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urna
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f
E
lect
rica
l En
g
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a
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Co
m
p
u
t
er
Science
Vo
l.
21
,
No
.
2
,
Feb
r
u
ar
y
2
0
2
1
,
p
p
.
7
3
5
~
7
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SS
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b
d
-
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elec
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ed
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m
1.
I
NT
RO
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UCT
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O
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C
r
y
p
to
g
r
ap
h
y
i
s
a
n
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m
p
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ta
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t
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r
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n
t
h
e
f
ield
o
f
i
n
f
o
r
m
atio
n
s
ec
u
r
it
y
[
1
]
.
T
h
e
m
o
s
t
co
m
m
o
n
l
y
u
s
ed
p
u
b
lic
-
k
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al
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ith
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s
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c
h
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Di
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A
l
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ith
m
(
DS
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a
n
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llip
tical
C
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p
to
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C
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p
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y
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m
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lar
m
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lt
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licatio
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[
2
]
.
T
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r
e,
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cr
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to
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ap
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s
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er
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lar
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lier
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is
th
e
w
id
el
y
u
s
ed
m
o
d
u
lar
m
u
lt
ip
lier
[
3
]
.
MM
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is
an
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f
ec
tiv
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tec
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n
iq
u
e
f
o
r
im
p
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m
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tin
g
th
e
m
o
d
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lar
m
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l
tip
licatio
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w
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th
lar
g
e
o
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d
s
in
h
i
g
h
-
p
er
f
o
r
m
a
n
ce
h
ar
d
w
ar
e
[
4
,
5]
.
Fo
r
s
ec
u
r
it
y
s
c
h
e
m
es
w
h
ic
h
ar
e
b
ased
o
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p
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k
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p
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it
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i
m
p
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t
an
t
to
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ar
d
w
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les
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a
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ce
.
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h
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s
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p
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s
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f
p
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lic
k
e
y
cr
y
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o
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est
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ith
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ct
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r
es
(
f
in
ite
f
ield
s
an
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g
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o
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p
s
)
[
2
]
.
Fu
r
t
h
er
m
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r
e,
th
e
s
e
o
p
er
atio
n
s
ar
e
co
n
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ted
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n
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m
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1
6
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2
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)
,
w
h
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m
a
k
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th
e
m
co
n
s
id
er
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l
y
ti
m
e
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s
u
m
i
n
g
o
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s
.
T
h
is
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s
s
u
e
h
as
m
o
ti
v
ated
th
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r
e
s
ea
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ch
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to
m
o
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e
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p
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o
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ac
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ler
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t
h
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co
m
p
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tatio
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t
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m
e
as th
e
f
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ca
l d
esig
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w
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m
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t th
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h
h
ar
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w
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r
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r
ce
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n
s
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m
p
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Ho
w
e
v
er
,
a
m
aj
o
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r
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b
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w
it
h
th
i
s
k
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n
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p
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s
is
t
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at
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th
e
e
m
b
ed
d
ed
s
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te
m
s
r
eq
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a
f
e
w
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ar
d
w
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r
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s
.
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h
er
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o
r
e,
th
is
asp
ec
t
is
o
n
e
o
f
th
e
g
r
ea
test
ch
a
llen
g
es
o
f
i
m
p
le
m
e
n
tatio
n
o
f
th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
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J
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C
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m
p
Sci,
Vo
l.
21
,
No
.
2
,
Feb
r
u
ar
y
2
0
2
1
:
7
3
5
-
7
4
3
736
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h
t
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ei
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cr
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p
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s
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te
m
s
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h
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lu
tio
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ly
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Field
P
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m
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Gate
A
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FP
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FP
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As
ar
e
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ailab
le
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t
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a
n
d
ar
e
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p
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ted
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o
m
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lar
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k
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Se
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Net
w
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I
n
ter
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t o
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T
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in
g
s
(
I
o
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)
[6
-
9]
.
T
h
is
r
esear
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p
r
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ts
a
n
e
w
a
p
p
r
o
ac
h
to
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o
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ith
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is
a
m
o
d
if
ica
tio
n
o
f
t
h
e
r
ad
ix
-
2
MM
M
s
tr
u
ct
u
r
e
to
im
p
r
o
v
e
th
e
ar
ea
an
d
th
e
ef
f
icie
n
c
y
,
n
o
s
u
b
tr
ac
tio
n
o
p
er
atio
n
s
ar
e
p
er
f
o
r
m
ed
.
T
h
e
p
r
ev
io
u
s
m
o
d
if
icat
io
n
o
f
M
M
M
to
d
is
ca
r
d
th
e
f
i
n
al
s
u
b
tr
ac
ti
o
n
p
r
esen
ted
a
n
e
w
s
et
o
f
p
ar
a
m
e
ter
s
an
d
m
o
r
e
c
y
cles
[
1
0
,
11]
.
T
h
is
m
e
th
o
d
i
n
v
o
l
v
ed
a
ca
lcu
latio
n
o
f
t
h
e
Mo
n
tg
o
m
er
y
p
r
o
d
u
ct
w
it
h
lo
n
g
er
o
p
er
an
d
s
an
d
m
o
r
e
iter
atio
n
s
,
w
h
ich
co
u
ld
r
ed
u
ce
p
er
f
o
r
m
an
ce
s
er
io
u
s
l
y
.
O
u
r
w
o
r
k
i
m
p
o
s
e
s
a
tig
h
ter
b
o
u
n
d
o
n
p
r
ev
io
u
s
a
s
s
u
m
p
tio
n
s
,
w
it
h
n
o
i
n
cr
ea
s
e
i
n
o
p
er
an
d
s
ize
o
r
n
u
m
b
er
o
f
iter
at
io
n
s
,
w
h
ich
e
n
ab
les
u
s
to
ad
v
a
n
ce
th
e
h
ar
d
w
ar
e
i
m
p
le
m
en
ta
t
io
n
e
f
f
icien
c
y
.
T
h
e
co
m
p
ac
t
s
tr
u
ct
u
r
e
o
f
t
h
e
p
r
o
p
o
s
ed
d
esig
n
o
v
er
t
h
e
o
th
er
co
m
p
etiti
v
e
d
esig
n
s
is
a
p
p
r
o
p
r
iate
f
o
r
em
b
ed
d
ed
s
y
s
te
m
s
an
d
h
ar
d
w
ar
e
ap
p
licatio
n
s
o
f
th
e
I
o
T
d
ev
ices.
T
h
e
p
r
o
p
o
s
ed
d
esig
n
h
a
s
b
ee
n
co
d
e
d
in
VHDL
,
an
d
tar
g
eted
Vir
tex
-
6
FP
GA
p
latf
o
r
m
.
T
h
e
p
r
o
p
o
s
ed
d
esig
n
h
as
b
ee
n
s
y
n
th
e
s
ized
u
tili
zi
n
g
Xilin
x
I
SE,
a
n
d
s
i
m
u
lated
u
ti
lizin
g
Mo
d
elSi
m
.
T
h
e
n
e
w
ap
p
r
o
ac
h
v
ar
io
u
s
b
it
-
len
g
th
(
1
6
0
-
1
0
2
4
)
h
as
b
ee
n
co
m
p
ar
ed
w
it
h
t
h
e
r
elate
d
w
o
r
k
s
in
ter
m
s
o
f
ar
ea
i
n
S
lice
L
UT
s
,
f
r
eq
u
e
n
c
y
i
n
MH
z
,
ti
m
e
in
,
th
r
o
u
g
h
p
u
t
in
Mb
p
s
,
A
r
ea
-
T
i
m
e
an
d
ef
f
ic
ien
c
y
(
Mb
p
s
p
er
FP
GA
L
UT
)
.
A
co
m
p
ar
is
o
n
o
f
th
e
r
esu
l
ts
r
ev
ea
ls
t
h
e
i
m
p
r
o
v
e
m
e
n
t o
f
t
h
e
u
tili
ze
d
h
ar
d
w
ar
e
r
eso
u
r
ce
s
o
f
th
e
p
r
o
p
o
s
ed
d
esig
n
o
v
er
th
e
r
elate
d
w
o
r
k
s
.
T
h
e
r
est
o
f
th
e
p
ap
er
is
o
r
g
an
i
ze
d
as
f
o
llo
w
s
.
Sectio
n
2
ex
p
l
o
r
er
s
th
e
class
ical
al
g
o
r
ith
m
o
f
Mo
d
u
lar
Mu
ltip
lier
R
ed
u
ctio
n
,
t
h
e
o
r
ig
i
n
al
MM
M,
an
d
r
ad
ix
-
2
MM
M
.
Sectio
n
3
d
escr
ib
es
th
e
al
g
o
r
ith
m
,
t
h
e
h
ar
d
w
ar
e
i
m
p
le
m
en
ta
tio
n
,
an
d
th
e
b
o
u
n
d
s
o
n
th
e
I
n
p
u
ts
/O
u
tp
u
ts
(
I
/O)
o
f
th
e
p
r
o
p
o
s
ed
d
esig
n
.
Sec
tio
n
4
p
r
o
v
id
es
th
e
s
i
m
u
lat
io
n
a
n
d
th
e
s
y
n
t
h
esi
s
r
esu
lt
s
o
f
t
h
e
i
m
p
le
m
e
n
tati
o
n
o
f
t
h
e
p
r
o
p
o
s
ed
d
esig
n
.
T
h
e
b
o
u
n
d
s
o
n
th
e
I
n
p
u
ts
/Ou
tp
u
ts
(
I
/O)
o
f
t
h
e
p
r
o
p
o
s
ed
d
esig
n
h
as
b
ee
n
an
al
y
z
ed
.
T
h
e
s
i
m
u
latio
n
o
f
t
h
e
p
r
o
p
o
s
ed
d
esig
n
an
d
t
h
e
ac
cu
r
ac
y
o
f
t
h
e
d
esi
g
n
h
as
b
ee
n
v
er
if
ied
u
s
in
g
Mo
d
elSi
m
.
T
h
e
s
y
n
t
h
esi
s
r
es
u
lts
o
f
i
m
p
le
m
en
ti
n
g
th
e
p
r
o
p
o
s
ed
d
esig
n
o
n
XI
L
I
NX
Vir
tex
-
6
F
P
GA
h
a
v
e
b
ee
n
p
r
o
v
id
ed
u
tili
zin
g
Xi
lin
x
I
SE
.
Fin
all
y
,
Secti
o
n
5
co
n
clu
d
es
o
u
r
co
n
tr
ib
u
tio
n
w
o
r
k
.
2.
B
ACK
G
RO
UND
2
.
1
.
Cla
s
s
ica
l
m
o
du
la
r
m
u
lt
i
pli
ca
t
io
n
A
l
g
o
r
ith
m
1
is
a
s
tr
aig
h
t
f
o
r
w
ar
d
alg
o
r
ith
m
to
co
m
p
u
te
t
h
e
m
o
d
u
lar
r
ed
u
ctio
n
o
f
t
w
o
m
u
ltip
lied
in
te
g
er
s
,
B
.
T
h
e
f
ir
s
t
s
tep
i
s
o
b
tain
in
g
th
e
p
r
o
d
u
ct
α
o
f
th
e
in
teg
er
n
u
m
b
er
s
.
T
h
e
r
ed
u
ctio
n
m
o
d
u
lo
s
tep
u
s
u
all
y
in
v
o
lv
e
s
a
d
iv
is
io
n
o
p
er
atio
n
o
f
α
b
y
th
e
m
o
d
u
lu
s
,
q
is
th
e
q
u
o
tien
t.
C
o
m
p
u
t
a
tio
n
o
f
th
e
q
u
o
tie
n
t
an
d
r
e
m
ai
n
d
er
w
h
e
n
is
d
iv
id
ed
b
y
i
s
d
e
m
o
n
s
tr
ated
i
n
[
1
2
]
.
T
h
e
th
ir
d
s
tep
is
o
b
tain
i
n
g
t
h
e
r
esid
u
e
o
f
th
e
d
iv
i
s
io
n
o
p
er
atio
n
as
t
h
e
r
esu
lt
o
f
m
o
d
u
lar
m
u
l
tip
licatio
n
r
ed
u
ctio
n
.
I
t
is
a
v
er
y
ti
m
e
-
co
n
s
u
m
in
g
o
p
er
atio
n
o
n
b
o
th
h
ar
d
w
ar
e
a
n
d
s
o
f
t
w
ar
e
p
latf
o
r
m
s
.
Algorithm
-
1
Classical Modular Multiplier
INPUT: Integers (
,
ℬ
,
)
OUTPUT:
=
×
ℬ
.
1.
=
×
ℬ
;
2.
=
/
=
+
;
3. Return (
).
2
.
2
.
M
o
ntg
o
m
er
y
m
o
du
la
r
m
u
lt
ipl
ier
a
lg
o
rit
h
m
Mo
n
tg
o
m
er
y
i
n
tr
o
d
u
ce
d
a
tech
n
iq
u
e
to
a
v
o
id
th
e
d
iv
i
s
io
n
p
r
o
ce
s
s
f
o
r
th
e
m
o
d
u
lar
r
ed
u
ctio
n
o
f
p
r
o
d
u
ct
o
f
t
w
o
in
te
g
er
s
[
3
]
.
Fo
r
th
e
th
r
ee
in
teg
er
s
(
,
ℬ
,
)
,
Mo
n
tg
o
m
er
y
s
u
b
s
tit
u
ted
th
e
d
iv
is
io
n
b
y
m
o
d
u
l
u
s
w
it
h
th
e
d
iv
is
io
n
b
y
m
o
d
u
l
u
s
.
T
h
e
Mo
n
tg
o
m
er
y
ap
p
r
o
ac
h
to
ca
lc
u
late
t
h
e
r
ed
u
ctio
n
p
r
o
d
u
ct
o
f
m
o
d
u
lo
f
o
r
t
w
o
in
te
g
er
s
an
d
ℬ
is
lis
ted
:
a)
C
h
o
o
s
i
n
g
R
>
.
R
i
s
a
n
in
teg
r
al
p
o
w
er
o
f
2
.
T
h
e
in
te
g
er
m
u
s
t
b
e
o
d
d
to
s
atis
f
y
th
e
co
n
d
itio
n
(
,
)
=
1
.
b)
P
r
ec
o
m
p
u
te
t
h
e
in
teg
er
s
−
1
̀
s
u
c
h
th
at:
−
1
−
̀
=
1
(
1
)
−
1
≡
1
(
2
)
̀
=
−
−
1
(
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4752
A
co
mp
a
ct
F
P
GA
-
b
a
s
ed
mo
n
tg
o
mery
mo
d
u
la
r
mu
ltip
lier
(
A
h
med
A
.
H.
A
b
d
-
E
lka
d
er
)
737
c)
T
r
an
s
f
o
r
m
th
e
o
p
er
an
d
to
its
Mo
n
tg
o
m
er
y
d
o
m
ai
n
,
w
h
ich
i
s
k
n
o
w
n
al
s
o
w
i
th
f
u
n
ctio
n
[
3
]
.
(
,
)
=
×
×
−
1
(
4
)
̅
=
(
,
2
)
=
×
(
5
)
ℬ
̅
=
(
ℬ
,
2
)
=
ℬ
×
(
6
)
Giv
i
n
g
th
e
p
r
o
d
u
ct
in
Mo
n
tg
o
m
er
y
d
o
m
ai
n
s
u
c
h
th
at
:
̅
=
(
̅
,
ℬ
̅
)
=
×
ℬ
×
R
(
7
)
If
̅
>
the
n
̅
=
̅
−
(
8
)
T
h
e
p
r
ev
io
u
s
s
tep
is
ex
p
en
s
i
v
e
d
u
e
to
its
r
ed
u
ctio
n
m
o
d
u
lo
.
T
h
er
ef
o
r
e,
Mo
n
tg
o
m
er
y
ap
p
lied
a
m
o
r
e
ef
f
icien
t
w
a
y
to
ca
lc
u
late
it.
(
,
)
=
(
+
̀
)
)
(
9
)
w
h
er
e
=
×
.
T
h
is
eq
u
atio
n
is
ap
p
lied
f
o
r
̅
,
ℬ
̅
,
̅
.
T
o
r
ev
ea
l
th
e
f
i
n
al
r
e
s
u
lt,
it
i
s
n
ec
es
s
ar
y
to
in
v
er
s
e
tr
an
s
f
o
r
m
atio
n
o
f
t
h
e
Mo
n
t
g
o
m
er
y
d
o
m
ai
n
to
its
o
r
ig
in
al
f
o
r
m
:
(
̅
,
1
)
=
(
̅
+
̅
̀
)
)
(
1
0
)
T
h
e
p
r
ec
ed
in
g
ap
p
r
o
ac
h
is
s
l
o
w
b
ec
au
s
e
o
f
a
lo
t
o
f
ad
d
itio
n
an
d
m
u
l
tip
licatio
n
o
p
er
ati
o
n
s
.
Mo
r
e
ef
f
icien
t
ap
p
r
o
ac
h
is
d
em
o
n
s
tr
ated
in
A
l
g
o
r
it
h
m
-
2
,
th
e
al
g
o
r
ith
m
Mo
n
t
g
o
m
er
y
Mo
d
u
lar
Mu
ltip
lier
(
d
en
o
ted
as
MM
M
alg
o
r
ith
m
)
.
Algorithm
-
2
Montgomery Modular Multiplier (MMM
)
Input: Integers (
,
ℬ
,
)
[
b
i
t
s
]
radix
representation
whe
r
e
0
≤
(
,
ℬ
)
<
,
R
=
k
,
(
,
)
=
1
,
̀
=
−
−
1
m
o
d
R
Output
:
=
×
ℬ
×
R
−
1
1.
=
0
;
2.
F
or
(
f
r
o
m
0
to
k
−
1
,
=
+
1
)
do
3
.
=
(
(
0
+
×
ℬ
0
)
×
̀
)
Mod
r
;
4.
=
(
+
×
ℬ
+
×
)
/
r
;
5.
Loop;
6.
If
>
t
he
n
=
−
;
end If
7. Return(
).
T
h
e
n
e
w
ap
p
r
o
ac
h
u
tili
ze
s
th
e
f
u
n
c
tio
n
f
o
r
t
w
o
in
te
g
er
o
p
e
r
an
d
s
,
ℬ
d
i
r
ec
tly
.
w
h
er
e
=
×
ℬ
.
=
(
,
ℬ
)
=
(
+
̀
)
)
(
1
1
)
Fo
r
MM
M
alg
o
r
ith
m
t
h
er
e
ar
e
iter
atio
n
s
,
w
h
er
e
is
t
h
e
b
it
len
g
th
o
f
t
h
e
m
o
d
u
lu
s
.
T
h
e
m
u
ltip
lie
r
b
its
ar
e
s
ca
n
n
ed
f
r
o
m
L
SB
to
MSB
,
0
ℬ
0
ar
e
th
e
L
SB
s
o
f
ℬ
r
esp
ec
tiv
el
y
.
T
h
e
s
tep
s
3
an
d
4
is
r
ep
ea
ted
f
o
r
ev
er
y
iter
atio
n
f
o
r
its
co
r
r
esp
o
n
d
in
g
an
d
f
o
r
its
ac
cu
m
u
lated
ac
co
r
d
in
g
to
th
e
s
tep
3
r
esu
lt 1
o
r
0
; m
o
d
u
l
u
s
ad
d
ed
o
r
n
o
t
to
s
tep
4
.
T
o
g
et
th
e
v
al
id
atio
n
r
esu
lt th
e
MM
M
al
g
o
r
it
h
m
i
s
u
s
ed
o
n
ce
m
o
r
e.
(
,
1
)
=
(
+
̀
)
)
(
1
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
2
-
4752
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
21
,
No
.
2
,
Feb
r
u
ar
y
2
0
2
1
:
7
3
5
-
7
4
3
738
Algorithm
-
3 Radix
-
2
(MMM)
Input: Integers (
,
ℬ
,
)
[
]
radix
-
2
representation.
whe
r
e
0
≤
(
,
ℬ
)
<
,
R
=
2
k
,
(
,
2
)
=
1
,
Output
:
=
×
ℬ
×
2
−
1.
=
0
;
2
.
or
(
f
r
o
m
0
to
k
−
1
,
=
+
1
)
do
3
.
=
(
0
+
×
ℬ
0
)
m
o
d
2
;
4.
=
(
+
×
ℬ
+
×
)
/
2
;
5.
Loop;
6.
If
>
t
he
n
=
−
;
end If
7. Return(
).
A
l
g
o
r
ith
m
-
3
d
e
m
o
n
s
tr
ates
th
e
R
ad
ix
-
2
v
er
s
io
n
o
f
MM
M,
w
h
er
e
=
2
,
[
1
3
,
14]
.
T
h
er
ef
o
r
,
̀
=
1
.
T
h
e
d
iv
is
io
n
b
y
2
in
s
tep
-
4
is
a
s
im
p
le
o
n
e
-
b
it
r
ig
h
t
s
h
if
t
o
p
er
atio
n
.
T
h
er
ef
o
r
e,
f
o
r
ea
ch
iter
atio
n
lo
o
p
tw
o
ad
d
itio
n
s
ar
e
r
eq
u
ir
ed
an
d
th
e
f
i
n
al
s
u
b
tr
ac
tio
n
(
s
tep
8
)
.
3.
RE
S
E
ARCH
M
E
T
H
O
D
3
.
1
.
P
r
o
po
s
ed
a
lg
o
rit
h
m
o
f
t
he
m
o
du
la
r
m
u
lt
ipl
ier
T
h
e
m
o
d
if
ied
MMM
R
ad
ix
-
2
m
o
d
u
lar
m
u
lt
ip
licatio
n
alg
o
r
ith
m
i
s
p
r
esen
ted
in
A
lg
o
r
it
h
m
-
4
.
T
h
is
alg
o
r
ith
m
is
t
h
e
m
o
d
i
f
icatio
n
o
f
A
l
g
o
r
it
h
m
-
3
w
it
h
o
u
t t
h
e
f
i
n
al
s
u
b
tr
ac
tio
n
s
tep
,
an
d
w
it
h
a
m
o
d
i
f
icat
io
n
o
f
t
h
e
co
n
s
tr
u
ct
io
n
to
r
ed
u
ce
t
h
e
ar
ea
an
d
en
h
an
ce
t
h
e
e
f
f
icie
n
c
y
.
I
n
A
l
g
o
r
ith
m
-
3
,
th
e
m
u
ltip
lier
co
r
r
esp
o
n
d
in
g
b
it
is
m
u
l
tip
lied
b
y
t
h
e
m
u
ltip
lica
n
d
L
e
s
t
Sig
n
i
f
ica
n
t B
it
(
L
SB
)
ℬ
0
in
s
tep
-
3
,
a
n
d
b
y
t
h
e
m
u
ltip
lic
an
d
ℬ
in
s
tep
-
4
.
In
Alg
o
r
it
h
m
-
3
,
th
e
m
u
ltip
lica
tio
n
in
s
tep
-
3
is
A
N
D
o
p
er
atio
n
,
an
d
in
s
tep
-
4
is
a
M
u
ltip
le
x
er
.
Algorithm
-
4 Proposed Modular Multiplier
Input: Integers (
,
ℬ
,
)
[
]
radix
2
representation.
ℎ
0
≤
(
,
ℬ
)
<
,
gc
d
(
,
2
)
=
1
.
Output
:
=
×
ℬ
×
2
−
1.
α
= 0;
2.
(
0
−
1
,
=
+
1
)
3. If
= '1' then
γ
=
ℬ
;
else
γ
= 0;
End If;
4. If
α
1
γ
0
= '1' then
Ȥ =
γ
+
;
else
Ȥ =
γ
;
End If;
5.
α = Ȥ +
;
6.
= α >>
1
;
7. Loop;
8.
=
9. Return
Nev
er
th
e
less
,
in
s
tep
-
3
o
f
A
l
g
o
r
ith
m
-
4
t
h
e
v
al
u
e
o
f
u
tili
ze
d
o
n
l
y
o
n
ce
an
d
s
elec
t
b
et
w
e
en
t
w
o
v
alu
e
s
; t
h
e
m
u
ltip
lica
n
d
v
al
u
e
ℬ
o
r
th
e
ze
r
o
v
alu
e.
T
h
e
f
i
n
al
s
u
b
tr
ac
tio
n
i
n
A
l
g
o
r
ith
m
-
3
is
a
s
o
u
r
ce
o
f
lea
k
a
g
e
an
d
h
ar
d
w
ar
e
co
n
s
u
m
p
tio
n
.
I
n
o
r
d
er
to
r
em
o
v
e
th
e
e
x
tr
a
s
u
b
tr
ac
tio
n
an
d
h
a
v
e
a
u
n
i
f
o
r
m
in
p
u
t
an
d
o
u
tp
u
t
r
a
n
g
e,
W
alter
[
1
0
]
p
r
o
p
o
s
ed
alter
in
g
th
e
r
an
g
e
o
f
,
ℬ
,
to
b
e
w
it
h
i
n
[
0
,
2
)
,
in
cr
ea
s
in
g
t
h
e
n
u
m
b
er
o
f
it
er
atio
n
s
f
r
o
m
to
+
2
,
an
d
s
etti
n
g
t
h
e
v
alu
e
o
f
to
2
+
2
[
1
5
,
1
6
]
.
I
n
A
lg
o
r
it
h
m
-
4
to
d
ec
r
ea
s
e
th
e
lea
k
ag
e
an
d
in
cr
ea
s
e
t
h
e
ef
f
icie
n
c
y
th
e
r
e
is
n
o
s
u
b
tr
ac
tio
n
o
p
er
atio
n
a
ls
o
,
an
d
th
e
o
u
tp
u
t
is
+
1
.
T
h
e
p
r
ec
o
n
d
itio
n
in
[
1
0
]
o
f
in
p
u
t
o
p
er
an
d
s
a
n
d
ℬ
is
<
2
a
n
d
ℬ
<
2
,
w
h
ich
ca
n
s
tr
ictl
y
d
eg
r
ad
e
p
er
f
o
r
m
an
ce
[
1
7
,
18]
.
B
y
co
n
tr
a
s
t,
th
e
n
e
w
m
o
d
i
f
ica
tio
n
in
t
h
is
w
o
r
k
p
r
ese
n
ted
b
y
A
l
g
o
r
ith
m
-
4
r
etai
n
i
n
g
t
h
e
r
an
g
e
o
f
o
p
er
an
d
s
an
d
m
o
d
u
l
u
s
w
i
th
i
n
[
0
,
)
,
th
at
h
a
s
b
ee
n
en
h
a
n
ce
d
th
e
p
er
f
o
r
m
a
n
ce
o
f
t
h
e
p
r
o
p
o
s
ed
d
esig
n
th
a
n
t
h
e
o
th
er
r
elate
d
w
o
r
k
s
.
Fu
r
t
h
er
m
o
r
e,
th
e
v
al
u
e
o
f
an
d
iter
atio
n
s
in
th
e
p
r
o
p
o
s
ed
A
lg
o
r
it
h
m
-
4
is
s
ti
ll
2
an
d
,
r
esp
ec
tiv
el
y
.
T
h
er
ef
o
r
e,
th
ese
m
o
d
if
icatio
n
s
i
m
p
r
o
v
e
th
e
p
er
f
o
r
m
a
n
ce
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
.
Fu
r
t
h
er
m
o
r
e,
t
h
e
h
ar
d
w
ar
e
r
eso
u
r
ce
s
an
d
i
n
ter
co
n
n
ec
ts
r
e
q
u
ir
ed
w
ill b
e
les
s
,
an
d
th
u
s
t
h
e
ar
ea
co
n
s
u
m
ed
is
also
h
a
s
b
een
r
ed
u
ce
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4752
A
co
mp
a
ct
F
P
GA
-
b
a
s
ed
mo
n
tg
o
mery
mo
d
u
la
r
mu
ltip
lier
(
A
h
med
A
.
H.
A
b
d
-
E
lka
d
er
)
739
3
.
2
.
P
r
o
po
s
ed
des
ig
n o
f
r
a
dix
-
2
MMM
T
h
e
h
ar
d
w
ar
e
cir
cu
it
f
o
r
A
lg
o
r
ith
m
-
4
i
s
r
ev
ea
led
i
n
Fig
u
r
e
1.
T
h
e
p
r
o
p
o
s
ed
d
esig
n
u
til
izes
a
co
u
n
ter
in
s
tead
o
f
u
s
i
n
g
lo
o
p
to
o
b
tain
a
r
ed
u
ce
d
ar
ea
,
it
n
ee
d
s
o
n
e
C
lo
ck
f
o
r
ea
ch
iter
atio
n
,
th
e
clo
ck
in
cr
ea
s
es
th
e
co
u
n
ter
b
y
o
n
e.
T
h
e
co
u
n
ter
c
o
u
n
t
s
iter
atio
n
s
.
D
u
e
to
th
e
r
eg
is
ter
s
u
s
ed
to
s
y
n
c
h
r
o
n
ize
i
m
p
le
m
en
ted
v
al
u
e
s
;
th
er
e
is
a
d
elay
o
f
o
n
e
m
o
r
e
clo
ck
,
th
er
ef
o
r
e,
th
e
co
u
n
ter
co
u
n
t
s
f
r
o
m
0
to
.
T
h
e
co
u
n
ter
is
a
p
ar
t
o
f
co
n
tr
o
l
u
n
i
t
r
esp
o
n
s
ib
le
f
o
r
s
ca
n
n
i
n
g
th
e
m
u
lt
ip
lier
f
r
o
m
L
SB
to
MSB
an
d
g
et
o
u
t
o
n
e
b
it
f
o
r
iter
atio
n
(
)
.
T
h
e
p
r
o
p
o
s
ed
d
esig
n
is
co
m
p
o
s
ed
o
f
o
n
e
+
1
b
it
A
d
d
er
,
o
n
e
+
2
b
it
A
d
d
er
,
an
d
t
w
o
Mu
ltip
le
x
er
s
.
I
n
itiall
y
r
eg
is
ter
i
s
lo
ad
ed
w
ith
ze
r
o
.
T
h
e
co
r
r
esp
o
n
d
in
g
b
it
s
elec
ts
b
et
w
ee
n
t
w
o
v
alu
e
s
,
t
h
e
m
u
l
ti
p
lican
d
ℬ
if
it
i
s
‘
1
’
o
r
ze
r
o
−
o
th
er
w
is
e.
T
h
e
o
u
t
p
u
t o
f
th
e
f
ir
s
t M
u
lt
ip
lex
er
is
d
en
o
ted
b
y
γ
.
T
h
e
f
ir
s
t
A
d
d
er
ex
ec
u
tes ad
d
it
io
n
o
f
m
o
d
u
l
u
s
a
n
d
γ
,
th
e
o
u
tp
u
t i
s
lo
ad
ed
to
an
in
p
u
t o
f
t
h
e
s
ec
o
n
d
Mu
ltip
le
x
er
,
th
e
o
th
er
i
n
p
u
t
i
s
γ
.
I
n
s
tep
-
4
o
f
A
lg
o
r
it
h
m
-
4
,
t
h
e
in
p
u
ts
o
f
th
e
XO
R
g
ate
ar
e
th
e
b
it
o
f
th
e
o
r
d
er
o
n
e
o
f
th
e
r
eg
i
s
ter
α
(
α
)
an
d
th
e
L
SB
o
f
th
e
r
eg
is
ter
γ
(
γ
0
)
.
Α
is
u
tili
z
ed
in
s
tead
o
f
0
to
r
ed
u
ce
th
e
cr
i
tical
p
ath
d
ela
y
.
T
h
e
o
u
tp
u
t
o
f
t
h
e
XOR
g
a
te
is
t
h
e
s
elec
to
r
o
f
t
h
e
s
ec
o
n
d
Mu
ltip
le
x
er
to
d
ef
in
e
ev
en
o
u
tp
u
t
(
Ȥ
)
o
f
th
e
Mu
ltip
lex
er
.
Step
-
4
d
ef
in
es
th
e
p
u
r
p
o
s
e
o
f
XO
R
g
ate
an
d
th
e
s
ec
o
n
d
Mu
ltip
lex
er
.
A
cc
o
r
d
in
g
to
th
e
co
n
d
itio
n
th
at
m
o
d
u
l
u
s
m
u
s
t
b
e
o
d
d
;
(
o
d
d
in
teg
er
+
)
i
s
an
ev
e
n
in
teg
er
.
I
f
γ
is
o
d
d
it
co
n
v
er
ted
to
ev
en
b
y
ad
d
i
n
g
m
o
d
u
l
u
s
to
it,
s
t
ep
-
4
.
T
h
e
s
ec
o
n
d
A
d
d
er
ex
ec
u
tes ad
d
itio
n
o
f
ac
c
u
m
u
la
to
r
an
d
Ȥ
,
th
e
o
u
tp
u
t i
s
lo
ad
ed
t
o
v
ar
iab
le
α
.
T
h
e
v
alu
e
o
f
α
is
s
h
i
f
ted
r
ig
h
t
o
n
e
b
it,
d
iv
id
ed
b
y
2
,
an
d
lo
a
d
ed
to
th
e
r
eg
is
ter
o
f
th
e
ac
cu
m
u
lated
v
al
u
e
.
T
h
o
s
e
p
r
o
ce
d
u
r
es
ar
e
r
eiter
ated
f
o
r
ea
ch
clo
ck
.
T
h
u
s
,
t
h
e
g
i
v
e
n
ar
c
h
itect
u
r
e
ta
k
es
+1
clo
ck
c
y
cle
to
m
a
k
e
o
u
t
th
e
ca
lcu
latio
n
p
r
o
ce
s
s
,
s
u
b
s
eq
u
en
tl
y
,
th
e
v
al
u
e
o
f
r
eg
is
ter
is
lo
ad
ed
to
th
e
o
u
tp
u
t
.
T
h
e
o
u
tp
u
t
,
+
1
,
is
th
e
r
ed
u
c
tio
n
p
r
o
d
u
ct
o
f
t
h
e
t
w
o
in
p
u
t
i
n
te
g
er
s
m
u
ltip
lied
b
y
2
−
.
T
h
e
s
ize
o
f
t
h
e
h
ar
d
w
ar
e
co
m
p
o
n
e
n
ts
d
ep
en
d
s
o
n
th
e
s
ize
o
f
th
e
o
p
er
an
d
s
an
d
th
e
m
o
d
u
l
u
s
f
o
r
th
e
m
u
lt
ip
lic
atio
n
p
r
o
ce
s
s
.
T
h
is
d
esig
n
ca
n
b
e
ea
s
il
y
s
ca
led
ac
co
r
d
in
g
to
r
eq
u
ir
e
m
e
n
t f
o
r
an
y
n
u
m
b
er
o
f
b
its
.
Fig
u
r
e
1
.
P
r
o
p
o
s
ed
m
o
d
u
lar
m
u
ltip
lier
cir
cu
i
t
3
.
3
.
B
o
un
ds
o
n t
he
I
/O
Ou
tp
u
ts
f
r
o
m
m
u
ltip
licat
io
n
s
ar
e
r
e
-
u
s
ed
as
i
n
p
u
t
s
t
h
r
o
u
g
h
o
u
t
t
h
e
cr
y
p
to
g
r
ap
h
ic
s
y
s
te
m
s
.
So
,
it
'
s
i
m
p
o
r
tan
t
to
k
ee
p
th
o
s
e
n
u
m
b
er
s
b
o
u
n
d
.
I
n
p
ar
ticu
lar
,
f
o
r
all
o
u
tp
u
ts
w
e
w
i
ll
s
h
o
w
t
h
at
<
2
is
m
ai
n
tai
n
ab
le.
T
h
e
f
o
u
r
v
ar
iab
les
in
t
h
e
A
lg
o
r
it
h
m
-
4
ar
e
γ
,
Ȥ
,
α
,
an
d
,
th
e
b
o
u
n
d
ed
b
it
-
le
n
g
th
o
f
it
ar
e
,
+
1
,
+
2
,
an
d
+
1
,
r
esp
ec
tiv
el
y
,
Fi
g
u
r
e
1
.
T
h
e
m
a
x
i
m
u
m
v
al
u
e
o
f
m
o
d
u
l
u
s
is
:
=
2
−
1
(
1
3
)
T
h
e
p
r
ec
o
n
d
itio
n
o
f
MM
M
is
ℬ
<
,
th
u
s
,
t
h
e
m
a
x
i
m
u
m
v
al
u
e
o
f
v
ar
iab
le
γ
is
t
h
e
m
ax
i
m
u
m
v
al
u
e
o
f
th
e
m
u
lt
ip
lican
d
ℬ
:
γ
=
ℬ
=
2
−
2
(
1
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
2
-
4752
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
21
,
No
.
2
,
Feb
r
u
ar
y
2
0
2
1
:
7
3
5
-
7
4
3
740
T
h
e
v
alu
e
o
f
v
ar
iab
le
Ȥ
is
:
Ȥ
=
γ
+
(
1
5
)
T
h
e
m
a
x
i
m
u
m
v
al
u
e
o
f
v
ar
iab
le
α
is
:
α
=
Ȥ
+
ℎ
−
1
(
1
6
)
w
h
er
e,
ℎ
is
th
e
iter
at
io
n
n
u
m
b
er
.
T
h
e
v
alu
e
o
f
th
e
ac
c
u
m
u
late
d
v
ar
iab
le
ℎ
is
:
=
2
(
1
7
)
A
fter
th
e
la
s
t iter
atio
n
ℎ
=
,
an
d
th
e
v
alu
e
o
f
th
e
o
u
tp
u
t
i
s
:
=
(
1
8
)
T
h
e
v
er
if
icatio
n
t
h
at
t
h
e
o
u
tp
u
t
an
d
o
th
er
v
ar
iab
les f
o
r
=
8
ar
e
b
elo
w
t
h
e
s
ize
o
f
t
h
eir
i
m
p
le
m
e
n
ted
h
ar
d
w
ar
e
v
ar
iab
le
s
s
h
o
w
n
i
n
T
ab
le
1
.
T
h
e
ce
ilin
g
f
u
n
ctio
n
m
ap
s
/
2
to
th
e
least
in
te
g
er
g
r
ea
ter
th
a
n
o
r
eq
u
al
to
/
2
.
I
n
g
en
er
al,
th
e
m
ax
i
m
u
m
v
alu
es
o
f
t
h
e
Alg
o
r
it
h
m
-
4
v
ar
ia
b
les
ar
e
as
f
o
llo
w
s
f
o
r
an
y
−
(
,
,
):
γ
=
2
−
2
(
1
9
)
Ȥ
=
γ
+
=
2
+
1
−
3
(
2
0
)
=
Ȥ
+
ℎ
−
1
=
2
+
2
−
2
−
(
ℎ
−
2
)
−
5
(
2
1
)
ℎ
=
⌈
2
⌉
=
2
+
1
−
2
−
(
ℎ
−
1
)
−
2
(
2
2
)
T
h
e
m
a
x
i
m
u
m
v
al
u
e
o
f
t
h
e
o
u
tp
u
t
i
s
:
=
=
2
+
1
−
4
T
h
er
ef
o
r
e,
th
e
b
it
-
len
g
t
h
o
f
t
h
e
o
u
tp
u
t
is
b
o
u
n
d
ed
o
n
+
1
b
its
.
T
ab
le
1
.
B
o
u
n
d
s
o
n
th
e
o
u
tp
u
t
o
f
th
e
p
r
o
p
o
s
ed
d
esig
n
ℎ
ℎ
−
1
<
2
+
1
γ
<
2
Ȥ
=
γ
+
<
2
+
1
=
Ȥ
+
ℎ
−
1
<
2
+
2
ℎ
=
⌈
/
2
⌉
<
2
+
1
1
0
2
−
2
2
+
1
−
3
2
+
1
−
3
2
−
1
2
2
−
1
2
−
2
2
+
1
−
3
2
+
2
−
2
−
4
2
+
1
−
2
−
1
−
2
3
2
+
1
−
2
−
1
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
1
−
5
2
+
1
−
2
−
2
−
2
4
2
+
1
−
2
−
2
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
2
−
5
2
+
1
−
2
−
3
−
2
5
2
+
1
−
2
−
3
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
3
−
5
2
+
1
−
2
−
4
−
2
6
2
+
1
−
2
−
4
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
4
−
5
2
+
1
−
2
−
5
−
2
7
2
+
1
−
2
−
5
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
5
−
5
2
+
1
−
2
−
6
−
2
8
2
+
1
−
2
−
6
−
2
2
−
2
2
+
1
−
3
2
+
2
−
2
−
6
−
5
2
+
1
−
2
−
7
−
2
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
h
ar
d
w
ar
e
ar
ch
itec
tu
r
es f
o
r
MM
M,
ex
i
s
tin
g
a
n
d
p
r
o
p
o
s
ed
m
et
h
o
d
,
h
as
b
ee
n
p
r
ese
n
ted
i
n
Sectio
n
s
2
an
d
3
,
r
esp
ec
tiv
el
y
.
T
h
is
ar
c
h
itect
u
r
e
o
f
th
e
p
r
o
p
o
s
ed
d
esig
n
h
as
b
ee
n
i
m
p
le
m
e
n
ted
i
n
V
HDL
.
T
h
i
s
w
o
r
k
h
a
s
b
ee
n
s
i
m
u
lated
a
n
d
s
y
n
t
h
esize
d
u
s
in
g
Mo
d
elSi
m
an
d
Xili
n
x
I
SE
to
o
ls
r
esp
ec
tiv
el
y
,
w
h
er
e
th
e
tar
g
et
d
e
v
ice
i
s
u
s
ed
Vir
te
x
-
6
d
ev
ice
X
C
6
V
L
X7
6
0
-
2
FF
1
7
6
0
FP
GA
.
4
.
1
.
Si
m
ula
t
io
n o
f
t
he
pro
po
s
ed
a
lg
o
rit
h
m
T
h
e
s
i
m
u
latio
n
a
n
d
v
er
i
f
icati
o
n
o
f
t
h
e
d
es
ig
n
i
s
ca
r
r
ied
o
u
t
u
s
i
n
g
Mo
d
elSi
m
,
Fi
g
u
r
e
2
.
As
a
n
ex
a
m
p
le
,
=
8
,
=
165
,
ℬ
=
231
,
an
d
th
e
m
o
d
u
l
u
s
=
245
.
A
s
e
x
p
ec
ted
,
th
e
f
i
n
al
r
esu
lt
is
=
280
,
w
h
er
e,
=
×
ℬ
×
2
−
=
165
∗
231
∗
2
−
8
245
=
280
.
T
h
e
r
esu
lt
is
+
1
b
its
as
a
co
n
s
eq
u
e
n
ce
o
f
er
ad
icatin
g
t
h
e
f
in
a
l su
b
tr
ac
tio
n
s
tep
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4752
A
co
mp
a
ct
F
P
GA
-
b
a
s
ed
mo
n
tg
o
mery
mo
d
u
la
r
mu
ltip
lier
(
A
h
med
A
.
H.
A
b
d
-
E
lka
d
er
)
741
Fig
u
r
e
2
.
Si
m
u
latio
n
o
f
th
e
p
r
o
p
o
s
ed
MM
M
4
.
2
.
I
m
ple
m
e
nta
t
io
n a
nd
co
m
pa
r
is
o
n
I
n
th
i
s
s
ec
tio
n
,
a
co
m
p
ar
is
o
n
o
f
s
tate
o
f
t
h
e
ar
t
(
)
h
ar
d
w
ar
e
Mo
n
tg
o
m
er
y
Mo
d
u
lar
M
u
ltip
licat
io
n
is
p
r
esen
ted
.
T
ab
le
2
s
h
o
w
s
th
at,
th
e
p
er
f
o
r
m
an
ce
an
al
y
s
is
o
f
o
u
r
w
o
r
k
w
i
th
e
x
i
s
ti
n
g
d
esi
g
n
s
.
T
h
e
m
etr
ic
s
u
s
ed
to
ev
a
lu
ate
th
e
p
r
o
p
o
s
ed
h
ar
d
w
ar
e
d
esig
n
s
f
o
r
v
ar
io
u
s
b
it
-
l
en
g
t
h
ar
e
ar
ea
in
Sli
ce
L
UT
s
,
f
r
eq
u
en
c
y
i
n
MH
z,
ti
m
e
i
n
,
th
r
o
u
g
h
p
u
t
i
n
Mb
p
s
,
A
r
ea
-
T
im
e
p
er
b
it
-
le
n
g
t
h
(
A
T
/b
)
,
an
d
ef
f
icien
c
y
(
Mb
p
s
p
er
FP
GA
L
UT
)
.
E
f
f
icien
c
y
m
e
tr
ic
h
as
b
ee
n
u
s
ed
in
p
r
ev
io
u
s
w
o
r
k
s
to
ev
alu
ate
th
e
ar
ea
r
eso
u
r
ce
s
u
s
e
d
an
d
p
er
f
o
r
m
an
ce
ac
h
iev
ed
i
n
cr
y
p
to
g
r
ap
h
ic
h
ar
d
w
ar
e
ar
ch
itect
u
r
es
[
1
9
]
.
K.
J
av
ee
d
[
1
4
]
p
r
esen
t
ed
a
r
ad
ix
-
2
i
m
p
le
m
e
n
tatio
n
o
f
t
h
e
M
MM
an
d
I
MM
al
g
o
r
ith
m
s
o
n
FP
GA
.
T
h
e
o
u
tco
m
es
in
d
icate
t
h
at
t
h
e
r
ad
ix
-
2
MM
M
d
esi
g
n
i
s
m
o
r
e
p
r
o
f
icien
t
in
ter
m
s
o
f
ca
lc
u
latio
n
ti
m
e,
FP
G
A
s
l
ice
ar
ea
an
d
th
r
o
u
g
h
p
u
t
as
co
m
p
ar
ed
to
th
e
r
a
d
ix
-
2
I
MM
d
esig
n
.
T
h
e
s
y
n
t
h
esi
s
r
esu
lt
s
o
f
i
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N.
G
.
C
h
o
ll
i,
“
A
n
e
ff
icie
n
t
a
p
p
r
o
a
c
h
f
o
r
se
c
u
re
d
c
o
m
m
u
n
ica
ti
o
n
i
n
w
irele
ss
se
n
so
r
n
e
t
w
o
rk
s,”
In
t.
J
.
El
e
c
tr.
Co
mp
u
t.
E
n
g
.
,
v
o
l
.
1
0
,
n
o
.
2
,
p
p
.
1
6
4
1
-
1
6
4
7
,
2
0
2
0
,
d
o
i:
1
0
.
1
1
5
9
1
/i
jec
e
.
v
1
0
i
2
.
[8
]
M
.
G
u
n
t
u
ri,
H.
D.
Ko
t
h
a
,
a
n
d
M
.
S
rin
iv
a
sa
Re
d
d
y
,
“
A
n
o
v
e
rv
ie
w
o
f
in
tern
e
t
o
f
th
i
n
g
s,”
J
.
A
d
v
.
Res
.
Dy
n
.
Co
n
tro
l
S
y
st.
,
v
o
l.
1
0
,
n
o
.
9
,
p
p
.
6
5
9
-
6
6
5
,
2
0
1
8
,
d
o
i:
1
0
.
1
2
9
2
8
/t
e
lk
o
m
n
ik
a
.
v
1
8
i
5
.
1
5
9
1
1
.
[9
]
A
.
Ka
ri
m
M
o
h
a
m
e
d
Ib
ra
h
im
,
R.
A
.
Ra
sh
id
,
A
.
H.
F
.
A
.
Ha
m
id
,
M
.
A
d
ib
S
a
rij
a
ri,
a
n
d
M
.
A
.
Ba
h
a
ru
d
i
n
,
“
L
ig
h
tw
e
ig
h
t
Io
T
m
id
d
lew
a
re
f
o
r
ra
p
id
a
p
p
li
c
a
ti
o
n
d
e
v
e
lo
p
m
e
n
t,
”
T
E
L
KOM
NIKA
(
T
e
lec
o
mm
u
n
ica
ti
o
n
,
Co
m
p
u
t
in
g
,
El
e
c
tro
n
ics
a
n
d
Co
n
tro
l
)
,
v
o
l.
1
7
,
n
o
.
3
,
p
p
.
1
3
8
5
-
1
3
9
2
,
2
0
1
9
,
d
o
i:
1
0
.
1
2
9
2
8
/T
EL
KO
M
NIK
A
.
V
1
7
I3
.
1
1
7
9
3
.
[1
0
]
C.
D.
W
a
lt
e
r,
“
M
o
n
tg
o
m
e
r
y
e
x
p
o
n
e
n
t
iatio
n
n
e
e
d
s
n
o
f
in
a
l
s
u
b
tra
c
ti
o
n
s,”
El
e
c
tro
n
.
L
e
tt
.
,
v
o
l.
3
5
,
n
o
.
2
1
,
p
p
.
1
8
3
1
-
1
8
3
2
,
1
9
9
9
,
d
o
i:
1
0
.
1
0
4
9
/el:1
9
9
9
1
2
3
0
.
[1
1
]
R.
L
iu
a
n
d
S
.
L
i,
“
A
d
e
sig
n
a
n
d
im
p
le
m
e
n
tatio
n
o
f
m
o
n
tg
o
m
e
r
y
m
o
d
u
lar
m
u
lt
ip
li
e
r,
”
in
Pro
c
e
e
d
in
g
s
-
IEE
E
In
ter
n
a
t
io
n
a
l
S
y
mp
o
siu
m
o
n
Circ
u
it
s
a
n
d
S
y
ste
ms
,
v
o
l.
2
0
1
9
-
M
a
y
,
n
o
.
1
,
p
p
.
1
-
4
,
2
0
1
9
,
d
o
i:
1
0
.
1
1
0
9
/I
S
CA
S
.
2
0
1
9
.
8
7
0
2
6
8
4
.
[1
2
]
B.
S
c
h
n
e
ier,
“
A
p
p
li
e
d
Cry
p
to
g
ra
p
h
y
,
”
El
e
c
tr.
En
g
.
,
v
o
l.
1
,
n
o
.
3
2
,
p
p
.
4
2
9
-
4
5
5
,
1
9
9
6
,
d
o
i:
1
0
.
1
.
1
.
9
9
.
2
8
3
8
.
[1
3
]
K.
P
ra
ti
b
h
a
a
n
d
R
.
M
u
t
h
a
iah
,
“
S
u
rv
e
y
o
n
Ha
rd
wa
re
I
m
p
le
m
e
n
tatio
n
o
f
M
o
n
tg
o
m
e
r
y
M
o
d
u
lar,”
In
t
.
J
.
Pu
re
Ap
p
l.
M
a
th
.
,
v
o
l.
1
1
9
,
n
o
.
1
2
,
p
p
.
1
3
4
3
7
-
1
3
4
5
2
,
2
0
1
8
.
[1
4
]
K.
Ja
v
e
e
d
,
D.
Irw
in
,
a
n
d
X
.
W
a
n
g
,
“
De
sig
n
a
n
d
p
e
rf
o
rm
a
n
c
e
c
o
m
p
a
riso
n
o
f
m
o
d
u
lar
m
u
lt
ip
li
e
rs
i
m
p
le
m
e
n
ted
o
n
F
P
GA
p
latf
o
r
m
,
”
L
e
c
t.
No
tes
Co
mp
u
t.
S
c
i.
(
in
c
lu
d
in
g
S
u
b
se
r.
L
e
c
t.
No
tes
Arti
f
.
In
tell
.
L
e
c
t.
N
o
tes
Bi
o
in
fo
rm
a
t
ics
)
,
v
o
l.
1
0
0
3
9
L
N
CS
,
p
p
.
2
5
1
-
2
6
0
,
2
0
1
6
,
d
o
i:
1
0
.
1
0
0
7
/9
7
8
-
3
-
3
1
9
-
4
8
6
7
1
-
0
_
2
3
.
[1
5
]
G
.
S
.
Y.
Ba
e
k
,
“
Un
if
o
r
m
M
o
n
tg
o
m
e
r
y
m
u
lt
ip
li
e
r,
”
J
.
Cry
p
to
g
r.
E
n
g
.
,
2
0
1
9
,
d
o
i:
1
0
.
1
0
0
7
/s1
3
3
8
9
-
0
1
9
-
0
0
2
1
3
-
7.
[1
6
]
M
.
De
r
S
h
ieh
,
J.
H.
C
h
e
n
,
W
.
C.
L
in
,
a
n
d
H.
H.
W
u
,
“
A
n
e
w
a
l
g
o
rit
h
m
f
o
r
h
ig
h
-
sp
e
e
d
m
o
d
u
lar
m
u
lt
ip
li
c
a
ti
o
n
d
e
sig
n
,
”
IEE
E
T
ra
n
s.
Circ
u
it
s S
y
st.
I
Reg
u
l.
Pa
p
.
,
v
o
l
.
5
6
,
n
o
.
9
,
p
p
.
2
0
0
9
-
2
0
1
9
,
2
0
0
9
,
d
o
i:
1
0
.
1
1
0
9
/T
CS
I.
2
0
0
8
.
2
0
1
1
5
8
5
.
[1
7
]
Z.
L
iu
a
n
d
J.
G
ro
ß
sc
h
ä
d
l,
“
Ne
w
sp
e
e
d
re
c
o
rd
s
f
o
r
m
o
n
tg
o
m
e
r
y
m
o
d
u
lar
m
u
lt
ip
li
c
a
ti
o
n
o
n
8
-
b
it
A
V
R
m
icro
c
o
n
tro
ll
e
rs,”
Pro
g
.
Cry
p
to
l
o
g
y
--
AF
RICA
CRY
PT
2
0
1
4
,
v
o
l
.
8
4
6
9
L
NCS,
p
p
.
2
1
5
-
2
3
4
,
2
0
1
4
,
d
o
i
:
1
0
.
1
0
0
7
/
9
7
8
-
3
-
3
1
9
-
0
6
7
3
4
-
6
_
1
4
.
[1
8
]
G
.
H
a
c
h
e
z
a
n
d
J.
J.
Qu
isq
u
a
ter,
“
M
o
n
tg
o
m
e
r
y
e
x
p
o
n
e
n
ti
a
ti
o
n
with
n
o
f
in
a
l
su
b
trac
ti
o
n
s:
Im
p
ro
v
e
d
re
su
lt
s,”
in
L
e
c
tu
re
No
tes
in
Co
mp
u
ter
S
c
ien
c
e
(
in
c
lu
d
in
g
s
u
b
se
rie
s
L
e
c
tu
re
No
tes
in
Arti
fi
c
ia
l
I
n
telli
g
e
n
c
e
a
n
d
L
e
c
tu
re
No
tes
in
Bi
o
in
fo
rm
a
t
ics
)
,
v
o
l.
1
9
6
5
L
NCS,
p
p
.
2
9
3
-
3
0
1
,
2
0
0
0
,
d
o
i:
1
0
.
1
0
0
7
/
3
-
5
4
0
-
4
4
4
9
9
-
8
-
23.
[1
9
]
L
.
Ro
d
ríg
u
e
z
-
F
lo
re
s,
M
.
M
o
ra
les
-
S
a
n
d
o
v
a
l,
R.
Cu
m
p
li
d
o
,
C
.
F
e
re
g
rin
o
-
Urib
e
,
a
n
d
I.
A
lg
re
d
o
-
Ba
d
i
ll
o
,
“
C
o
m
p
a
c
t
F
P
GA
h
a
rd
w
a
re
a
rc
h
it
e
c
tu
re
f
o
r
p
u
b
li
c
k
e
y
e
n
c
r
y
p
ti
o
n
i
n
e
m
b
e
d
d
e
d
d
e
v
ice
s,”
PL
o
S
O
n
e
,
v
o
l.
1
3
,
n
o
.
1
,
p
p
.
1
-
2
1
,
2
0
1
8
,
d
o
i:
1
0
.
1
3
7
1
/
jo
u
rn
a
l.
p
o
n
e
.
0
1
9
0
9
3
9
.
[2
0
]
S
.
G
h
o
sh
,
D.
M
u
k
h
o
p
a
d
h
y
a
y
,
a
n
d
D.
R.
Ch
o
w
d
h
u
ry
,
“
Hig
h
S
p
e
e
d
F
p
M
u
lt
i
p
li
e
rs
a
n
d
A
d
d
e
rs
o
n
F
P
G
A
P
latf
o
rm
,
”
in
De
sig
n
a
n
d
Arc
h
it
e
c
tu
re
s fo
r
S
ig
n
a
l
a
n
d
Ima
g
e
Pro
c
e
ss
in
g
(
DAS
IP)
,
p
p
.
2
1
-
26
,
2
0
1
0
.
[2
1
]
K.
Ja
v
e
e
d
a
n
d
X
.
W
a
n
g
,
“
Ra
d
ix
-
4
a
n
d
ra
d
ix
-
8
b
o
o
th
e
n
c
o
d
e
d
i
n
terle
a
v
e
d
m
o
d
u
lar
m
u
lt
ip
l
iers
o
v
e
r
g
e
n
e
ra
l
F
p
,
”
C
o
n
f.
Dig
.
-
2
4
t
h
I
n
t.
C
o
n
f
.
F.
Pro
g
ra
m.
L
o
g
.
A
p
p
l
.
FP
L
2
0
1
4
,
2
0
1
4
,
d
o
i:
1
0
.
1
1
0
9
/
F
P
L
.
2
0
1
4
.
6
9
2
7
4
5
2
.
[2
2
]
J.
Din
g
a
n
d
S
.
L
i,
“
Bro
k
e
n
-
Ka
ra
tsu
b
a
m
u
lt
ip
l
ica
ti
o
n
a
n
d
i
ts
a
p
p
li
c
a
ti
o
n
to
M
o
n
tg
o
m
e
r
y
m
o
d
u
lar
m
u
lt
ip
li
c
a
ti
o
n
,
”
i
n
2
0
1
7
2
7
th
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
Fi
e
ld
Pro
g
ra
mm
a
b
le L
o
g
i
c
a
n
d
A
p
p
l
ica
ti
o
n
s,
FP
L
2
0
1
7
,
p
p
.
5
-
8
,
2
0
1
7
,
d
o
i:
1
0
.
2
3
9
1
9
/F
P
L
.
2
0
1
7
.
8
0
5
6
7
6
9
.
[2
3
]
G
.
C.
T
.
Ch
o
w
,
K.
Eg
u
ro
t,
W
.
Lu
k
,
a
n
d
P
.
L
e
o
n
g
,
“
A
k
a
r
a
tsu
b
a
-
b
a
se
d
m
o
n
tg
o
m
e
r
y
m
u
lt
ip
li
e
r,
”
Pro
c
.
-
2
0
1
0
I
n
t.
Co
n
f.
F.
Pr
o
g
r
a
m.
L
o
g
.
Ap
p
l.
FP
L
2
0
1
0
,
p
p
.
4
3
4
-
4
3
7
,
2
0
1
0
,
d
o
i:
1
0
.
1
1
0
9
/F
P
L
.
2
0
1
0
.
8
9
.
[2
4
]
X
.
Ya
n
,
G
.
W
u
,
D.
W
u
,
F
.
Z
h
e
n
g
,
a
n
d
X
.
X
ie,
“
A
n
im
p
le
m
e
n
tatio
n
o
f
m
o
n
tg
o
m
e
r
y
m
o
d
u
lar
m
u
lt
ip
li
c
a
ti
o
n
o
n
F
P
GA
s,”
in
Pro
c
e
e
d
in
g
s
-
2
0
1
3
I
n
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
In
f
o
rm
a
ti
o
n
S
c
ien
c
e
a
n
d
Cl
o
u
d
Co
m
p
u
t
i
n
g
,
IS
CC
2
0
1
3
,
p
p
.
3
2
-
38
,
2
0
1
4
,
d
o
i:
1
0
.
1
1
0
9
/I
S
CC.2
0
1
3
.
1
9
.
[2
5
]
K.
Ja
v
e
e
d
,
X
.
W
a
n
g
,
a
n
d
M
.
S
c
o
tt
,
“
S
e
rial
a
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