TELKOM
NIKA
, Vol.12, No
.3, Septembe
r 2014, pp. 6
51~656
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v12i3.83
651
Re
cei
v
ed Ma
rch 2
9
201
4; Re
vised July
24, 2014; Accepted Augu
st
10, 2014
The Implementation of Henon Map Algorithm
for Digital Imag
e Encryption
Edi Sukirman
1
, Sur
y
adi
MT
*2
, M. Ag
us Mubarak
3
1
Jurusan Siste
m
Informasi, Universit
a
s Gun
adarm
a
, Depo
k 1642
4, Indon
esia
2
Departeme
n Matematika, U
n
iversit
a
s Indo
nesi
a
, Dep
o
k 1
642
4, Indon
esi
a
3
Jurusan T
e
knik Informati
ka, Univers
i
tas Gu
nad
arma, Dep
o
k 164
24, Indo
nesi
a
*Corres
p
o
ndi
n
g
author, em
ail
:
yadi.mt@sc
i.ui.ac
.
id
A
b
st
r
a
ct
Security infor
m
ati
on
is a v
e
r
y
importa
nt as
pect t
hat
must
be n
o
tice
d. Informatio
n
not
o
n
ly i
n
the
form
of text bu
t also i
n
for
m
of data
i
m
ag
e.
Usin
g
d
a
ta e
n
c
riptio
n to se
n
d
priv
ate i
n
for
m
ati
on
hav
e b
een
w
i
dely us
e. Bu
t still n
eed
to i
m
pr
ove th
e
en
dura
n
ce fro
m
bruto forc
e att
a
ck. One w
a
ys
to i
m
pr
ove
it, is by
usin
g cha
o
s th
eory w
i
th hen
om
alg
o
rith
m.
T
e
st resu
lt ga
ve the al
gh
orit
m ca
n encry
pt image
data fr
om
graysca
le typ
e
to colorfull
type. Encryption an
d
desc
r
yption ti
me
prop
ortion
al to
the si
z
e
i
m
age.
Co
mp
ositio
n and variety cou
l
or does
n
’
t effect
the time. this alg
o
rith
m has key space
of
10
and key
sensitivity up
t
o
10
. So, it can
be co
nclu
de
d that, the al
gorit
hm
is very d
i
fficult to be cr
ac
ked by
brute
force attack.
Ke
y
w
ords
: en
cryption a
l
gor
ih
tm, chaos, di
git
a
l i
m
a
ge, he
no
n ma
p
1. Introduc
tion
No
w a days using
comp
uters to sen
d
any kind
s of information thorou
g
h
internet
con
n
e
c
tion a
r
e commo
n. By using p
ublic p
a
th p
eople from
arou
nd the
worl
d can send
informatio
n e
v
en though it has very lo
w safety level.
Information
secu
rity is a
n
asp
e
ct th
at
is ve
ry imp
o
rtant a
nd u
r
gent to
be
notice
d
.
Information
o
n
con
c
e
r
nin
g
the interest
s o
f
privat
e, insti
t
utional and
corpo
r
ate
ne
cessarily h
a
ve
a
high value a
n
d
sho
u
ld be
kept confid
enti
a
l. Confid
e
n
tial informatio
n
must be in g
r
eat dema
nd for
variou
s pu
rpo
s
e
s
an
d mu
st carefully gua
rded
.
T
h
is i
n
formatio
n not
only in the form of text data,
but also in th
e form of ima
ge data that is highly confi
dential.
By encryptin
g data so th
at only the reci
pi
ent ca
n
only decryp
t
the data is one of
s
o
lutions
that many enggineers
do. Some enc
r
yption algorithms
s
u
c
h
as
DES
,
AES, RSA,
and
others have been
wi
dely use
d
to en
crypt
the
im
a
g
e
data,
but t
hese al
gorith
m
s
still mu
st
be
impr
oved durability of var
i
ous
attacks
, s
u
c
h
as
brute for
c
e
attack
s
[1]. Many r
e
s
e
ar
c
h
have been
done in ho
w
to improve th
e dura
b
ility of the algor
ith
m
s used in the en
cryption
process fro
m
a
brute force
attack, provide a good
combi
nati
on
of speed, hi
gh se
cu
rity,
compl
e
xity,
and
c
o
mputational power, etc [2],[3].
One of them
is
us
ing the
c
h
aos
theory
. Chaos
-
based
encryption al
so be
en extensively
studie
d
by rese
archers be
cau
s
e
of its superi
o
r in safety a
n
d
compl
e
xity [4]. One
algo
rithm
whi
c
h
impleme
n
ts t
he the
o
ry of
ch
ao
s i
s
th
e Heno
n ma
p
algorith
m
, this al
gorithm
i
m
pleme
n
ts
chao
s t
heo
ry
by gene
ratin
g
ra
ndom
nu
mbers
with t
w
o
initial values. The algo
rithm
implement
s the ch
ao
s th
e
o
ry that has
sensitivity to small cha
nge
s in
initial para
m
e
t
er values
an
d has a hi
gh l
e
vel of
s
e
curity from brute forc
e attack
s
[5]-[11].
Based
on the
above expla
nation, this
rese
ar
ch i
s
ab
out impleme
n
t
ing the algo
rithm on
the Hen
on m
ap for the en
cryption and d
e
cryptio
n
of digital image
s informatio
n.
2. Rese
arch
Metho
d
Cha
o
s th
eory come
s fro
m
the theo
ry of sy
stem
s
that exhibit irre
gula
r
ap
p
eara
n
ce,
despite the fact that this theory is u
s
e
d
to ex
plain the occu
rre
nce of r
andom
data. Inventor of
cha
o
s th
eory
is a mete
oro
l
ogist, Edward Lorent
z, in
1960
whe
n
h
e
made
a m
odel of weat
her
forecast
s.by Iterating weat
her math
ema
t
ical model
to
obtain weath
e
r fore
ca
sts i
n
the future. The
longe
r time weath
e
r fore
ca
sts
are co
mputed, the longe
r iterat
i
on to be do
ne. By chan
ging
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 3, September 20
14: 65
1 – 656
652
slightly the i
n
itial value o
f
0.00012
7 iteration
only,
he foun
d th
at the weath
e
r forecast
h
ad
gene
rated g
r
eat diverge
n
ce [12].
Hen
on equ
ation is a dynamic sy
stem
t
hat implements a discrete system. Hen
o
n
equation usi
n
g a point (x,y) in
an equation and mapped to a new point with the equation [9],[13]:
1
,
(1)
(2)
Stages in u
s
i
ng He
non e
q
uation is divi
ded into
two
stage
s, whi
c
h are
key stream and
encryption/de
cryption stag
e.
In the key stream g
ene
rati
on ph
ase is
done
by usi
n
g algo
rithm
s
Hen
on ma
p,
the two
keys are requ
ired to
be
alg
o
rithm
s
g
ene
rate serie
s
of
numbe
rs of p
s
eu
do-ra
ndo
m re
al nu
mb
ers,
so that
seq
u
ence num
be
rs can b
e
u
s
ed a
s
a
key
stre
am, the
n
these nu
m
bers mu
st be
conve
r
ted to an intege
r array with a ran
ge between 0
to 255.
The p
r
o
c
e
s
s is d
one
by absolut
seq
uen
ce n
u
mb
ers (Xn
)
, ea
ch of th
ese
numbe
rs
multiplied by
1000. M
a
the
m
atically the
integer
co
nv
e
r
sio
n
fun
c
tion
s can b
e
writ
ten as foll
ows:
the pro
c
e
ss i
s
done by ab
solut seque
n
c
e num
bers
(Xn), each of these n
u
mbe
r
s multiplie
d by
1000. Mathe
m
atically the integer
conve
r
si
on fun
c
tion
s can be
writte
n as follo
ws:
‖
1000
‖
Then the ro
undin
g
do
wn
(floor) re
sul
t
ing in
teger
(Fn). Havin
g
obtaine
d the
integer
seri
es, the
se
ries i
s
ma
ppe
d to the ran
g
e
[0,
255]. Mathematically, the mappi
ng
function
can
be
written as
follows
K
n
= F
n
m
od 256
Encrptyon st
age i
s
the
sta
ge w
here the
origin
al imag
e or pl
ain im
age (P
n) i
s
converte
d
into cip
her im
age (Cn
)
by
XOR the pixe
ls of plai
n im
age (P
n) of t
he keystre
a
m
(Kn)
whi
c
h h
a
s
been raised. Mathemati
c
al
ly this encrypt
i
on functio
n
can be written as follo
ws:
C
n
= P
n
K
n
(3)
whe
r
e :
C
n
: Ciphe
r im
age (e
ncrypti
on image
).
P
n
: Plain image (p
reviou
s i
m
age
).
K
n
: Keys
tream.
The
de
crypti
on p
h
a
s
e
ha
s the
same
pro
c
e
s
s
with
en
cryptio
n
stage, it's ju
st usi
n
g
a
ciph
er en
cryp
ted
imag
e (
C
n
) a
s
the i
npu
t image. T
o
d
e
crypt th
e o
r
i
g
inal im
age
f
r
om th
e
ciph
er
image (
C
n
), b
y
XOR opera
t
ion for the pixel-pixel ima
ge cip
her
(
C
n
) of the keyst
r
eam
(
K
n
) whi
c
h
has b
een raised. The process ca
n be d
e
scrib
ed by the algo
rithm
sho
w
n in Fig
u
re 1.
Algorithm
Henon map
Initial state
: orginal image (P)
Final state
: encrypted image (C)
Input
Key A, B
Input
Image P(1..mxn)
Initial value
x(0), y(0)
Loop
i = 0
to
(mxn)-1
x(i+1) = y(i)+1-a*x(i)*x(i)
y(i+1) = b*x(i)
E(i+1) = |x(i+1)|*10000
F(i+1) = floor(E(i+1))
K(i+1) = mod(F(i+1), 256)
C(i+1) = P(i+1)
K(i+1)
Next loop
Output
Image C(1..mxn)
Figure 1. Image Encryption
Algorithm
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
The Im
plem
e
n
tation of He
non Map Alg
o
rithm
for Dig
i
tal Im
age En
cryption
(Edi Sukirm
an
)
653
3. Results a
nd Analy
s
is
Implementati
on of the app
lication i
s
do
ne on a
com
puter
with ha
rdware sp
ecs: Intel ®
Core ™
pro
c
e
s
sor B
940
(2.0G
H
z, 2
M
B L3
ca
ch
e), DDR3
RAM (2GB
),
VGA Intel ®
HD
Grap
hics, 32
0 GB HDD, monitor, keyb
o
a
rd an
d mou
s
e.
In the test p
hase, the en
cryption
and
decry
ption p
r
oce
s
s u
s
e a
numbe
rs of
image
s.
Images d
a
ta use
d
in this st
udy can b
e
seen in Tabl
e 1.
Table 1. Imag
e Data Te
stin
g
File Na
me
Origi
n
al Im
age
Image
T
y
p
e
File Size
Image Size
Clock. bmp
G
r
ay
scale
(8 bits/ pixel)
192 kb
256 x 2
5
6
Aerial. bmp
G
r
ay
scale
(8 bits/ pixel)
768 kb
512 x 5
1
2
Airport. bmp
G
r
ay
scale
(8 bits/ pixel)
300 kb
1024 x
1024
Gi
rl
.bmp
Color
(24 bits/ pixel)
192 kb
256 x 2
5
6
Lena. bmp
Color
(24 bits/ pixel)
768 kb
512 x 5
1
2
SteelSea. bmp
Color
(24 bits/ pixel)
300 kb
1024 x
1024
3.1. Analy
s
is
of Ke
y
Space
Key spa
c
e i
s
the total nu
mber
of diffe
rent
keys th
a
t
can
be u
s
e
d
for e
n
crypti
on an
d
decryption. T
o
de
al
with
a
brute
force
at
tack,
cryptog
r
aphi
c al
gorith
m
s
sh
ould
ha
ve a l
a
rg
e
ke
y
spa
c
e, then the longe
r the
time it
takes
to brea
k the lock of the algorithm. Key param
eters used
in the en
cryption alg
o
rithm
are t
w
o, na
m
e
ly key A an
d
key B, ea
ch
data in d
oubl
e type. Do
ubl
e-
prec
is
ion computing for prec
is
ion acc
o
rding
to s
t
andard 64-bit IEEE floating-point is
10
-15
. So
the numbe
r o
f
possi
ble values
of each key is 10
-15
, th
en the possib
l
e combi
natio
ns of two key
s
k
e
y is
R
(A, B) =
10
-15
× 10
-1
5
=
10
30
Time requi
re
d to try all
co
mbination
s
of ke
y
s
(exhau
stive key sea
r
ch
) [1
4] can
be
see
n
in Table 2.
Table 2. Time
Requi
red to
Try exhau
stive Key Search
Ke
y
Space
Experime
nts/s
e
c
Time need
ed
Secon
d
Years
10
30
10
6
10
24
3,215 × 10
16
10
12
10
18
321502057
61
10
18
10
12
32150,2057
6
10
24
10
6
0,03215020
6
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 3, September 20
14: 65
1 – 656
654
Data in
Tabl
e 2
sho
w
s that
it take
s
approximatel
y 3.215
× 1
0
16
years to
try all
combi
nations of keys
with a co
mputer
that can do
1 million expe
riments per
second. So it is
kno
w
n th
at it take
s sub
s
ta
ntial time to solve two
key
combi
nation t
hat led to
a b
r
ute force
attack
is not efficient
.
3.2. Image Similarit
y
Analy
s
is
Testing i
s
do
ne by com
p
a
r
ing the ima
g
e
of the begi
nning of the
encrypted im
age an
d
decrypted im
age on a n
u
m
ber of test im
age
s.
The re
sults are
sho
w
n in Table 3.
Table 3. Re
sult Test Similarity Image
Image na
me
Origi
n
al Im
age
Encr
y
p
ti
on i
m
a
g
e
Descr
y
p
tio
n
im
age
Clock.bmp
256 x 2
5
6
192 kb
192 kb
192 kb
Aerial.bmp
512 x 5
1
2
768 kb
768 kb
768 kb
Airport.bmp
1024 x
1024
3000 kb
3000 kb
3000 kb
Gi
rl
.bmp
256 x 2
5
6
192 kb
192 kb
192 kb
Lena.bmp
512 x 5
1
2
768 kb
768 kb
768 kb
Steel Sea.bmp
1024 x
1024
3000 kb
3000 kb
3000 kb
Tabel
3
sh
ows that
the file
si
ze
and
dim
ens
i
o
n
s
of th
e o
r
iginal
ima
ge,en
crypted
imag
e
and d
e
crypte
d imag
e is e
s
sentially the
sam
e
a
s
the
en
cryption
a
nd de
cryption
pro
c
e
s
s in t
h
is
study only ch
ange the valu
es
of the pixels of t
he imag
e usin
g XOR
operation
s
for key bits wit
h
image pixel
s
.
3.3. Anal
y
s
is
of Parameter Sensitivit
y
Ke
y
The test is p
e
rform
ed by comp
ari
ng th
e im
age of the de
cryption
on a numbe
r of test
image
s that
have be
en e
n
crypte
d with
a key valu
e
of 1.4 and
a
value of 0.3
B key with
a
very
small
ch
ang
e
of value
on
one
or two
key pi
eces
.
Re
sults of te
sting
don
e b
y
the de
cryption
pro
c
e
ss
uses a different
key value, ke
y A with a value 1.4
+ 1
0
-16
, whose result
s app
ea
r in
Table 4.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
The Im
plem
e
n
tation of He
non Map Alg
o
rithm
for Dig
i
tal Im
age En
cryption
(Edi Sukirm
an
)
655
Table 4. Key Sentitivity Te
st Re
sult
Image na
mes
E
n
c
r
y
p
t
i
o
n
im
ag
e
Ke
y
A
=
1.4
Ke
y
B = 0.3
Descr
y
p
tio
n
im
age
ke
y
A
= 1.4 + 10
-1
6
ke
y
B = 0.3
Descr
y
p
tio
n
im
age
Ke
y
A
= 1.4 = 1.
4 + 10
-1
7
ke
y
B = 0.3
Clock.bmp
Aerial.bmp
Airport.bmp
Gi
rl
.bmp
Lena.bmp
Steel Sea.bmp
Table 4, sho
w
s the p
r
o
c
e
ss of de
cryption of
the encrypted imag
e
with little different in
one of the ke
y, encryption
algorith
m
ha
s key sen
s
itivity that reache
s
10
.
3.4. Analy
s
is
Process En
cr
y
p
tion and Decr
y
p
tion Time
The test i
s
pe
rforme
d by calcul
ating the
pro
c
e
ss tim
e
of encryption
and d
e
cryption on
a
numbe
r of test images. The
test results shown in Tabl
e 5.
Table 5. Time
Test Re
sult for Encryption
and De
crypti
on Pro
c
e
s
s
Image na
me
Image t
y
p
e
File
size
Image
dime
nsio
ns
(pixel
)
Encr
y
p
ti
on tim
e
proces
(sec
ond
)
Descr
y
p
tio
n
time proces
(seco
nd
)
Clock.bmp
G
r
ay
scale
(8 bits/ pixel)
192 kb
256 x 2
5
6
2.684
2.638
Aerial.bmp
G
r
ay
scale
(8 bits/ pixel)
768 kb
512 x 5
1
2
40.867
40.613
Airport.bmp
G
r
ay
scale
(8 bits/ pixel)
300 kb
1024 x
1024
866.170
816.802
Gi
rl
.bmp
Color
(24 bits/pixel)
192 kb
256 x 2
5
6
2.653
2.649
Lena.bmp
Color
(24 bits/pixel)
768 kb
512 x 5
1
2
40.725
40.587
Steel Sea.bmp
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 3, September 20
14: 65
1 – 656
656
Test
result d
a
ta in Ta
ble
5, sho
w
s that
the en
crypti
on an
d de
cryption p
r
o
c
e
s
sing time
prop
ortio
nal t
o
the
size
of the ima
ge dim
ensi
o
n
s
, the
greate
r
the
di
mensi
o
n
s
of
an ima
ge of t
he
longe
r time re
quire
d to en
crypt the image
. Table
5 al
so
sho
w
s th
at the co
mpo
s
iti
on and
diversity
of colo
rs tha
t
make
up t
he ima
ge di
d not
si
gnifi
cantly affect
time imag
e
encryption
and
decryption p
r
oce
s
s.
4. Conclusio
n
Hen
on m
a
p
algorith
m
im
plementatio
n
on
the
pro
c
ess of
en
cry
p
tion a
nd
de
cryption
of
digital
imag
e has bee
n
succe
ssfully ca
rri
ed
out on
the
appli
c
ation
a
nd teste
d
o
n
several ima
g
e
s.
The experi
m
ental re
sults sho
w
th
at
th
e
alg
o
rithm
Hen
on map
can
en
crypt and de
crypt
i
m
age
with exactly the sam
e
as t
he origi
nal im
age,
and
can
be ded
uced from this stu
d
y that :
a.
Encryptio
n
a
nd de
cryptio
n
pro
c
e
ssi
ng ti
me propo
rtio
nal to the si
ze of the imag
e dimen
s
io
ns,
the greater the dim
e
n
s
ion
s
of a
n
ima
g
e
of t
he l
ong
er time
re
qui
red to
en
cry
p
t the ima
g
e
becau
se the
bigger the
dimen
s
ion
s
of the image
, the bigger
pixels of the
image to be
pro
c
e
s
sed an
d vice versa.
b.
Comp
ositio
n and diversity of colors of image di
d n
o
t signifi
cantly affect the time
of encryptio
n
and de
cryptio
n
pro
c
e
s
ses.
t the image which h
a
s the
comp
ositio
n and diversity of high col
o
r
with the
ima
ge that
ha
s t
he
comp
ositi
on a
nd
diversity of l
o
w
colo
r h
a
s tim
e
en
cryption
proc
es
s
is
relatively the s
a
me.
c.
Encryptio
n
al
gorithm h
a
s
key sp
ace for
10
and key se
nsitivity that reache
s
10
, s
o
the
algorith
m
is very difficult to be cra
c
ked b
y
brute force attack.
Thus the
digit
a
l imag
e e
n
cryption algo
rit
h
m is
very dif
f
icult to b
e
so
lved with
brute force
at
t
a
ck
s.
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