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1
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k
i
n
g
o
n
t
h
is
s
u
b
j
e
ct
s
u
c
h
li
k
e
Si
n
g
h
[
1
4
]
,
in
2
0
1
5
u
s
ed
i
m
a
g
e
e
n
cr
y
p
tio
n
u
s
i
n
g
ellip
tic
c
u
r
v
e
cr
y
p
to
g
r
a
p
h
y
.
T
h
e
y
i
m
p
le
m
en
t
th
e
elli
p
tic
cu
r
v
e
cr
y
p
to
g
r
ap
h
y
to
en
cr
y
p
t,
d
ec
r
y
p
t
a
n
d
d
ig
itall
y
s
ig
n
th
e
cip
h
er
i
m
a
g
e
to
p
r
o
v
id
e
a
u
t
h
e
n
ticit
y
a
n
d
in
te
g
r
it
y
.
Ah
m
ed
[
1
5
]
,
in
2
0
1
3
u
s
ed
a
h
y
b
r
id
ch
ao
tic
s
y
s
te
m
a
n
d
c
y
c
lic
elli
p
tic
cu
r
v
e
f
o
r
i
m
a
g
e
e
n
cr
y
p
tio
n
.
T
h
e
n
e
w
s
ch
e
m
e
g
e
n
er
ates
an
i
n
itial
k
e
y
s
t
r
e
a
m
b
a
s
e
d
o
n
c
h
a
o
t
i
c
s
y
s
t
e
m
a
n
d
a
n
e
x
t
e
r
n
a
l
s
e
c
r
e
t
k
e
y
o
f
2
5
6
-
b
i
t
i
n
a
f
e
e
d
b
a
c
k
m
a
n
n
e
r
.
T
h
e
n
,
t
h
e
g
e
n
e
r
a
t
e
d
k
e
y
s
t
r
e
a
m
a
r
e
m
i
x
e
d
w
i
t
h
k
e
y
s
e
q
u
e
n
c
e
s
d
e
r
i
v
e
d
f
r
o
m
t
h
e
c
y
c
l
i
c
e
l
l
i
p
t
i
c
c
u
r
v
e
p
o
i
n
t
s
.
T
h
o
r
o
u
g
h
en
cr
y
p
t
io
n
p
er
f
o
r
m
a
n
ce
a
n
d
s
ec
u
r
it
y
a
n
al
y
s
i
s
ascer
tai
n
s
e
f
f
icac
y
o
f
t
h
e
p
r
o
p
o
s
ed
en
cr
y
p
tio
n
s
c
h
e
m
e.
Nag
ar
aj
[
1
6
]
th
e
y
p
r
o
p
o
s
e
in
2
0
1
5
a
n
e
w
e
n
cr
y
p
tio
n
tech
n
iq
u
e
u
s
i
n
g
el
lip
tic
c
u
r
v
e
cr
y
p
to
g
r
ap
h
y
w
it
h
a
m
ag
ic
m
atr
i
x
o
p
er
atio
n
s
f
o
r
s
ec
u
r
i
n
g
i
m
ag
e
s
t
h
at
tr
an
s
m
it
s
o
v
er
a
p
u
b
lic
u
n
s
ec
u
r
ed
ch
an
n
el.
T
h
er
e
ar
e
t
w
o
m
o
s
t
i
m
p
o
r
ta
n
t
g
r
o
u
p
s
o
f
i
m
a
g
e
e
n
cr
y
p
tio
n
alg
o
r
it
h
m
s
:
s
o
m
e
ar
e
n
o
n
c
h
ao
s
-
b
a
s
ed
s
elec
ti
v
e
m
et
h
o
d
s
an
d
c
h
a
o
s
-
b
ased
s
elec
ti
v
e
m
et
h
o
d
s
.
T
h
e
m
aj
o
r
ity
o
f
t
h
ese
al
g
o
r
ith
m
s
is
p
la
n
n
ed
f
o
r
a
s
p
ec
if
ic
i
m
a
g
e
f
o
r
m
at,
co
m
p
r
es
s
ed
o
r
u
n
co
m
p
r
e
s
s
ed
.
2.
E
L
L
I
P
T
I
C
C
URVE F
U
NCT
I
O
N
E
llip
tic
cu
r
v
e
cr
y
p
to
s
y
s
te
m
(
E
C
C
)
it
i
s
a
r
ea
s
o
n
ab
le
e
n
cr
y
p
tio
n
s
tr
ate
g
y
to
b
e
u
ti
li
ze
d
in
f
o
r
ex
a
m
p
le:
e
m
b
ed
d
ed
s
y
s
te
m
s
a
n
d
m
o
b
ile
d
e
v
ices,
th
at
is
b
ec
au
s
e
it c
a
n
p
r
o
v
id
e
h
ig
h
s
ec
u
r
i
t
y
w
i
th
s
m
aller
k
e
y
s
ize
an
d
f
e
w
er
co
m
p
u
tatio
n
s
w
it
h
le
s
s
m
e
m
o
r
y
u
s
ag
e
a
n
d
lo
w
er
p
o
w
er
co
n
s
u
m
p
tio
n
s
[
1
7
]
.
D
ef
i
n
itio
n
:
An
ellip
tic
cu
r
v
e
o
v
er
a
p
r
im
e
f
ield
is
d
ef
in
ed
b
y
(
,
)
:
2
=
3
+
+
,
w
h
er
e
>
3
,
,
∈
an
d
s
atis
f
y
th
e
co
n
d
itio
n
4
3
+
27
2
≢
0
.
T
h
e
ellip
tic
cu
r
v
e
g
r
o
u
p
(
)
co
n
s
is
t
s
o
f
all
p
o
in
ts
(
,
)
th
at
s
ati
s
f
y
th
e
ell
ip
tic
cu
r
v
e
(
,
)
an
d
th
e
p
o
in
t
at
th
e
in
f
i
n
it
y
∞
[
1
8
,
1
9
]
.
E
llip
tic
cu
r
v
e
o
p
er
atio
n
s
:
T
h
e
p
r
i
m
ar
y
o
p
er
atio
n
s
r
elate
d
to
ellip
tic
c
u
r
v
e
f
u
n
ctio
n
is
t
h
e
ellip
tic
cu
r
v
e
s
ca
lar
m
u
ltip
licat
io
n
w
h
ic
h
i
s
th
e
m
ai
n
o
p
er
atio
n
o
n
t
h
e
ellip
tic
cu
r
v
e
th
a
t
co
n
s
u
m
es
m
o
r
e
t
i
m
e
in
en
cr
y
p
tio
n
a
n
d
d
ec
r
y
p
tio
n
o
p
er
atio
n
s
.
T
w
o
o
p
er
atio
n
s
ar
e
in
v
o
l
v
ed
in
ca
lc
u
lati
n
g
t
h
e
ellip
tic
cu
r
v
e
s
ca
lar
m
u
ltip
licatio
n
,
th
e
y
ar
e
p
o
in
t a
d
d
itio
n
an
d
p
o
in
t d
o
u
b
lin
g
[
2
0
]
.
P
o
in
t
ad
d
itio
n
:
Su
p
p
o
s
e
1
=
(
1
,
1
)
an
d
2
=
(
2
,
2
)
,
w
h
er
e
1
≠
2
,
ar
e
t
w
o
p
o
in
ts
lie
o
n
a
n
ellip
tic
cu
r
v
e
(
,
)
.
A
d
d
in
g
t
h
e
t
w
o
p
o
in
ts
1
an
d
2
g
i
v
i
n
g
a
t
h
ir
d
p
o
in
t
3
=
(
3
,
3
)
,
as
3
≡
(
2
−
1
−
2
)
,
3
≡
(
(
1
−
3
)
−
1
)
)
an
d
=
2
−
1
2
−
1
.
3
s
h
o
u
ld
lie
o
n
t
h
e
s
a
m
e
c
u
r
v
e
(
,
)
.
P
o
in
t
d
o
u
b
lin
g
:
S
u
p
p
o
s
e
=
(
1
,
1
)
is
a
p
o
in
t
o
n
an
ellip
tic
cu
r
v
e
(
,
)
,
th
e
p
o
in
t
=
2
=
(
2
,
2
)
th
at
r
es
u
lts
f
r
o
m
d
o
u
b
lin
g
t
h
e
p
o
in
t
as
2
≡
(
2
−
2
1
)
,
2
≡
(
(
1
−
2
)
−
1
)
)
an
d
=
3
1
2
+
2
1
.
is
also
p
o
in
t o
n
an
ellip
t
ic
cu
r
v
e
(
,
)
.
E
llip
tic
cu
r
v
e
s
ca
lar
m
u
ltip
li
ca
tio
n
:
L
et
b
e
an
y
p
o
in
t
o
n
th
e
ellip
tic
cu
r
v
e
(
,
)
.
E
llip
tic
c
u
r
v
e
s
ca
lar
m
u
ltip
licatio
n
o
p
er
atio
n
o
v
er
is
d
ef
in
ed
b
y
t
h
e
r
ep
ea
ted
ad
d
itio
n
=
+
+
⋯
+
⏟
t
i
m
es
.
3.
H
I
L
B
E
R
T
M
AT
RIX
Hilb
er
t
[
2
1
]
in
1
8
9
4
p
r
esen
ted
a
s
q
u
ar
e
m
atr
i
x
w
it
h
e
n
tr
ie
s
b
ein
g
th
e
u
n
it
f
r
ac
tio
n
s
.
T
h
at
is
m
ea
n
ea
ch
ele
m
e
n
t
o
f
t
h
is
m
atr
i
x
s
a
y
h
ij
w
i
ll
b
e
w
r
itte
n
as
1
i
+
j
−
1
.
Dav
i
d
Hilb
er
t
n
a
m
ed
th
i
s
t
y
p
e
o
f
m
atr
ices
as
Hilb
er
t
Ma
tr
ix
.
Fo
r
th
e
p
u
r
p
o
s
es
o
f
th
e
w
o
r
k
in
t
h
i
s
s
t
u
d
y
,
w
e
w
il
l
u
s
e
n
×
n
-
Hilb
e
r
t
m
atr
i
x
w
i
th
f
o
r
m
o
d
u
lo
2n
−
1
,
s
o
n
o
d
en
o
m
i
n
ato
r
w
ill
ap
p
ea
r
in
an
y
ele
m
en
t
o
f
t
h
is
m
a
tr
ix
.
So
b
y
co
m
p
u
ti
n
g
t
h
e
i
n
v
er
s
e
o
f
ea
c
h
d
en
o
m
i
n
ato
r
m
o
d
u
lo
(
i
+
j
−
1
)
to
cr
ea
te
a
n
e
w
Hilb
er
t
m
atr
i
x
.
Fo
r
ex
a
m
p
le,
4
×
4
−
Hilb
er
t
m
atr
i
x
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
I
ma
g
e
en
cryp
tio
n
b
a
s
ed
o
n
ell
ip
tic
cu
r
ve
cryp
to
s
ystem
(
Za
h
r
a
a
K
a
d
h
im
Ob
a
id
a
n
d
)
1295
[
1
1
2
1
2
1
3
1
3
1
4
1
4
1
5
1
3
1
4
1
4
1
5
1
5
1
6
1
6
1
7
]
ca
n
b
e
ex
ch
a
n
g
e
to
4
×
4
-
Hilb
er
t m
atr
i
x
m
o
d
u
lo
7
,
as
[
1
4
4
5
5
2
2
3
5
2
2
3
3
6
6
7
]
.
I
n
v
o
l
v
ed
Hilb
er
t
m
atr
i
x
w
i
th
cr
y
p
to
g
r
ap
h
y
is
n
e
w
s
u
b
j
ec
t
d
u
s
c
u
d
e
r
ec
en
t
l
y
b
y
R
o
o
p
ae
i
[
2
2
]
.
I
n
th
i
s
w
o
r
k
,
w
e
w
ill
tr
y
to
u
s
e
th
i
s
m
atr
i
x
to
h
ig
h
le
v
el
o
f
s
ec
u
r
i
t
y
to
e
n
cr
y
p
te
i
m
a
g
es.
I
n
d
ee
d
,
th
er
e
ar
e
an
o
t
h
er
t
y
p
es o
f
i
n
v
er
tib
le
m
atr
i
x
[
2
3
,
2
4
]
ca
n
b
e
s
u
b
j
ec
t to
f
u
t
u
r
e
s
t
u
d
ies.
4.
M
AT
E
RIAL
S AN
D
M
E
T
H
O
DS
4
.
1
.
P
ro
po
s
ed
a
lg
o
rit
h
m
s
H
y
b
r
id
th
e
E
C
C
a
n
d
Hilb
er
t
m
atr
i
x
is
ap
p
r
o
ac
h
to
en
cr
y
p
t
th
e
i
m
ag
e
is
a
tec
h
n
iq
u
e
h
as
b
ee
n
in
tr
o
d
u
ce
d
in
th
is
s
ec
tio
n
i
n
t
w
o
ca
s
e
s
.
T
h
ese
tec
h
n
iq
u
es
i
n
cr
ea
s
e
t
h
e
s
ec
u
r
it
y
a
n
d
m
a
k
e
th
e
s
y
s
te
m
m
o
r
e
ef
f
icien
t,
also
it
s
p
ee
d
s
u
p
th
e
d
ec
r
y
p
tio
n
co
m
p
u
tat
io
n
s
s
i
n
ce
it
d
o
es
n
o
t
n
ee
d
th
e
co
m
p
u
tatio
n
o
f
th
e
k
e
y
m
atr
i
x
in
v
er
s
e
f
o
r
th
e
f
ir
s
t c
as
e.
Su
p
p
o
s
e
th
e
s
en
d
er
(
Us
er
A
)
w
a
n
ts
to
s
en
d
a
n
i
m
ag
e
M
to
th
e
o
th
er
p
ar
t
y
(
U
s
er
B
)
u
s
i
n
g
t
h
i
s
tech
n
iq
u
e
o
v
er
a
n
in
s
ec
u
r
e
c
h
an
n
el.
Fir
s
tl
y
,
th
e
y
s
h
o
u
ld
a
g
r
ee
o
n
th
e
e
llip
tic
cu
r
v
e
E
p
(
a
,
b
)
an
d
s
h
ar
e
th
e
d
o
m
ai
n
p
ar
a
m
eter
s
{
a
,
b
,
p
,
G
}
,
w
h
er
e
G
is
th
e
g
e
n
er
ato
r
p
o
in
t.
T
h
en
ea
c
h
p
ar
t
y
n
ee
d
s
to
ch
o
o
s
e
r
an
d
o
m
l
y
h
i
s
p
r
iv
ate
k
e
y
f
r
o
m
th
e
in
ter
v
al
[
1
,
p
−
1
]
;
n
A
f
o
r
User
A
an
d
n
B
f
o
r
User
B
,
an
d
g
e
n
er
ates
h
is
p
u
b
lic
k
e
y
a
s
P
A
=
n
A
∙
G
an
d
P
B
=
n
B
∙
G
.
E
ac
h
u
s
er
m
u
ltip
lies
h
i
s
p
r
iv
ate
k
e
y
b
y
t
h
e
p
u
b
lic
k
e
y
o
f
th
e
o
th
er
u
s
er
to
g
et
t
h
e
in
itial
k
e
y
a
s
:
K
=
n
A
∙
P
B
=
n
B
∙
P
A
=
n
A
∙
n
B
∙
G
=
(
x
,
y
)
t
h
en
co
m
p
u
te
s
;
K
1
=
x
.
G
=
(
k
11
,
k
12
)
K
2
=
y
.
G
=
(
k
21
,
k
22
)
T
h
e
n
ex
t
s
tep
is
g
en
er
ati
n
g
t
h
e
s
ec
r
et
k
e
y
m
atr
ix
K
m
b
y
s
e
n
d
er
an
d
r
ec
eiv
er
.
T
h
e
in
v
er
s
e
o
f
t
h
e
k
e
y
m
atr
i
x
d
o
es n
o
t a
l
w
a
y
s
e
x
is
t.
So
,
if
t
h
e
k
e
y
m
atr
ix
is
n
o
t in
v
er
tib
le,
th
e
r
ec
ip
ien
t c
a
n
n
o
t d
e
cr
y
p
t t
h
e
e
n
cr
y
p
ted
d
ata.
T
o
s
o
lv
e
th
i
s
p
r
o
b
le
m
,
t
h
e
s
elf
i
n
v
er
tib
le
k
e
y
m
atr
i
x
[
2
5
]
w
ill
b
e
g
e
n
er
ated
,
a
n
d
t
h
e
s
a
m
e
k
e
y
w
il
l
b
e
u
s
ed
f
o
r
en
cr
y
p
tio
n
a
n
d
d
ec
r
y
p
tio
n
an
d
n
o
n
ee
d
to
f
i
n
d
th
e
i
n
v
er
s
e
k
e
y
m
atr
ix
i
n
t
h
e
f
ir
s
t c
ase.
B
u
t th
e
s
ec
o
n
d
ca
s
e,
w
e
n
ee
d
co
m
p
u
te
i
n
v
er
s
e
f
o
r
th
e
k
e
y
m
a
tr
ix
.
T
h
e
f
ir
s
t
ca
s
e
an
d
th
e
s
ec
o
n
d
ca
s
e
w
ill
b
e
i
m
p
l
e
m
en
ted
o
n
g
r
a
y
s
ca
le
i
m
a
g
es.
T
h
e
i
m
ag
e
w
ill
b
e
d
iv
id
ed
in
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
p
ix
el
v
al
u
es.
So
,
ea
ch
p
ar
t
y
p
r
o
d
u
ce
s
th
e
4
×4
k
e
y
m
a
tr
ix
K
m
,
w
h
er
e
K
m
b
e
a
s
elf
in
v
er
tib
le
m
atr
ix
p
ar
titi
o
n
ed
as
f
o
u
r
s
q
u
ar
e
m
a
tr
ices:
K
11
,
K
12
,
K
21
an
d
K
22
.
So
,
w
e
ca
n
r
e
w
r
i
te
K
m
as
[
K
11
K
12
K
21
K
22
]
.
No
w
,
if
w
e
co
n
s
id
er
K
11
=
[
k
11
k
12
k
21
k
22
]
w
e
ca
n
ca
lcu
late
t
h
e
o
th
er
s
s
q
u
ar
e
m
a
tr
ices a
s
:
K
12
=
I
−
K
11
,
K
21
=
I
+
K
11
,
K
11
+
K
22
=
0
,
w
h
er
e
I
is
th
e
id
e
n
tit
y
m
a
tr
ix
.
4
.
2
.
F
irst
ca
s
e
I
n
th
i
s
ca
s
e
g
e
n
er
atin
g
K
m
w
ill
b
e
as
f
o
llo
w
s
:
S
u
p
p
o
s
e
th
at
K
11
=
[
k
11
k
12
k
21
k
22
]
+
H
2
×
2
,
w
h
er
e
H
is
Hilb
er
t
m
atr
i
x
w
it
h
d
i
m
e
n
s
io
n
2
×
2
.
So
,
if
w
e
co
n
s
id
er
K
m
=
[
K
11
K
12
K
21
K
22
]
b
e
a
s
elf
i
n
v
er
tib
le
m
atr
i
x
,
th
e
n
o
th
er
p
ar
titi
o
n
s
o
f
t
h
e
s
ec
r
et
m
atr
i
x
k
e
y
K
m
is
o
b
tain
ed
b
y
s
o
lv
in
g
K
12
=
I
−
K
11
,
K
21
=
I
+
K
11
,
K
11
+
K
22
=
0
,
w
h
er
e
I
is
t
h
e
id
en
tit
y
m
atr
ix
.
No
w
,
s
ep
ar
ate
th
e
i
m
ag
e
p
i
x
el
v
al
u
es
i
n
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
,
ea
c
h
b
lo
ck
w
ill
b
e
co
n
v
er
ted
to
a
v
ec
to
r
o
f
s
ize
4
×
1
:
P
1
,
P
2
,
P
3
,
…
.
Nex
t
s
tep
w
il
l
b
e
ca
lc
u
lati
n
g
t
h
e
cip
h
er
ed
v
ec
to
r
s
C
1
,
C
2
,
C
3
,
…
as:
C
i
=
K
m
.
P
i
mod
256
,
i
=
1
,
2
,
3
,
…
.
T
h
e
last
s
tep
f
o
r
th
e
e
n
cr
y
p
tio
n
al
g
o
r
ith
m
i
s
r
ec
o
n
s
tr
u
ct
t
h
e
cip
h
er
ed
i
m
a
g
e
f
r
o
m
t
h
e
v
al
u
es
o
f
C
i
an
d
s
e
n
d
it
to
t
h
e
o
th
er
p
ar
t
y
B
.
Fo
r
th
e
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
es,
p
ar
ty
B
w
ill
s
ep
ar
ate
th
e
cip
h
er
ed
im
a
g
e
p
ix
el
v
al
u
es
i
n
to
b
lo
ck
s
o
f
4
×
1
:
C
1
,
C
2
,
C
3
,
…
.
T
h
e
n
ex
t
s
tep
is
co
m
p
u
t
in
g
P
1
,
P
2
,
P
3
,
…
as:
P
i
=
K
m
.
C
i
mod
256
,
i
=
1
,
2
,
3
,
…
.
T
h
e
last
s
tep
f
o
r
th
e
d
ec
r
y
p
tio
n
alg
o
r
ith
m
is
r
ec
o
n
s
tr
u
c
t th
e
p
l
ain
i
m
ag
e
f
r
o
m
t
h
e
v
al
u
e
s
o
f
P
i
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
2
,
A
p
r
il 2
0
2
1
:
1
2
9
3
-
1302
1296
4
.
3
.
Seco
nd
ca
s
e
I
n
th
is
ca
s
e
g
e
n
er
ati
n
g
K
m
̌
w
ill
b
e
as
f
o
llo
w
s
:
K
m
̌
=
K
m
+
H
4
×
4
,
,
w
h
er
e
K
m
=
[
K
11
K
12
K
21
K
22
]
an
d
H
4
×
4
,
w
h
er
e
H
is
Hilb
er
t
m
atr
i
x
w
it
h
d
i
m
e
n
s
io
n
4
×
4
.
No
w
,
s
ep
ar
ate
t
h
e
i
m
a
g
e
p
ix
e
l
v
al
u
es
i
n
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
,
ea
ch
b
lo
ck
w
ill
b
e
co
n
v
er
ted
to
a
v
ec
to
r
o
f
s
ize
4
×
1
:
P
1
,
P
2
,
P
3
,
…
.
Nex
t
s
tep
w
ill
b
e
ca
lcu
lati
n
g
th
e
cip
h
er
ed
v
ec
to
r
s
C
1
,
C
2
,
C
3
,
…
as:
C
i
=
K
m
.
P
i
m
od
256
,
i
=
1
,
2
,
3
,
…
.
T
h
e
last
s
tep
f
o
r
th
e
en
cr
y
p
t
io
n
alg
o
r
ith
m
i
s
r
ec
o
n
s
tr
u
ct
th
e
c
ip
h
er
ed
i
m
ag
e
f
r
o
m
t
h
e
v
alu
e
s
o
f
C
i
an
d
s
en
d
it
to
th
e
o
t
h
er
p
ar
ty
B
.
Fo
r
th
e
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
es,
p
ar
t
y
B
w
ill
s
ep
ar
ate
th
e
cip
h
er
ed
im
a
g
e
p
ix
el
v
alu
es
i
n
to
b
lo
ck
s
o
f
4
×
1
:
C
1
,
C
2
,
C
3
,
…
.
T
h
e
n
ex
t
s
tep
is
co
m
p
u
ti
n
g
P
1
,
P
2
,
P
3
,
…
as:
P
i
=
K
M
̌
−
1
.
C
i
mod
256
,
i
=
1
,
2
,
3
,
…
.
T
h
e
last
s
tep
f
o
r
th
e
d
ec
r
y
p
tio
n
al
g
o
r
ith
m
i
s
r
ec
o
n
s
tr
u
ct
th
e
p
lai
n
i
m
ag
e
f
r
o
m
t
h
e
v
alu
e
s
o
f
P
i
.
5.
I
M
P
L
E
M
E
NT
AT
I
O
N
O
F
T
H
E
P
RO
P
O
SE
D
A
L
G
O
R
I
T
H
M
S
M
AT
E
RIA
L
S A
ND
M
E
T
H
O
DS
Ass
u
m
e
th
at
p
ar
t
y
A
w
a
n
ts
t
o
s
en
d
an
i
m
a
g
e
"
C
A
T
I
M
A
G
E
"
to
p
ar
ty
B
u
s
in
g
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
.
T
h
e
y
w
ill
ag
r
ee
d
to
u
s
e
an
ellip
tic
cu
r
v
e
s
a
y
E
37
(
1
,
3
)
:
y
2
=
x
3
+
x
+
3
mod
37
,
w
h
er
e
a
3
+
27
b
2
modp
=
247
mod
37
=
25
≠
0
.
T
h
e
p
o
in
ts
th
at
s
a
tis
f
y
in
g
E
37
(
1
,
3
)
ar
e:
(
0
,
1
5
)
,
(
0
,
2
2
)
,
(
3
,
1
2
)
,
(
3
,
2
5
)
,
(
4
,
1
6
)
,
(
4
,
2
1
)
,
(
6
,
1
5
)
,
(
6
,
2
2
)
,
(
9
,
1
)
,
(
9
,
3
6
)
,
(
1
2
,
2
)
,
(
1
2
,
3
5
)
,
(
1
3
,
1
7
)
,
(
1
3
,
2
0
)
,
(
1
5
,
1
0
)
,
(
1
5
,
2
7
)
,
(
1
7
,
7
)
,
(
1
7
,
3
0
)
,
(
1
8
,
9
)
,
(
1
8
,
2
8
)
,
(
1
9
,
6
)
,
(
1
9
,
3
1
)
,
(
2
6
,
1
7
)
,
(
2
6
,
2
0
)
,
(
2
9
,
1
)
,
(
2
9
,
3
6
)
,
(
3
1
,
1
5
)
,
(
3
1
,
2
2
)
,
(
3
2
,
1
3
)
,
(
3
2
,
2
4
)
,
(
3
3
,
3
)
,
(
3
3
,
3
4
)
,
(
3
4
,
1
1
)
,
(
3
4
,
2
6
)
,
(
3
5
,
1
7
)
,
(
3
5
,
2
0
)
,
(
3
6
,
1
)
an
d
(
3
6
,
3
6
)
.
So
#
E
37
(
1
,
3
)
=
39
.
So
,
if
w
e
ch
o
o
s
e
G
=
(
0
,
15
)
,
th
e
d
o
m
a
in
p
ar
a
m
e
ter
s
f
o
r
E
37
(
1
,
3
)
a
r
e
{
a
,
b
,
P
,
G
}
=
{
1
,
3
,
37
,
(
0
,
15
)
}
.
Fig
u
r
es
1
an
d
2
s
h
o
w
t
h
e
o
r
i
g
in
a
l
i
m
a
g
e,
cip
h
er
ed
i
m
a
g
e,
an
d
d
ec
ip
h
er
ed
i
m
ag
e
f
o
r
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
f
ir
s
t
ca
s
e
a
n
d
s
ec
o
n
d
ca
s
e
r
esp
ec
ti
v
el
y
.
M
A
T
L
A
B
R
2
0
1
4
a,
6
4
-
b
it
s
o
f
t
w
ar
e
o
n
C
o
r
e
i5
co
m
p
u
ter
w
it
h
C
P
U
@
1
.
8
0
GHz
2
.
3
0
G
Hz
an
d
R
A
M
6
GB
is
u
s
ed
f
o
r
en
cr
y
p
tio
n
a
n
d
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
es.
No
w
,
to
ap
p
l
y
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
f
ir
s
t c
ase
w
e
w
i
ll d
o
th
e
f
o
llo
w
i
n
g
s
tep
s
:
Step
1
: “
Gen
er
at
in
g
o
f
k
e
y
s
”
P
ar
ty
C
h
o
o
s
es t
h
e
p
r
iv
ate
k
e
y
=
11
∈
[
1
,
36
]
He
co
m
p
u
tes t
h
e
p
u
b
lic
k
e
y
=
∙
=
11
(
0
,
15
)
=
(
3
,
25
)
He
co
m
p
u
tes t
h
e
=
.
=
11
(
26
,
20
)
=
(
26
,
17
)
=
(
,
)
He
co
m
p
u
tes
1
=
.
=
26
(
0
,
15
)
=
(
26
,
17
)
=
(
11
,
12
)
an
d
2
=
.
=
17
(
0
,
15
)
=
(
19
,
31
)
=
(
21
,
22
)
He
co
n
s
tr
u
ct
s
11
=
[
11
12
21
22
]
+
2
×
2
=
[
26
17
19
31
]
+
[
1
2
2
3
]
=
[
27
19
21
34
]
He
ca
lcu
late
s
12
=
(
−
11
)
256
=
(
[
1
0
0
1
]
−
[
27
19
21
34
]
)
256
=
[
230
237
235
223
]
He
ca
lcu
late
s
21
=
(
+
11
)
256
=
(
[
1
0
0
1
]
+
[
27
19
21
34
]
)
256
=
[
28
19
21
35
]
He
s
o
lv
es
11
+
22
=
0
,
s
o
22
=
(
−
11
)
256
=
[
229
237
235
222
]
Fin
all
y
,
h
e
co
n
s
tr
u
cts
t
h
e
s
el
f
i
n
v
er
tib
le
k
e
y
m
atr
i
x
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
I
n
th
e
o
th
er
s
id
e:
P
ar
ty
C
h
o
o
s
es t
h
e
p
r
iv
ate
k
e
y
=
13
∈
[
1
,
36
]
He
co
m
p
u
tes t
h
e
p
u
b
lic
k
e
y
=
.
=
13
(
0
,
15
)
=
(
26
,
20
)
Hec
o
m
p
u
te
s
=
.
=
13
(
3
,
25
)
=
(
26
,
17
)
=
(
,
)
He
co
m
p
u
tes
1
=
.
=
26
(
0
,
15
)
=
(
26
,
17
)
=
(
11
,
12
)
and
2
=
.
=
17
(
0
,
15
)
=
(
19
,
31
)
=
(
21
,
22
)
H
e
w
i
ll d
o
th
e
o
th
er
s
tep
s
a
m
e
lik
e
p
ar
t
y
to
co
n
s
tr
u
ct
th
e
s
el
f
in
v
er
tib
le
k
e
y
m
atr
ix
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
I
ma
g
e
en
cryp
tio
n
b
a
s
ed
o
n
ell
ip
tic
cu
r
ve
cryp
to
s
ystem
(
Za
h
r
a
a
K
a
d
h
im
Ob
a
id
a
n
d
)
1297
Step
2
: “
E
n
cr
y
p
t
io
n
b
y
p
ar
t
y
”
He
s
ep
ar
ates th
e
p
ix
e
l v
al
u
es
o
f
“C
A
T
I
MA
GE
”
i
n
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
as:
1
2
3
4
5
6
7
8
9
1
2
4
9
2
4
6
2
4
4
2
4
5
2
4
7
2
4
4
2
3
7
2
3
1
…
2
2
4
9
2
4
6
2
4
4
2
4
5
2
4
6
2
4
4
2
3
7
2
3
0
…
3
2
4
9
2
4
6
2
4
3
2
4
4
2
4
6
2
4
4
2
3
7
2
3
0
…
4
2
4
8
2
4
5
2
4
3
2
4
4
2
4
6
2
4
3
2
3
6
2
3
0
…
5
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
So
,
1
=
[
249
246
244
245
]
,
2
=
[
247
244
237
231
]
,
3
=
[
249
246
244
245
]
, …
He
co
m
p
u
tes t
h
e
v
alu
e
s
o
f
1
,
2
,
3
,
…
as:
1
=
∙
1
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
249
246
244
245
]
256
=
[
142
128
147
129
]
,
2
=
∙
2
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
247
244
237
231
]
256
=
[
242
115
252
128
]
,
3
=
∙
3
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
249
246
244
245
]
256
=
[
142
128
147
129
]
, …
He
r
ec
o
n
s
tr
u
ct
s
th
e
cip
h
er
ed
im
ag
e
f
r
o
m
t
h
e
v
a
lu
e
s
o
f
as:
1
2
3
4
5
6
7
8
9
1
1
4
2
1
2
8
1
4
7
1
2
9
2
4
2
1
1
5
2
5
2
1
2
8
…
2
1
4
2
1
2
8
1
4
7
1
2
9
2
3
4
1
2
7
2
4
3
1
4
1
…
3
1
8
7
1
8
2
1
9
3
1
8
4
2
3
4
1
2
7
2
4
3
1
4
1
…
4
1
4
1
1
2
7
1
4
6
1
2
8
2
4
1
1
1
4
2
5
1
1
2
7
…
5
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
He
s
en
d
s
it to
t
h
e
o
th
er
p
ar
t
y
.
Step
3
: “
Dec
r
y
p
tio
n
b
y
p
ar
t
y
B”
He
s
ep
ar
ates th
e
p
ix
e
l v
al
u
es
o
f
cip
h
er
ed
i
m
a
g
e
in
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
as:
1
=
[
142
128
147
129
]
,
2
=
[
242
115
252
128
]
,
3
=
[
142
128
147
129
]
, …
He
co
m
p
u
tes t
h
e
v
alu
e
s
o
f
1
,
2
,
3
,
…
as:
1
=
∙
1
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
142
128
147
129
]
256
=
[
249
246
244
245
]
,
2
=
∙
2
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
242
115
252
128
]
256
=
[
247
244
237
231
]
,
3
=
∙
3
=
[
27
19
21
34
230
237
235
223
28
19
21
35
229
237
235
222
]
[
142
128
147
129
]
256
=
[
249
246
244
245
]
, …
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
2
,
A
p
r
il 2
0
2
1
:
1
2
9
3
-
1302
1298
He
r
ec
o
n
s
tr
u
ct
s
th
e
p
lai
n
i
m
ag
e
f
r
o
m
th
e
v
al
u
es o
f
as:
1
2
3
4
5
6
7
8
9
1
2
4
9
2
4
6
2
4
4
2
4
5
2
4
7
2
4
4
2
3
7
2
3
1
…
2
2
4
9
2
4
6
2
4
4
2
4
5
2
4
6
2
4
4
2
3
7
2
3
0
…
3
2
4
9
2
4
6
2
4
3
2
4
4
2
4
6
2
4
4
2
3
7
2
3
0
…
4
2
4
8
2
4
5
2
4
3
2
4
4
2
4
6
2
4
3
2
3
6
2
3
0
…
5
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
(
a)
(
b
)
(
c)
Fig
u
r
e
1
.
(
a)
P
lain
i
m
a
g
e,
(
b
)
cip
h
er
ed
i
m
ag
e
,
a
n
d
(
c)
d
ec
ip
h
er
ed
i
m
a
g
e
f
o
r
ca
t
i
m
a
g
e
/
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
f
ir
s
t c
ase
T
h
e
s
ec
o
n
d
ca
s
e
f
o
r
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
w
ill ap
p
l
y
b
y
d
o
in
g
t
h
e
f
o
llo
w
in
g
s
tep
s
:
Step
1
: “
Gen
er
at
in
g
o
f
k
e
y
s
”
P
ar
ty
C
h
o
o
s
es t
h
e
p
r
iv
ate
k
e
y
=
11
∈
[
1
,
36
]
He
co
m
p
u
tes t
h
e
p
u
b
lic
k
e
y
=
∙
=
11
(
0
,
15
)
=
(
3
,
25
)
He
co
m
p
u
tes t
h
e
=
.
=
11
(
26
,
20
)
=
(
26
,
17
)
=
(
,
)
He
co
m
p
u
tes
1
=
.
=
26
(
0
,
15
)
=
(
26
,
17
)
=
(
11
,
12
)
an
d
2
=
.
=
17
(
0
,
15
)
=
(
19
,
31
)
=
(
21
,
22
)
He
co
n
s
tr
u
ct
s
11
=
[
11
12
21
22
]
=
[
26
17
19
31
]
He
ca
lcu
late
s
12
=
(
−
11
)
256
=
(
[
1
0
0
1
]
−
[
26
17
19
31
]
)
256
=
[
231
239
237
226
]
He
ca
lcu
late
s
21
=
(
+
11
)
256
=
(
[
1
0
0
1
]
+
[
26
17
19
31
]
)
256
=
[
27
17
19
32
]
He
s
o
lv
es
11
+
22
=
0
,
s
o
22
=
(
−
11
)
256
=
[
230
239
237
225
]
He
co
n
s
tr
u
ct
s
t
h
e
s
el
f
in
v
er
tib
l
e
k
e
y
m
atr
i
x
=
[
26
17
19
31
231
239
237
226
27
17
19
32
230
239
237
225
]
Fin
all
y
,
h
e
ca
lc
u
late
s
̌
=
+
4
×
4
=
(
[
26
17
19
31
231
239
237
226
27
17
19
32
230
239
237
225
]
+
[
1
4
4
5
5
2
2
3
5
2
2
3
3
6
6
7
]
)
256
=
[
27
21
23
36
236
241
239
229
32
19
21
35
233
245
243
232
]
.
I
n
th
e
o
th
er
s
id
e:
P
ar
ty
C
h
o
o
s
es t
h
e
p
r
iv
ate
k
e
y
=
13
∈
[
1
,
36
]
He
co
m
p
u
tes t
h
e
p
u
b
lic
k
e
y
=
.
=
13
(
0
,
15
)
=
(
26
,
20
)
Hec
o
m
p
u
te
s
=
.
=
13
(
3
,
25
)
=
(
26
,
17
)
=
(
,
)
He
co
m
p
u
tes
1
=
.
=
26
(
0
,
15
)
=
(
26
,
17
)
=
(
11
,
12
)
an
d
2
=
.
=
17
(
0
,
15
)
=
(
19
,
31
)
=
(
21
,
22
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
I
ma
g
e
en
cryp
tio
n
b
a
s
ed
o
n
ell
ip
tic
cu
r
ve
cryp
to
s
ystem
(
Za
h
r
a
a
K
a
d
h
im
Ob
a
id
a
n
d
)
1299
He
w
i
ll d
o
th
e
o
th
er
s
tep
s
a
m
e
lik
e
p
ar
t
y
to
co
n
s
tr
u
ct
th
e
m
a
tr
ix
̌
=
[
27
21
23
36
236
241
239
229
32
19
21
35
233
245
243
232
]
Fin
all
y
,
h
e
co
m
p
u
te
s
̌
−
1
=
[
97
235
57
118
48
69
141
119
136
53
73
125
43
177
107
106
]
.
Step
2
: “
E
n
cr
y
p
t
io
n
b
y
p
ar
t
y
”
He
s
ep
ar
ates
th
e
p
ix
el
v
al
u
es
o
f
“
C
A
T
I
MA
GE
”
i
n
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
as
h
e
d
id
in
th
e
p
r
o
p
o
s
e
d
alg
o
r
ith
m
f
ir
s
t c
ase
:
He
co
m
p
u
tes t
h
e
v
alu
e
s
o
f
1
,
2
,
3
,
…
as:
1
=
̌
∙
1
=
[
27
21
23
36
236
241
239
229
32
19
21
35
233
245
243
232
]
[
249
246
244
245
]
256
=
[
6
236
239
179
]
,
2
=
̌
∙
2
=
[
27
21
23
36
236
241
239
229
32
19
21
35
233
245
243
232
]
[
247
244
237
231
]
256
=
[
4
103
196
238
]
,
3
=
̌
∙
3
=
[
27
21
23
36
236
241
239
229
32
19
21
35
233
245
243
232
]
[
249
246
244
245
]
256
=
[
6
236
239
179
]
, …
He
r
ec
o
n
s
tr
u
ct
s
th
e
cip
h
er
ed
im
ag
e
f
r
o
m
t
h
e
v
a
lu
e
s
o
f
as:
1
2
3
4
5
6
7
8
9
1
6
2
3
6
2
3
9
1
7
9
4
1
0
3
1
9
6
2
3
8
…
2
6
2
3
6
2
3
9
1
7
9
2
4
8
1
0
7
1
7
5
2
4
1
…
3
41
24
17
2
1
6
2
4
8
1
0
7
1
7
5
2
4
1
…
4
2
4
9
2
2
1
2
2
2
1
6
0
2
4
7
88
1
7
9
2
1
9
…
5
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
He
s
en
d
s
it to
t
h
e
o
th
er
p
ar
t
y
.
Step
3
: “
Dec
r
y
p
tio
n
b
y
p
ar
t
y
B”
He
s
ep
ar
ates th
e
p
ix
e
l v
al
u
es
o
f
cip
h
er
ed
i
m
a
g
e
in
to
b
lo
ck
s
o
f
s
ize
f
o
u
r
as:
1
=
[
6
236
239
179
]
,
2
=
[
4
103
196
238
]
,
3
=
[
6
236
239
179
]
, …
He
co
m
p
u
tes t
h
e
v
alu
e
s
o
f
1
,
2
,
3
,
…
as:
̌
−
1
=
[
97
235
57
118
48
69
141
119
136
53
73
125
43
177
107
106
]
1
=
̌
−
1
∙
1
=
[
97
235
57
118
48
69
141
119
136
53
73
125
43
177
107
106
]
[
6
236
239
179
]
256
=
[
249
246
244
245
]
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
2
,
A
p
r
il 2
0
2
1
:
1
2
9
3
-
1302
1300
2
=
̌
−
1
∙
2
=
[
97
235
57
118
48
69
141
119
136
53
73
125
43
177
107
106
]
[
4
103
196
238
]
256
=
[
247
244
237
231
]
,
3
=
̌
−
1
∙
3
=
[
97
235
57
118
48
69
141
119
136
53
73
125
43
177
107
106
]
[
6
236
239
179
]
256
=
[
249
246
244
245
]
, …
He
r
ec
o
n
s
tr
u
ct
s
th
e
p
lai
n
i
m
ag
e
f
r
o
m
th
e
v
al
u
es o
f
as:
1
2
3
4
5
6
7
8
9
1
2
4
9
2
4
6
2
4
4
2
4
5
2
4
7
2
4
4
2
3
7
2
3
1
…
2
2
4
9
2
4
6
2
4
4
2
4
5
2
4
6
2
4
4
2
3
7
2
3
0
…
3
2
4
9
2
4
6
2
4
3
2
4
4
2
4
6
2
4
4
2
3
7
2
3
0
…
4
2
4
8
2
4
5
2
4
3
2
4
4
2
4
6
2
4
3
2
3
6
2
3
0
…
5
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
(
a)
(
b
)
(
c)
Fig
u
r
e
2.
(
a)
P
lain
i
m
a
g
e,
(
b
)
cip
h
er
ed
i
m
ag
e,
a
n
d
(
c)
d
ec
ip
h
er
ed
i
m
a
g
e
f
o
r
ca
t
i
m
a
g
e/
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
ec
o
n
d
ca
s
e
6.
SE
CUR
I
T
Y
ANA
L
YS
I
S
A
s
ec
u
r
it
y
a
n
al
y
s
i
s
o
f
t
h
e
cr
y
p
to
g
r
ap
h
ic
alg
o
r
it
h
m
s
i
s
a
b
as
ic
p
r
o
ce
d
u
r
e
to
g
u
ar
an
tee
t
h
e
q
u
alit
y
o
f
cr
y
p
to
g
r
ap
h
ic
ca
lc
u
latio
n
[2
6
]
.
So
,
to
d
em
o
n
s
tr
ate
t
h
e
s
tr
en
g
t
h
o
f
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
w
e
w
ill
d
i
s
cu
s
s
t
w
o
a
n
al
y
ze
s
PS
N
R
an
d
UACI
.
6
.
1
.
P
ea
k
s
ig
na
l t
o
no
is
e
ra
t
io
(
)
I
t
is
an
ac
ce
s
s
ib
ilit
y
e
s
ti
m
ati
o
n
o
f
w
h
eth
er
a
cr
itical
u
n
iq
u
e
i
m
a
g
e
in
f
o
r
m
atio
n
is
in
s
ta
lled
in
th
e
cip
h
er
ed
i
m
a
g
e.
A
ctu
a
ll
y
,
i
t
ca
n
b
e
d
ef
i
n
ed
b
y
t
h
e
co
m
p
u
tin
g
t
h
e
m
ea
n
s
q
u
ar
ed
er
r
o
r
(
M
SE
)
[2
7
]
.
I
n
th
i
s
w
o
r
k
,
PS
N
R
h
as
b
ee
n
ca
lc
u
lated
f
o
r
th
e
cip
h
er
ed
i
m
a
g
e
an
d
p
lai
n
i
m
ag
e
th
at
ar
e
s
h
o
w
n
i
n
Fi
g
u
r
e
s
1
an
d
2
w
er
e
7
.
8
6
5
4
an
d
7
.
6
5
6
8
r
esp
e
ctiv
el
y
,
w
h
ic
h
m
ea
n
s
th
a
t t
h
e
cip
h
er
e
d
i
m
a
g
e
is
n
o
t
lik
e
t
h
e
p
lain
i
m
a
g
e,
s
o
it
i
s
s
o
h
ar
d
f
o
r
an
ag
g
r
ess
o
r
to
r
e
co
v
er
th
e
p
lain
i
m
a
g
e.
T
h
e
eq
u
atio
n
o
f
t
h
e
PS
N
R
is
as
PS
N
R
=
10
l
og
10
255
∙
255
MS
E
w
h
er
e
M
SE
=
1
M
∗
N
∑
∑
(
X
(
i
,
j
)
−
Y
(
i
,
j
)
)
2
M
j
=
1
N
i
=
1
,
X
(
i
,
j
)
an
d
Y
(
i
,
j
)
ar
e
th
e
p
ix
el
v
alu
e
o
f
p
lai
n
i
m
a
g
e
an
d
cip
h
er
ed
i
m
ag
e
r
esp
ec
ti
v
el
y
.
6
.
2
.
Unifie
d a
v
er
a
g
e
cha
ng
ing
inte
ns
it
y
(
)
I
t
is
o
n
e
o
f
d
i
f
f
er
en
tia
l
an
a
l
y
s
e
s
u
s
ed
to
e
v
alu
a
te
t
h
e
s
tr
en
g
t
h
o
f
i
m
a
g
e
e
n
cr
y
p
tio
n
,
w
h
er
e
it
is
esti
m
ated
th
e
co
n
tr
ast
b
et
w
e
en
th
e
cip
h
er
ed
i
m
a
g
e
an
d
p
lain
i
m
a
g
e.
T
h
e
h
ig
h
est
v
a
lu
e
o
f
th
e
U
AC
I
(a
p
p
r
o
x
im
atel
y
3
3
.
4
6
%)
i
m
p
li
es
th
a
t
t
h
e
p
r
o
p
o
s
ed
p
r
o
ce
d
u
r
e
is
s
a
f
e
a
g
ai
n
s
t
d
if
f
er
en
t
ial
a
s
s
a
u
lts
[2
8
]
.
I
n
t
h
is
w
o
r
k
w
e
g
et
3
1
.
6
9
1
2
an
d
3
4
.
0
9
9
8
f
o
r
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
f
ir
s
t
a
n
d
s
ec
o
n
d
ca
s
e
r
esp
ec
tiv
el
y
,
it
is
s
o
h
ar
d
f
o
r
an
attac
k
er
to
r
ec
o
v
er
th
e
p
lain
i
m
ag
e.
I
t
ca
n
b
e
ca
lcu
l
ated
b
y
th
e
UACI
=
1
256
∗
256
∑
∑
X
(
i
,
j
)
−
Y
(
i
,
j
)
255
∙
256
j
=
1
256
i
=
1
100%
,
X
(
i
,
j
)
an
d
Y
(
i
,
j
)
ar
e
th
e
p
ix
el
v
al
u
e
o
f
p
lain
i
m
a
g
e
a
n
d
cip
h
er
ed
i
m
ag
e
r
esp
ec
tiv
el
y
.
T
h
e
f
ir
s
t
ca
s
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
I
ma
g
e
en
cryp
tio
n
b
a
s
ed
o
n
ell
ip
tic
cu
r
ve
cryp
to
s
ystem
(
Za
h
r
a
a
K
a
d
h
im
Ob
a
id
a
n
d
)
1301
an
d
s
ec
o
n
d
ca
s
e
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
ar
e
te
s
ted
o
n
f
o
r
ca
t
i
m
a
g
e,
le
n
a
i
m
a
g
e,
b
ab
o
o
n
i
m
a
g
e
a
n
d
ca
r
a
m
en
i
m
a
g
e
an
d
th
e
r
es
u
lt
s
ar
e
s
u
m
m
ar
ized
in
T
ab
le
1
.
T
ab
le
1
.
PS
N
R
,
an
d
UACI
T
h
e
P
r
o
p
o
se
d
A
l
g
o
r
i
t
h
m
C
a
t
I
mag
e
L
e
n
a
I
mag
e
B
a
b
o
o
n
I
mag
e
C
a
r
a
me
n
I
mag
e
P
S
N
R
U
A
C
I
P
S
N
R
U
A
C
I
P
S
N
R
U
A
C
I
P
S
N
R
U
A
C
I
F
i
r
st
C
a
se
7
.
8
6
5
4
3
1
.
6
9
1
2
9
.
2
9
9
6
2
8
.
1
8
5
4
9
.
8
0
1
5
2
7
.
0
9
0
6
9
.
0
0
9
7
2
6
.
9
8
9
7
S
e
c
o
n
d
C
a
se
7
.
6
5
6
8
3
4
.
0
9
9
8
9
.
2
5
4
9
2
8
.
5
2
9
9
.
7
7
6
5
2
7
.
1
8
8
8
.
0
9
2
6
3
2
.
0
9
2
7.
CO
NCLU
SI
O
N
E
C
C
p
r
o
v
id
es
eq
u
i
v
ale
n
t
s
ec
u
r
it
y
w
i
th
les
s
k
e
y
s
ize,
lo
w
m
ath
e
m
atica
l
r
at
h
er
t
h
a
n
t
h
e
G
l
o
b
al
R
S
A
cr
y
p
to
s
y
s
te
m
.
Alg
o
r
it
h
m
s
f
o
r
i
m
ag
e
e
n
cr
y
p
tio
n
b
ased
o
n
ellip
tic
c
u
r
v
e
cr
y
p
to
s
y
s
te
m
i
n
t
w
o
ca
s
es
ar
e
p
r
o
p
o
s
ed
in
t
h
is
w
o
r
k
.
Hi
lb
er
t
m
atr
i
x
i
s
i
n
v
o
lv
ed
i
n
t
h
e
f
ir
s
t
an
d
t
h
e
s
ec
o
n
d
ca
s
e
f
o
r
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
w
it
h
d
i
m
e
n
s
io
n
2
×2
a
n
d
4
×4
r
esp
ec
tiv
el
y
.
T
ab
le
1
s
h
o
w
s
t
h
at
t
h
e
p
r
o
p
o
s
ed
alg
o
r
it
h
m
s
f
o
r
b
o
th
ca
s
es
o
n
ca
t
i
m
a
g
e
256×
2
5
6
g
iv
es
g
r
ea
t
o
u
tco
m
e
s
P
SNR
,
an
d
U
A
C
I
.
P
SNR
h
a
s
b
ee
n
co
m
p
u
ted
f
o
r
th
e
cip
h
er
ed
i
m
a
g
e
an
d
p
lain
i
m
a
g
e
u
s
i
n
g
th
e
f
ir
s
t
an
d
s
ec
o
n
d
p
r
o
p
o
s
ed
alg
o
r
ith
m
w
er
e
7
.
8
6
5
4
an
d
7
.
6
5
6
8
r
esp
ec
tiv
el
y
,
w
h
ic
h
m
ea
n
s
it
i
s
h
ar
d
f
o
r
an
a
g
g
r
e
s
s
o
r
to
r
ec
o
v
er
th
e
p
lain
i
m
a
g
e
.
I
n
th
e
o
th
er
h
an
d
,
U
AC
I
h
as
b
ee
n
co
m
p
u
ted
f
o
r
th
e
cip
h
er
ed
i
m
a
g
e
a
n
d
p
lai
n
i
m
ag
e
u
s
i
n
g
t
h
e
f
ir
s
t
a
n
d
s
ec
o
n
d
p
r
o
p
o
s
ed
alg
o
r
it
h
m
w
er
e
3
1
.
6
9
1
2
an
d
3
4
.
0
9
9
8
r
esp
ec
tiv
el
y
,
w
h
ic
h
m
ea
n
s
it i
s
h
ar
d
f
o
r
an
ag
g
r
es
s
o
r
to
r
ec
o
v
er
th
e
p
lai
n
i
m
a
g
e.
RE
F
E
R
E
NC
E
S
[1
]
F
a
h
rn
b
e
rg
e
r,
G
.
,
“
Ed
it
in
g
En
c
ry
p
te
d
M
e
ss
a
g
e
s
w
it
h
o
u
t
De
c
r
y
p
ti
n
g
o
r
Un
d
e
rsta
n
d
i
n
g
th
e
m
,
”
P
h
.
D.
th
e
sis
,
Un
iv
e
rsit
y
o
f
Ha
g
e
n
,
2
0
1
9
.
[2
]
A
.
J.
M
e
n
e
z
e
s,
e
t
a
l.
,
“
Ha
n
d
b
o
o
k
o
f
A
p
p
li
e
d
Cry
p
to
g
ra
p
h
y
,
”
CRC
Pre
ss
,
1
9
9
7
.
[3
]
Ju
risic,
A
.
a
n
d
M
e
n
e
z
e
s,
A
.
,
“
El
li
p
ti
c
Cu
rv
e
s a
n
d
Cry
p
to
g
ra
p
h
y
,
”
Dr.
Do
b
b
’s
J
o
u
r
n
a
l
,
p
p
.
2
6
-
3
6
,
1
9
9
7
.
[4
]
Ko
b
li
tz,
N.,
“
El
li
p
ti
c
C
u
rv
e
Cry
p
t
o
sy
st
e
m
s,”
M
a
th
e
ma
ti
c
s o
f
Co
m
p
u
ta
ti
o
n
,
v
o
l.
4
8
,
n
o
.
1
7
7
,
p
p
.
2
0
3
-
2
0
9
,
1
9
8
7
.
[5
]
M
il
ler,
V.
S
.
,
“
Us
e
o
f
E
ll
ip
ti
c
Cu
rv
e
s
in
Cr
y
p
to
g
ra
p
h
y
,
”
C
o
n
fer
e
n
c
e
o
n
t
h
e
T
h
e
o
ry
a
n
d
Ap
p
li
c
a
ti
o
n
o
f
Cry
p
to
g
ra
p
h
ic T
e
c
h
n
i
q
u
e
s
,
v
o
l.
2
1
8
,
1
9
8
6
,
p
p
.
4
1
7
-
4
2
6
.
[6
]
Riv
e
st,
R.
L
.
,
S
h
a
m
ir,
A
.
a
n
d
A
d
le
m
a
n
,
L
.
,
“
A
M
e
th
o
d
f
o
r
Ob
tain
i
n
g
Dig
it
a
l
S
ig
n
a
tu
re
s
a
n
d
P
u
b
li
c
-
Ke
y
Cr
y
p
to
sy
ste
m
s,”
Co
mm
u
n
ica
ti
o
n
s o
f
t
h
e
ACM
,
v
o
l.
2
1
,
n
o
.
2
,
p
p
.
1
2
0
-
1
2
6
,
1
9
7
8
.
[7
]
Kh
a
n
,
M
.
A
.
,
e
t
a
l.
,
“
A
n
e
ff
icie
n
t
a
n
d
p
r
o
v
a
b
ly
se
c
u
re
c
e
rti
f
ica
t
e
les
s
b
li
n
d
sig
n
a
tu
re
sc
h
e
m
e
f
o
r
f
l
y
in
g
a
d
-
h
o
c
n
e
tw
o
rk
b
a
se
d
o
n
m
u
lt
i
-
a
c
c
e
ss
e
d
g
e
c
o
m
p
u
ti
n
g
,
”
El
e
c
tro
n
ics
,
v
o
l.
9
,
n
o
.
1
,
p
p
.
1
-
22
,
2
0
2
0
.
[8
]
Ko
b
li
tz,
N.,
M
e
n
e
z
e
s,
A
.
a
n
d
Va
n
sto
n
e
,
S
.
,
“
T
h
e
S
tate
o
f
El
li
p
ti
c
Cu
rv
e
Cr
y
p
to
g
ra
p
h
y
,”
De
sig
n
s,
Co
d
e
s
a
n
d
Cry
p
to
g
ra
p
h
y
,
v
o
l.
1
9
,
p
p
.
1
03
-
1
2
3
,
2
0
0
0
.
[9
]
A
li
S
o
le
y
m
a
n
i,
e
t
a
l.
,
“
A
su
rv
e
y
o
n
p
rin
c
ip
a
l
a
sp
e
c
t
s
o
f
se
c
u
re
ima
g
e
tran
s
m
is
sio
n
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Co
mp
u
ter
,
El
e
c
trica
l,
A
u
t
o
ma
ti
o
n
,
Co
n
tro
l
a
n
d
I
n
fo
rm
a
ti
o
n
E
n
g
i
n
e
e
rin
g
,
v
o
l.
6
,
n
o
.
6
,
p
p
.
7
8
0
-
7
8
7
,
2
0
1
2
.
[1
0
]
G
u
p
ta,
A
n
v
it
a
,
e
t
a
l.
,
“
A
n
e
ff
icie
n
t
im
a
g
e
e
n
c
r
y
p
ti
o
n
u
sin
g
n
o
n
-
d
o
m
in
a
ted
so
rti
n
g
g
e
n
e
ti
c
a
lg
o
ri
t
hm
-
III
b
a
se
d
4
-
D
c
h
a
o
ti
c
m
a
p
s,”
J
o
u
rn
a
l
o
f
Amb
ien
t
In
telli
g
e
n
c
e
a
n
d
H
u
ma
n
ize
d
Co
mp
u
ti
n
g
,
v
o
l.
1
1
,
n
o
.
3
,
p
p
.
1
3
0
9
-
1
3
2
4
,
2
0
2
0
.
[1
1
]
Ja
sra
,
Bh
a
t,
a
n
d
Ay
a
z
H
a
ss
a
n
M
o
o
n
,
“
Im
a
g
e
En
c
r
y
p
ti
o
n
tec
h
n
i
q
u
e
s:
A
Re
v
ie
w
,
”
2020
1
0
t
h
In
ter
n
a
t
io
n
a
l
Co
n
fer
e
n
c
e
o
n
Clo
u
d
Co
mp
u
ti
n
g
,
Da
ta
S
c
ien
c
e
&
En
g
in
e
e
rin
g
(
Co
n
fl
u
e
n
c
e
)
,
No
id
a
,
In
d
ia
,
2
0
2
0
,
p
p
.
2
2
1
-
2
2
6
.
[1
2
]
Na
ji
,
M
a
it
h
a
m
A
li
,
e
t
a
l.
,
“
Cr
y
p
tan
a
ly
sis
c
ip
h
e
r
te
x
t
u
sin
g
n
e
w
m
o
d
e
li
n
g
:
Tex
t
e
n
c
r
y
p
ti
o
n
u
si
n
g
e
ll
ip
ti
c
c
u
rv
e
c
r
y
p
to
g
ra
p
h
y
,
"
A
IP
Co
n
fer
e
n
c
e
Pro
c
e
e
d
in
g
s
,
v
o
l.
2
2
0
3
,
n
o
.
1
,
2
0
2
0
,
p
p
.
1
-
9
.
[1
3
]
Bo
isv
e
rt,
R.
,
“
T
h
e
Qu
a
li
ty
o
f
Nu
m
e
rica
l
S
o
f
t
w
a
r
e
,
As
se
s
s
m
e
n
t
a
n
d
En
h
a
n
c
e
m
e
n
t,
”
Qu
a
li
ty
o
f
N
u
me
ric
a
l
S
o
ft
w
a
re
,
p
p
.
3
7
5
-
3
8
0
,
1
9
9
7
.
[1
4
]
S
in
g
h
,
L
a
ip
h
ra
k
p
a
m
Do
len
d
ro
a
n
d
Kh
u
m
a
n
th
e
m
M
a
n
g
le
m
S
in
g
h
,
“
Im
a
g
e
e
n
c
r
y
p
ti
o
n
u
si
n
g
e
ll
ip
ti
c
c
u
rv
e
c
r
y
p
to
g
ra
p
h
y
,
"
Pro
ced
ia
Co
mp
u
t
e
r S
c
ien
c
e
,
v
o
l.
5
4
,
p
p
.
4
7
2
-
4
8
1
,
2
0
1
5
.
[1
5
]
A
.
A
h
m
e
d
,
A
b
d
El
-
L
a
ti
f
a
n
d
X
iam
u
Niu
A
.
,
“
H
y
b
rid
Ch
a
o
ti
c
S
y
ste
m
a
n
d
C
y
c
li
c
El
li
p
ti
c
C
u
rv
e
f
o
r
I
m
a
g
e
En
c
ry
p
ti
o
n
,
”
In
AEU
-
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
tro
n
ics
a
n
d
C
o
m
mu
n
ica
ti
o
n
s
,
v
o
l.
6
7
,
n
o
.
2
,
p
p
.
1
3
6
-
1
4
3
,
2
0
1
3
.
[1
6
]
S
.
Na
g
a
ra
j,
e
t
a
l.
,
“
I
m
a
g
e
e
n
c
r
y
p
ti
o
n
u
sin
g
e
ll
ip
ti
c
c
u
rv
e
c
r
y
p
to
g
ra
p
h
y
a
n
d
m
a
tri
x
,”
Pro
c
e
d
ia
Co
mp
u
ter
S
c
ien
c
e
,
v
o
l.
4
8
,
p
p
.
2
7
6
-
2
8
1
,
2
0
1
5
.
[1
7
]
S
il
v
e
r
m
a
n
,
J.
H.,
“
T
h
e
A
rit
h
m
e
ti
c
o
f
El
li
p
ti
c
Cu
rv
e
s:
G
ra
d
u
a
te
Tex
ts
in
M
a
th
e
m
a
ti
c
s
1
0
6
,
”
2
n
d
e
d
n
.
Ne
w
Y
o
rk
:
S
p
rin
g
e
r
,
v
o
l.
1
0
6
,
2
0
0
9
.
[1
8
]
Na
jl
a
e
H
a
m
e
e
d
A
l
-
S
a
f
fa
r
a
n
d
M
.
Ru
sh
d
a
n
,
“
Hig
h
P
e
rf
o
rm
a
n
c
e
M
e
th
o
d
s
o
f
El
li
p
t
ic
Cu
rv
e
S
c
a
lar
M
u
lt
i
p
li
c
a
ti
o
n
,
”
In
ter
n
a
ti
o
n
a
l
j
o
u
r
n
a
l
o
f
c
o
mp
u
ter
a
p
p
li
c
a
t
io
n
s
,
v
o
l.
1
0
8
,
n
o
.
2
0
,
p
p
.
3
9
-
4
5
,
2
0
1
4
.
[1
9
]
S
h
o
m
e
n
De
b
.
a
n
d
M
d
.
M
o
k
a
m
m
e
l
Ha
q
u
e
,
“
El
li
p
ti
c
c
u
rv
e
a
n
d
p
se
u
d
o
-
in
v
e
rse
m
a
tri
x
b
a
se
d
c
ry
p
to
s
y
ste
m
f
o
r
w
irele
ss
se
n
so
r
n
e
tw
o
rk
s,”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
E
lec
trica
l
a
n
d
C
o
mp
u
ter
En
g
in
e
e
rin
g
(
IJ
ECE
)
,
v
o
l.
9
,
n
o
.
5
,
p
p
.
4
4
7
9
-
4
4
9
2
,
2
0
1
9
.
[2
0
]
F
a
z
-
He
rn
á
n
d
e
z
,
e
t
a
l.
,
“
Hig
h
-
p
e
rf
o
rm
a
n
c
e
i
m
p
le
m
e
n
tatio
n
o
f
e
ll
ip
t
ic
c
u
rv
e
c
ry
p
to
g
r
a
p
h
y
u
sin
g
v
e
c
to
r
i
n
stru
c
ti
o
n
s,”
ACM
T
ra
n
sa
c
ti
o
n
s
o
n
M
a
th
e
ma
ti
c
a
l
S
o
ft
wa
re
,
v
o
l.
4
5
,
n
o
.
3
,
p
p
.
1
-
3
5
,
2
0
1
9
.
[
2
1
]
H
i
l
b
e
r
t
,
D
a
v
i
d
,
“
E
i
n
B
e
i
t
r
a
g
z
u
r
T
h
e
o
r
i
e
d
e
s
L
e
g
e
n
d
r
e
'
s
c
h
e
n
P
o
l
y
n
o
m
s
,
”
A
c
t
a
M
a
t
h
e
m
a
t
i
c
a
,
v
o
l
.
1
8
,
p
p
.
1
5
5
-
1
5
9
,
1
8
9
4
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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1302
1302
[2
2
]
Ro
o
p
a
e
i,
Ha
d
i
,
“
No
rm
o
f
Hilb
e
rt
o
p
e
ra
to
r
o
n
se
q
u
e
n
c
e
sp
a
c
e
s,”
J
o
u
rn
a
l
o
f
I
n
e
q
u
a
li
t
ies
a
n
d
A
p
p
l
i
c
a
ti
o
n
s
,
v
o
l.
1
,
n
o
.
2
0
2
0
,
p
p
.
1
1
3
,
2
0
2
0
.
[2
3
]
Jo
se
,
S
e
lb
y
,
a
n
d
V
ij
a
y
T
i
w
a
ri.
,
“
S
tu
d
y
o
f
2
×
n
rig
h
t
in
v
e
rti
b
le
m
a
t
rix
g
ro
u
p
v
ia
S
u
slin
m
a
tri
c
e
s,
”
J
o
u
rn
a
l
o
f
X
i'a
n
Un
ive
rs
it
y
o
f
Arc
h
it
e
c
tu
re
&
T
e
c
h
n
o
lo
g
y
,
v
o
l.
1
2
,
n
o
.
2
,
p
p
.
6
7
1
-
6
7
8
,
2
0
2
0
.
[2
4
]
Co
sta
ra
,
Co
n
sta
n
ti
n
,
“
No
n
li
n
e
a
r
in
v
e
rti
b
il
it
y
p
re
se
r
v
in
g
m
a
p
s
o
n
m
a
tri
x
a
lg
e
b
ra
s,”
L
in
e
a
r
Al
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8
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