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Dif
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m
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[
1
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i
n
tr
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d
P
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Ke
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r
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p
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)
[
1
-
2
]
.
T
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s
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an
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m
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[
3
]
.
Sev
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p
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1
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K.
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Mc
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[
4
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.
T
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Har
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[
5
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p
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p
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ct
cr
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6
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I
n
t
h
is
p
ap
er
,
w
e
p
r
o
p
o
s
e
a
cr
y
p
to
-
s
y
s
te
m
p
r
o
to
co
l
th
at
is
b
ased
o
n
b
o
th
o
f
ch
ao
tic
m
ap
s
a
n
d
f
ac
to
r
izatio
n
p
r
o
b
le
m
s
.
T
h
e
n
e
w
p
r
o
to
co
l
i
m
p
r
o
v
es
t
h
e
o
v
er
all
s
ec
u
r
it
y
,
a
n
d
n
ee
d
s
a
lo
w
er
n
u
m
b
er
o
f
o
p
er
atio
n
s
in
b
o
th
o
f
th
e
en
cr
y
p
t
io
n
an
d
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
es.
T
h
er
ef
o
r
e,
th
e
p
r
o
p
o
s
ed
cr
y
p
to
-
s
y
s
te
m
i
s
m
o
r
e
p
r
ac
tical
f
o
r
r
ea
l
is
tic
ap
p
licatio
n
s
.
T
h
e
f
as
h
io
n
i
n
to
w
h
ic
h
t
h
e
r
est
o
f
t
h
is
p
ap
er
is
ar
r
an
g
e
d
in
to
is
as
f
o
llo
w
s
:
I
n
Sectio
n
2
,
w
e
b
r
ief
l
y
i
n
tr
o
d
u
ce
t
h
e
n
ec
e
s
s
ar
y
m
ath
e
m
at
ic
al
f
r
a
m
e
w
o
r
k
u
s
ed
i
n
t
h
e
p
ap
er
.
I
n
th
e
s
ec
tio
n
3
,
th
e
n
e
w
p
r
o
p
o
s
ed
en
cr
y
p
ti
o
n
s
ch
e
m
e
is
in
tr
o
d
u
ce
d
.
I
n
Se
ctio
n
s
4
,
5
a
n
d
6
,
w
e
a
n
al
y
z
e
th
e
s
ec
u
r
it
y
an
d
ef
f
icien
c
y
o
f
t
h
e
p
r
o
p
o
s
ed
s
ch
e
m
e.
W
e
f
i
n
all
y
co
n
clu
d
e
i
n
S
ec
tio
n
7
.
2.
CH
AO
T
I
C
M
AP
S
C
h
ao
tic
t
h
eo
r
y
h
as
b
ee
n
h
ea
v
il
y
u
s
ed
in
d
esi
g
n
in
g
s
ec
u
r
e
co
m
m
u
n
icatio
n
p
r
o
to
co
ls
s
in
ce
th
e
1
9
9
0
s
[
10
-
15
]
,
w
h
ile
c
h
a
o
ti
c
m
ap
s
h
av
e
b
ee
n
u
tili
ze
d
in
t
h
e
d
esi
g
n
o
f
s
y
m
m
etr
ic
e
n
cr
y
p
tio
n
p
r
o
to
co
ls
in
[
16
-
19
]
.
Desig
n
i
n
g
a
ch
ao
tic
m
ap
s
e
tti
n
g
i
s
u
s
u
all
y
d
if
f
ic
u
lt,
b
u
t
g
e
n
er
all
y
cr
ea
te
s
s
ec
u
r
e
an
d
e
f
f
icien
t
p
r
o
to
co
ls
.
T
h
at
is
b
ec
au
s
e
c
h
ao
tic
m
ap
-
b
ased
p
r
o
to
co
ls
h
a
v
e
lo
w
co
m
p
u
tatio
n
al
co
s
ts
wh
en
co
m
p
ar
ed
w
it
h
o
th
er
m
o
d
u
lar
ex
p
o
n
en
t
ial
co
m
p
u
ti
n
g
b
ased
p
r
o
to
c
o
ls
o
r
p
r
o
to
co
ls
th
at
ar
e
b
ased
o
n
s
ca
lar
m
u
ltip
licatio
n
o
n
ellip
tic
cu
r
v
e
s
.
2
.
1
.
Cheby
s
hev
m
a
p
s
A
m
ap
o
f
a
C
h
eb
y
s
h
e
v
p
o
l
y
n
o
m
ial
,
:
→
o
f
d
eg
r
ee
,
ca
n
b
e
d
ef
in
ed
w
i
th
th
e
s
u
b
s
eq
u
e
n
t
r
ec
u
r
r
en
t
r
elatio
n
[
20
]
:
+
1
(
)
=
2
(
)
−
−
1
(
)
,
(
1
)
w
it
h
0
(
)
=
1
,
an
d
1
(
)
=
,
th
e
h
ea
d
m
o
s
t
C
h
e
b
y
s
h
e
v
p
o
l
y
n
o
m
ia
ls
ar
e
,
2
(
)
=
2
2
−
1
,
(
2
)
3
3
(
)
=
4
3
−
3
,
(
3
)
4
(
)
=
8
4
−
8
2
+
1
`
(
4
)
A
s
ig
n
i
f
ica
n
t p
r
o
p
er
ty
o
f
C
h
eb
y
s
h
ev
p
o
l
y
n
o
m
ials
i
s
th
e
s
e
m
i
-
g
r
o
u
p
p
r
o
p
er
ty
:
(
(
)
)
=
(
)
(
5
)
An
i
n
s
tan
t
s
eq
u
e
l
o
f
th
e
ab
o
v
e
p
r
o
p
er
ty
is
t
h
at
C
h
eb
y
s
h
ev
p
o
l
y
n
o
m
ial
s
u
n
d
er
co
m
p
o
s
i
tio
n
co
m
m
u
te,
i.e
.
,
(
)
=
(
)
.
U
n
d
er
t
h
e
ac
tio
n
o
f
th
e
m
ap
:
(
[
−
1
,
1
]
)
=
[
−
1
,
1
]
,
t
h
e
in
ter
v
al
[
−
1
,
1
]
is
in
v
ar
iab
le
.
T
h
u
s
,
a
C
h
eb
y
s
h
e
v
p
o
ly
n
o
m
ial
co
n
f
i
n
ed
to
th
e
in
ter
v
al
[
−
1
,
1
]
w
i
ll
b
e
th
e
p
r
o
m
i
n
e
n
t
ch
ao
tic
m
ap
f
o
r
all
>
1
.
I
t
h
as
a
u
n
iq
u
e
i
n
v
ar
ian
t
m
ea
s
u
r
e
(
)
=
√
1
−
2
,
w
h
ic
h
i
s
a
b
s
o
lu
tel
y
co
n
ti
n
u
o
u
s
w
it
h
p
o
s
itiv
e
L
y
ap
u
n
o
v
e
x
p
o
n
en
t
=
.
T
h
e
C
h
eb
y
s
h
ev
m
ap
,
f
o
r
,
=
2
,
r
ed
u
ce
s
to
th
e
f
a
m
iliar
lo
g
i
s
tic
m
ap
.
T
w
o
p
r
esu
m
ab
l
y
i
n
tr
ac
tab
le
p
r
o
b
lem
s
r
elate
d
to
C
h
eb
y
s
h
e
v
p
o
ly
n
o
m
ia
ls
[
21
]
ar
e
:
Def
ini
t
io
n
1
.
C
h
ao
tic
m
ap
s
d
is
cr
ete
lo
g
ar
ith
m
(
C
MD
L
)
p
r
o
b
lem
:
Gi
v
e
n
a
r
an
d
o
m
n
u
m
b
er
∈
ℤ
∗
,
an
d
an
ele
m
e
n
t
∈
ℤ
,
th
e
task
o
f
t
h
e
C
MD
L
p
r
o
b
lem
i
s
to
f
i
n
d
an
i
n
te
g
er
s
u
c
h
th
at
=
(
)
(
)
.
Def
ini
t
io
n
2
.
C
h
ao
tic
m
ap
s
D
if
f
ie
–
Hell
m
an
(
C
MD
H)
p
r
o
b
le
m
:
Gi
v
e
n
a
r
an
d
o
m
n
u
m
b
er
∈
ℤ
p
∗
,
an
d
t
w
o
ele
m
e
n
ts
,
r
(
)
an
d
s
(
)
,
f
o
r
u
n
k
n
o
wn
v
a
lu
e
s
an
d
,
th
e
tas
k
o
f
th
e
C
MD
H
p
r
o
b
le
m
is
to
co
m
p
u
t
e
rs
(
)
.
2
.
2
.
P
ub
lic
-
k
ey
encr
y
ptio
n w
it
h
C
heby
s
hev
po
ly
no
m
ia
l
S
y
s
te
m
b
ased
o
n
ch
ao
tic
t
h
eo
r
y
is
u
s
u
al
l
y
d
ef
i
n
ed
o
n
r
ea
l
n
u
m
b
er
s
.
I
n
f
ac
t,
a
n
y
en
cr
y
p
tio
n
alg
o
r
ith
m
,
w
h
ich
u
til
izes
c
h
a
o
tic
m
ap
s
,
u
p
o
n
it
s
i
m
p
le
m
e
n
tatio
n
o
n
a
co
m
p
u
ter
(
e.
g
.
,
f
i
n
ite
-
s
tate
m
ac
h
i
n
e)
,
it
tu
r
n
s
i
n
to
a
tr
an
s
f
o
r
m
at
io
n
o
n
to
its
el
f
f
r
o
m
a
f
i
n
ite
s
et
.
B
ec
au
s
e
f
lo
ati
n
g
-
p
o
in
t
h
a
s
a
w
id
e
d
y
n
a
m
ic
r
ag
e,
its
i
m
p
le
m
en
ta
tio
n
s
ee
m
s
ap
p
licab
le
f
o
r
s
o
f
t
w
ar
e
i
m
p
le
m
e
n
tatio
n
o
f
C
h
eb
y
s
h
e
v
p
o
l
y
n
o
m
ial
s
.
Ne
v
er
th
ele
s
s
,
f
lo
ati
n
g
-
p
o
in
t c
a
n
n
o
t
b
e
u
s
ed
in
p
u
b
lic
-
k
e
y
en
cr
y
p
tio
n
f
o
r
th
e
f
o
llo
w
in
g
r
ea
s
o
n
s
:
‒
T
h
er
e
is
n
o
u
n
i
f
o
r
m
d
i
s
tr
ib
u
ti
o
n
f
o
r
f
lo
ati
n
g
-
p
o
in
t
n
u
m
b
er
s
,
o
n
t
h
e
r
ea
l
a
x
i
s
,
o
v
er
a
n
y
g
iv
en
i
n
ter
v
al.
Mo
r
eo
v
er
,
th
er
e
is
an
ex
is
te
n
c
e
o
f
r
ed
u
n
d
an
t
n
u
m
b
er
r
ep
r
esen
tatio
n
s
i
n
f
lo
ati
n
g
-
p
o
i
n
t
ar
it
h
m
e
tic
ca
u
s
ed
b
y
n
o
r
m
alize
d
ca
lcu
la
tio
n
s
.
As
th
e
s
a
m
e
r
ea
l
s
ig
n
al
v
a
l
u
e
is
r
ep
r
esen
ted
b
y
s
o
m
e
f
lo
ati
n
g
-
p
o
in
t
n
u
m
b
er
s
[
22
]
.
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.
10
,
No
.
2
,
A
p
r
il 2
0
2
0
:
1
4
3
0
-
1437
1432
‒
T
h
er
e
is
a
r
estrictio
n
o
n
t
h
e
m
es
s
ag
e
len
g
t
h
b
ec
au
s
e
a
C
h
eb
y
s
h
ev
p
o
l
y
n
o
m
ial
i
s
a
n
o
n
-
i
n
v
er
tib
le.
I
n
[
23
]
,
th
e
p
u
b
lic
k
e
y
en
c
r
y
p
tio
n
p
r
o
to
co
l
u
s
e
s
C
h
eb
y
s
h
e
v
p
o
l
y
n
o
m
ials
.
T
h
i
s
al
g
o
r
ith
m
ca
n
b
e
ex
p
lain
ed
as
f
o
llo
w
s
:
L
et
a
lar
g
e
in
te
g
er
s
et
s
b
e
g
en
er
ated
b
y
T
h
o
m
a
s
,
th
en
let
a
n
u
m
b
er
∈
[
−
1
,
1
]
b
e
g
en
er
ated
r
an
d
o
m
l
y
,
a
n
d
let
(
)
b
e
co
m
p
u
ted
.
T
h
o
m
as
’
s
p
u
b
li
c
k
e
y
is
(
,
(
)
)
,
h
i
s
p
r
iv
ate
k
e
y
is
.
B
o
b
d
en
o
tes
th
e
m
es
s
ag
e
as
n
u
m
b
er
∈
[
−
1
,
1
]
,
th
en
cr
ea
tes
a
lar
g
e
in
teg
er
an
d
ca
lcu
lates
(
)
,
(
)
=
(
(
)
)
,
an
d
=
(
)
.
B
o
b
r
elay
s
t
h
e
cip
h
er
-
tex
t
=
(
(
)
,
)
to
T
h
o
m
as.
T
o
r
ec
o
v
er
p
lain
-
tex
t
f
r
o
m
,
T
h
o
m
a
s
u
til
izes
th
e
p
r
iv
ate
k
e
y
to
co
m
p
u
te
(
)
=
(
(
)
)
,
an
d
r
ec
o
v
er
s
th
e
tex
t
b
y
ca
lc
u
lati
n
g
=
∕
(
)
.
L
et
,
,
b
e
th
e
len
g
t
h
s
(
in
b
its
)
o
f
,
an
d
,
r
esp
ec
tiv
el
y
,
a
n
d
let
-
b
it
p
r
ec
is
io
n
ar
ith
m
etic
b
e
e
m
p
lo
y
ed
in
th
e
alg
o
r
it
h
m
s
o
f
t
w
ar
e
i
m
p
l
e
m
en
tatio
n
.
T
h
en
≤
−
−
[
12
,
23
]
.
‒
W
h
en
f
lo
atin
g
-
p
o
in
t
r
ep
r
esen
t
atio
n
is
u
s
ed
to
i
m
p
le
m
en
t
ch
ao
tic
m
ap
s
,
it
is
h
ar
d
to
i
m
p
lem
en
t
to
o
ls
f
o
r
th
e
p
u
r
p
o
s
e
o
f
a
n
al
y
s
i
n
g
t
h
e
s
tr
u
ctu
r
e
o
f
th
e
p
er
io
d
icit
y
o
f
t
h
e
p
er
io
d
ic
o
r
b
its
.
F
u
r
th
er
m
o
r
e,
th
er
e
i
s
n
o
h
o
p
e
in
estab
lis
h
i
n
g
a
lin
k
b
et
w
ee
n
th
e
n
u
m
b
er
an
d
c
h
ao
s
th
eo
r
y
.
2
.
3
.
M
o
dified
C
heby
s
he
v
po
ly
no
m
ia
l
s
T
h
e
f
o
llo
w
i
n
g
m
ap
w
ill
b
e
u
s
ed
to
s
h
o
w
a
n
E
lGa
m
al
an
d
R
S
A
p
u
b
lic
-
k
e
y
alg
o
r
it
h
m
s
to
C
h
eb
y
s
h
ev
m
ap
s
:
:
{
0
,
1
,
…
,
−
1
}
→
:
{
0
,
1
,
…
,
−
1
}
d
ef
i
n
ed
as
=
(
)
(
mod
)
,
w
h
er
e
a
n
d
ar
e
in
teg
er
s
.
W
e
w
il
l
ca
ll
=
(
)
(
mod
)
as
m
o
d
if
ied
C
h
eb
y
s
h
e
v
p
o
l
y
n
o
m
ial.
T
h
is
c
an
r
ep
lace
t
h
e
p
o
w
er
i
n
b
o
t
h
alg
o
r
ith
m
s
of
E
lGa
m
al
an
d
R
S
A
p
u
b
lic
-
k
e
y
,
i
f
an
d
o
n
l
y
i
f
,
s
u
b
s
tit
u
tio
n
is
p
o
s
s
ib
le
u
n
d
er
co
m
p
o
s
itio
n
,
an
d
th
e
ir
o
r
b
its
p
er
io
d
ca
n
b
e
co
m
p
u
ted
.
T
h
e
p
r
o
p
e
r
ties
o
f
th
e
m
o
d
if
ied
C
h
eb
y
s
h
e
v
p
o
l
y
n
o
m
ials
ar
e
s
h
o
w
n
i
n
th
e
f
o
llo
w
i
n
g
th
eo
r
e
m
s
:
T
heo
re
m
2
.
3
.
1
Mo
d
if
ied
C
h
e
b
y
s
h
e
v
p
o
l
y
n
o
m
ia
ls
co
m
m
u
te
u
n
d
er
co
m
p
o
s
itio
n
,
th
at
i
s
,
(
(
)
mod
)
=
(
)
(
mod
)
(
6
)
T
heo
re
m
2
.
3
.
2
L
et
b
e
an
o
d
d
p
r
im
e
an
d
let
∈
ℤ
s
u
c
h
th
at
0
≤
<
.
T
h
en
t
h
e
p
er
io
d
o
f
th
e
s
eq
u
en
c
e
(
)
(
)
f
o
r
=
01
,
2
,
…
,
is
a
d
iv
is
o
r
o
f
2
−
1
.
3.
T
H
E
P
RO
P
O
SE
D
P
UB
L
I
C
K
E
Y
E
NC
RYP
T
I
O
N
W
e
p
r
o
p
o
s
e
in
t
h
is
s
ec
tio
n
o
u
r
n
e
w
p
r
o
to
co
l,
w
h
ic
h
is
b
ased
o
n
ch
ao
tic
m
ap
s
a
n
d
f
ac
to
r
in
g
p
r
o
b
lem
s
.
T
h
e
n
e
w
p
r
o
to
co
l c
o
m
p
r
i
s
es t
h
r
ee
p
ar
ts
: k
e
y
g
e
n
e
r
atio
n
,
en
cr
y
p
tio
n
,
a
n
d
d
ec
r
y
p
tio
n
.
3
.
1
.
K
ey
g
ener
a
t
io
n
I
n
g
e
n
er
al
,
i
t
i
s
as
s
u
m
e
d
t
h
at
it
i
s
d
esire
d
to
j
o
in
th
e
p
r
o
p
o
s
ed
cr
y
p
to
-
s
y
s
te
m
as
en
ti
t
y
A
.
Fo
r
k
e
y
g
en
er
atio
n
p
u
r
p
o
s
es
,
th
e
cr
ea
tio
n
o
f
a
p
u
b
lic
an
d
a
p
r
iv
ate
k
e
y
r
eq
u
ir
es
p
er
f
o
r
m
i
n
g
a
s
et
o
f
p
r
o
ce
s
s
es
.
W
e
d
escr
ib
e
th
ese
p
r
o
ce
s
s
es in
th
e
f
o
llo
w
i
n
g
s
tep
s
:
Step
s
1
:
Select
t
w
o
lar
g
e
r
an
d
o
m
p
r
i
m
es
an
d
o
f
al
m
o
s
t sa
m
e
s
ize.
Step
s
2
:
C
o
m
p
u
te
=
an
d
=
(
2
−
1
)
(
2
−
1
)
.
Step
s
3
:
C
h
o
o
s
e
a
r
an
d
o
m
i
n
te
g
er
,
1
<
<
(
)
s
u
ch
th
at
gc
d
(
,
(
)
)
=
1
.
Step
s
4
:
C
alcu
late
th
e
u
n
iq
u
e
i
n
teg
er
,
1
<
<
(
)
,
s
u
c
h
th
a
t
≡
1
(
mod
(
)
)
.
Step
s
5
:
C
h
o
o
s
e
t
w
o
r
an
d
o
m
i
n
te
g
er
s
,
s
u
c
h
th
a
t
0
≤
,
≤
(
)
−
1
.
Step
s
6
:
C
h
o
o
s
e
,
∈
ℤ
∗
an
d
co
m
p
u
te.
1
=
2
(
)
(
mod
)
2
=
2
(
)
(
mod
)
T
h
e
p
u
b
lic
k
e
y
o
f
is
(
,
,
1
,
2
,
,
)
an
d
th
e
co
r
r
esp
o
n
d
in
g
p
r
iv
ate
k
e
y
i
s
(
,
,
,
,
)
.
3
.
2
.
E
ncry
ptio
n
E
n
cr
y
p
tio
n
al
g
o
r
ith
m
s
ar
e
n
o
r
m
al
l
y
in
v
o
l
v
ed
in
t
h
e
cr
y
p
to
g
r
ap
h
ic
p
r
o
ce
s
s
.
Ma
n
y
iter
atio
n
s
th
a
t
in
cl
u
d
e
s
u
b
s
titu
tio
n
s
an
d
tr
a
n
s
f
o
r
m
atio
n
s
ar
e
p
er
f
o
r
m
ed
in
t
h
ese
al
g
o
r
ith
m
s
o
n
o
r
i
g
i
n
al
d
ata
(
k
n
o
w
n
as
p
lain
tex
t)
.
T
h
is
is
d
o
n
e
s
o
as
to
m
a
k
e
t
h
e
p
r
o
ce
s
s
o
f
id
en
ti
f
y
i
n
g
t
h
e
d
ata
b
y
a
h
ac
k
er
o
r
in
tr
u
d
er
co
m
p
lica
ted
[
2
4
]
.
I
n
th
is
p
ap
er
,
w
e
c
o
n
s
id
er
th
e
p
lai
n
te
x
t
s
p
ac
e
as
ℤ
n
.
A
s
s
u
m
e
t
h
at
a
u
s
er
ℬ
wis
h
e
s
to
s
e
n
d
a
m
es
s
ag
e
∈
ℤ
n
to
u
s
i
n
g
’
s
p
u
b
lic
k
e
y
.
T
h
en
ℬ
h
as to
ca
r
r
y
-
o
u
t
t
h
e
f
o
llo
w
in
g
s
tep
s
:
S
tep
s
1
:
Select
∈
ℤ
∗
an
d
f
i
n
d
1
=
(
)
(
mo
d
)
.
Step
s
2
:
Gen
er
ate
t
w
o
r
an
d
o
m
n
o
n
-
n
e
g
ativ
e
i
n
teg
er
s
,
∈
ℤ
an
d
co
m
p
u
te
:
2
=
(
)
(
mod
)
3
=
(
)
(
mod
)
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
A
n
ew R
S
A
p
u
b
lic
ke
y
en
cryp
t
io
n
s
ch
eme
w
ith
ch
a
o
tic
ma
p
s
(
N
ed
a
l Ta
h
a
t)
1433
Step
s
3
:
C
o
m
p
u
te
4
=
(
1
)
(
2
)
(
+
1
)
(
mod
)
.
T
h
en
,
ℬ
s
en
d
to
th
e
en
cr
y
p
ted
m
es
s
ag
e
(
1
,
2
,
3
,
4
)
3
.
3
.
Dec
ry
pti
o
n
Gen
er
all
y
,
th
e
p
r
o
ce
s
s
o
f
d
ec
r
y
p
tio
n
i
s
r
ev
er
s
in
g
all
o
p
er
atio
n
s
ca
r
r
ied
-
out
t
o
p
er
f
o
r
m
th
e
en
cr
y
p
t
io
n
[
25
]
.
I
t
en
tails
tr
an
s
f
o
r
m
i
n
g
t
h
e
en
cr
y
p
ted
d
ata
b
ac
k
to
th
e
o
r
ig
in
al
f
o
r
m
in
o
r
d
er
to
allo
w
th
e
r
ec
ei
v
er
to
u
n
d
er
s
ta
n
d
it.
I
n
t
h
is
p
ap
er
,
to
r
ec
o
v
er
th
e
m
ess
a
g
e
f
r
o
m
(
s
1
,
s
2
,
s
3
,
s
4
)
,
s
h
o
u
ld
ca
r
r
y
-
o
u
t
th
e
f
o
llo
w
i
n
g
:
Step
s
1
:
C
o
m
p
u
te
=
(
1
)
(
mo
d
)
.
Step
s
2
:
C
o
m
p
u
te
=
4
−
1
(
+
1
)
(
mod
)
.
Step
s
3
:
C
o
m
p
u
te
(
)
+
2
(
2
)
mod
=
2
(
2
)
mod
=
2
(
)
=
(
1
)
(
mod
)
.
Step
s
4
:
C
o
m
p
u
te
(
)
+
2
(
3
)
mod
=
2
(
3
)
mod
=
2
(
)
=
(
2
)
(
mod
)
.
Step
s
5
:
C
o
m
p
u
te
=
−
1
(
1
)
−
1
(
2
)
(
mo
d
)
.
T
o
ac
h
iev
e
a
s
u
cc
e
s
s
f
u
l
d
ec
r
y
p
t
io
n
p
r
o
ce
s
s
,
th
e
ac
cu
r
ac
y
ca
n
n
o
t
b
e
co
m
p
r
o
m
is
ed
in
p
er
f
o
r
m
i
n
g
d
ec
r
y
p
tio
n
.
T
heo
re
m
:
I
f
th
e
i
n
it
ializatio
n
an
d
en
cr
y
p
t
io
n
al
g
o
r
ith
m
s
ar
e
ex
e
cu
ted
co
r
r
ec
tly
,
t
h
e
n
it
i
s
g
u
ar
an
teed
to
g
et
th
e
o
r
ig
i
n
al
te
x
t b
y
u
s
i
n
g
t
h
e
d
ec
r
y
p
tio
n
alg
o
r
it
h
m
.
P
ro
o
f
:
Fr
o
m
t
h
e
r
elatio
n
−
1
(
1
)
−
1
(
2
)
(
)
=
,
w
e
h
av
e
−
1
(
1
)
−
1
(
2
)
=
4
−
1
(
+
1
)
−
1
(
1
)
−
1
(
2
)
=
4
−
1
(
+
1
)
(
1
)
(
2
)
=
(
1
)
(
2
)
(
+
1
)
−
1
(
+
1
)
(
1
)
(
2
)
=
(
mod
)
.
(
7
)
No
te
th
at
,
in
R
S
A
k
e
y
g
e
n
er
atio
n
,
th
e
t
w
o
in
te
g
er
s
an
d
ar
e
ca
lled
,
r
esp
ec
tiv
el
y
,
th
e
en
c
r
y
p
tio
n
ex
p
o
n
en
t
,
a
n
d
th
e
d
e
cr
y
p
tio
n
ex
p
o
n
e
n
t.
W
h
ile
is
ca
lled
th
e
m
o
d
u
lu
s
.
I
t
w
as
s
h
o
w
n
in
Sectio
n
3
.
2
th
at
1
(
)
≡
(
mod
)
.
B
y
t
h
e
s
a
m
e
ar
g
u
m
e
n
t,
(
(
)
)
≡
(
)
≡
1
+
(
)
≡
1
(
)
≡
(
mod
)
(
8
)
L
aste
l
y
,
s
in
ce
an
d
ar
e
d
is
tin
c
t p
r
im
e
s
,
th
e
C
h
in
e
s
e
r
e
m
ai
n
d
er
th
eo
r
e
m
m
a
y
be
u
s
e
to
s
h
o
w
t
h
at:
(
(
)
)
≡
(
)
≡
1
+
(
)
≡
1
(
)
≡
(
mod
)
(
9
)
4.
E
XAM
P
L
E
T
o
illu
s
tr
ate
th
e
i
m
p
ac
t
o
f
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e,
w
e
h
av
e
u
s
ed
ar
tif
icial
l
y
s
m
all
p
ar
a
m
eter
s
in
t
o
a
r
ep
r
esen
tativ
e
e
x
a
m
p
le
a
s
f
o
llo
w
s
:
‒
Ke
y
g
e
n
er
atio
n
:
T
h
e
u
s
er
ch
o
o
s
e
p
=
13
,
q
=
17
an
d
co
m
p
u
te
n
=
221
,
φ
=
43384
.
s
elec
t
s
a
r
an
d
o
m
i
n
te
g
er
e
=
317
,
an
d
f
in
d
th
e
u
n
iq
u
e
in
te
g
er
,
d
≡
e
−
1
mod
φ
≡
(
317
)
−
1
mod
43384
≡
12821
(
1
0
)
C
h
o
o
s
es
t
w
o
r
an
d
o
m
in
te
g
er
s
a
=
211
an
d
b
=
311
s
u
ch
th
at
0
≤
a
,
b
≤
φ
(
n
)
−
1
,
an
d
h
e
a
ls
o
ch
o
o
s
es
α
=
107
,
β
=
179
∈
ℤ
n
∗
an
d
co
m
p
u
tes:
y
1
=
T
(
211
)
2
(
107
)
≡
T
100
(
107
)
mod
(
221
)
=
199
(
1
1
)
y
2
=
T
(
311
)
2
(
179
)
=
T
144
(
179
)
mod
(
221
)
=
18
(
1
2
)
T
h
en
,
th
e
u
s
er
p
u
b
lic
k
ey
is
(
n
,
e
,
y
1
,
y
2
,
α
,
β
)
,
an
d
(
p
,
q
,
a
,
b
,
d
)
r
ep
r
esen
ts
th
e
co
r
r
esp
o
n
d
in
g
p
r
iv
ate
k
e
y
.
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.
10
,
No
.
2
,
A
p
r
il 2
0
2
0
:
1
4
3
0
-
1437
1434
‒
E
n
cr
y
p
tio
n
: T
o
en
cr
y
p
t a
m
e
s
s
ag
e
m
=
155
.
ℬ
ch
o
o
s
es
r
=
173
∈
ℤ
n
∗
an
d
co
m
p
u
te
:
s
1
=
T
317
(
173
)
mod
221
=
31
(
1
3
)
A
u
s
er
ℬ
ch
o
o
s
es t
w
o
r
an
d
o
m
n
o
n
-
n
eg
a
tiv
e
i
n
te
g
er
s
c
=
127
,
t
=
123
∈
ℤ
n
an
d
co
m
p
u
tes
:
s
2
=
T
127
(
107
)
mod
(
221
)
=
72
(
1
4
)
s
3
=
T
123
(
179
)
mod
(
221
)
=
135
(
1
5
)
s
4
=
155
T
127
(
199
)
T
123
(
18
)
T
317
(
174
)
(
1
6
)
=
155
(
199
)
(
69
)
(
23
)
(
mod
221
)
=
178
(
1
7
)
ℬ
s
en
d
s
to
th
e
en
cr
y
p
ted
m
e
s
s
a
g
e
(
s
1
,
s
2
,
s
3
,
s
4
)
.
‒
Dec
r
y
p
tio
n
: T
o
r
ec
o
v
er
th
e
m
ess
a
g
e
f
r
o
m
(
s
1
,
s
2
,
s
3
,
s
4
)
,
co
m
p
u
tes:
r
=
T
12821
(
31
)
(
mod
221
)
=
173
(
1
8
)
R
=
178
(
T
317
(
174
)
)
−
1
mod
221
=
75
(
1
9
)
T
a
φ
(
n
)
+
2
(
s
2
)
mod
n
=
T
c
(
y
1
)
(
mod
n
)
=
199
(
2
0
)
T
b
φ
(
n
)
+
2
(
s
3
)
mod
n
=
T
t
(
y
2
)
(
mod
n
)
=
69
(
2
1
)
=
75
(
199
)
−
1
(
69
)
−
1
mod
221
=
75
(
10
)
(
205
)
mod
221
=
155
(
2
2
)
5.
SE
CUR
I
T
Y
T
he
p
r
o
p
o
s
ed
c
r
y
p
to
-
s
y
s
te
m
’
s
ec
u
r
it
y
is
f
o
u
n
d
o
n
f
ac
to
r
in
g
an
d
ch
ao
tic
m
ap
.
T
o
d
e
p
ict
th
e
h
eu
r
is
tic
s
ec
u
r
it
y
at
o
u
r
s
c
h
e
m
e,
a
co
lle
ctio
n
o
f
co
m
m
o
n
at
tack
s
w
er
e
co
n
s
id
er
ed
in
th
e
f
o
llo
w
in
g
:
At
t
a
ck
1
:
A
s
s
u
m
e
t
h
at
an
attac
k
er
d
esire
s
to
r
ec
o
v
er
all
s
e
cr
et
v
alu
es
(
,
,
,
,
)
,
u
tili
zin
g
all
ac
ce
s
s
ib
l
e
s
y
s
te
m
i
n
f
o
r
m
atio
n
.
I
n
t
h
is
s
ce
n
ar
io
,
t
h
e
attac
k
er
h
a
s
to
co
n
d
u
ct
f
ac
to
r
in
g
a
n
d
ch
ao
tic
m
ap
s
s
o
l
u
tio
n
s
.
S/h
e
n
ee
d
s
to
f
i
n
d
th
e
p
r
i
m
es
o
f
f
o
r
f
ac
to
r
in
g
,
w
h
ich
ca
n
u
s
u
all
y
b
e
s
o
lv
ed
u
s
in
g
th
e
n
u
m
b
er
f
ield
s
ie
v
e
m
et
h
o
d
[
9
]
.
Nev
er
th
eless
,
t
h
e
s
ize
o
f
m
o
d
u
l
u
s
in
f
lu
e
n
ce
s
th
i
s
m
et
h
o
d
,
an
d
co
m
p
u
tatio
n
all
y
ca
n
n
o
t
f
ac
to
r
a
n
in
te
g
er
o
f
s
ize
1
0
2
4
-
b
it
an
d
ab
o
v
e.
I
f
th
e
t
w
o
p
r
i
m
e
n
u
m
b
er
s
p
an
d
q
ar
e
ch
o
s
en
w
el
l,
it
w
ill
d
e
f
i
n
itel
y
in
cr
ea
s
e
t
h
e
r
e
s
is
ta
n
ce
o
f
t
h
e
s
ch
e
m
e
to
a
ttack
b
y
th
e
s
p
ec
i
al
-
p
u
r
p
o
s
e
f
ac
to
r
iza
tio
n
alg
o
r
ith
m
s
.
Fo
r
ch
ao
tic
m
ap
s
to
f
in
d
an
d
f
r
o
m
1
=
2
(
)
(
mod
)
an
d
2
=
2
(
)
(
mod
)
,
an
d
if
th
e
s
a
m
e
le
v
el
o
f
s
ec
u
r
i
t
y
is
u
s
ed
o
v
er
p
r
i
m
e
s
,
t
h
en
t
h
e
attac
k
er
h
as
to
s
o
l
v
e
in
teg
er
f
ac
to
r
i
za
tio
n
p
r
o
b
lem
an
d
ch
ao
tic
m
ap
.
A
l
s
o
,
th
e
i
n
te
g
er
s
an
d
m
u
s
t
b
e
lar
g
e
to
p
r
ev
en
t
ex
h
a
u
s
ti
v
e
s
ea
r
ch
attac
k
.
On
e
o
b
v
io
u
s
e
n
c
r
y
p
tio
n
p
r
ac
tice
is
to
u
s
e
d
if
f
er
e
n
t
p
ar
am
e
ter
s
,
an
d
f
o
r
d
if
f
er
en
t
m
e
s
s
a
g
es,
b
ec
au
s
e
if
a
s
en
d
er
u
s
ed
th
e
s
a
m
e
p
ar
am
e
ter
s
f
o
r
en
cr
y
p
tio
n
o
f
t
w
o
m
e
s
s
a
g
e
s
a
y
1
an
d
2
,
th
en
s
/
h
e
w
o
u
ld
o
b
tain
4
=
(
1
)
(
2
)
(
+
1
)
(
m
od
)
an
d
′
4
=
(
1
)
(
2
)
(
+
1
)
(
mo
d
)
.
So
,
f
r
o
m
t
h
e
r
elatio
n
2
=
′
4
4
1
,
an
attac
k
er
w
h
o
k
n
o
w
s
th
e
m
e
s
s
a
g
e
1
ca
n
r
ec
o
v
er
2
.
No
te,
th
e
n
e
w
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
r
an
d
o
m
ized
,
p
ar
am
eter
s
,
an
d
ar
e
r
an
d
o
m
l
y
ch
o
s
e
n
b
y
th
e
s
e
n
d
er
.
A
ls
o
,
it
ca
n
b
e
p
r
o
v
ed
t
h
at
an
attac
k
er
ca
n
n
o
t
f
i
n
d
th
e
c
i
p
h
er
tex
t
o
f
1
2
ev
en
i
f
h
e
k
n
o
w
s
t
h
e
co
r
r
esp
o
n
d
in
g
cip
h
er
te
x
t o
f
m
es
s
ag
e
s
1
an
d
2
.
At
t
a
ck
2
:
I
f
th
e
attac
k
er
m
a
n
ag
es
to
f
ac
to
r
t
h
e
m
o
d
u
l
u
s
,
t
h
en
,
h
e
ca
n
u
s
e
an
d
to
ca
lcu
late
th
e
v
al
u
e
=
(
1
)
(
mod
)
an
d
=
4
−
1
(
+
1
)
(
mod
)
=
(
1
)
(
2
)
(
mo
d
)
.
T
o
r
ec
o
v
er
th
e
m
e
s
s
a
g
e
f
r
o
m
(
1
)
(
2
)
(
mod
)
,
h
e
h
as
to
f
i
n
d
an
d
.
A
n
d
t
h
at
is
t
h
e
co
m
p
u
ta
tio
n
all
y
i
n
f
ea
s
ib
l
e
ass
u
m
p
tio
n
o
f
t
h
e
ch
ao
tic
m
ap
s
.
A
ttac
k
3
:
Ass
u
m
e
t
h
at
th
e
att
ac
k
er
is
ab
le
to
s
o
lv
e
th
e
ch
ao
tic
m
ap
s
p
r
o
b
le
m
,
an
d
th
u
s
o
b
tain
th
e
in
te
g
er
s
2
an
d
2
.
T
h
en
,
h
e
w
ill
k
n
o
w
2
(
2
)
m
od
=
2
(
)
=
(
1
)
(
m
od
)
a
n
d
2
(
3
)
m
od
=
2
(
)
=
(
2
)
,
w
h
ic
h
is
n
o
t
en
o
u
g
h
to
r
ec
o
v
er
th
e
m
e
s
s
a
g
e.
T
h
e
attac
k
er
s
till
h
as
to
co
m
p
u
te
=
(
1
)
(
mod
)
to
f
i
n
d
=
4
−
1
(
+
1
)
(
mod
)
,
an
d
s
in
ce
th
e
f
ac
to
r
izatio
n
o
f
is
n
o
t
k
n
o
w
n
,
i
t
is
in
f
ea
s
ib
le
to
co
m
p
u
tatio
n
all
y
co
m
p
u
te
.
At
t
a
ck
4
:
No
w
,
let
u
s
a
s
s
u
m
e
th
at
an
o
r
ac
le
w
h
i
ch
ca
n
b
r
ea
k
th
e
p
r
o
p
o
s
e
d
s
ch
e
m
e
ex
i
s
ts
(
i.e
.
,
th
e
co
r
r
esp
o
n
d
in
g
cip
h
er
-
te
x
t
is
o
b
tain
ed
th
r
o
u
g
h
f
r
o
m
t
h
e
m
e
s
s
a
g
e)
.
No
w
,
w
e
ca
n
s
h
o
w
t
h
e
s
ec
u
r
it
y
o
f
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e
b
y
t
h
e
f
o
llo
w
in
g
t
h
e
th
eo
r
e
m
.
T
heo
re
m
:
I
f
t
h
er
e
ex
i
s
t
s
a
n
o
r
ac
le
th
at
is
ab
le
to
b
r
ea
k
th
e
s
u
g
g
e
s
ted
s
ch
e
m
e
,
t
h
en
i
t
i
s
also
ab
le
to
b
r
ea
k
th
e
DR
S
A
an
d
C
M.
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
A
n
ew R
S
A
p
u
b
lic
ke
y
en
cryp
t
io
n
s
ch
eme
w
ith
ch
a
o
tic
ma
p
s
(
N
ed
a
l Ta
h
a
t)
1435
P
ro
o
f
:
If
=
0
=
,
th
en
1
=
2
(
)
=
1
=
2
(
)
an
d
s
o
to
b
e
a
p
ar
ticu
lar
ca
s
e
o
f
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e
is
s
atis
f
ied
b
y
th
e
d
ep
en
d
en
t
R
S
A
cr
y
p
to
-
s
y
s
te
m
.
T
h
er
ef
o
r
e
,
if
an
o
r
ac
le
ex
is
t
s
s
u
c
h
t
h
at
it
is
ca
p
ab
le
o
f
b
r
ea
k
in
g
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e
,
th
en
it
i
s
ca
p
ab
le
also
of
b
r
ea
k
in
g
t
h
e
d
ep
en
d
en
t
R
S
A
s
c
h
e
m
e.
Ass
u
m
e
th
at
th
er
e
is
an
o
r
ac
le
th
at
is
ca
p
ab
le
o
f
b
r
ea
k
in
g
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e.
W
e
w
ill
s
h
o
w
t
h
at
c
an
al
s
o
b
r
ea
k
C
M.
Gi
v
en
th
at
(
,
,
)
is
th
e
p
u
b
lic
k
e
y
an
d
as
s
u
m
e
t
h
at
is
t
h
e
p
r
i
v
ate
k
e
y
o
f
th
e
C
M,
w
it
h
=
(
)
(
mod
)
A
s
s
u
m
e
th
a
t
a
cip
h
er
tex
t
,
(
,
)
w
as
ca
p
tu
r
ed
b
y
an
attac
k
er
,
w
h
ic
h
is
en
cr
y
p
ted
b
y
th
e
C
M
s
c
h
e
m
e
,
an
d
s/
h
e
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k
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t
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p
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s
ch
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m
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n
t
h
e
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p
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6.
P
E
RF
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RM
ANCE E
VA
L
U
AT
I
O
N
I
n
t
h
is
s
ec
tio
n
,
ev
a
lu
at
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n
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f
th
e
n
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w
p
r
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p
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s
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f
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ter
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p
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m
m
u
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co
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t
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r
r
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t
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h
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n
o
t
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w
h
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ar
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s
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in
t
h
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p
ap
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d
d
ef
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i
n
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ab
le
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a
b
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th
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p
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r
o
m
t
h
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ta
in
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T
ab
le
2
,
it
is
c
lear
th
at
t
h
e
p
r
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ed
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ch
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m
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ased
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ch
ao
tic
m
ap
s
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n
d
f
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to
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p
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le
m
s
h
as
b
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t
en
th
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tr
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ial
DR
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A
a
n
d
QE
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ch
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m
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in
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t
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also
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o
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n
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h
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tr
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n
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ab
le
1
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tatio
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s
o
f
th
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p
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m
a
n
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an
a
l
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t
i
me
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r
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t
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n
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l
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r
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a
t
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r
a
t
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≈
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.
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t
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me
f
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l
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≈
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me
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h
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i
me
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o
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p
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o
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l
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s
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u
a
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p
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t
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t
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o
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p
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t
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mo
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l
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r
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n
v
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se
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o
mp
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t
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t
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n
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≈
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T
ab
le
2
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A
C
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m
p
ar
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b
et
w
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en
th
e
n
e
w
p
r
o
p
o
s
ed
s
ch
e
m
e
s
w
it
h
t
w
o
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th
er
s
c
h
e
m
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ter
m
s
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lex
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r
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t
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D
e
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p
t
i
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n
T
o
t
a
l
(
i
n
se
c
o
n
d
s
)
H
a
r
d
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r
o
b
l
e
ms
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o
s
w
a
mi
e
t
a
l
.
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9
]
6
+
3
4
+
3
+
3
4
4
.
7
7
D
L
,
F
A
C
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o
u
l
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k
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s
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8
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6
+
4
3
+
2
+
2
4
8
.
3
7
D
L
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8
F
A
C
,
C
M
D
L
7.
CO
NCLU
SI
O
N
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n
co
n
cl
u
s
io
n
,
t
h
i
s
p
ap
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p
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p
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s
ed
a
n
e
w
cr
y
p
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s
y
s
te
m
b
a
s
ed
o
n
i
n
te
g
er
f
ac
to
r
izatio
n
a
n
d
ch
ao
ti
c
m
ap
s
d
is
cr
ete
lo
g
ar
ith
m
(
C
M
DL
)
p
r
o
b
lem
s
.
T
h
e
n
e
w
cr
y
p
t
o
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s
y
s
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a
s
en
h
a
n
ce
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th
e
o
v
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all
s
ec
u
r
it
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w
h
en
co
m
p
ar
ed
w
it
h
o
t
h
er
m
aj
o
r
p
u
b
lic
k
e
y
cr
y
p
to
-
s
y
s
te
m
s
al
g
o
r
ith
m
s
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T
h
e
s
u
g
g
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ted
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h
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m
e
n
ee
d
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m
i
n
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m
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m
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o
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tio
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r
y
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t
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o
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h
m
s
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w
h
ic
h
m
a
k
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it
v
er
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ef
f
icie
n
t.
W
e
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e
p
r
o
v
ed
t
h
at
t
h
e
n
e
w
p
r
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p
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d
s
a
m
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m
p
u
tatio
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al
co
s
t
t
h
an
o
t
h
er
s
ch
e
m
es.
W
e
h
a
v
e
p
r
o
v
ed
th
at
o
u
r
s
ch
e
m
e
is
r
o
b
u
s
t a
g
ai
n
s
t
s
ev
er
al
attac
k
s
.
Hen
ce
,
o
u
r
p
r
o
p
o
s
ed
s
ch
e
m
e
i
s
as
s
ec
u
r
e
as
R
S
A
al
g
o
r
ith
m
.
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[7
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D.
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m
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sh
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(
HU
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,
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re
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2
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n
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,
re
sp
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ti
v
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p
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M
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(
2
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–
2
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1
).
His
c
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rr
e
n
t
re
se
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tere
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lu
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p
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ls f
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ted
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e
m
s,
a
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d
m
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lt
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d
ia se
c
u
rit
y
.
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