Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
21,
No.
2,
September
2021,
pp.
1103
1112
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v21i2.pp1103-1112
r
1103
An
image
encryption
algorithm
with
a
no
v
el
chaotic
coupled
mapped
lattice
and
chaotic
image
scrambling
technique
Behrang
Chaboki,
Ali
Shakiba
Department
of
Computer
Science,
V
ali-e-Asr
Uni
v
ersity
of
Rafsanjan,
Iran
Article
Inf
o
Article
history:
Recei
v
ed
Jun
13,
2020
Re
vised
Aug
11,
2020
Accepted
Aug
21,
2020
K
eyw
ords:
Chaotic
coupled
lattice
mapping
Chaotic
image
encryption
Chaotic
image
scrambling
ABSTRA
CT
In
this
paper
,
we
b
uild
a
no
v
el
chaotic
coupled
lattice
mapping
with
positi
v
e
L
ya-
puno
v
e
xponent,
and
introduce
a
no
v
el
chaotic
image
scrambling
mechanism.
Then,
we
propose
a
chaotic
image
encryption
algorithm
which
uses
the
introduced
chaotic
coupled
lattice
mapping
to
apply
permutation
by
iterati
v
ely
applying
the
introduced
chaotic
image
scrambling
mechanism,
and
dif
fusing
the
pix
el
v
alues.
W
e
use
a
sorting
approach
rather
than
quantizing
the
chaotic
floating-point
v
alues
to
construct
the
dif
fu-
sion
matrix.
W
e
also
study
the
security
of
the
proposed
algorithm
concerning
se
v
eral
security
measures
including
brute-force
attack,
dif
ferential
attack,
k
e
y
sensiti
vity
,
and
statistical
attacks.
Moreo
v
er
,
the
proposed
algorithm
is
rob
ust
ag
ainst
data
loss
and
noise
attacks.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Ali
Shakiba
Department
of
Computer
Scince
V
ali-e-Asr
Uni
v
ersity
of
Rafsanjan,
Iran
Email:
ali.shakiba@vru.ac.ir
1.
INTR
ODUCTION
In
the
current
w
orld,
with
the
staggering
speed
of
technology
de
v
elopment
in
the
era
of
digital
com-
puting
and
with
the
widespread
application
of
the
internet
in
our
life,
the
application
of
digital
color
images
become
more
and
more
ine
vitable.
F
or
instance,
the
application
of
digital
images
in
medical
imaging
[6],
social
and
personal
li
fe,
military
and
other
applications
are
almost
clear
to
e
v
eryone.
So,
in
some
situations,
such
as
medical
images
and
military
applications,
the
concept
of
pri
v
ac
y
and
security
become
one
of
the
important
challenges.
One
strate
gy
for
solving
this
problem
is
to
use
some
encryption
techniques
so
that
the
image
becom
es
unreadable
for
an
unauthorized
person.
F
or
approaching
this
problem,
man
y
encryption
techniques
ha
v
e
been
proposed,
such
as
compressi
v
e
sensing
[1],
quantum
theory
[2],
DN
A
coding
[3],
transform
domains
[4],
ma-
trix
transforms
[5]
and
chaotic
systems
[7–9].
In
the
last
20
years,
the
chaotic
system
encryption
algorithms
ha
v
e
been
the
attention
of
researchers
due
to
its
immense
inherent
aspects
such
as
initial
criteria
sensiti
vity
,
un-
predictability
,
and
pseudorandomness.
Ho
we
v
er
,
image
encryption
techniques
that
are
solely
based
on
chaotic
systems,
ha
v
e
been
sho
wn
to
be
vulnerable
ag
ainst
cipherte
xt-only
and
chosen
plainte
xt
attacks.
Thus,
the
per
-
formance
of
a
chaotic
system
and
structure
of
encrypt
ion
algorithm
plays
an
important
role
in
the
resistance
of
an
encryption
scheme
to
the
common
attacks.
By
combining
an
appropriate
chaotic-based
system,
in
terms
of
initial
sensiti
vity
,
unpredictability
,
and
randomness
with
proper
techniques
of
confusion
and
dif
fusion,
a
rob
ust
encryption
algorithm
will
be
acquired.
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1104
r
ISSN:
2502-4752
The
rest
of
the
paper
is
or
g
anized
as
follo
ws:
In
Section
2,
we
firs
t
construct
a
no
v
el
chaotic
coupled
mapped
lattice
using
tw
o
one-dimensional
chaotic
mappings
and
study
its
chaotic
properties.
Then,
we
gen-
eralize
an
image
scrambling
method
and
mak
e
it
chaotic.
These
tw
o
primiti
v
es
are
combined
to
propose
an
image
encryption
algorithm.
The
security
of
the
proposed
method
is
studied
with
respect
to
se
v
eral
security
measures
in
Section
3.
Finally
,
a
conclusion
is
dra
wn
in
Section
4.
2.
METHOD
In
this
paper
we
use
the
logistic
mapping
in
combination
with
the
Chebyshe
v
mapping
of
the
first
type,
which
is
abbre
viat
ed
as
Chebyshe
v
mapping.
These
mappings
are
combined
with
coupled
mapped
lattices,
or
CML
for
s
hort,
to
construct
a
no
v
el
chaotic
mapping
with
greater
chaotic
performance.
The
logistic
mapping
is
a
polynomial
defined
as
x
i
+1
=
L
(
;
x
i
)
=
x
i
(1
x
i
)
where
x
i
2
[0
;
1]
for
i
=
0
;
1
;
:
:
:
and
2
(0
;
4]
.
This
mapping
sho
ws
a
chaotic
beha
vior
for
2
(3
:
569
945
6
;
4]
and
this
beha
vior
impro
v
es
as
gets
closer
to
4
.
The
Chebyshe
v
mapping
is
another
chaotic
mapping
which
is
defined
mathematically
as
T
n
(
x
)
=
cos
(
n
arccos(
x
))
where
x
2
[
1
;
1]
.
Moreo
v
er
,
there
is
an
equi
v
alence
recurrence
relation
used
to
define
Chebyshe
v
mappings
as
T
n
(
x
)
=
2
xT
n
(
x
)
T
n
1
(
x
)
with
initial
conditions
T
0
(
x
)
=
T
1
(
x
)
=
1
.
The
CML
originally
uses
the
logistic
mapping
to
generate
sequences
with
the
follo
wing
relation
X
(
i
)
n
+1
=
(1
"
)
f
[
x
n
(
i
)]
+
"
2
f
f
[
x
n
(
i
+
1)]
+
f
[
x
n
(
i
1)]
g
where
"
is
the
coupling
parameter
and
f
[
x
]
denotes
the
logistic
map
[10].
In
this
paper
,
we
propose
a
no
v
el
chaotic
map
with
e
xcellent
chaotic
beha
vior
using
the
CML
with
the
logistic
and
the
Chebyshe
v
mappings.
This
no
v
el
mapping
is
defined
as
y
(
i
)
n
+1
=
(1
"
)
L
;
T
n
+1
x
(
i
)
+
"
2
L
;
T
n
+1
x
(
i
+1)
+
L
;
T
n
+1
x
(
i
1)
where
the
sequences
x
(
i
)
j
are
computed
as
follo
ws:
-
A
random
initial
v
alue
s
0
2
(
1
;
1)
is
generated
-
A
sequence
of
chaotic
v
alues
s
i
=
T
`
(
s
i
1
)
are
generated
for
i
=
1
;
:
:
:
;
m
+
m
0
-
W
e
use
the
last
m
v
alues
of
this
sequence
as
the
sequence
x
0
,
i.e.
x
(
i
)
0
=
s
m
0
+
i
for
i
=
1
;
:
:
:
;
m
and
x
i
-
F
or
n
1
,
we
set
x
(
i
)
n
=
y
(
i
)
n
1+max
y
(
i
)
n
,
i.e.
we
normalize
the
sequence
y
n
to
f
all
within
the
range
[
1
;
1]
The
proposed
chaotic
system
has
good
chaotic
properties,
as
it
can
be
observ
ed
in
Fig.
1
which
illustrates
its
the
L
yaponuv
e
xponent.
Note
that
the
L
yapuno
v
e
xponent
describes
the
rate
of
the
con
v
er
gence
or
di
v
er
gence
of
trajectories
and
positi
v
e
v
alues
of
L
yapuno
v
e
xponent
sho
w
chaotic
beha
vior
[11].
Figure
1.
Bifurcation
diagram
of
the
proposed
mapping
with
starting
point
x
0
=
1
10
5
and
n
2
[0
;
1
10
5
]
.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
September
2021
:
1103
–
1112
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1105
Ne
xt,
we
describe
our
chaotic
permutation
algorithm
which
is
a
chaotic
e
xtension
to
an
im
age
sc
ram-
bling
procedure,
which
is
introduced
in
[12].
First
of
all,
the
original
image
scrambling
algorithm
of
[12]
is
as
follo
ws.
The
input
to
the
original
image
scrambling
algorithm
is
an
image
with
a
starting
pix
el,
I
P
.
As
sho
wn
in
Fig.
2
(a),
suppose
that
the
start
position
(
S
P
)
for
scrambling
of
the
image
is
the
point
(5
;
3)
,
pix
el
with
v
alue
35
.
Based
on
this
point,
we
di
vide
the
image
into
sub-blocks
Fig.
2
(b).
After
this
partitioning,
those
sub-blocks
are
con
v
erted
into
linear
sequences
by
the
follo
wing
procedure.
F
or
the
S
B
1
and
S
B
2
,
we
scan
them
do
wn
column-wise
from
left
to
right
and
right
to
left,
respecti
v
ely
.
The
S
B
3
and
S
B
4
are
scanned
upw
ard
column-wise
from
left
to
right
and
right
to
left,
respecti
v
ely
.
The
S
B
5
is
scanned
from
left
to
right
in
a
column-wise
f
ashion
and
the
S
B
6
is
scanned
from
top
to
bottom,
in
a
ro
w-wise
f
ashion.
This
is
illustrated
in
Fig.
2
(c).
Then,
all
of
the
arrays
are
concatenated
as
I
P
,
S
B
6
,
S
B
5
,
S
B
4
,
S
B
3
,
S
B
2
and
S
B
1
to
form
a
single
array
.
Finally
,
the
permuted
image
is
re-constructed
from
this
array
in
a
ro
w-wise
f
ashion
Fig.
2
(d).
T
o
generalize
this
scrambling
algorithm
and
mak
e
it
chaotic,
we
mak
e
the
follo
wing
changes:
-
As
we
iterat
i
v
ely
apply
this
image
scrambling
technique
for
se
v
eral
times,
we
add
an
inde
x
v
alue
i
starting
from
zero.
-
The
ro
ws
and
the
columns
of
each
of
the
submatrices
S
B
1
to
S
B
8
are
permuted
based
on
the
chaotic
v
ectors
R
and
C
,
respecti
v
ely
.
T
o
be
precise,
at
each
iterati
on
i
,
the
ro
ws
and
columns
of
a
sub-matrix
S
B
j
of
size
r
(
j
)
c
(
j
)
are
permuted
with
respect
to
the
non-decreasing
order
of
the
numbers
R
[
i
:
i
+
r
(
j
)
]
and
R
[
i
:
i
+
c
(
j
)
]
,
respecti
v
ely
.
-
Then,
if
we
are
in
an
e
v
en
iteration,
we
con
v
ert
S
B
1
and
S
B
6
sub-matrices
to
v
ectors
by
appending
their
ro
ws,
and
con
v
ert
S
B
3
and
S
B
8
sub-matrices
to
v
ectors
by
appending
their
columns.
Moreo
v
er
,
in
e
v
en
iterations,
we
combine
S
B
1
through
S
B
8
and
the
starting
pix
el
to
obtain
I
P
.
In
odd
iterations,
we
con
v
ert
S
B
1
and
S
B
6
to
v
ectors
column-wise
and
S
B
3
and
S
B
8
to
v
ectors
ro
w-wise.
Moreo
v
er
,
the
I
P
is
constructed
by
concatenating
S
B
1
,
S
B
4
,
S
B
6
,
S
B
2
,
S
B
7
,
S
B
3
,
S
B
5
,
S
B
8
,
and
the
starting
pix
el.
1
2
9
10
17
18
25
26
3
11
19
27
4
5
6
7
8
12
13
14
15
16
20
21
22
23
24
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
49
50
57
58
43
51
59
44
45
46
47
48
52
53
54
55
56
60
61
62
63
64
5
3
1
7
6
8
4
2
8
1
5
7
3
2
6
4
C(
i
)
R(
i
)
S
B
1
S
B
2
S
B
3
S
B
4
S
B
6
18
17
10
9
2
1
26
25
23
15
7
31
22
14
6
30
20
12
4
28
24
16
8
32
21
13
5
29
63
62
60
64
61
47
46
44
48
45
55
54
52
56
53
58
42
50
57
41
49
27
3
11
19
37
40
36
38
39
59
43
51
34
33
S
B
1
S
B
2
S
B
3
S
B
4
S
B
5
S
B
6
(a)
(b)
18
17
10
9
2
1
26
25
23
15
7
31
22
14
6
30
20
12
4
28
24
16
8
32
21
13
5
29
63
62
60
64
61
47
46
44
48
45
55
54
52
56
53
58
42
50
57
41
49
27
3
11
19
37
40
36
38
39
59
43
51
34
33
35
(c)
S
B
5
Figure
2.
Generalized
chaotic
image
scrambling
technique.
No
w
,
we
are
ready
to
describe
the
proposed
encryption
algorithm.
T
o
be
concrete,
the
v
ariables
used
in
the
description
of
the
proposed
algorithm
are
as
follo
ws:
-
"
denotes
the
Coupling
parameter
in
the
CML
-
is
the
logistic
mapping
parameter
in
the
proposed
CML
-
`
is
the
Chebyshe
v
mapping
parameter
to
generate
initial
sequence
x
0
-
s
0
is
the
initial
v
alue
used
to
generate
initial
sequence
x
0
-
m
0
is
the
number
of
initial
iterations
of
the
Chebyshe
v
mapping
in
generating
initial
sequence
to
a
v
oid
transient
ef
fect
-
m
is
the
length
of
sequences
generated
at
each
le
v
el
-
n
0
is
the
number
of
initial
iterations
of
the
CML
in
generating
sequences
to
a
v
oid
transient
ef
fect
-
n
is
the
total
number
of
sequences
of
length
m
to
generate
by
iterating
the
CML
-
r
is
the
number
of
ro
ws
of
the
plain
image
-
c
is
the
number
of
columns
of
the
plain
image
-
K
is
a
symmetric
secret
k
e
y
used
to
encrypt
the
image
-
X
denotes
the
plain
image
-
Y
denotes
the
encrypted
image
-
h
is
the
hash
v
alue
of
the
plain
image
-
R
is
the
sequence
used
to
permute
ro
ws
of
the
plain
image
Ima
g
e
encryption
with
a
no
vel
CCML
and
c
haotic
ima
g
e
scr
ambling
tec
hnique
(Behr
ang
Chaboki)
Evaluation Warning : The document was created with Spire.PDF for Python.
1106
r
ISSN:
2502-4752
-
C
is
the
sequence
used
to
permute
the
columns
of
the
plain
image
-
D
is
the
sequence
used
to
dif
fuse
the
pix
els
in
the
plain
image.
The
k
e
y
K
and
the
hash
v
alue
of
the
input
image
h
are
gi
v
en
as
inputs
to
the
E
X
T
R
A
C
T
P
A
R
A
M
E
T
E
R
S
algorithm
to
obtain
the
required
encryption
parameters
as
follo
ws:
-
K
0
K
+
h
mo
d
2
256
,
-
"
K
0
2
256
,
-
Let
K
0
=
k
0
32
k
0
31
:
:
:
k
0
2
k
0
1
2
be
the
binary
representation
of
the
modified
k
e
y
in
32
Bytes,
where
k
0
i
2
f
0
;
1
g
8
for
i
=
1
;
:
:
:
;
32
and
the
plain
(encrypted)
image
be
of
size
r
c
,
-
n
max
k
0
31
k
0
29
:
:
:
k
0
3
k
0
1
2
mo
d
2
128
(
mo
d
r
)
;
4
,
-
n
0
k
0
15
k
0
13
:
:
:
k
0
1
2
,
-
m
l
4
r
c
n
2
m
+
1
,
-
m
0
k
0
31
k
0
29
:
:
:
k
0
17
2
+
k
0
32
k
0
30
:
:
:
k
0
18
2
,
-
`
k
0
16
k
0
15
:
:
:
k
0
1
2
+
k
0
32
k
0
31
:
:
:
k
0
17
2
,
-
s
0
`
2
15
1
,
and
(10)
(
K
0
mo
d
2
64
)
+1
2
64
0
:
43
+
3
:
57
.
The
E
X
T
R
A
C
T
P
A
R
A
M
E
T
E
R
S
algorithm
is
of
constant
tim
e
comple
xity
,
gi
v
en
that
the
k
e
y
length
is
fix
ed.
More
precisely
,
i
t
is
linear
in
the
k
e
y
length
in
bits.
W
e
also
use
the
G
E
N
E
R
A
T
E
S
E
Q
U
E
N
C
E
S
algorithm
to
generate
chaotic
sequences,
which
is
described
as
follo
ws:
A
chaotic
matrix
S
eq
M
at
of
size
n
0
+
n
by
m
0
+
m
chaotic
v
alues
are
generated
by
the
proposed
CML
using
the
parameters
of
the
Algorithm
E
X
T
R
A
C
T
-
P
A
R
A
M
E
T
E
R
S
.
The
output
of
the
algorithm
is
a
n
m
sub-matrix
of
S
eq
M
at
by
considering
its
last
n
ro
ws
and
its
last
m
columns,
i.e.
S
eq
M
at
[
n
:
;
m
:]
in
Python’
s
notation.
The
G
E
N
E
R
A
T
E
S
E
Q
U
E
N
C
E
S
algorithm
requires
(
n
0
+
n
)
(
m
0
+
m
)
iterations
of
the
proposed
CML,
e.g.
is
of
time
comple
xity
O
(
m
n
)
considering
e
v
aluation
of
the
proposed
CML
requires
constant
time.
The
E
N
C
R
Y
P
T
I
O
N
algorithm
is
as
follo
ws:
The
hash
v
alue
of
the
input
image
X
is
computed
by
the
SHA-256
algorithm
and
is
denoted
as
h
.
The
encryption
parameters
"
,
,
n
,
n
0
,
m
,
m
0
,
`
,
and
s
0
are
e
xtracted
from
the
combination
of
the
k
e
y
K
and
h
with
the
E
X
T
R
A
C
T
P
A
R
A
M
E
T
E
R
S
algorithm.
The
algorithm
G
E
N
E
R
A
T
E
S
E
Q
U
E
N
C
E
S
is
used
to
generate
three
chaotic
sequences
R
,
C
and
D
of
sizes
2
r
,
2
c
and
(
n
2)
m
,
respecti
v
ely
,
as
follo
ws:
(3-a)
The
first
2
r
elements
are
sorted
in
a
non-decreasing
order
and
their
corresponding
sorted
indices
are
assigned
to
the
sequence
R
.
(3-b)
The
ne
xt
2
c
elements
are
sorted
in
a
non-decreasing
order
and
their
corresponding
sorted
indices
are
assigned
to
the
sequence
C
.
(3-c)
The
ne
xt
r
c
elements
ar
e
sorted
in
a
non-decreasing
order
and
their
corresponding
sorted
indices
are
assigned
to
the
matrix
D
,
which
is
filled
ro
w-wise
and
is
of
shape
r
c
.
The
sequences
R
and
C
are
used
to
permute
the
input
plain
image
according
to
the
Step
4
and
the
sequence
D
is
used
to
dif
fuse
the
input
plain
image
according
to
the
Step
5.
The
sequences
R
and
C
are
used
to
permute
the
input
image
using
the
chaotic
e
xtension
of
the
image
se
gmentation
method
of
[12]
by
X
(
a
)
P
E
R
M
U
T
E
I
M
A
G
E
(
X
(
a
1)
;
i;
j
)
,
for
(
R
(
i
)
;
C
(
i
))
where
R
(
i
)
and
C
(
i
)
denote
the
i
th
element
of
the
sequences
R
and
C
,
respecti
v
ely
,
and
a
=
1
;
:
:
:
;
min
r
;
c
.
Let
X
(
m
)
be
the
output
of
thi
s
step.
-
The
sequence
D
is
used
to
dif
fuse
the
permuted
image
X
(
m
)
by
X
D
X
(
m
)
+
D
mo
d
256
.
pair
Y
=
(
X
D
;
h
)
is
the
output
of
the
E
N
C
R
Y
P
T
I
O
N
.
The
first
and
the
second
steps
of
the
encryption
algorithm
can
be
considered
of
constant
time
com-
ple
xity
.
In
the
third
step
of
the
encryption
algorithm,
we
need
to
generate
m
n
chaotic
elements
from
the
CML
sequence
and
then,
sorting
them
which
requires
O
((
m
n
)
log
(
m
n
)
+
m
n
)
operations.
In
the
fourth
step,
the
input
image
is
scrambled
for
min
r
;
c
steps,
where
each
step
requires
r
c
operations.
Finally
,
the
last
step
can
be
accomplished
with
r
c
operations.
Therefore,
the
time
comple
xity
of
this
algorithm
is
O
((
m
n
)
log
(
m
n
)
+
r
c
min(
r
;
c
)
+
r
c
+
1)
,
e.g.
it
is
polynomial
in
terms
of
the
size
of
the
input
image.
Finally
,
an
encrypted
image
can
be
decrypted
by
re
v
ersing
the
encryption
process.
More
precisely
,
the
D
E
C
R
Y
P
T
I
O
N
is
as
follo
ws:
Let
the
input
of
the
algorithm
be
the
pair
Y
=
(
X
D
;
h
)
and
the
secret
k
e
y
K
.
The
encryption
parameters
"
,
,
n
,
n
0
,
m
,
m
0
,
`
,
and
s
0
are
e
xtracted
from
the
combination
of
the
k
e
y
K
and
h
with
the
E
X
T
R
A
C
T
P
A
R
A
M
E
T
E
R
S
algorithm.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
September
2021
:
1103
–
1112
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1107
The
chaotic
sequences
R
,
C
and
D
are
generated
with
the
same
procedure
as
in
the
third
step
of
the
encryption
algorithm.
The
dif
fusion
is
re
v
ersed
by
computing
Y
D
=
Y
D
mo
d
256
.
The
permutation
is
re
v
ersed
by
re
v
ersing
each
step
of
the
chaotic
image
scrambling
using
sequences
R
and
C
.
It
is
easy
to
v
erify
that
the
time
comple
xity
of
the
encryption
algorithm
and
the
decryption
algorithm
are
the
same,
i.e.
polynomial
in
the
size
of
the
image.
3.
RESUL
TS
AND
DISCUSSION
An
ef
ficient
encryption
algorithm
must
demonstrate
a
proper
security
performance
in
e
n
c
ou
nt
ering
dif
ferent
security
attack
and
could
withstand
them
rigorous
ly
.
T
o
test
the
security
of
our
encryption
algorithm,
we
analyzing
common
f
actors,
such
as
k
e
y
space
analysis,
k
e
y
and
plainte
xt
sensiti
vity
,
histogram
analysis
of
encrypted
image,
entrop
y
analysis,
and
correlation
coef
ficient
analysis.
The
proposed
algorithm
is
implemented
in
Python
3.6.7
and
the
results
are
obtained
on
an
Intel
C
orei3-
350
M
Processor
2.26
GHz
running
Ub
untu
18.04
64-bit.
The
running
t
ime
of
the
proposed
algorithm
is
1.44
seconds
on
a
v
erage
for
plainte
xt
images
of
size
512
512
using
256
-bits
k
e
y
with
this
implementation.
Our
proposed
algorithm
is
secure
ag
ainst
the
brute-force
attack,
since
its
k
e
y
space
is
of
size
2
256
.
Sensiti
vity
ag
ainst
the
encryption/decryption
k
e
y
is
a
requirement
for
a
rob
ust
image
encryption
algo-
rithm.
There
are
three
types
of
k
e
y
sensiti
vity:
(1)
Decryption
of
an
encrypted
image
with
single
bit
perturbation
in
the
le
gitimate
k
e
y
,
(2)
Decryption
of
an
encrypted
image
with
ille
g
al
k
e
ys,
and
(3)
Encryption
of
the
same
image
with
single
bit
perturbation
in
the
k
e
y
.
It
is
required
that
the
result
of
each
sensiti
vity
analysis
be
dif
ferent
to
each
other
as
much
as
possible.
The
results
of
these
tests
for
the
proposed
algorithm
are
reported
in
T
ables
1-(a),
1-(b),
and
1-(c),
respecti
v
ely
.
As
it
can
be
observ
ed
from
these
tables,
the
proposed
algorithm
pro
vides
acceptable
sensiti
vity
to
t
he
encryption
k
e
y
,
as
all
these
v
alues
are
v
ery
close
to
their
ideal
v
alues.
Moreo
v
er
,
the
encryption
sensiti
vity
of
the
proposed
algorithm
in
the
k
e
y
is
much
higher
t
han
some
recent
chaotic
image
encryption
algorithms,
as
it
can
be
observ
ed
from
T
ables
1-(a)
to
1-(c).
The
main
purpose
of
these
tests
is
to
ensure
the
dif
fusion
propert
y
of
an
encryption
algorithm,
which
means
that
the
smallest
change
in
plain
image
should
ha
v
e
huge
consequences
in
cipher
image.
The
Uni-
fied
A
v
erage
Changing
Intensity
(U
A
CI)
and
the
Number
of
Pix
els
Change
Rate
(NPCR)
are
tw
o
of
the
most
standard
(or
another
synon
ym)
tests
that
are
used
by
researchers
for
testing
the
resistance
of
their
encryption
algorithm
ag
ainst
the
dif
ferential
attacks
(or
chosen-plain
te
xt
attacks
).The
NPCR
is
used
to
calculate
the
per
-
centage
of
dif
ference
between
to
image
pix
el
numbers,
on
the
other
hand
the
U
A
CI
is
used
to
measure
the
a
v
erage
se
v
erit
y
of
dif
ferences
between
the
tw
o
images.
The
ideal
v
alues
of
NPCR
and
U
A
CI
based
on
the
e
xpectations
of
[13]
are
99
:
6094%
and
33
:
4635%
,
respecti
v
ely
.
The
NPCR
and
U
A
CI
tests
for
tw
o
images
C
1
and
C
2
of
size
M
N
are
defined
as
the
follo
wing
equations
NPCR
=
P
M
i
=1
P
N
j
=1
D
(
i;j
)
M
N
100%
;
and
U
A
CI
=
P
M
i
=1
P
N
j
=1
j
C
1
(
i;j
)
C
2
(
i;j
)
j
256
M
N
100%
;
respecti
v
ely
,
where
D
(
i;
j
)
=
1
if
C
1
(
i;
j
)
=
C
2
(
i;
j
)
,
otherwise
D
(
i;
j
)
=
0
.
In
some
cases,
the
NPCR
and
U
A
CI
cannot
accurately
detect
the
visual
dif
ferences
between
the
plain-image
and
cipher
-image,
so
to
compensate
these
criteria,
we
use
B
A
CI
(Block
A
v
erage
Changing
Intensity)
test
for
analyzing
our
encryption
algorithm.
This
test
quantit
ati
v
ely
e
v
aluates
the
dif
ferences
between
our
plain
and
cipher
-images.
Let
M
and
P
be
tw
o
images
of
the
size
r
c
3
.
Then,
the
v
alue
of
their
PSNR
is
calculated
as
PSNR
(
M
;
P
)
=
20
log
255
MSE
where
MSE
(
M
;
P
)
=
1
r
c
3
P
r
i
=1
P
c
j
=1
P
3
k
=1
(
M
(
i;
j
;
k
)
P
(
i;
j
;
k
))
2
.
Note
that
high
numerical
dif
ferences
between
the
plain
and
the
decrypted
image
results
in
lo
wer
v
alues
of
PSNR.
In
a
dif
ferential
attack,
an
attack
er
slightly
changes
the
plain
image
and
encrypts
the
original
plai
n
and
the
modified
images
with
the
same
k
e
y
.
Then,
he
tries
to
trace
the
dif
ferences
between
the
tw
o
encrypted
images
and
use
this
kno
wledge
to
crack
it.
Our
proposed
algorithm
pro
vides
acceptable
sensiti
vity
with
respect
to
dif
ferential
attack
as
is
illustrated
in
T
able
1-(d).
Moreo
v
er
,
our
proposed
algorithm
outperforms
most
of
the
recent
image
encryption
algorithms,
as
it
is
sho
wn
in
T
able
1-(d).
In
image
processing
and
image
encryption
c
o
nt
e
x
t
,
the
histogram
is
a
statistical
feature
of
image
that
sho
ws
the
frequenc
y
of
each
pix
el
intensity
in
grayscale
or
e
v
en
color
image.
F
or
grayscale
images
that
are
presented
in
this
article,
distrib
ution
of
256
dif
ferent
possible
combinations
are
sho
w
graphically
by
image
histogram.
As
depicted
in
the
Figure
3,
the
uniform
distrib
ution
of
cipher
image
histogram
sho
ws
that
our
algorithm
resistance
ag
ainst
statistical
attacks.
Ima
g
e
encryption
with
a
no
vel
CCML
and
c
haotic
ima
g
e
scr
ambling
tec
hnique
(Behr
ang
Chaboki)
Evaluation Warning : The document was created with Spire.PDF for Python.
1108
r
ISSN:
2502-4752
T
able
1.
Sensiti
vity
analysis
where
“—”
denotes
that
the
corresponding
v
alue
is
not
a
v
ailable.
(a)
Image
Decryption
sensiti
vity
in
the
secret
k
e
y
NPCR
U
A
CI
B
A
CI
PSNR
Pepper
99.61578
30.69388
0.23172
8.53623
Lena
99.61231
28.98183
0.21558
9.09798
Baboon
99.60728
27.95127
0.20694
9.48089
Ship
99.60918
28.44964
0.20764
9.29461
Ceremon
y
99.60678
32.66921
0.25568
7.95964
Cameraman
99.61529
28.85233
0.21541
9.15081
Girl
99.60777
32.39105
0.25241
8.03456
Home
99.60735
27.61322
0.19721
9.61956
A
v
erage
99.61022
29.70030
0.22282
8.89678
[14]
99.60860
28.69690
0.21382
—
[15]
99.60890
28.62900
0.21325
—
[16]
99.60960
28.6321
0.21321
—
(b)
Image
Decryption
sensiti
vi
ty
in
ille
g
al
k
e
ys
NPCR
U
A
CI
B
A
CI
PNSR
Pepper
99.60403
33.45612
0.26634
7.75197
Lena
99.61967
33.43398
0.26623
7.75487
Baboon
99.61891
33.37835
0.26645
7.76399
Ship
99.63074
33.46454
0.26656
7.75178
Ceremon
y
99.60785
33.47861
0.26727
7.74123
Cameraman
99.61967
33.41536
0.26648
7.75820
Girl
99.62425
33.43350
0.26633
7.75333
Home
99.60136
33.44234
0.26623
7.75573
A
v
erage
99.61581
33.43875
0.26648
7.75388
[17]
63.62070
29.21160
0.34080
—
[15]
99.60760
33.47100
0.26779
—
[16]
99.60740
33.46030
0.26777
—
(c)
Image
Encryption
sensi
ti
vity
in
the
secret
k
e
y
NPCR
U
A
CI
B
A
CI
PSNR
Pepper
99.61578
33.49226
0.26669
7.74241
Lena
99.61231
33.48726
0.26704
7.73877
Baboon
99.60728
33.45494
0.26674
7.74823
Ship
99.60918
33.43114
0.26643
7.75608
Ceremon
y
99.60678
33.51472
0.26706
7.73511
Cameraman
99.61529
33.46900
0.26647
7.74836
Girl
99.60777
33.45926
0.26671
7.74803
Home
99.60735
33.48952
0.26667
7.74308
A
v
erage
99.61022
33.47476
0.26672
7.74501
[18]
99.3097
33.4553
—–
—–
[19]
99.60269
33.49419
—–
—–
[20]
99.418
33.39
—–
—–
[21]
99.6074
33.4570
—–
—-
[22]
99.61596
33.43452
—–
—–
[23]
99.60867
33.49567
—–
—–
(d)
Image
Encrypti
on
of
P
erturbed
Image
with
Same
K
e
y
NPCR
U
A
CI
B
A
CI
Pepper
99.60964
33.44334
0.26650
Lena
99.61067
33.46390
0.26670
Baboon
99.61662
33.46825
0.26680
Ship
99.61140
33.43944
0.26657
Ceremon
y
99.60907
33.46023
0.26691
Cameraman
99.60968
33.45641
0.26670
Girl
99.60316
33.48295
0.26665
Home
99.61449
33.46452
0.26667
A
v
erage
99.61059
33.45988
0.26669
[24]
99.61
33.53
—
[25]
99.61
33.43
—
[14]
99.6092
33.4668
0.2678
[21]
99.6074
33.4570
—
[26]
99.61
33.33
—
[20]
99.61
33.45
—
In
a
natural
image,
the
correlation
between
adjacent
pix
els
is
usually
high.
Due
to
this
high
correlation
between
tw
o
adjacent
pix
els,
the
encryption
algorithm
could
be
vulnerable
ag
ainst
statistical
attacks.
So,
an
encryption
algorithm
is
as
rob
ust
as
possible
if
it
could
break
do
wn
this
relationship
as
much
as
possible.
T
o
measure
the
correlation
of
pix
els
in
plain
and
cipher
-images
we
randomly
choose
3
000
pix
els
form
original
and
corresponding
cipher
images
in
the
v
ertical
,
horizontal
and
diagonal
direction,
and
then
compute
the
correlation
coef
ficients.
The
results
are
gi
v
en
in
T
able
2-(a).
As
it
can
be
observ
ed
from
thi
s
data,
our
proposed
algorithm
outperforms
four
out
of
se
v
en
recent
image
encryption
algorithms
in
all
three
directions,
and
outperforms
the
rest
of
them
in
at
least
one
direction.
Thus,
it
pro
vides
acceptable
security
in
terms
of
correlation
analysis.
The
concept
of
informat
ion
entrop
y
is
a
solution
for
finding
a
de
gree
of
randomness
and
uncertainty
in
the
image.
As
the
information
entrop
y
of
an
image
becomes
close
to
its
ideal
v
alue,
it
suggests
that
the
infor
-
mation
is
distrib
uted
randomly
throughout
the
image.
F
or
grayscale
images,
the
optimum
v
alue
for
information
entrop
y
is
8,
so
as
f
ar
as
our
analysis
results
become
closer
to
this
v
alue,
we
can
conclude
that
our
encryption
algorithm
has
a
better
uniform
distrib
ution
of
information.
The
entrop
y
of
a
grayscale
image
P
is
calculated
by
the
follo
wing
equations:
H
(
P
)
=
P
255
i
=0
p
(
i
)
log
2
p
(
i
)
;
where
p
(
i
)
is
the
fraction
of
pix
els
with
color
i
in
the
image
P
.
As
it
can
be
v
erified
in
T
able
2b,
our
proposed
method
pro
vides
acceptable
security
in
terms
of
entrop
y
analysis
compared
with
se
v
eral
recent
image
encryption
algorithms.
It
is
vital
for
a
secure
image
encryption
algorithm
to
be
rob
ust
ag
ainst
data
loss
and
noise
attacks,
since
the
presence
of
noise
and
data
loss
is
quite
usual
in
real
w
orld
scenarios.
The
proposed
algorithm
satisfies
this
requirement
as
we
ha
v
e
tested
it
by
cropping
200
200
random
square
from
an
encrypted
image
and
then
decrypted
it.
Moreo
v
er
,
we
ha
v
e
also
added
a
Salt
&
Pepper
noise
of
0
:
1%
,
1%
and
5%
to
the
encrypted
image
and
decrypted
it.
As
it
can
be
visually
observ
ed
in
Figures
4
and
5,
the
proposed
encryption
scheme
pro
vides
an
acceptable
rob
ustness
ag
ainst
noise
and
data
loss
attacks.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
September
2021
:
1103
–
1112
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1109
(a)
Plainte
xt
image.
(b)
Encrypted
image.
(c)
Plainte
xt
image
histogram.
(d)
Cipherte
xt
image
histogram.
Figure
3.
Histogram
analysis
for
the
proposed
algorithm
for
a
sample
image.
(a)
(b)
(c)
(d)
Figure
4.
Crop
Attack
200
200
.
(a)
Plainte
xt
image,
(b)
Encrypted
image,
(c)
Cropped
encrypted
image,
(d)
Decrypted
image
of
the
cropped
encrypted
image.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure
5.
Noise
attacks:
(a)
Plainte
xt
image,
(b)
Encrypted
image,
(c)
Salt
and
Pepper
noise
of
0
:
01
added
to
encrypted
image,
(d)
Decryption
of
encrypted
image
with
Salt
and
Pepper
noise
of
0
:
01
added,
(e)
Salt
and
Pepper
noise
of
0
:
1
added
to
encrypted
image,
(f)
Decryption
of
encrypted
image
with
Salt
and
Pepper
noise
of
0
:
1
added,
(g)
Salt
and
Pepper
noise
of
0
:
5
added
to
encrypted
image,
(h)
Decryption
of
encrypted
image
with
Salt
and
Pepper
noise
of
0
:
5
added,
(i)
Gaussian
noise
of
mean
0
and
v
ariance
0
:
01
added
to
encrypted
image,
(j)
Decryption
of
the
encrypted
image
with
Gaussian
noise
of
mean
0
and
v
ariance
of
0
:
01
added.
Ima
g
e
encryption
with
a
no
vel
CCML
and
c
haotic
ima
g
e
scr
ambling
tec
hnique
(Behr
ang
Chaboki)
Evaluation Warning : The document was created with Spire.PDF for Python.
1110
r
ISSN:
2502-4752
T
able
2.
Correlation
and
entrop
y
analysis
for
plain
and
encrypted
images
with
the
proposed
algorithm
for
all
images.
(a)
Correlation
analysis
Image
Correlation
direction
Horizontal
V
ertical
Diagonal
Pepper
0.01403
0.00189
-0.00372
Lena
-0.03244
0.00820
0.04202
Baboon
0.02080
0.01370
0.00952
Ship
0.00194
0.01289
-0.01126
Ceremon
y
0.00687
0.01787
0.04778
Cameraman
0.00559
0.00523
0.00272
Girl
-0.00811
0.01079
-0.00783
Home
-0.01582
-0.00224
0.05586
A
v
erage
0.01320
0.00910
0.02259
[18]
-0.09736
-0.07068
0.04844
[19]
0.02424
0.02608
0.02446
[20]
0.0493
0.0172
0.043375
[27]
-0.01055
0.0141
0.0081
[22]
0.039886
0.034475
0.001949
[23]
0.07700
-0.07236
-0.06153
[28]
0.0120
0.0917
0.1019
(b)
Entrop
y
for
all
images
Image
Entrop
y
Plain
Encrypted
Pepper
7.59373
7.99939
Lenna
7.44507
7.99934
Baboon
7.35815
7.99931
Ship
7.19137
7.99933
Ceremon
y
6.70385
7.99927
Cameraman
7.15270
7.99934
Girl
7.48451
7.99936
Home
7.20101
7.99929
A
v
erage
7.39129
7.99933
[19]
5.570216
7.99881
[20]
——–
7.99714
[21]
7.569453
7.989367
[27]
6.501775
7.9972
[22]
7.350343
7.998740
4.
CONCLUSION
In
this
paper
,
we
ha
v
e
constructed
a
no
v
el
chaotic
coupled
mapped
lattice
using
the
chaotic
Logistic
mapping
and
the
Chebyshe
v
mapping.
W
e
ha
v
e
also
studied
the
chaotic
beha
vior
of
the
proposed
chaotic
mapping
in
terms
of
its
L
yapuno
v
e
xponent,
which
w
as
greater
t
han
zero.
W
e
ha
v
e
also
generalized
an
image
scrambling
method
proposed
in
[12]
by
making
it
chaotic
and
iterati
v
e.
Then,
we
ha
v
e
combined
these
tw
o
primiti
v
es
to
encrypt
images
by
generating
chaotic
sequences
from
the
no
v
el
proposed
chaotic
mapping
and
then,
applying
permutation
using
the
proposed
generalized
chaotic
image
scrambling
method.
T
o
construct
the
dif
fusion
matrix,
we
ha
v
e
used
a
sorting
approach
rather
than
quantizing
the
chaotic
floating-point
v
alues.
W
e
ha
v
e
al
so
studied
the
security
performance
of
the
proposed
encryption
algorithm
concerning
brute-force
attacks,
k
e
y
sensiti
vity
attacks,
dif
ferential
attack,
and
statistical
analysis.
Moreo
v
er
,
the
rob
ustness
of
the
proposed
method
is
tested.
The
results
of
all
these
tests
sho
wed
acceptable
security
in
comparison
with
se
v
eral
recent
image
encryption
algorithms.
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Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1111
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BIOGRAPHIES
OF
A
UTHORS
Behrang
Chaboki
is
a
lecturer
of
Computer
Science
at
the
V
ali-e-Asr
Uni
v
ersity
of
Rafsanjan.
His
research
interests
include
Algorithms
and
Computational
Comple
xity
.
He
holds
a
M.Sc.
in
Computer
Science
from
Sharif
Uni
v
ersity
of
T
echnology
.
Ima
g
e
encryption
with
a
no
vel
CCML
and
c
haotic
ima
g
e
scr
ambling
tec
hnique
(Behr
ang
Chaboki)
Evaluation Warning : The document was created with Spire.PDF for Python.
1112
r
ISSN:
2502-4752
Ali
Shakiba
is
an
assistant
professor
of
Computer
Science
a
t
the
V
ali-e-Asr
Uni
v
ersity
of
Rafsanjan.
His
research
interest
inc
lude
Cryptograph
y
,
Machine
Learning
and
Chaotic
Systems.
He
holds
a
Ph.D.
in
Computer
Science
from
Y
azd
Uni
v
ersity
.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
September
2021
:
1103
–
1112
Evaluation Warning : The document was created with Spire.PDF for Python.