
onal
seque
nce
D1.
The
po
sitio
n
of
element
s in g
r
een
com
pon
ent seq
uen
ce
has chan
ged
following the
element
s in chaotic a
r
ray I;
(4) We had a
doubl
e encryption on the i
m
age
of
the
blue
com
ponent,first it made a
gray
encryption
(
th
e method
is t
he same a
s
t
he en
cryptio
n
algorith
m
of
red
com
pon
ent),then m
a
de a
positio
n en
cryption(the me
thod is the sa
me as t
he en
cryption al
go
rithm of green
compo
nent
).
3.2. Decry
p
tion
Scheme
(1)
Rea
d
the
s
e en
cryped
image
s of th
e red, g
r
ee
n
and blu
e
co
mpone
nt,
sa
ved their
value in th
ree
two-dimen
s
i
onal a
r
rays
resp
ectively, then
conve
r
te
d them to
3 l
ength of
256*
256
one-dime
nsio
nal array A1, A2 and A3;
(2) T
he de
cry
p
tion sceme
of the red co
mpone
nt ima
ge wa
s to ha
ve its value on array
L1 rem
a
ind
e
r operatio
n, then co
nvert it to a 2-dim
e
n
s
i
onal sequ
en
ce;
(3) Th
e de
cryption scem
e of the gre
en co
m
pon
e
n
t image: Built a two-di
mensi
onal
matrix E, put the eleme
n
ts of a
rray I
on the
first row of the m
a
trix P, put these num
be
rs
1,2,3...256*2
56 on the se
cond line,
then
sorted the el
ements in the
array I, took the second li
ne
of sorte
d
mat
r
ix E, we got
a one
-dime
n
sional sequ
en
ce Q. Built a t
w
o-dime
nsio
nal matrix K, put
the elem
ents
of array A2 o
n
the first ro
w of the matrix
K,and put th
e elem
ents
of array Q o
n
the
se
con
d
ro
w o
f
the matrix K, sorted th
e e
l
ements i
n
th
e se
quen
ce
Q, took the fi
rst line
of so
rted
matrix K, con
v
erted it to
a
2-dim
e
n
s
iona
l matr
ix G,
m
a
trix G i
s
th
e
decrypted i
m
age
of the
green
comp
one
nt;
(4) Th
e d
e
cryption
scem
e of th
e bl
u
e
comp
one
n
t
image: fi
rst we
mad
e
a g
r
ay
decryption (t
he metho
d
is the sam
e
a
s
the de
cryption algo
rithm
of red comp
onent), the
n
we
made a
po
si
tion de
crypti
on(the
metho
d
is the
sam
e
as th
e de
cryption
algo
rithm of gre
en
comp
one
nt);
(5) Put these three de
crypted comp
o
nent in
a 3-dimen
s
ion
a
l matrix, comp
ose
d
the
three comp
on
ents of image
to a color im
age
s.
3.3. Experimental Results
In this pape
r we u
s
ed the
doubl
e encry
ption app
roa
c
h to encrypt i
m
age
s, and the initial
value a
nd th
e
co
ntrol
pa
ra
meters
were:
23
.
0
,
22
.
0
,
9
.
0
,
01
.
1
,
2
.
0
,
1
.
0
1
1
2
1
y
x
b
a
r
r
. We
have
use
d
the
USC-SIPI imag
e datab
ase
whi
c
h i
s
refe
rre
d in
Reference [13] (freely availabl
e at
http://sipi.usc.
edu/database/). Fi
gure 3(a)
and Figure
3(b) are the color plai
n image and t
h
e
gray plain im
age.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8229 – 82
37
8232
Figure 3(a
)
. Plain image
Figure 3(b
)
. Plain gray ima
g
e
Figure 4(b, d) are the
en
cr
ypted image
s of red a
nd g
r
een
comp
one
nt, Figure
4(f, g) a
r
e
the encrypted
images of g
r
een compo
n
e
n
t.
(a) Ima
ge of the red
com
p
o
nent (b
) Encr
yped image o
f
the red com
pone
nt(c) Image of
the gree
n co
mpone
nt (d)
Encryp
ed ima
ge of the gre
en com
pon
en
t(e) Imag
e of the
blue compo
n
ent (f) The first encryped i
m
age of the b
l
ue com
pon
e
n
t (g)T
he second
encryped im
a
ge of the blue
compo
nent
Figure 4. Cip
her ima
g
e
s
The de
crypte
d image
s of t
he re
d, green
, blue
compo
nent are sho
w
n in Fi
gure
5(h, j, k),
the comp
osit
ed image i
s
shown in Figu
re 5(k).
(h)The de
crypted image of
the red comp
onent (i
) The
decrypt
ed im
age of the green
comp
one
nt (j) The de
crypte
d image of the blue compo
nent (k) Th
e comp
osite
d
image
Figure 5. De
crypted imag
e
s
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A New Im
age
Encryption A
l
gorithm
Base
d on Two
-
dim
ensi
onal
Cou
p
led Chaoti
c
Map (Li Tu
)
8233
4. Performan
ce and Secu
rit
y
Analy
s
is
All the secu
ri
ty analysis h
a
s be
en don
e on MA
TLA
B
7.0 by Intel Pentium 64 X2 Dual
Core processor 2.0G
Hz p
e
rsonal
com
p
uter.
4.1. Histo
g
r
a
m
Gray hi
stog
ram is
a fun
c
tion of grayscale,
it de
scribes th
e num
ber of
gray l
e
vels of
pixels i
n
a
n
i
m
age, a
nd it
reflect
s
th
e freque
ncy
of
g
r
ay value
in
a
n
imag
e. Its
absci
ssa i
s
gray
level, its ordi
nate is the freque
nc
y of the gray level. Figure 6
i
s
the histog
ra
m of the three
comp
one
nts.
(a)
Histo
g
ra
m
of the image of red compo
nent(a
) Hi
sto
g
ram of the i
m
age of gree
n
comp
one
nt(a
) Histo
g
ram o
f
the image of
blue com
pon
ent(d
) Hi
stogram of the
decrypted im
age of red
co
mpone
nt(e
) Hist
ogram of the decry
pted i
m
age of gree
n
comp
one
nt(f) Histog
ram of
the
decrypted image of bl
ue com
pon
en
t
Figure 6. The
histogram of the three
com
pone
nts
Figure 6
sho
w
s, b
e
fore
en
cryption th
e ri
se a
nd
fall of
the histo
g
ra
m
s
are very la
rge, the
distrib
u
tion i
s
not uniform,
and after e
n
c
ryption
th
e histog
ram of
the image of
red comp
on
ent
and g
r
ee
n
compon
ent a
r
e co
mplan
a
te, the gray value of en
crypted ima
g
e
is in
uniform
distrib
u
tion. This sh
ows that
in the
range
of (0,2
55), the
p
r
o
bability of th
e pixel valu
e in
encrypted im
age is e
qual.
The statistical cha
r
a
c
te
ri
stics of en
cryp
ted image a
r
e quite different
from that
of t
he pl
ain im
ag
e. The
stati
s
tical
ch
ara
c
te
ri
stics
of plain
image
s sp
rea
d
to
e
n
crypte
d
image
s evenl
y, this red
u
ce
s thei
r correl
ation g
r
eat
ly.while it o
n
ly made
a po
siti
on en
cryption
on
the image of gree
n co
mpo
nent, its histo
g
ram d
o
e
s
no
t chang
e.
4.2.
Correla
tion Analy
s
is of T
w
o
Adjacen
t Pixels
The sub
s
tan
t
ive characte
ristics of a
digi
tal imag
e
determi
ne t
hat there i
s
stron
g
correl
ation
a
m
ong
adja
c
e
n
t pixels.
Thi
s
co
rrelation
m
a
ke
s th
e
cont
ent of th
e ima
ge i
s
e
a
sy
to
be
identified [1
0]. We
cal
c
ul
ated the
pixel
correlation
u
s
ing the
follo
wing formula
(3
) an
d fo
rmula
(4
)
[11]:
)))
(
))(
(
((
)
,
cov(
y
E
y
x
E
x
E
y
x
(
3
)
)
(
)
(
)
,
cov(
y
D
x
D
y
x
R
xy
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8229 – 82
37
8234
Here x and
y are the gray values of
two
adja
c
e
n
t pixels in the image, E
(
x) is a
mathemati
c
al
expectatio
n
, D(x) i
s
the
varian
ce of
x
,
cov(x,y) is t
he po
pulatio
n cova
rian
ce.
In
orde
r to dest
r
oy the statistical attacking,
we mu
st
red
u
ce the corre
l
ation of adja
c
ent pixels. T
h
e
lowe
r the
co
rrelation
coefficient, the
bet
ter the e
n
cry
p
tion effect. I
n
the p
r
o
c
e
s
s of
cal
c
ulati
o
n
,
we u
s
e form
u
l
a (5-
7
).
N
i
i
x
N
x
E
1
1
)
(
(5)
N
i
i
x
E
x
N
x
D
1
2
))
(
(
1
)
(
(6)
))
(
)
(
(
))
(
(
1
)
,
cov(
1
y
E
i
y
x
E
x
N
y
x
N
i
i
(
7
)
The follo
wing
step
s are
pe
rforme
d to evaluat
e an im
a
ge’s
co
rrel
a
tion prope
rty: (1) 20
00
pixels a
r
e ra
ndomly sele
cted as
sampl
e
s, (2
) t
he
correlation
s
b
e
twee
n two
adja
c
ent pixe
ls in
hori
z
ontal, ve
rtical
or
diago
nal di
re
ctions are
cal
c
ul
ate
d
by the fo
rm
ula ab
ove. T
heir
distri
buti
on
is sh
own in Figure 7, Fig
u
re 8 and Figu
re 9.
Figure 7 is th
e correl
ation
of adjacent pi
xels of red
co
mpone
nt.
(a)
Correl
atio
n of level adjace
n
t pixels o
f
the
image of red co
mpo
n
e
n
t (b) Correla
t
ion of level
adja
c
ent pixel
s
of the encry
ped imag
e of red
comp
one
nt (c)
Co
rrel
a
tion of diago
n
a
l adja
c
ent
pixels of the image of re
d compon
ent (k) Correlatio
n o
f
diagonal a
d
j
a
ce
nt pixels o
f
the
encryped im
a
ge of red
com
pone
nt
Figure 7. Correlation of adj
ace
n
t pixels o
f
red com
pon
ent
Figure 8 is th
e correl
ation
of adjacent pi
xels of gree
n comp
one
nt.
(e)
Correl
atio
n of level adjace
n
t pixels o
f
the image of green
com
p
o
nent (f) Corre
l
ation of level
adja
c
ent pixel
s
of the encry
ped imag
e of gr
ee
n co
mpo
nent (g
) Co
rrelation of dia
gonal
adja
c
ent pixel
s
of the imag
e of green
co
mpone
nt (h)
Correl
ation of
diagon
al adj
ace
n
t pixels o
f
the encryped
image of gree
n comp
one
nt
Figure 8. Correlation of adj
ace
n
t pixels o
f
green
comp
onent
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A New Im
age
Encryption A
l
gorithm
Base
d on Two
-
dim
ensi
onal
Cou
p
led Chaoti
c
Map (Li Tu
)
8235
Figure 7 is th
e correl
ation
of adjacent pi
xels of blue compon
ent.
(i) Correlatio
n
of level adjacent pixels of the
image of b
l
ue com
pon
e
n
t (j) Co
rrelati
on of level
adja
c
ent pixel
s
of the encry
ped imag
e of blue co
mpo
n
ent (k) Co
rrel
ation of hori
z
ontal adja
c
e
n
t
pixels of the image of blue
comp
one
nt (l) Correl
atio
n o
f
horizo
n
tal a
d
jacent pixels of the
encryped im
a
ge of blue co
mpone
nt
Figure 9. Correlation of adj
ace
n
t pixels o
f
blue comp
o
nent
The mo
re o
b
v
ious the
scrambling
deg
ree of im
ag
es were, the b
e
tter the effe
ct of the
encryption i
s
. The co
rrela
t
ion amon
g the plain
i
m
a
ge pixel
s
sh
ows a line
a
r distrib
u
tion,the
correl
ation a
m
ong th
e en
crypted
imag
e pixels is
a random
di
strib
u
tion. It can
be seen
from
th
e
Figures a
bov
e that the deg
ree of imag
e scram
b
ling i
s
very significa
nt.
4.3. MSE
MSE (Me
an
Square Erro
r) is u
s
ed
to m
easure
the
pe
rforma
nc
e of encryption,
th
e
big
g
e
r
the the value of mean sq
ua
re erro
r, the better t
he effect of encryptio
n. The formul
a of MSE is:
2
11
,
,
*
1
M
i
N
i
j
i
P
j
i
D
N
M
MSE
(8)
Whe
r
e
pa
ra
meter
M, N
are
the
gray
level of i
m
a
ges,
pa
ramet
e
r
D i
s
th
e
grayscal
e of
the
encrypted im
age, an
d pa
rameter P i
s
t
he g
r
ayscale
of the plain
i
m
age.
Ta
ble
1 is th
e MSE
value
of the encryped image of the red, g
r
ee
n
and
blue
co
mpone
nts an
d their plain i
m
age
s:
Table 1. MSE value
Image
MSE
the encr
y
ped ima
ge of the r
ed co
mponent
and the plain image of red comp
o
nent
12113
the encr
y
ped ima
ge of the g
r
een
component and t
he plain image of green
component
2167.6
the encr
y
ped ima
ge of the blue co
mponent
and the plain image of blue compo
nent
17068
the decr
y
pted im
age of red com
p
onent and
the plain image of red componen
t
0
the decr
y
pted im
age of gre
en co
mponent
and the plain image of gree
n com
ponent
0
the decr
y
pted im
age of blue comp
onent and
the plain image of blue component
0
Table 2. MSE value
component
Entrop
y
of
infor
m
ation
the plain image of red
component
4070662
the encr
y
ped ima
ge of
red component
7.954878
the plain image of
green compon
en
t
5.737772
the encr
y
pted im
age of
green compon
en
t
5.737772
the plain image of blue
component
3.372893
the double encr
y
pted
image of blue
component
7.993671
4.4.
Informa
t
ion Entropy
Analy
s
is
Information
e
n
tropy i
s
on
e
of the criteri
a
to mea
s
u
r
e the
stren
g
th of a
crypto
system,
whi
c
h
wa
s firstly prop
osed
by Shanno
n
in 194
9 [
11]. Informatio
n en
tropy of imag
e de
scribe
s t
he
distrib
u
tion of
grey value [12], its formula
is:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8229 – 82
37
8236
)]
(
[
log
2
1
2
1
i
i
i
S
P
S
P
s
H
n
(9)
Whe
r
e P
(
S
j
) i
s
the
proba
bil
i
ty of symbol
S
i
, 2
n
is the
total num
be
r
of state
of inf
o
rmatio
n
sou
r
ce
S. The info
rm
ation e
n
tropy
is u
s
e
d
to
an
alyze th
e p
e
rf
orma
nce of
e
n
cryptio
n
m
e
thod.
Whe
n
th
e
image pixel i
s
unifo
rmly di
stribute
d
, the
prob
abilit
ie
s
of grey value
are b
a
si
cally
equal, e
n
tro
p
y
can
achieve
the maximu
m, it sho
w
s t
hat, the mo
re disperse
d the grey valu
e, the better the
perfo
rman
ce
of en
cryption.
A 256
level
of gray im
ag
e ha
s 2
8
kind
s of
possibl
e
pixel value
s
,
so
its ideal information entro
p
y
should be 8
.
If the
information entro
py of
a 256 level gray encrypted
image i
s
clo
s
e to 8, the
cip
her im
age
clo
s
e
s
to
the ra
ndom distri
bu
tion.
The
info
rmation entro
py
obtaine
d from
simulation ex
perim
ent is shown in Tabl
e 2.
Table 2
sho
w
s that, it made a g
r
ay e
n
cryptio
n
on
the image of
red
comp
one
nt, and it
made a dou
b
l
e encryption
on on the image of blue
compo
nent, their inform
ation entro
py had
cha
nge
d a lo
t, their performance of en
cryption m
e
th
od is very go
od, it is hard
to be de
cryp
ed.
And it mad
e
a po
sition e
n
c
ryption
on th
e imag
e
of g
r
een
com
pone
nt, its inform
a
t
ion entropy d
i
d
not cha
nge, this mea
n
s th
at, position e
n
cryptio
n
onl
y chang
es th
e positio
n of the pixel, it d
o
e
s
not cha
nge it
s inform
ation
entropy.
4.5.
Ke
y
Sp
ace An
aly
s
is
Key space si
ze is the total
number of di
ffer
ent keys t
hat can be u
s
ed in the en
cryption
[13]. There
a
r
e six p
a
ra
m
e
ters i
n
the i
m
prove
d
cha
o
tic eq
uation,
in theory, th
e key
spa
c
e
o
f
each pa
rame
ter is 10
14,
due to the a
c
tual p
r
e
c
isi
o
n of comp
uter, the key spa
c
e of e
a
c
h
para
m
eter
was 10
6
, so, the key sp
ace
of the two-di
m
ensi
onal
co
upled chaoti
c
map is 1.0*1
0
36
.
It has obvio
us
sup
e
rio
r
it
y, and it is
easi
e
r to
im
plement the
algorith
m
by
usin
g ha
rd
ware.
Simulation
re
sults sho
w
th
at, even un
d
e
r the
conditi
on of
existing
com
pute
r
p
r
eci
s
ion, th
e key
spa
c
e
is la
rg
e en
oug
h. An
d 10
35
=21
1
7
,
it mean
s that
, an
attacker
need
s
a 1
1
7
-
bit co
mpute
r
to
decode
the al
gorit
hm. If he
u
s
e th
e
violen
ce
attack method
s,
10
36
=2
117
/365
/24/60/60/2.6
G
=1.2
192*
10
18
, it means, if an attacker de
co
de th
e algo
rithm
by
usin
g a 2.6G
HZ freq
uen
cy
of compute
r
, he nee
ds 1.2
192*1
0
18
years.
5. Conclusio
n
In this
wo
rk
a kin
d
of t
w
o-dim
e
n
s
iona
l cou
p
led
ch
aotic m
ap
b
a
se
d on
Fei
genb
aum
transce
nde
ntal eq
uation
is propo
sed,
th
e be
havior of
this method
is
s
i
milar to the
s
u
bs
titution
box like en
cryption algorit
hms. The
results sh
ow tha
t
the encrypti
on algo
rithm is ea
sy to real
ize,
the pixels
of
encrypted im
age h
a
s
ch
aracteri
stics
of statistical di
st
ribution, a
nd
the algo
rithm
is
sen
s
itive e
n
o
ugh to
the
ke
ys, The
key space i
s
la
rg
e
enou
gh, the
correl
ation
of adja
c
e
n
t pix
e
ls
of encrypte
d
image
s is cl
o
s
e to 0, the algorit
h
m
is
more
se
cure and he
nce more suitable f
o
r
image e
n
cryp
tion for appli
c
ations. As fut
u
re
work,
the
diffusion effici
ency of this a
l
gorithm n
eed
s
to be improve
d
.
Ackn
o
w
l
e
dg
ements
This
work
was supp
orte
d
by Scientific
Re
se
arch
Fund of Hu
nan Provin
ci
al Scien
c
e
Dep
a
rtme
nt unde
r G
r
ant
(No: 2
013
FJ3087, 2
013F
J3
0
86, 20
12
FJ43
29
), Sci
entific Resea
r
ch
Fund of Hu
na
n Provinci
al Education De
p
a
rtment un
de
r Gra
n
t (No: 12C0
573
).
Referen
ces
[
1
]
Liu YJ,
Che
n CPL,
Wen GX,
T
ong SC.
Ada
p
t
i
ve ne
ural o
u
t
put
f
eedb
ack t
r
ackin
g
cont
rol
f
o
r a class of
uncert
a
in discr
et
e-t
i
me no
nli
n
ear s
y
st
ems.
I
EEE Trans.
Neural N
e
t
w
.
2011;
22:
116
2–1
1
67.
[
2
]
Li H,
Wang Y
.
I
n
f
o
rmat
ion
securit
y
s
y
st
e
m
base
d
on it
erat
ive mu
lt
ip
l
e
-ph
a
se ret
r
i
e
val in
g
y
rat
o
r
doma
i
n.
Opt
Laser T
e
chn
o
l
.
200
8;
40:
962
–
966.
[
3
]
Meng
XF
,
Cai
LZ
,
Wang YR,
Yang
XL,
Xu
XF
,
Dong GY,
et
al.
Digit
a
l ima
ge s
y
nt
h
e
sis a
nd mult
i
p
l
e
-
imag
e encr
y
pt
i
on b
a
se
d on
p
a
ramet
e
r m
u
lt
i
p
le
xi
ng
an
d ph
ase-sh
if
t
i
ng i
n
t
e
rf
eromet
r
y
.
O
p
t
Lasers E
n
g
.
200
9;
47:
96–
1
02.
[
4
]
Liu Z
J
,
Guo Q,
et
al.
Doub
le
image
encr
y
pt
ion b
y
usin
g it
erat
ive ra
nd
om
binar
y
enc
odi
ng in
g
y
rat
o
r
doma
i
ns.
Opt
Express
.
201
0;
18:
120
33–
12
043.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A New Im
age
Encryption A
l
gorithm
Base
d on Two
-
dim
ensi
onal
Cou
p
led Chaoti
c
Map (Li Tu
)
8237
[
5
]
Jin J.
An ima
g
e
encr
y
pt
ion
b
a
sed
on e
l
eme
n
t
a
r
y
c
e
ll
ul
ar a
u
t
o
mat
a
.
Opt Lasers Eng
.
201
2;
50:
18
36
–
184
3.
[
6
]
Wang Y,
Wong
K W,
Liao X,
et
al.
.
A ne
w
c
haos-
base
d
f
a
st
image encr
y
pt
ion a
l
gor
it
hm
.
Applie
d Sof
t
Co
mp
ut
ing.
2
0
11;
11(1):
5
14-
522.
[
7
]
Liu
Z
,
Li
S,
Li
u
W,
Wang Y,
Liu
S.
I
m
ag
e e
n
cr
ypt
i
o
n
a
l
g
o
r
i
t
h
m b
y
usi
n
g
f
r
act
i
on
al F
o
uri
e
r t
r
ansf
o
r
m
and
pi
xe
l scra
m
blin
g o
perat
i
on b
a
sed
on
d
oub
le ra
nd
om
phas
e enc
od
in
g.
Opt
Lasers
Eng
.
20
13;
5
1
:
8–1
4.
[
8
]
R Bro
w
n,
LO
Chu
a.
Clar
if
yin
g
cha
o
s:
e
x
am
ples
an
d cou
n
t
e
re
xam
p
les.
I
n
t
.
J.
Bif
u
rcat
Chaos.
19
96
; 6
:
219
–2
42.
[
9
]
Belmo
uho
ub I
,
Djemai M,
Barbot
JP.
Cr
ypt
ogr
aph
y
b
y
discr
et
e-t
i
me h
y
p
e
rcha
ot
ic s
y
st
ems.
Procee
din
g
s of
42nd I
EEE Co
nf
erenc
e on D
e
cisio
n
an
d Co
nt
rol.
200
3;
2:
190
2-19
07.
[
10]
I
q
t
adar Huss
a
i
n
,
T
a
riq Shah
Muhamm
ad A
s
if
Gondal,
Ha
san Mahm
ood.
A novel ima
g
e
encr
y
pt
i
o
n
alg
o
rit
h
m
bas
e
d
o
n
ch
aot
ic
m
aps
an
d GF
(2
8
) e
x
p
one
nt
t
r
an
sf
ormat
i
on.
No
nlin
ear
Dyn
.
20
13;
7
2
:
3
99-
406.
[
11]
Shan
no
n CE.
Commun
i
cat
i
o
n
t
heor
y of
sec
r
ec
y
s
y
st
em.
Bell Syst Tech J.
1949;
2
8
:
656-
715.
[
12]
Wei Z
hang,
Kw
o
k
-
w
o Won
g
,
Hai Yu ,
Z
h
i-liang Z
hu.
An i
m
age e
n
cr
ypt
i
on schem
e usi
ng revers
e 2
-
dime
nsio
nal c
h
aot
ic map
an
d
dep
end
ent
d
i
f
f
u
sio
n
.
Co
mmu
n
Non
li
near Sc
i Nu
mer Si
mul
a
t
.
2013;
1
8
:
206
6-20
80.
[
13]
Shub
o Li
u,
Jin
g
Sun,
Z
hen
g
qua
n Xu.
An I
m
prove
d
I
m
ag
e Encr
ypt
i
o
n
Algorit
hm bas
ed on C
haot
ic
Sy
s
t
e
m
.
Jour
n
a
l of
Co
mp
ut
er
s.
2009;
4(1
1
):
109
1-11
00
Evaluation Warning : The document was created with Spire.PDF for Python.