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th
r
ee
m
ai
n
p
r
o
ce
s
s
es
f
o
r
th
is
tech
n
iq
u
e.
T
h
e
f
ir
s
t
p
r
o
ce
s
s
is
a
k
e
y
g
en
er
atio
n
an
d
t
h
er
e
ar
e
th
r
ee
s
tep
s
to
f
i
n
i
s
h
t
h
i
s
p
r
o
ce
s
s
.
Step
1
is
to
g
en
er
ate
t
w
o
p
r
i
m
e
n
u
m
b
er
s
r
an
d
o
m
l
y
,
p
a
n
d
q
,
an
d
t
h
en
co
m
p
u
te
m
o
d
u
l
u
s
,
n
=
p
*
q
,
an
d
eu
ler
to
tie
n
t
v
alu
e,
(
n
)
=
(p
-
1
)
*
(
q
-
1
)
.
T
h
e
n
ex
t
s
tep
i
s
to
s
e
lect
a
p
u
b
lic
k
e
y
,
t,
w
it
h
t
h
e
f
o
llo
w
i
n
g
co
n
d
itio
n
,
1
<
t
<
(
n
)
an
d
g
cd
(
t,
(
n
)
)
=
1
.
Af
ter
th
at,
a
p
r
iv
ate
k
e
y
,
h
,
ca
n
b
e
co
m
p
u
ted
f
r
o
m
t
*
h
m
o
d
(
n
)
=
1
b
y
u
s
in
g
s
o
m
e
o
f
e
x
te
n
d
ed
eu
clid
ea
n
alg
o
r
ith
m
s
[
1
6
-
1
9
]
.
T
h
e
s
e
co
n
d
p
r
o
ce
s
s
is
an
en
cr
y
p
t
io
n
p
r
o
ce
s
s
to
co
n
v
er
t
o
r
ig
in
a
l
p
lain
te
x
t,
m
,
as
cip
h
er
te
x
t,
c,
f
r
o
m
th
e
eq
u
atio
n
:
c
=
m
t
m
o
d
n
.
T
h
e
la
s
t
p
r
o
ce
s
s
is
a
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
to
r
ec
o
v
er
m
b
y
u
s
i
n
g
t
h
e
eq
u
atio
n
:
m
=
c
h
m
o
d
n
.
Ge
n
er
all
y
,
it
is
v
er
y
d
i
f
f
ic
u
lt
to
b
r
ea
k
t
h
i
s
s
y
s
te
m
w
h
e
n
b
it
-
le
n
g
t
h
o
f
n
is
at
least
1
0
2
4
an
d
all
p
ar
a
m
eter
s
ar
e
s
tr
o
n
g
.
I
n
co
n
tr
a
s
t,
R
S
A
b
ec
o
m
e
s
ea
s
il
y
attac
k
ed
w
h
e
n
s
o
m
e
o
f
p
ar
a
m
eter
s
ar
e
w
ea
k
.
On
e
o
f
t
h
e
w
ea
k
p
ar
am
eter
s
is
t
h
e
s
m
al
l
v
al
u
e
o
f
p
-
q
.
T
h
er
e
ar
e
v
ar
io
u
s
t
ec
n
iq
u
e
s
w
h
ich
ar
e
s
u
itab
le
f
o
r
th
is
co
n
d
itio
n
.
O
n
e
o
f
th
e
m
is
V
Facto
r
w
h
ic
h
is
a
t
y
p
e
o
f
in
te
g
er
f
ac
to
r
izatio
n
alg
o
r
ith
m
.
2
.
2
.
VF
a
ct
o
r
a
nd
i
m
pro
v
e
ment
VFacto
r
is
o
n
e
o
f
in
te
g
er
f
ac
to
r
izatio
n
al
g
o
r
ith
m
s
.
T
h
is
alg
o
r
ith
m
w
h
ich
w
a
s
p
r
o
p
o
s
ed
b
y
Sh
ar
m
a
et
al.
,
h
as
v
er
y
h
i
g
h
p
er
f
o
r
m
a
n
ce
w
h
e
n
t
h
e
r
es
u
lt
o
f
p
-
q
is
v
er
y
clo
s
e
to
0
.
T
w
o
o
d
d
in
teg
er
s
ar
e
ch
o
s
en
as
t
h
e
in
i
tial
v
al
u
es.
T
h
e
f
i
r
s
t
v
al
u
e
is
y
=
⌊
√
⌋
b
u
t
y
m
a
y
b
e
d
ec
r
ea
s
ed
b
y
1
to
en
s
u
r
e
th
at
it
is
an
o
d
d
n
u
m
b
er
w
h
e
n
it
is
a
n
ev
e
n
n
u
m
b
er
.
T
h
e
o
th
er
v
alu
e
i
s
x
=
y
+
2
.
T
h
e
m
ai
n
p
r
o
ce
s
s
is
t
o
co
m
p
u
te
m
=
x
*
y
.
I
n
f
ac
t,
if
m
=
n
,
t
h
en
it
i
m
p
lie
s
th
a
t
x
an
d
y
ar
e
t
w
o
lar
g
e
p
r
i
m
e
f
ac
to
r
s
o
f
n
.
Ho
w
e
v
er
,
it
is
d
iv
id
ed
in
to
t
w
o
ca
s
es.
T
h
e
f
ir
s
t
ca
s
e
is
m
>
n
w
h
ile
y
is
to
o
lar
g
e,
t
h
e
n
y
h
as
to
b
e
d
ec
r
ea
s
ed
b
y
2
.
On
t
h
e
o
t
h
er
h
an
d
,
th
e
s
ec
o
n
d
ca
s
e
is
o
cc
u
r
r
ed
w
h
en
m
<
n
,
x
is
to
o
s
m
all
a
n
d
it
m
u
s
t
b
e
in
cr
ea
s
ed
b
y
2
.
I
n
f
ac
t,
th
e
p
r
o
ce
s
s
is
co
n
tin
u
o
u
s
l
y
r
ep
ea
ted
u
n
til
m
=
n
is
f
o
u
n
d
.
Mo
r
eo
v
er
,
th
e
m
o
d
if
ied
alg
o
r
ith
m
s
o
f
V
Facto
r
w
er
e
p
r
o
p
o
s
ed
to
r
e
m
o
v
e
s
o
m
e
lo
o
p
s
a
n
d
ti
m
e.
MV
Facto
r
[
2
0
]
is
th
e
tech
n
iq
u
e
to
d
ec
r
ea
s
e
b
o
th
o
f
x
an
d
y
o
u
t
o
f
th
e
co
m
p
u
ta
tio
n
w
h
en
t
h
e
last
d
ig
it
is
eq
u
al
to
5
.
I
n
f
ac
t,
th
e
o
d
d
in
teg
er
s
w
h
ich
t
h
e
last
d
ig
it
is
eq
u
al
to
5
,
ex
ce
p
t
5
,
ar
e
n
o
t
ce
r
tain
l
y
a
p
r
im
e
n
u
m
b
er
,
b
ec
au
s
e
5
d
iv
id
es
all
o
f
th
e
m
.
L
ater
,
MV
F
ac
to
r
V2
[
2
1
]
w
a
s
p
r
o
p
o
s
ed
.
T
h
e
k
ey
is
to
ch
o
o
s
e
o
n
l
y
x
a
n
d
y
w
h
ic
h
m
u
s
t
b
e
w
r
it
ten
i
n
th
e
f
o
llo
w
i
n
g
f
o
r
m
:
6
k
+
1
o
r
6
k
-
1
,
w
h
er
e
k
.
Mo
r
eo
v
er
,
th
e
last
d
ig
it
o
f
t
h
e
m
m
u
s
t
n
o
t
b
e
eq
u
al
to
5
.
T
h
er
ef
o
r
e,
th
e
o
d
d
i
n
teg
er
s
w
h
ic
h
th
e
la
s
t
d
i
g
it
i
s
5
a
n
d
ca
n
n
o
t
b
e
w
r
itte
n
a
s
t
h
e
f
o
r
m
6
k
+
1
o
r
6
k
-
1
ar
e
ce
r
tai
n
l
y
n
o
t
a
p
r
i
m
e
n
u
m
b
er
.
T
ab
le
1
is
s
h
o
w
n
t
h
e
s
tep
s
o
f
in
cr
ea
s
in
g
t
h
e
o
d
d
in
teg
er
to
s
k
ip
u
n
r
ela
ted
v
al
u
es.
Fu
r
t
h
o
r
m
o
r
e,
th
e
in
f
o
r
m
a
tio
n
i
n
th
e
tab
le
i
s
also
s
elec
ted
to
co
n
s
id
er
th
e
d
ec
r
ea
s
in
g
s
tep
s
.
T
ab
le
1
.
T
h
e
i
n
cr
ea
s
in
g
s
tep
s
o
f
th
e
o
d
d
in
teg
er
t
h
at
m
a
y
b
e
a
p
r
im
e
n
u
m
b
er
R
o
w
L
S
G
(
n
)
n
mo
d
6
I
n
c
r
e
a
si
n
g
S
t
e
p
s
1
1
5
0
2
3
1
2
3
5
3
N
o
n
e
4
7
5
4
5
9
1
2
6
1
3
N
o
n
e
7
3
5
4
8
5
1
N
o
n
e
9
7
3
N
o
n
e
10
9
5
6
11
1
1
2
12
3
3
N
o
n
e
13
5
5
N
o
n
e
14
7
1
6
15
9
3
N
o
n
e
16
1
5
4
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
.
6
,
Dec
em
b
er
2
0
2
0
:
6
4
4
6
-
6
4
5
2
6448
T
h
e
in
f
o
r
m
atio
n
in
T
ab
le
1
s
h
o
w
s
t
h
e
in
cr
ea
s
i
n
g
s
tep
s
o
f
th
e
o
d
d
in
teg
er
th
at
m
a
y
b
e
a
p
r
im
e
n
u
m
b
er
.
All
p
r
i
m
e
n
u
m
b
er
s
,
ex
ce
p
t
2
an
d
3
,
m
u
s
t
b
e
u
s
u
al
l
y
r
e
w
r
itte
n
as
t
w
o
f
o
r
m
s
co
n
s
is
ti
n
g
o
f
6
k
-
1
an
d
6k
+
1
.
T
h
at
m
ea
n
th
e
i
n
te
g
er
w
h
ic
h
it
s
co
n
d
itio
n
is
eq
u
al
to
th
e
d
ata
i
n
r
o
w
3
rd
,
6
th
,
8
th
,
9
th
,
1
2
th
,
1
3
th
an
d
15
th
o
f
th
i
s
tab
le
i
s
n
o
t
ce
r
tai
n
l
y
a
p
r
im
e.
T
h
e
r
ea
s
o
n
is
t
h
at
t
h
e
f
o
r
m
o
f
s
o
m
e
o
f
th
e
m
is
6
k
+
3
o
r
th
e
last
d
ig
it
i
s
5
.
T
h
er
ef
o
r
e,
if
x
h
as
to
b
e
i
n
cr
ea
s
ed
o
r
y
h
a
s
to
b
e
d
ec
r
ea
s
ed
,
th
en
t
h
e
i
n
cr
ea
s
i
n
g
s
tep
s
i
n
t
h
i
s
tab
le
ca
n
b
e
ch
o
s
en
to
r
e
m
o
v
e
th
e
o
d
d
i
n
teg
er
s
w
h
ich
ar
e
n
o
t
ce
r
tai
n
l
y
a
p
r
i
m
e
n
u
m
b
er
.
Fo
r
ex
a
m
p
le,
ass
u
m
e
t
h
at
th
e
la
s
tes
t
v
al
u
e
o
f
x
h
as
th
e
last
2
d
i
g
it
s
as
6
3
an
d
th
e
co
n
d
itio
n
is
i
n
7
th
r
o
w
,
t
h
e
n
e
x
t
v
al
u
e
s
h
o
u
ld
h
a
v
e
th
e
last
2
d
ig
i
ts
as 6
9
.
2
.
3
.
Ana
ly
zing
t
he
la
s
t
m
di
g
it
s
o
f
p a
nd
q
I
n
2
0
1
7
,
[
2
2
]
th
e
tech
n
iq
u
e
to
an
al
y
ze
al
l
last
m
d
ig
its
o
f
p
an
d
q
w
as
p
r
o
p
o
s
ed
.
Af
ter
f
i
n
d
in
g
all
o
f
th
e
m
,
m
an
y
u
n
r
elate
d
in
teg
er
s
ar
e
r
e
m
o
v
ed
o
u
t
o
f
t
h
e
co
m
p
u
tatio
n
.
I
n
f
ac
t,
t
h
e
y
ar
e
c
h
o
s
en
to
leav
e
s
o
m
e
lo
o
p
s
o
f
FF
A.
Ass
u
m
in
g
s
o
m
e
v
al
u
es
w
h
ich
m
a
y
b
e
th
e
las
t
m
d
ig
i
ts
o
f
p
a
n
d
q
ar
e
d
is
c
l
o
s
ed
.
T
h
er
e
ar
e
t
w
o
r
u
les
f
o
r
an
al
y
zi
n
g
th
e
o
th
er
s
w
h
ich
m
a
y
b
e
also
t
h
e
last
m
d
ig
it
s
o
f
p
an
d
q
as
f
o
llo
w
s
:
(
Ass
i
g
n
in
g
p
m
is
r
ep
r
esen
ted
as th
e
las
t
m
d
i
g
it
s
o
f
p
an
d
q
m
is
r
ep
r
ese
n
ted
as th
e
last
m
d
ig
its
o
f
q
.
R
u
le
1
:
I
f
t
h
e
la
s
t
d
ig
i
t
o
f
p
a
n
d
q
ar
e
s
a
m
e,
t
h
e
o
t
h
er
p
air
s
ca
n
b
e
co
m
p
u
ted
f
r
o
m
p
m'
=
(p
m
+
10
m
-
1
)
m
o
d
1
0
m
a
n
d
q
m'
=
(q
m
+
9
*
1
0
m
-
1
)
m
o
d
1
0
m
E
xa
mp
le
1
A
s
s
u
m
i
n
g
p
2
=
1
1
,
q
2
=
3
1
(
th
e
last
2
d
i
g
its
o
f
1
1
*
31
=
4
1
)
,
th
en
p
2'
=
2
1
a
n
d
q
2'
=
2
1
(
th
e
la
s
t
2
d
ig
its
o
f
2
1
*
21
=
41)
R
u
le
2
:
I
f
th
e
la
s
t
d
ig
it
o
f
p
an
d
q
ar
e
d
if
f
er
e
n
t,
th
e
o
th
er
p
air
s
ca
n
b
e
co
m
p
u
ted
f
r
o
m
p
m'
=
(p
m
+
k
1
*
1
0
m
-
1
)
m
o
d
1
0
m
a
n
d
q
m'
=
(q
m
+
k
2
*
1
0
m
-
1
)
m
o
d
1
0
m
,
w
h
er
e
(
(
p
m
m
o
d
1
0
)
*
k
2
+
(q
m
m
o
d
1
0
)
*
k
1
)
m
o
d
10
=
0
I
n
f
ac
t,
a
f
ter
f
in
d
i
n
g
all
las
t
m
d
ig
i
ts
o
f
p
an
d
q
,
all
p
o
s
s
i
b
le
r
esu
lts
o
f
t
h
e
last
m
d
i
g
it
o
f
p
+
q
an
d
p
-
q
ar
e
also
d
is
clo
s
ed
.
A
s
s
u
m
i
n
g
U
i
s
r
ep
r
esen
ted
as
t
h
e
s
et
o
f
all
p
o
s
s
ib
le
v
a
lu
e
s
o
f
th
e
last
m
d
ig
i
ts
o
f
p
+
q
an
d
V
is
r
ep
r
ese
n
ted
as
t
h
e
s
et
o
f
a
l
l
p
o
s
s
ib
le
v
al
u
es
o
f
t
h
e
la
s
t
m
d
i
g
its
o
f
p
-
q
.
E
x
a
m
p
le
2
is
s
h
o
w
n
th
e
w
a
y
to
f
i
n
d
all
m
e
m
b
er
s
f
o
r
b
o
th
o
f
th
e
m
.
E
xa
mp
le
2
Fin
d
in
g
U
a
n
d
V
f
o
r
all
v
alu
es o
f
n
t
h
at
t
h
e
last
2
d
ig
its
ar
e
8
3
S
o
l.
I
n
g
en
er
al,
b
o
th
o
f
r
u
le
1
an
d
r
u
le
2
ar
e
th
e
k
e
y
to
f
i
n
d
all
m
e
m
b
er
s
o
f
U
an
d
V.
I
n
f
a
ct,
all
p
air
s
o
f
th
e
la
s
t
2
d
ig
it
s
o
f
p
a
n
d
q
ar
e
as
f
o
llo
w
s
:
(
1
1
,
5
3
)
,
(
2
1
,
2
3
)
,
(
3
1
,
9
3
)
,
(
4
1
,
6
3
)
,
(
5
1
,
3
3
)
,
(
6
1
,
3
)
,
(
7
1
,
7
3
)
,
(
8
1
,
4
3
)
,
(
9
1
,
1
3
)
,
(
1
,
8
3
)
,
(
1
7
,
9
9
)
,
(
2
7
,
2
9
)
,
(
3
7
,
5
9
)
,
(
4
7
,
8
9
)
,
(
5
7
,
1
9
)
,
(
6
7
,
4
9
)
,
(
7
7
,
7
9
)
,
(
8
7
,
9
)
,
(
9
7
,
3
9
)
an
d
(
7
,
6
9
)
.
T
h
er
ef
o
r
e,
U
an
d
V
ar
e
as f
o
llo
w
s
:
U
=
{0
4
,
1
6
,
2
4
,
3
6
,
4
4
,
5
6
,
6
4
,
7
6
,
8
4
,
9
6
}
V
=
{0
2
,
1
8
,
2
2
,
3
8
,
4
2
,
5
8
,
6
2
,
7
8
,
8
2
,
9
8
}
I
n
ad
d
itio
n
,
af
ter
U
an
d
V
ar
e
f
o
u
n
d
,
th
e
in
i
tial
v
al
u
e
o
f
u
,
u
i
,
w
h
ic
h
i
s
b
eg
u
n
as
2
⌈
√
⌉
ca
n
b
e
r
ee
s
ti
m
ated
.
T
h
e
last
m
d
ig
it
s
o
f
u
i
s
h
o
u
ld
b
e
o
n
e
o
f
th
e
m
e
m
b
er
s
i
n
U.
T
h
at
m
ea
n
s
i
t
ca
n
b
e
in
cr
ea
s
ed
w
h
e
n
ev
er
t
h
e
r
esu
l
t is st
ill n
o
t
a
m
e
m
b
er
o
f
U.
2
.
4
.
Ana
ly
zing
t
he
ini
t
ia
l v
a
l
ue
o
f
p
-
q
T
h
e
in
itial
v
al
u
e
o
f
p
-
q
s
h
o
u
l
d
b
e
u
s
u
all
y
b
eg
u
n
as
0
,
p
=
q
.
Ho
w
ev
er
,
r
ea
l
v
al
u
e
o
f
p
-
q
is
v
er
y
f
ar
f
r
o
m
t
h
e
in
i
tial
v
al
u
e.
I
n
2
0
1
8
,
[
2
3
]
th
e
eq
u
atio
n
to
e
s
ti
m
ate
th
e
n
e
w
in
itial
v
al
u
e
o
f
v
,
v
i
,
w
a
s
p
r
o
p
o
s
ed
.
I
n
f
ac
t,
b
e
f
o
r
e
u
s
i
n
g
th
e
eq
u
atio
n
,
all
las
t
m
d
i
g
it
s
o
f
p
an
d
q
m
u
s
t
b
e
d
is
clo
s
ed
.
I
n
ad
d
itio
n
,
v
i
ca
n
b
e
co
m
p
u
ted
f
r
o
m
t
h
e
f
o
llo
w
i
n
g
eq
u
atio
n
:
v
i
=
⌈
√
2
∗
4
⌉
,
w
h
er
e
d
is
th
e
d
is
tan
ce
b
et
w
ee
n
th
e
tr
ad
itio
n
al
v
alu
e
o
f
u
i
a
n
d
th
e
n
e
w
v
al
u
e
o
f
u
i
.
I
n
ad
d
itio
n
,
th
e
n
e
w
v
al
u
es
o
f
u
i
an
d
v
i
ca
n
b
e
also
s
elec
ted
t
o
d
ec
r
ea
s
e
ti
m
e
f
o
r
s
o
m
e
o
th
er
f
ac
to
r
izatio
n
al
g
o
r
tih
m
s
.
Fo
r
ex
a
m
p
l
e,
in
2
0
1
9
,
th
is
tec
h
n
iq
u
e
i
s
ch
o
s
e
n
to
co
m
b
i
n
e
w
it
h
tr
ial
d
iv
itio
n
al
g
o
r
ith
m
(
T
DA
)
[
2
4
]
.
B
ef
o
r
e
a
p
p
ly
i
n
g
t
h
is
m
et
h
o
d
w
ith
T
DA
,
t
h
e
f
ir
s
t
d
iv
is
o
r
is
u
s
u
all
y
b
eg
u
n
a
s
⌊
√
⌋
.
On
th
e
o
t
h
er
h
an
d
,
it
m
a
y
b
e
ass
i
g
n
ed
as
t
h
e
in
teg
er
w
h
i
ch
is
les
s
t
h
an
t
h
i
s
v
al
u
e
w
h
e
n
it
is
ap
p
l
ied
w
it
h
T
DA
.
2
.
5
.
Ana
ly
zing
t
he
re
m
a
in
de
r
o
f
(
p
+
q)
m
o
d 8
I
n
[
2
5
]
,
it
is
f
o
u
n
d
th
a
t
if
t
h
e
r
esu
lt
o
f
(
n
+
1
)
m
o
d
8
=
0
,
th
en
th
e
r
esu
lt
o
f
(
p
+
q
)
m
o
d
8
m
u
s
t
b
e
al
w
a
y
s
eq
u
al
to
0
.
T
h
er
ef
o
r
e,
o
n
l
y
p
at
ter
n
o
f
p
+
q
w
h
ich
is
i
n
t
h
e
co
n
d
itio
n
w
ill
b
e
i
n
c
lu
d
ed
to
r
e
m
o
v
e
s
o
m
e
lo
o
p
s
o
f
th
e
co
m
p
u
tatio
n
.
3.
T
H
E
P
RO
P
O
SE
D
M
E
T
H
O
D
I
n
t
h
is
s
ec
tio
n
,
th
e
n
e
w
in
iti
al
v
al
u
es
to
b
o
th
o
f
x
an
d
y
f
o
r
V
f
ac
to
r
ar
e
p
r
o
p
o
s
ed
to
d
ec
r
ea
s
e
th
e
ce
r
tai
n
l
y
u
n
r
elate
d
v
al
u
e
s
o
u
t
o
f
t
h
e
co
m
p
u
ta
tio
n
.
I
n
g
en
er
al,
th
e
tr
ad
itio
n
al
i
n
i
tial
v
a
lu
e
o
f
x
i
s
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
mp
r
o
vin
g
th
e
in
itia
l v
a
lu
es o
f
V
F
a
cto
r
s
u
ita
b
le
fo
r
b
a
l
a
n
ce
d
mo
d
u
lu
s
(
K
r
its
a
n
a
p
o
n
g
S
o
m
s
u
k)
6449
th
e
m
in
i
m
u
m
o
d
d
in
teg
er
wh
ich
i
s
lar
g
er
t
h
an
n
.
On
th
e
o
th
er
h
an
d
,
af
ter
t
h
e
last
m
d
ig
its
o
f
n
ar
e
an
al
y
ze
d
,
it
ca
n
b
e
esti
m
ated
as
q
i
w
h
e
n
q
i
i
s
an
o
d
d
n
u
m
b
e
r
o
r
q
i
+
1
w
h
e
n
q
i
is
an
ev
e
n
n
u
m
b
er
.
I
n
ad
d
itio
n
,
th
e
tr
ad
itio
n
al
in
i
tial
v
alu
e
o
f
y
is
th
e
m
a
x
i
m
u
m
o
d
d
in
te
g
er
w
h
ic
h
is
s
till
les
s
th
a
n
n
.
T
h
e
s
am
e
r
ea
s
o
n
w
it
h
ab
o
v
e
co
n
d
itio
n
,
t
h
e
n
e
w
v
a
lu
e
ca
n
b
e
esti
m
ated
as
th
e
m
a
x
i
m
u
m
o
d
d
in
te
g
er
w
h
ic
h
is
les
s
th
a
n
.
Fu
r
t
h
er
m
o
r
e,
if
t
h
e
co
n
ce
p
ts
o
f
MV
Facto
r
an
d
MV
Facto
r
V2
ar
e
also
in
clu
d
ed
,
th
e
n
t
h
e
las
t
d
ig
it
m
u
s
t
n
o
t
b
e
eq
u
al
to
5
an
d
th
e
f
o
r
m
s
o
f
t
h
e
m
m
u
s
t b
e
al
w
a
y
s
6
k
+
1
o
r
6
k
-
1.
A
l
g
o
r
ith
m
1
:
T
h
e
n
e
w
i
n
itial
v
alu
es o
f
x
an
d
y
Input:
n, u
i
, v
i
1. q
i
ii
u
+
v
2
2. IF q
i
%2 == 0 Then
3. q
i
q
i
+
1
4. End IF
5. x
q
i
6. x
10
x % 10
7. x
6
x % 6
8. IF (x
10
== 5 and x
6
== 3) OR (x
10
== 1 AND x
6
== 3) OR (x
10
== 7 AND x
6
== 3) OR (x
10
==5
AND x
6
==5) OR (x
10
== 9 AND x
6
== 3) Then
9. x
x
+
2
10.Else IF (x
10
== 5 AND x
6
== 1) OR (x
10
== 3 AND x
6
== 3) Then
11. x
x
+
4
12.End IF
13. y
n
x
14. IF y%2 == 0 Then
15. y
y
-
1
16. End IF
17. y
10
y %10
18. y
6
y % 6
19. IF (y
10
== 5 and y
6
== 3) OR (y
10
== 1 AND y
6
== 3) OR (y
10
== 5 AND y
6
== 1) OR (y
10
==3
AND y
6
==3) OR (y
10
== 9 AND y
6
== 3) Then
20. y
y
-
2
21. Else IF (y
10
== 7 AND y
6
== 3) OR (y
10
== 5 AND y
6
== 5) Then
22.
y
y
-
4
23. End IF
Output:
The new initial values of x and y
A
l
g
o
r
ith
m
2
:
T
h
e
n
e
w
i
n
itial
v
alu
es o
f
u
i
a
n
d
v
i
Input:
n (n
+
1 mod 8
=
0), U, V
1. u
i
2
n
2. t
u
i
3. IF the last m digits of u
i
is not a member of U Then
4. Increaning u
i
until the last two digits are equal to one of the members in U
5. End IF
6. While u
i
mod 8 is not equal to 0 do
7. Replacing the last m digits of u
i
by the next member of U
8. End While
9. d
u
i
-
t
10. v
i
4
2
d
n
*
11. IF the last m digits of v
i
is not a member of V Then
12. Increaning v
i
until the last two digits are equal to one of the members in V
13. End IF
Output:
The new initial values of u
i
and v
i
E
xa
mp
le
3
:
Fin
d
in
g
th
e
n
e
w
in
itial
v
alu
e
s
o
f
x
a
n
d
y
wh
en
n
=
2
6
2
0
3
6
1
0
8
3
(
5
6
5
3
3
*
4
6
3
5
1
)
b
y
co
n
s
id
er
in
g
th
e
la
s
t 2
d
ig
it
s
o
f
n
=
8
3
an
d
u
s
i
n
g
Alg
o
r
it
h
m
1
S
o
l.
B
ef
o
r
e
u
s
in
g
A
l
g
o
r
ith
m
1
,
u
i
an
d
v
i
m
u
s
t
b
e
co
m
p
u
ted
.
Usu
al
l
y
u
i
=
2
⌈
√
2620361083
⌉
=
1
0
2
3
8
0
.
Ho
w
e
v
er
,
th
e
last
2
d
i
g
it
s
is
8
0
w
h
ich
is
n
o
t
a
m
e
m
b
er
o
f
U.
T
h
er
ef
o
r
e,
u
i
ca
n
b
e
in
cr
ea
s
ed
as
1
0
2
3
8
4
,
an
d
th
en
d
=
4
.
I
n
ad
d
itio
n
,
v
i
=
2
⌈
√
4
2
∗
2620361083
4
⌉
=
9
0
6
.
Nev
er
th
eless
,
th
e
l
ast
2
d
ig
its
is
0
6
w
h
ich
is
n
o
t
a
m
e
m
b
er
o
f
V.
T
h
en
,
v
i
ca
n
b
e
in
cr
ea
s
ed
as 9
1
8
.
T
h
er
ef
o
r
e,
ea
ch
s
tep
in
Alg
o
r
it
h
m
1
is
a
s
f
o
llo
w
s
:
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
.
6
,
Dec
em
b
er
2
0
2
0
:
6
4
4
6
-
6
4
5
2
6450
Step
1
: q
i
=
102
384
918
2
=
51651
Step
2
-
4
: q
i
is
n
o
t c
h
a
n
g
ed
,
b
ec
au
s
e
q
i
% 2
=
1
Step
5
: x
=
51651
Step
6
-
7
: x
10
=
1
an
d
x
6
=
3
Step
8
-
1
2
: x
=
51653
Step
1
3
: y
=
⌊
26203610
83
51653
⌋
=
5
0
7
3
0
Step
1
4
-
1
6
: y
i
s
ch
a
n
g
ed
as 5
0
7
2
9
,
b
ec
au
s
e
y
% 2
=
0
Step
17
-
1
8
: y
10
=
9
an
d
y
6
=
5
Step
1
9
-
2
3
: y
i
s
n
o
t c
h
a
n
g
ed
,
b
ec
au
s
e
b
o
th
o
f
y
10
an
d
y
6
ar
e
n
o
t
m
atc
h
ed
w
i
t
h
th
e
co
n
d
it
io
n
s
.
T
h
er
ef
o
r
e,
th
e
n
e
w
in
i
tial v
a
lu
es a
r
e
x
=
5
1
6
5
1
an
d
y
=
50729
I
n
f
ac
t,
th
e
tr
ad
itio
n
al
in
itial
v
alu
e
s
i
n
ex
a
m
p
le
3
ar
e
as
f
o
llo
w
s
:
=
⌊
√
26203610
83
⌋
=
51189
an
d
x
=
y
+
2
=
5
1
1
9
1
.
T
h
en
,
x
’
s
lo
o
p
s
ar
e
d
ec
r
ea
s
ed
as
51653
−
5
1
1
91
2
=
231
a
n
d
y
’
s
lo
o
p
s
ar
e
d
ec
r
ea
s
ed
as
51189
−
50729
2
=
230
.
T
h
er
ef
o
r
e,
to
tal
lo
o
p
s
ar
e
lef
t
o
u
t
t
h
e
co
m
p
u
tatio
n
ab
o
u
t
4
6
1
w
h
e
n
ev
er
t
h
e
n
e
w
i
n
itia
l
v
alu
e
s
o
f
x
an
d
y
ar
e
ch
o
s
en
i
n
s
tead
o
f
t
h
e
tr
ad
itio
n
v
alu
e
s
.
Fu
r
th
er
m
o
r
e,
to
tal
lo
o
p
s
ar
e
m
o
r
e
d
ec
r
ea
s
ed
w
h
en
m
is
lar
g
e.
T
h
e
r
ea
s
o
n
is
th
at
th
e
ch
ar
ac
ter
is
tic
o
f
n
is
an
aly
ze
d
m
o
r
e
d
ee
p
ly
.
I
n
co
n
tr
ast,
lo
o
p
s
ar
e
n
o
t
r
ed
u
ce
d
w
h
en
th
e
last
m
d
ig
its
o
f
2
⌈
√
⌉
is
th
e
m
em
b
er
o
f
U,
b
ec
au
s
e
d
is
eq
u
al
to
0
.
T
h
er
ef
o
r
e,
b
o
th
o
f
u
i
an
d
v
i
ar
e
n
o
t c
h
an
g
ed
.
Mo
r
eo
v
er
,
th
e
id
ea
in
[
2
5
]
ca
n
b
e
s
elec
ted
to
ap
p
ly
w
ith
th
e
p
r
o
p
o
s
ed
m
eth
o
d
w
h
en
th
e
r
esu
lt
o
f
(n
+
1
)
m
o
d
8
is
eq
u
al
to
0
.
I
n
f
ac
t,
d
is
ex
p
an
d
ed
,
b
ec
au
s
e
u
i
ca
n
b
e
m
o
d
if
ied
in
th
e
co
n
d
itio
n
s
o
f
U
an
d
u
m
o
d
8
=
0
.
Ho
w
ev
er
,
b
o
th
o
f
u
i
an
d
v
i
m
u
s
t
b
e
im
p
r
o
v
ed
b
ef
o
r
e
u
s
in
g
A
lg
o
r
ith
m
1
.
Fo
r
A
lg
o
r
tith
m
2
,
it sh
o
w
s
th
e
p
r
o
ce
s
s
to
im
p
r
o
v
e
u
i
an
d
v
i
w
h
en
(
n
+
1
)
m
o
d
8
is
eq
u
al
to
0
.
E
xa
mp
le
4
:
Fin
d
in
g
th
e
n
ew
in
itial
v
alu
e
o
f
x
an
d
y
w
h
en
n
=
3801472783
(
6
3
0
7
3
*
6
0
2
7
1
)
b
y
co
n
s
id
er
in
g
th
e
last
2
d
ig
its
o
f
n
=
8
3
an
d
u
s
in
g
A
lg
o
r
ith
m
1
an
d
A
lg
o
r
ith
m
2
S
o
l.
First,
th
e
r
esu
lt
o
f
(n
+
1
)
m
o
d
8
h
av
e
to
b
e
d
eter
m
i
n
d
ed
.
B
ec
au
s
e
t
h
e
r
es
u
lt
is
0
,
th
e
n
th
e
p
atter
n
o
f
(
p
+
q
)
m
o
d
8
m
u
s
t b
e
also
0
.
T
h
e
p
r
o
ce
s
s
to
f
in
d
th
e
n
ew
v
alu
es
o
f
u
i
an
d
v
i
b
y
u
s
in
g
A
lg
o
r
ith
m
2
is
as
f
o
llo
w
s
:
Usu
ally
u
i
=
2
3801472783
=
1
2
3
3
1
4
.
Ho
w
ev
er
,
th
e
last
2
d
ig
its
is
1
4
w
h
ich
is
n
o
t
a
m
em
b
er
o
f
U.
T
h
er
ef
o
r
e,
u
i
ca
n
b
e
in
cr
ea
s
ed
as
1
2
3
3
1
6
.
I
n
co
n
tr
ast,
1
2
3
3
1
6
m
o
d
8
=
4
0
,
th
en
n
ex
t
v
alu
e
o
f
u
i
s
h
o
u
ld
b
e
1
2
3
3
2
4
.
Ho
w
ev
er
,
1
2
3
3
2
4
m
o
d
8
=
4
0
,
th
en
n
ex
t
v
alu
e
o
f
u
i
s
h
o
u
ld
b
e
ass
ig
n
ed
as
1
2
3
3
3
6
.
B
ec
au
s
e
1
2
3
3
3
6
m
o
d
8
=
0
,
it
is
th
e
n
ew
v
alu
e
o
f
u
i
,
d
=
2
2
.
I
n
ad
d
itio
n
,
=
2
⌈
√
22
2
∗
3801472783
4
⌉
=
2330
.
Ho
w
ev
er
,
th
e
last
2
d
ig
its
is
3
0
w
h
ich
is
n
o
t
a
m
em
b
er
o
f
V.
T
h
er
ef
o
r
e,
v
i
ca
n
b
e
in
cr
ea
s
ed
as
2
3
3
8
.
T
h
er
ef
o
r
e,
ea
ch
s
tep
in
A
lg
o
r
ith
m
1
is
as f
o
llo
w
in
g
:
Step
1
: q
i
=
123336
+
2338
2
=
62837
Step
2
-
4
: q
i
is
n
o
t c
h
a
n
g
ed
,
b
ec
au
s
e
q
i
% 2
=
1
Step
5
: x
=
62837
Step
6
-
7
: x
10
=
7
an
d
x
6
=
5
Step
8
-
1
2
: x
is
n
o
t c
h
an
g
ed
,
b
ec
au
s
e
b
o
th
o
f
x
10
an
d
x
6
ar
e
n
o
t m
a
tch
ed
w
it
h
t
h
e
co
n
d
itio
n
s
.
Step
1
3
: y
=
3801472783
62837
=
6
0
4
9
7
Step
1
4
-
1
6
: y
i
s
n
o
t c
h
a
n
g
ed
,
b
ec
au
s
e
y
% 2
=
1
Step
1
7
-
1
8
: y
10
=
7
an
d
y
6
=
5
Step
1
9
-
2
2
: y
i
s
n
o
t c
h
a
n
g
ed
,
b
ec
au
s
e
b
o
th
o
f
y
10
an
d
y
6
ar
e
n
o
t
m
atc
h
ed
w
i
t
h
th
e
co
n
d
it
io
n
s
.
T
h
er
ef
o
r
e,
th
e
n
e
w
in
i
tial v
a
lu
es a
r
e
x
=
6
2
8
3
7
an
d
y
=
60497
.
T
h
e
t
r
a
d
i
t
i
o
n
a
l
i
n
i
t
i
a
l
v
a
l
u
e
s
i
n
e
x
a
m
p
l
e
4
a
r
e
u
s
u
a
l
l
y
a
s
f
o
l
l
o
w
i
n
g
:
y
=
⌊
√
3801472783
⌋
=
61656
=
61655
(y
i
s
a
l
w
a
y
s
a
n
o
d
d
n
u
m
b
e
r
)
a
n
d
x
=
y
+
2
=
6
1
6
5
7
.
T
h
e
n
,
x
’
s
l
o
o
p
s
a
r
e
d
e
c
r
e
a
s
e
d
a
s
62837
−
61655
2
=
590
a
n
d
y
’
s
l
o
o
p
s
a
r
e
d
e
c
r
e
a
s
e
d
a
s
61655
−
6
0
4
97
2
=
579
.
T
h
e
r
e
f
o
r
e
,
t
o
t
a
l
l
o
o
p
s
a
r
e
l
e
f
t
o
u
t
t
h
e
c
o
m
p
u
t
a
t
i
o
n
a
b
o
u
t
1
1
6
9
w
h
e
n
e
v
e
r
t
h
e
n
e
w
i
n
i
t
i
a
l
v
a
l
u
e
s
o
f
x
a
n
d
y
a
r
e
c
h
o
s
e
n
i
n
s
t
e
a
d
o
f
t
h
e
t
r
a
d
i
t
i
o
n
v
a
l
u
e
s
.
I
n
f
a
c
t
,
i
n
t
h
i
s
e
x
a
m
p
l
e
,
a
s
s
u
m
i
n
g
t
h
e
c
o
n
c
e
p
t
i
n
[
2
5
]
i
s
n
o
t
c
h
o
s
e
n
t
o
c
o
m
b
i
n
e
w
i
t
h
t
h
e
p
r
o
p
o
s
e
d
m
e
t
h
o
d
,
u
i
=
1
2
3
3
1
6
,
d
=
2
a
n
d
v
i
=
7
1
8
.
H
e
n
c
e
,
t
h
e
t
o
t
a
l
l
o
o
p
s
a
r
e
d
e
c
r
e
a
s
e
d
o
n
l
y
3
5
9
.
T
h
e
r
e
f
o
r
e
,
l
o
o
p
s
a
r
e
m
o
r
e
d
e
c
r
e
s
e
d
a
b
o
u
t
t
h
r
e
e
t
i
m
e
s
.
H
o
w
e
v
e
r
,
t
h
i
s
t
e
c
h
n
i
q
u
e
c
a
n
n
o
t
b
e
a
p
p
i
e
d
w
i
t
h
n
i
n
e
x
a
m
p
l
e
3
,
b
e
c
a
u
s
e
n
+
1
m
o
d
8
=
4
0.
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
mp
r
o
vin
g
th
e
in
itia
l v
a
lu
es o
f
V
F
a
cto
r
s
u
ita
b
le
fo
r
b
a
l
a
n
ce
d
mo
d
u
lu
s
(
K
r
its
a
n
a
p
o
n
g
S
o
m
s
u
k)
64
51
4.
E
XP
E
R
I
M
E
NT
A
L
RE
SUL
T
S AN
D
D
I
SC
USS
I
O
N
I
n
th
is
s
ec
tio
n
,
ex
p
er
im
en
tal
r
esu
lts
an
d
d
is
cu
s
s
io
n
w
ill
b
e
p
r
esen
ted
.
I
n
f
ac
t,
th
e
last
2
d
ig
its
o
f
all
v
alu
es
o
f
n
in
th
e
ex
p
er
im
en
t
ar
e
8
3
,
b
ec
au
s
e
U
an
d
V
ar
e
alr
ea
d
y
co
n
s
id
er
ed
in
r
elate
d
w
o
r
k
s
s
ec
tio
n
to
s
k
ip
th
is
p
r
o
ce
s
s
.
Ho
w
ev
er
,
if
th
e
o
th
er
ca
s
es
o
f
n
ar
e
o
cc
u
r
r
ed
,
b
o
th
o
f
U
an
d
V
m
u
s
t
b
e
co
n
s
id
er
ed
at
f
ir
s
t.
Fu
r
th
er
m
o
r
e,
n
+
1
m
o
d
8
=
0
f
o
r
all
v
alu
es
o
f
n
in
th
is
s
ec
tio
n
is
s
ele
cted
to
in
clu
d
e
th
e
id
ea
b
eh
in
d
A
lg
o
r
ith
m
2
.
T
h
e
ex
p
er
im
en
t
is
ab
o
u
t
th
e
co
m
p
ar
is
o
n
o
f
d
ec
r
ea
s
in
g
lo
o
p
s
b
etw
ee
n
u
s
in
g
o
n
ly
A
lg
o
r
ith
m
1
an
d
th
e
co
m
b
in
atio
n
b
etw
ee
n
A
lg
o
r
ith
m
1
w
ith
A
lg
o
r
ith
m
2
.
I
n
ad
d
itio
n
,
b
it
-
len
g
th
o
f
n
w
h
ich
is
r
an
d
o
m
ly
ch
o
s
en
in
th
is
ex
p
er
im
en
t
co
n
s
is
t
o
f
3
2
,
6
4
,
1
2
8
,
2
5
6
,
5
1
2
an
d
1
0
2
4
.
Mo
r
eo
v
er
,
5
0
v
alu
es
o
f
n
ar
e
ch
o
s
en
f
o
r
th
e
s
am
e
b
it
-
len
g
th
to
f
in
d
th
e
av
er
ag
e.
Ho
w
ev
er
,
th
e
co
n
d
itio
n
o
f
n
in
th
is
s
ess
io
n
is
th
at
d
m
u
s
t
n
o
t
b
e
eq
u
al
to
0
.
T
h
e
r
ea
s
o
n
is
th
at
th
e
n
ew
in
itial
v
alu
es
f
o
r
b
o
th
o
f
A
lg
o
r
ith
m
1
an
d
th
e
co
m
b
in
atio
n
b
etw
ee
n
A
lg
o
r
ith
m
1
an
d
A
lg
o
r
ith
m
2
ar
e
s
till
eq
u
al
to
th
e
tr
ad
itio
n
in
itial
v
alu
es.
T
h
e
in
f
o
r
m
atio
n
in
T
ab
le
2
s
h
o
w
s
th
at
if
(
n
+
1
)
m
o
d
8
=
0
an
d
d
0
,
d
ec
r
ea
s
in
g
lo
o
p
s
o
f
th
e
co
m
p
u
tatio
n
b
y
u
s
in
g
th
e
co
m
b
in
atio
n
b
etw
ee
n
A
lg
o
r
ith
m
1
an
d
A
lg
o
r
ith
m
2
ar
e
m
u
ch
h
ig
h
er
th
an
u
s
in
g
A
lg
o
r
ith
m
1
o
n
ly
.
I
n
ad
d
itio
n
,
it
is
lar
g
er
th
an
th
e
o
th
er
ab
o
u
t
tw
o
tim
es.
T
h
er
ef
o
r
e,
to
en
s
u
r
e
th
at
all
h
id
d
en
p
ar
am
eter
s
ar
e
s
tr
o
n
g
,
th
e
r
esu
lt
o
f
(
n
+1
)
m
o
d
8
s
h
o
u
ld
n
o
t
b
e
eq
u
al
to
0
.
I
n
f
ac
t,
th
e
p
r
o
b
ab
ilit
y
is
eq
u
al
to
0
.
2
5
th
at
th
e
r
esu
lt
o
f
(
n
+
1
)
m
o
d
8
=
0
,
n
is
s
elec
ted
r
an
d
o
m
ly
.
T
ab
le
2
.
T
h
e
co
m
p
ar
io
n
o
f
d
ec
r
ea
s
in
g
lo
o
p
s
b
et
w
ee
n
t
w
o
p
r
o
p
o
s
ed
tech
n
iq
u
es
f
o
r
(
n
+
1
)
m
o
d
8
=
0
Bit
-
l
e
n
g
t
h
o
f
n
D
e
c
r
e
a
si
n
g
L
o
o
p
s
A
l
g
o
r
i
t
h
m
1
A
l
g
o
r
i
t
h
m
1
+
A
l
g
o
r
i
t
h
m
2
32
2
8
7
7
5
2
64
7
7
6
8
3
2
5
7
6
1
2
1
2
8
9
6
5
6
6
4
3
9
6
2
1
8
0
6
5
9
2
6
6
0
1
2
5
6
4
.
3
1
*
1
0
19
8
.
0
7
*
1
0
19
5
1
2
7
.
2
3
*
1
0
38
1
.
3
5
*
1
0
39
1
0
2
4
3
.
0
6
*
1
0
77
5
.
7
2
*
1
0
77
Ho
w
ev
er
,
if
n
is
lar
g
er
th
an
1
0
2
4
b
its
an
d
all
h
id
d
en
p
ar
am
eter
s
ar
e
s
tr
o
n
g
,
VFacto
r
an
d
all
im
p
r
o
v
in
g
alg
o
r
th
m
s
,
in
clu
d
in
g
th
e
p
r
o
p
o
s
ed
m
eth
o
d
s
an
d
th
e
r
esu
lt
o
f
(
n
+
1
)
m
o
d
8
=
0
,
d
o
n
o
t
s
till
b
r
ea
k
R
SA
w
ith
in
a
p
o
ly
n
o
m
ial
tim
e.
T
h
e
ex
am
p
le
is
s
h
o
w
n
as
f
o
llo
w
s
:
A
s
s
u
m
in
g
n
=
293060910868290979627266785232142097857
*
205030072726927862555415759877785028319
=
60086299868745423605959016054076895558512635625836613350661870819438814212383
(256
bi
t
s
-
len
g
th
)
,
af
ter
esti
m
atin
g
th
e
n
ew
in
itial
v
alu
es,
th
e
d
ec
r
ea
s
in
g
lo
o
p
s
ar
e
ab
o
u
t
7
.
9
8
*
1
0
19
.
Ho
w
ev
er
,
th
e
to
tal
lo
o
p
s
ar
e
2
.
3
4
*
10
37
.
T
h
er
ef
o
r
e,
af
ter
u
s
in
g
th
e
p
r
o
p
o
s
ed
m
eth
o
d
,
lo
o
p
s
ar
e
d
ec
r
ea
s
ed
o
n
ly
3
.
1
4
*
1
0
-
11
%
th
at
is
v
er
y
to
o
s
m
all.
I
n
co
n
tr
ast,
th
e
p
r
o
p
o
s
ed
m
eth
o
d
b
ec
o
m
e
h
ig
h
p
er
f
o
r
m
an
ce
w
h
en
p
is
clo
s
e
to
q
.
T
h
e
e
x
am
p
le
is
s
h
o
w
n
as
f
o
llo
w
s
:
A
s
s
u
m
in
g
n
=
194456630408620613527183578802116928289
*
194456630408620613127183578802116928247
=
37813381109874874999245217886281867608252777770
855434333178732422091857479383
(256
bi
t
s
-
len
g
th
)
,
af
ter
esti
m
atin
g
th
e
n
ew
in
itial
v
alu
es
,
th
e
d
ec
r
ea
s
in
g
lo
o
p
s
ar
e
ab
o
u
t
2
.
7
8
*
1
0
19
.
Ho
w
ev
er
,
th
e
to
tal
lo
o
p
s
ar
e
1
.
0
6
*
10
20
.
T
h
er
ef
o
r
e,
af
ter
u
s
in
g
th
e
p
r
o
p
o
s
ed
m
eth
o
d
,
lo
o
p
s
ar
e
d
ec
r
ea
s
ed
ab
o
u
t
2
6
%
th
at
ar
e
v
er
y
h
ig
h
.
T
h
er
ef
o
r
e,
th
e
r
atio
o
f
th
e
d
ec
r
ea
s
in
g
lo
o
p
s
is
b
ased
o
n
th
e
ch
ar
ac
ter
is
tics
o
f
p
an
d
q
an
d
th
e
p
r
o
p
o
s
ed
m
eth
o
d
is
s
u
itab
le
f
o
r
a
s
m
all
r
esu
lt
o
f
p
an
d
q
.
5.
CO
NCLU
SI
O
N
Th
e
n
e
w
i
n
itial
v
alu
e
s
f
o
r
VF
ac
to
r
ar
e
ass
ig
n
ed
to
leav
e
th
e
u
n
r
elate
d
v
al
u
es
o
u
t
o
f
th
e
co
m
p
u
tat
io
n
.
T
h
e
k
e
y
is
to
ch
o
o
s
e
th
e
co
n
c
ep
t
o
f
t
h
e
co
n
s
id
er
atio
n
o
f
t
h
e
l
ast
m
d
ig
it
s
o
f
p
a
n
d
q
.
I
n
f
a
ct,
af
ter
a
ll
o
f
t
h
e
m
ar
e
f
o
u
n
d
,
t
h
e
p
atter
n
s
o
f
t
h
e
last
m
d
ig
i
ts
o
f
p
+
q
a
n
d
p
-
q
ar
e
also
d
is
c
lo
s
ed
.
B
o
th
o
f
t
h
e
m
ar
e
t
h
e
k
e
y
s
to
esti
m
ate
th
e
n
e
w
in
itial
v
a
lu
es
f
o
r
th
i
s
m
et
h
o
d
.
Mo
r
eo
v
er
,
th
is
tech
n
iq
u
e
i
s
also
in
c
lu
d
ed
w
i
th
t
h
e
o
th
er
p
atter
n
o
f
p
+
q
th
at
th
e
r
esu
lt
o
f
(
p
+
q
)
m
o
d
8
is
al
w
a
y
s
e
q
u
al
to
0
w
h
en
t
h
en
r
es
u
lt
o
f
(
n
+
1
)
m
o
d
8
is
0
.
T
w
o
alg
o
r
it
h
m
s
ar
e
p
r
o
p
o
s
ed
in
th
i
s
p
ap
er
.
T
h
e
f
ir
s
t
is
ca
lled
A
l
g
o
r
ith
m
1
w
h
ic
h
ca
n
b
e
ap
p
lied
w
it
h
all
v
alu
e
s
o
f
n
.
Ho
w
ev
er
,
b
ef
o
r
e
u
s
i
n
g
th
i
s
alg
o
r
it
h
m
,
U
an
d
V
m
u
s
t
b
e
ca
lcu
lated
.
A
n
o
th
er
o
n
e
is
ca
lled
A
l
g
o
r
ith
m
2
.
T
h
is
al
g
o
r
ith
m
is
c
h
o
s
e
n
to
s
u
p
p
o
r
t
A
l
g
o
r
ith
m
1
w
h
e
n
t
h
e
r
es
u
lt
o
f
(
n
+
1
)
m
o
d
8
=
0.
T
h
e
ex
p
er
im
e
n
tal
r
esu
lts
s
h
o
w
t
h
at
if
(
n
+
1
)
m
o
d
8
=
0
,
th
e
d
ec
r
ea
s
in
g
lo
o
p
s
o
f
th
e
co
m
p
u
tat
io
n
b
y
u
s
in
g
th
e
co
m
b
i
n
atio
n
b
et
w
ee
n
Alg
o
r
ith
m
1
a
n
d
A
l
g
o
r
ith
m
2
ar
e
h
i
g
h
er
t
h
a
n
u
s
i
n
g
o
n
l
y
A
l
g
o
r
ith
m
1
ab
o
u
t
t
w
o
ti
m
e
s
.
F
u
r
th
o
r
m
o
r
e,
i
n
e
x
p
er
im
en
tal
r
es
u
lt
s
,
it
i
s
s
h
o
w
n
t
h
a
t
th
e
lo
o
p
s
ca
n
b
e
d
ec
r
ea
s
ed
2
6
%
in
th
e
e
x
a
m
p
le
o
f
2
5
6
b
its
-
len
g
t
h
o
f
n
w
h
e
n
t
h
e
d
if
f
er
en
ce
b
et
w
ee
n
p
r
i
m
e
f
ac
to
r
s
is
s
m
a
ll.
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
.
6
,
Dec
em
b
er
2
0
2
0
:
6
4
4
6
-
6
4
5
2
6452
RE
F
E
R
E
NC
E
S
[1
]
R.
L.
Riv
e
st,
e
t
a
l.
,
“
A
m
e
th
o
d
f
o
r
o
b
tain
in
g
d
ig
it
a
l
sig
n
a
tu
re
s
a
n
d
p
u
b
li
c
k
e
y
c
ry
p
to
sy
ste
m
s
,
”
Co
mm
u
n
ica
ti
o
n
s
o
f
ACM
,
v
o
l.
2
1
,
p
p
.
1
2
0
-
1
2
6
,
1
9
7
8
.
[2
]
W
.
Dif
f
ie
a
n
d
M.
He
ll
m
a
n
,
“
Ne
w
d
irec
ti
o
n
s
in
c
ry
p
to
g
ra
p
h
y
,
”
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
In
fo
rm
a
ti
o
n
T
h
e
o
ry
,
v
o
l.
2
2
,
n
o
.
6
,
p
p
.
6
4
4
-
6
5
4
,
1
9
7
6
.
[3
]
Y.
K
.
Ku
m
a
r
a
n
d
R.
M
.
S
h
a
f
i,
“
A
n
e
f
f
icie
n
t
a
n
d
se
c
u
re
d
a
ta
sto
ra
g
e
in
c
lo
u
d
c
o
m
p
u
ti
n
g
u
sin
g
m
o
d
if
ied
RS
A
p
u
b
li
c
k
e
y
c
ry
p
to
sy
ste
m
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
tr
ica
l
a
n
d
Co
mp
u
ter
En
g
in
e
e
rin
g
(
IJ
ECE
)
,
v
o
l.
1
0
,
n
o
.
1
,
p
p
.
5
3
0
-
5
3
7
,
2
0
2
0
.
[4
]
M.
J.
W
ien
e
r,
“
Cr
y
p
tan
a
l
y
sis
o
f
sh
o
rt
RS
A
se
c
r
e
t
e
x
p
o
n
e
n
ts
,
”
IE
EE
T
ra
n
sa
c
ti
o
n
s
o
n
In
f
o
rm
a
ti
o
n
T
h
e
o
ry
,
v
o
l.
3
6
,
n
o
.
6
,
p
p
.
5
5
3
-
5
5
8
,
1
9
9
0
.
[5
]
D.
Bo
n
e
h
a
n
d
G
.
Du
rf
e
e
,
“
Cr
y
p
ta
n
a
ly
sis
o
f
RS
A
w
it
h
P
r
iv
a
te
Ke
y
d
les
s
th
a
n
N
0.
292
,
”
I
n
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
th
e
T
h
e
o
ry
a
n
d
A
p
p
l
ica
ti
o
n
s
o
f
C
ry
p
to
g
ra
p
h
ic T
e
c
h
n
i
q
u
e
s
,
p
p
.
1
-
1
1
,
1
9
9
9
.
[6
]
M.
E.
W
u
,
e
t
a
l.
,
“
On
th
e
Im
p
ro
v
e
m
e
n
t
o
f
W
ien
e
r
A
tt
a
c
k
o
n
RS
A
w
it
h
S
m
a
ll
P
riv
a
te
Ex
p
o
n
e
n
t
,
”
T
h
e
S
c
ien
ti
fi
c
W
o
rld
J
o
u
r
n
a
l
,
v
o
l.
2
0
1
4
,
p
p
.
1
-
9
,
2
0
1
4
.
[7
]
K.
S
o
m
su
k
,
“
T
h
e
N
e
w
Eq
u
a
ti
o
n
f
o
r
RS
A
'
s
De
c
r
y
p
ti
o
n
P
ro
c
e
ss
A
p
p
ro
p
riate
w
it
h
Hig
h
P
riv
a
te
Ke
y
E
x
p
o
n
e
n
t
,
”
Pro
c
e
e
d
in
g
s
o
f
I
n
ter
n
a
ti
o
n
a
l
Co
mp
u
ter
S
c
ien
c
e
a
n
d
En
g
i
n
e
e
rin
g
Co
n
fer
e
n
c
e
,
p
p
.
1
-
5
,
2
0
1
7
.
[8
]
M
.
S
a
h
i
n
,
“
G
e
n
e
ra
li
z
e
d
T
rial
D
iv
isio
n
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
Co
n
tem
p
o
ra
ry
M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
,
v
o
l.
6
,
n
o
.
2
,
p
p
.
5
9
-
6
4
,
2
0
1
1
.
[9
]
N.
L
a
l,
e
t
a
l.
,
“
M
o
d
if
ied
tri
a
l
d
iv
isio
n
a
lg
o
rit
h
m
u
sin
g
KN
J
-
f
a
c
to
riza
ti
o
n
m
e
th
o
d
t
o
f
a
c
to
rize
RS
A
p
u
b
li
c
k
e
y
e
n
c
r
y
p
ti
o
n
,
”
Pr
o
c
e
e
d
in
g
s
of
IE
EE
In
ter
n
a
ti
o
n
a
l
C
o
n
fer
e
n
c
e
o
n
Co
n
tem
p
o
ra
ry
Co
mp
u
ti
n
g
a
n
d
In
fo
rm
a
ti
c
s
,
p
p
.
9
9
2
-
9
9
5
,
2
0
1
4
.
[1
0
]
S
.
S
a
rn
a
ik
,
e
t
a
l.
,
“
Co
m
p
a
ra
ti
v
e
stu
d
y
o
n
In
teg
e
r
F
a
c
to
riza
ti
o
n
A
lg
o
rit
h
m
-
P
o
ll
a
rd
'
s
RHO
a
n
d
P
o
ll
a
rd
'
s
P
-
1
,
”
Pro
c
e
e
d
in
g
s
o
f
t
h
e
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
Co
mp
u
ti
n
g
fo
r
S
u
sta
in
a
b
le
Glo
b
a
l
De
v
e
lo
p
me
n
t
,
p
p
.
6
7
7
-
6
7
9
,
2
0
1
5
.
[1
1
]
M.
E.
W
u
,
e
t
a
l.
,
“
On
th
e
im
p
ro
v
e
m
e
n
t
o
f
F
e
rm
a
t
f
a
c
to
riza
ti
o
n
u
sin
g
a
c
o
n
ti
n
u
e
d
f
ra
c
ti
o
n
tec
h
n
iq
u
e
,
”
Fu
tu
re
Ge
n
e
ra
ti
o
n
Co
mp
u
ter
S
y
ste
ms
,
v
o
l.
3
0
,
n
o
.
1
,
p
p
.
162
-
1
6
8
,
2
0
1
4
.
[1
2
]
K.
Om
a
r
a
n
d
L
.
S
z
a
lay
,
“
S
u
f
fi
c
ien
t
c
o
n
d
it
io
n
s
f
o
r
f
a
c
to
rin
g
a
c
las
s
o
f
larg
e
in
teg
e
rs
,
”
J
o
u
rn
a
l
o
f
In
ter
d
isc
ip
li
n
a
ry
M
a
th
e
ma
ti
c
a
l
,
v
o
l.
1
3
,
n
o
.
1
,
p
p
.
9
5
-
1
0
3
,
2
0
1
0
.
[1
3
]
S
.
V
y
n
n
y
c
h
u
k
,
e
t
a
l.
,
“
A
p
p
li
c
a
ti
o
n
o
f
th
e
b
a
sic
m
o
d
u
le'
s
f
o
u
n
d
a
ti
o
n
f
o
r
f
a
c
to
riza
ti
o
n
o
f
b
ig
n
u
m
b
e
rs
b
y
th
e
f
е
rm
а
t
m
e
th
o
d
,
”
Ea
ste
rn
Eu
ro
p
e
a
n
J
o
u
rn
a
l
o
f
En
ter
p
rise
T
e
c
h
n
o
lo
g
ies
,
v
o
l.
6
,
p
p
.
1
4
-
2
3
,
2
0
1
8
.
[1
4
]
J.
M
c
k
e
e
,
“
S
p
e
e
d
in
g
F
e
rm
a
t’s
f
a
c
to
rin
g
m
e
th
o
d
,
”
M
a
th
e
ma
ti
c
s
o
f
Co
mp
u
ta
ti
o
n
,
v
o
l.
6
8
,
n
o
.
2
2
8
,
p
p
.
1
7
2
9
-
1
7
3
7
,
1999.
[1
5
]
P
.
S
h
a
rm
a
,
e
t
a
l.
,
“
No
ti
c
e
o
f
V
io
latio
n
o
f
IEE
E
P
u
b
li
c
a
ti
o
n
P
rin
c
i
p
les
:
M
o
d
if
ied
In
teg
e
r
F
a
c
to
riza
ti
o
n
A
lg
o
rit
h
m
Us
in
g
V
-
F
a
c
to
r
M
e
th
o
d
,
”
Pro
c
e
e
d
in
g
s
o
f
In
ter
n
a
ti
o
n
a
l
C
o
n
fer
e
n
c
e
o
n
Ad
v
a
n
c
e
d
Co
m
p
u
ti
n
g
&
Co
mm
u
n
ica
ti
o
n
T
e
c
h
n
o
l
o
g
ies
,
p
p
.
4
2
3
-
4
2
5
,
2
0
1
2
.
[1
6
]
D.
P
h
iam
p
h
u
a
n
d
P
.
S
a
h
a
,
“
Re
d
e
sig
n
e
d
th
e
A
rc
h
it
e
c
tu
re
o
f
Ex
ten
d
e
d
-
Eu
c
li
d
e
a
n
A
lg
o
rit
h
m
f
o
r
M
o
d
u
lar
M
u
lt
ip
li
c
a
ti
v
e
In
v
e
rse
a
n
d
Ja
c
o
b
i
S
y
m
b
o
l
,
”
Pro
c
e
e
d
in
g
s
o
f
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
T
re
n
d
s
in
El
e
c
tro
n
ics
a
n
d
In
fo
rm
a
ti
c
s
,
p
p
.
1
3
4
5
-
1349
,
2018.
[1
7
]
J
.
Zh
o
u
,
e
t
a
l.
,
“
Ex
ten
d
e
d
Eu
c
li
d
a
lg
o
ri
th
m
a
n
d
i
ts
a
p
p
li
c
a
ti
o
n
in
RS
A
,
”
Pro
c
e
e
d
in
g
s
o
f
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
In
fo
rm
a
ti
o
n
S
c
ien
c
e
a
n
d
En
g
in
e
e
rin
g
,
p
p
.
2
0
7
9
-
2
0
8
1
,
2010.
[1
8
]
M.
M
.
A
sa
d
,
e
t
a
l.
,
“
P
e
rf
o
rm
a
n
c
e
A
n
a
ly
sis
o
f
1
2
8
-
b
it
M
o
d
u
lar
In
v
e
rse
Ba
se
d
Ex
ten
d
e
d
Eu
c
li
d
e
a
n
Us
in
g
A
lt
e
ra
F
P
G
A
Kit
,
”
Pro
c
e
d
ia
Co
mp
u
ter
S
c
ien
c
e
,
v
o
l.
1
6
0
,
p
p
.
5
4
3
-
5
4
8
,
2
0
1
9
.
[1
9
]
Q.
Zh
o
u
,
e
t
a
l.
,
“
Ho
w
to
se
c
u
re
ly
o
u
tso
u
rc
e
th
e
e
x
ten
d
e
d
e
u
c
li
d
e
a
n
a
lg
o
rit
h
m
f
o
r
larg
e
-
sc
a
le
p
o
ly
n
o
m
ials
o
v
e
r
f
in
it
e
f
ield
s
,
”
In
fo
rm
a
ti
o
n
S
c
ien
c
e
s
,
v
o
l.
5
1
2
,
p
p
.
6
4
1
-
6
6
0
,
2
0
2
0
.
[2
0
]
K.
S
o
m
su
k
a
n
d
S
.
Ka
se
m
v
il
a
s,
“
M
V
F
a
c
to
r:
A
m
e
th
o
d
to
d
e
c
re
a
se
p
ro
c
e
ss
in
g
ti
m
e
f
o
r
f
a
c
to
riza
ti
o
n
a
lg
o
rit
h
m
,
”
Pro
c
e
e
d
in
g
s o
f
In
ter
n
a
ti
o
n
a
l
Co
mp
u
ter
S
c
ien
c
e
a
n
d
En
g
in
e
e
rin
g
Co
n
fer
e
n
c
e
,
p
p
.
3
3
9
-
342,
2013
.
[2
1
]
K.
S
o
m
su
k
,
“
M
V
F
a
c
to
rV
2
:
A
n
im
p
ro
v
e
d
in
teg
e
r
f
a
c
to
riza
ti
o
n
a
lg
o
rit
h
m
to
sp
e
e
d
u
p
c
o
m
p
u
t
a
ti
o
n
ti
m
e
,
”
Pro
c
e
e
d
in
g
s o
f
In
ter
n
a
ti
o
n
a
l
Co
mp
u
ter
S
c
ien
c
e
a
n
d
En
g
in
e
e
rin
g
Co
n
fer
e
n
c
e
,
p
p
.
3
0
8
-
311
,
2
0
1
4
.
[2
2
]
K.
S
o
m
su
k
a
n
d
K.
T
ien
tan
o
p
a
jai,
“
A
n
Im
p
ro
v
e
m
e
n
t
o
f
F
e
rm
a
t'
s
F
a
c
to
riza
ti
o
n
b
y
Co
n
sid
e
rin
g
th
e
L
a
st
m
Dig
it
s
o
f
M
o
d
u
lu
s
to
De
c
re
a
se
Co
m
p
u
tatio
n
T
im
e
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Ne
two
rk
S
e
c
u
rity
,
v
o
l.
1
9
,
n
o
.
1
,
p
p
.
9
9
-
1
1
1
,
2
0
1
7
.
[2
3
]
K.
S
o
m
su
k
,
“
T
h
e
i
m
p
ro
v
e
m
e
n
t
o
f
in
it
ial
v
a
lu
e
c
lo
se
r
to
t
h
e
targ
e
t
f
o
r
F
e
r
m
a
t’s
f
a
c
to
riza
ti
o
n
a
lg
o
rit
h
m
,
”
J
o
u
rn
a
l
o
f
Disc
re
te M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
s a
n
d
Cry
p
t
o
g
r
a
p
h
y
,
v
o
l.
2
1
,
n
o
.
7
-
8
,
p
p
.
1
5
7
3
-
1
5
8
0
,
2
0
1
8
.
[2
4
]
K.
S
o
m
su
k
,
e
t
a
l.
,
“
Esti
m
a
ti
n
g
th
e
n
e
w In
it
ial
V
a
lu
e
o
f
T
rial
Di
v
is
io
n
A
lg
o
rit
h
m
f
o
r
Ba
lan
c
e
d
M
o
d
u
lu
s to
De
c
re
a
se
Co
m
p
u
tatio
n
L
o
o
p
s
,
”
Pro
c
e
e
d
i
n
g
s
o
f
I
n
ter
n
a
ti
o
n
a
l
J
o
i
n
t
C
o
n
fer
e
n
c
e
o
n
Co
mp
u
ter
S
c
ie
n
c
e
a
n
d
S
o
f
twa
re
En
g
i
n
e
e
rin
g
,
p
p
.
1
4
3
-
1
4
7
,
2
0
1
9
.
[2
5
]
Y.
B.
Ha
m
m
a
d
,
e
t
a
l.
,
“
RAK
f
a
c
to
rin
g
a
lg
o
rit
h
m
,
”
Au
stra
la
si
a
n
J
o
u
rn
a
l
o
f
C
o
mb
i
n
a
to
ric
s
,
v
o
l.
33
,
n
o
.
1,
pp.
2
9
1
-
3
0
5
,
2
0
0
5
.
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