I
n
t
e
r
n
at
io
n
al
Jou
r
n
al
of
A
d
van
c
e
s
i
n
A
p
p
li
e
d
S
c
ie
n
c
e
s
(
I
JA
A
S
)
V
ol
.
15
, N
o.
1
,
M
a
r
c
h
20
26
, pp.
293
~
302
I
S
S
N
:
2252
-
8814
,
D
O
I
:
10.11591/
ij
a
a
s
.
v15.
i
1
.
pp
293
-
302
293
Jou
r
n
al
h
om
e
page
:
ht
tp
:
//
ij
aas
.i
ae
s
c
or
e
.c
om
A
n
ove
l
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l
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t
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ase
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M
c
E
l
i
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e
f
r
am
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w
or
k
f
or
s
e
c
u
r
e
d
i
gi
t
al
c
om
m
u
n
i
c
at
i
on
R
avi
k
u
m
ar
I
n
ak
ot
i
1
, Jam
e
s
S
t
e
p
h
e
n
M
e
k
a
2
, P
ad
al
a V
e
n
k
at
a G
op
al
a D
u
r
ga P
r
as
ad
R
e
d
d
y
1
1
D
e
pa
r
t
m
e
nt
of
C
om
put
e
r
S
c
i
e
nc
e
a
nd S
ys
t
e
m
s
E
ngi
ne
e
r
i
ng, A
ndhr
a
U
ni
ve
r
s
i
t
y,
V
i
s
a
kha
pa
t
na
m
, I
ndi
a
2
D
r
. B
. R
. A
m
be
dka
r
C
ha
i
r
, A
ndhr
a
U
ni
ve
r
s
i
t
y, V
i
s
a
kh
a
pa
t
na
m
, I
ndi
a
A
r
t
ic
le
I
n
f
o
A
B
S
T
R
A
C
T
A
r
ti
c
le
h
is
to
r
y
:
R
e
c
e
iv
e
d
A
pr
24, 2025
R
e
vi
s
e
d
D
e
c
23, 2025
A
c
c
e
pt
e
d
J
a
n 1, 2026
McEliece
cryptosy
stem
is
old
and
well
-
explored
post
-
quantum
cryptography
system
that
offers
superior
security
against
quantum
attacks.
Thou
gh
the
system
holds
great
potential
and
superior
security,
the
challenge
ass
ociated
with
large
key
sizes
has
made
system
impractical
for
most
application
s.
The
first
challenge
against
McEliece
cryptosystem
remains
its
large
key
sizes,
which
make
system
impractical,
especially
when
implementing
inte
rnet
of
things
(
IoT
)
and
mobile
communication
applications.
Overc
oming
challenge
s
and
retaining
superior
security
still
remains
an
issue
to
e
xplore.
This
paper
presents
investigation
into
use
of
circulant
matrices
for
Mc
Eliece
encryptio
n
system
to
achieve
a
considerab
le
reduction
in
key
siz
es
and
enhance
fast
encryptio
n
processes.
The
use
of
circulant
matri
ces’
in
herent
properties
boosts
performance
without
focusing
much
on
system’s
se
curity.
In
addition,
the
paper
presents
security
evaluation
process
for
m
odified
communi
cation
system
to
determin
e
and
mitig
ate
weaknesses
that
might
arise
,
considering
use
of
sophisticated
encryption
systems.
Findin
gs
and
results
explore
use
of
circulant
matrices,
which
achieve
great
reducti
ons
in
key
sizes
and
improve
efficiency
of
process.
Security
evaluation
repor
ts
that
proper
scrambling
techniques
are
efficient
at
mending
the
vulnera
bilities
associated
with
circulant
matrix
structures
.
A
modifi
ed
Mc
Eliece
cryptosy
stem
using
circulant
matrices
offers
superior
data
communi
cation,
balancing
both
strong
security
and
efficient
computational
pro
cesses,
making
system ideal for use
in recent c
ommunication systems.
K
e
y
w
o
r
d
s
:
C
ir
c
ul
a
nt
m
a
tr
ix
C
ode
-
ba
s
e
d c
r
ypt
os
y
s
te
m
C
r
ypt
ogr
a
phy
D
a
ta
c
om
m
uni
c
a
ti
on
M
c
E
li
e
c
e
This is an
open
acce
ss artic
le unde
r the
CC BY
-
SA
license.
C
or
r
e
s
pon
di
n
g A
u
th
or
:
R
a
vi
kum
a
r
I
na
kot
i
D
e
pa
r
tm
e
nt
of
C
om
put
e
r
S
c
ie
nc
e
a
nd S
ys
te
m
s
E
ngi
ne
e
r
in
g, A
n
dhr
a
U
ni
ve
r
s
it
y
V
is
a
kha
pa
tn
a
m
,
A
ndhr
a
P
r
a
de
s
h, I
ndi
a
E
m
a
il
:
r
a
vi
r
k1228@
gm
a
il
.c
om
1.
I
N
T
R
O
D
U
C
T
I
O
N
T
he
s
e
c
ur
it
y
of
da
ta
is
pa
r
a
m
ount
dur
in
g
e
nd
-
to
-
e
nd
da
ta
c
om
m
uni
c
a
ti
on
a
nd
da
ta
s
to
r
a
ge
.
T
o
e
ns
ur
e
s
a
f
e
de
li
ve
r
y
of
da
ta
tr
a
ns
f
e
r
r
e
d
ove
r
th
e
in
te
r
ne
t,
c
r
ypt
ogr
a
phy
ha
s
be
e
n
w
id
e
ly
us
e
d
to
tr
a
ns
f
or
m
th
e
da
ta
in
to
a
non
-
r
e
a
da
bl
e
c
ont
e
nt
th
a
t
c
a
n
onl
y
be
r
e
ve
r
te
d
t
o
it
s
in
it
ia
l
by
a
n
a
ut
ho
r
iz
e
d
us
e
r
.
W
it
h
th
e
c
ont
in
uous
a
dopt
io
n
of
i
nt
e
r
ne
t
te
c
hnol
ogy,
da
ta
c
om
m
uni
c
a
ti
on
e
xpe
r
ie
nc
e
s
a
la
r
ge
in
c
r
e
a
s
e
in
s
e
c
ur
it
y
a
tt
a
c
k
e
s
pe
c
ia
ll
y
w
he
n
w
ir
e
le
s
s
c
ha
nne
ls
a
r
e
e
m
pl
oye
d
f
or
c
om
m
uni
c
a
ti
on.
T
o
a
ddr
e
s
s
th
e
s
e
c
ur
it
y
c
ha
ll
e
nge
s
in
da
ta
c
om
m
uni
c
a
ti
ons
a
nd
e
ns
ur
e
da
ta
in
te
gr
it
y,
num
e
r
ous
c
r
ypt
ogr
a
phi
c
a
lg
or
i
th
m
s
w
e
r
e
de
ve
lo
pe
d
[
1]
. T
he
s
e
a
lg
or
it
hm
s
ha
ve
pr
ove
d pr
om
is
in
g i
n pr
e
ve
nt
in
g va
r
io
us
f
or
m
s
of
a
tt
a
c
ks
.
H
ow
e
ve
r
,
m
a
jo
r
it
y
of
th
e
s
e
c
r
ypt
os
y
s
te
m
s
c
a
n
be
e
a
s
il
y
br
oke
n
by
th
e
e
xi
s
t
e
nc
e
of
qua
nt
um
c
om
put
in
g.
Q
ua
nt
um
c
om
put
e
r
s
a
r
e
hi
ghl
y
c
om
put
a
ti
on
-
in
te
ns
iv
e
a
nd
c
a
pa
bl
e
of
e
m
pl
oyi
ng
a
lg
o
r
it
hm
s
li
ke
S
hor
’
s
a
nd
G
r
ove
r
’
s
[
2]
,
[
3
]
to
a
c
c
e
le
r
a
te
th
e
e
xe
c
ut
io
n
of
ta
s
ks
.
W
it
h
th
e
e
vol
ut
io
n
a
nd
a
dva
nc
e
m
e
nt
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8814
I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
. 15, No. 1, M
a
r
c
h 2026
:
293
-
302
294
qua
nt
um
c
om
put
in
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pr
om
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c
r
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R
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s
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s
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ut
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m
a
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ti
ll
be
ve
r
y di
f
f
ic
ul
t
f
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th
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qua
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um
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om
put
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t
o s
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. C
ode
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ba
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d c
r
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phy, mul
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tt
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e
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y t
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put
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r
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ode
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ba
s
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d
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n
c
r
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a
ppr
oa
c
h
r
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pr
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s
e
nt
s
one
of
th
e
m
o
s
t
vi
a
bl
e
opt
io
ns
f
or
c
r
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ogr
a
phy
f
ol
lo
w
in
g
qua
nt
um
c
om
put
in
g,
or
e
nc
r
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io
n
s
ys
te
m
s
im
m
une
to
a
tt
a
c
ks
by
qua
nt
um
c
om
put
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r
s
.
T
he
M
c
E
li
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c
e
a
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N
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de
r
r
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it
e
r
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nc
r
ypt
io
n
s
c
he
m
e
s
a
r
e
two
e
xa
m
p
le
s
[
4]
,
[
5]
.
I
t
ha
s
be
e
n
e
s
ta
bl
is
he
d
in
[
6]
th
a
t
th
e
f
unda
m
e
nt
a
l
is
s
ue
i
n t
he
s
c
he
m
e
s
i
s
t
he
e
f
f
ic
ie
nc
y of
de
c
odi
ng t
he
l
in
e
a
r
bl
oc
k c
ode
s
, w
hi
c
h i
s
c
ons
id
e
r
e
d
a
non
de
te
r
m
in
is
ti
c
pol
ynomi
a
l
-
time
(
NP
)
-
c
om
pl
e
te
pr
obl
e
m
.
I
n
c
ode
-
ba
s
e
d
c
r
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a
phy,
th
e
c
om
m
on
e
nc
r
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io
n
s
c
he
m
e
s
a
r
e
M
c
E
li
e
c
e
e
nc
r
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s
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m
e
,
N
ie
de
r
r
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it
e
r
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nc
r
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io
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s
c
he
m
e
.
R
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e
nt
ly
,
th
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hybr
id
M
c
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c
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e
n
c
r
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m
e
(
H
yM
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S
)
w
a
s
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xpl
a
in
e
d
[
7]
.
L
ik
e
th
e
R
S
A
e
n
c
r
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io
n
s
c
he
m
e
,
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
t
e
m
of
in
f
or
m
a
ti
on
e
nc
r
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io
n
f
or
s
e
c
ur
e
in
f
or
m
a
ti
on
c
om
m
uni
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a
ti
on
f
a
il
s
to
c
a
pt
ur
e
th
e
a
tt
e
nt
i
on
of
r
e
s
e
a
r
c
he
r
s
,
u
s
e
r
s
,
a
nd
in
dus
tr
ie
s
,
a
s
th
e
m
a
gni
tu
de
of
th
e
ge
ne
r
a
ti
on
m
a
tr
ix
us
e
d
in
th
e
publ
ic
ke
y
of
th
e
a
ppr
oa
c
h
is
r
e
la
ti
ve
ly
e
nor
m
ous
.
H
ow
e
ve
r
,
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vul
ne
r
a
bi
li
ty
a
s
s
oc
ia
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d
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c
onve
nt
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na
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nc
r
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n s
c
he
m
e
s
l
ik
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R
S
A
a
nd D
i
f
f
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-
He
l
lm
a
n
a
lg
or
it
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s
w
he
n us
e
d on qua
nt
um
c
om
put
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ha
s
m
a
de
i
t
one
of
t
he
f
oc
us
e
s
of
r
e
s
e
a
r
c
h i
n qu
a
nt
um
s
e
c
ur
it
y.
O
ne
of
th
e
m
a
jo
r
f
oc
us
e
s
on
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
te
m
of
e
nc
r
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is
r
e
duc
ti
on
in
th
e
m
a
gni
tu
de
of
th
e
s
ys
te
m
ke
y.
F
or
e
xa
m
pl
e
,
F
a
th
a
ll
a
a
nd
A
z
a
b
[
8]
in
ve
s
ti
ga
te
d
th
e
not
io
n
of
e
m
pl
oyi
n
g
c
om
pa
c
t
r
e
pr
e
s
e
nt
a
ti
on
of
th
e
s
ha
r
e
d
m
a
tr
ix
o
f
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
te
m
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n.
T
he
r
e
s
e
a
r
c
h i
n
[
9]
, [
10]
pr
opos
e
d
qua
s
i
-
c
yc
li
c
(
QC
)
a
lt
e
r
na
nt
a
lo
ng w
it
h t
he
qua
s
i
-
dya
di
c
(
QD
)
G
oppa
c
ode
s
to
r
e
duc
e
th
e
m
a
gni
tu
de
of
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
c
h
e
m
e
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n,
w
hi
c
h
th
e
y
s
uc
c
e
e
d
e
d
in
r
e
du
c
in
g
it
f
r
om
s
e
ve
r
a
l
hundr
e
d
th
ous
a
nd
bi
ts
to
20
ki
lo
bi
ts
.
T
h
e
pur
pos
e
of
th
os
e
c
on
s
tr
uc
ts
i
s
to
us
e
f
ir
s
t
r
ow
pe
r
m
ut
a
ti
ons
to
p
r
oduc
e
th
e
e
nt
ir
e
m
a
t
r
ix
.
A
d
di
ti
ona
ll
y,
th
os
e
c
ons
tr
uc
ti
ons
e
na
bl
e
m
e
s
s
a
ge
e
nc
r
ypt
io
n
us
in
g
onl
y
th
e
in
it
ia
l
r
ow
o
f
th
e
m
a
tr
ix
r
a
th
e
r
th
a
n
t
he
e
nt
ir
e
m
a
tr
ix
.
I
n
bot
h
s
c
e
na
r
io
s
,
th
e
bi
na
r
y
pa
r
a
m
e
te
r
s
r
e
m
a
in
s
e
c
ur
e
e
ve
n a
f
te
r
m
ul
ti
pl
e
a
tt
a
c
ks
.
O
th
e
r
m
e
th
ods
pr
opos
e
d
to
de
pl
e
te
th
e
m
a
gni
tu
de
of
th
e
s
h
a
r
e
d
ke
y
of
th
e
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n
s
c
he
m
e
in
c
lu
de
“
a
lg
e
br
a
ic
ge
om
e
tr
ic
(
A
G
)
c
ode
s
”
[
11]
,
“
ge
ne
r
a
li
z
e
d
R
e
e
d
-
S
ol
om
on
(
G
R
S
)
c
ode
s
”
[
12]
,
“
lo
w
-
de
ns
it
y
pa
r
it
y
c
he
c
k
(
L
D
P
C
)
c
o
de
s
”
[
13]
,
“
R
e
e
d
-
M
ul
le
r
(
R
M
)
c
ode
s
”
[
14]
,
“
lo
w
-
r
a
nk
pa
r
it
y
c
he
c
k
(
L
R
P
C
)
c
ode
s
”
[
15]
,
a
nd
m
a
ny
m
or
e
th
a
t
a
ll
ow
f
or
s
hor
te
r
s
ha
r
e
d
ke
ys
,
ha
v
e
be
e
n
de
m
ons
tr
a
te
d
th
r
ough
va
r
io
us
c
ode
s
.
T
he
m
a
jo
r
it
y
of
th
e
s
e
va
r
ia
ti
ons
ha
ve
be
e
n
s
uc
c
e
s
s
f
ul
ly
c
r
ypt
a
na
ly
z
e
d,
e
ve
n
th
ough
th
e
or
ig
in
a
l
M
c
E
li
e
c
e
c
r
ypt
os
y
s
te
m
is
s
ti
ll
s
e
c
u
r
e
[
16]
,
[
17]
.
T
he
a
lt
e
r
na
ti
ve
c
ode
s
m
us
t
b
e
tr
e
a
te
d
w
it
h
c
a
ut
io
n
be
c
a
us
e
of
th
e
ir
e
xc
e
s
s
iv
e
s
tr
uc
tu
r
e
,
e
ve
n
w
it
h
th
e
ir
pr
om
is
in
g
f
e
a
tu
r
e
s
.
P
ol
a
r
c
odi
n
g
w
a
s
i
ni
ti
a
ll
y i
nt
e
nde
d t
o be
a
m
e
th
od, a
ki
n
t
o p
r
e
vi
ous
P
in
s
ke
r
a
nd M
a
s
s
e
y s
c
he
m
e
s
, f
or
i
nc
r
e
a
s
in
g t
he
c
ut
of
f
r
a
te
of
s
e
que
nt
ia
l
de
c
odi
ng.
T
he
s
e
c
r
e
t
to
r
a
is
in
g
th
e
c
ut
of
f
r
a
te
is
to
ta
ke
a
v
e
c
to
r
c
ha
nne
l
(
w
he
th
e
r
it
’
s
na
tu
r
a
ll
y
oc
c
ur
r
in
g
or
pur
pos
e
f
ul
ly
p
r
oduc
e
d)
,
s
pl
it
it
up
in
to
m
ul
ti
pl
e
in
te
r
r
e
la
te
d
s
ubs
id
ia
r
y
c
ha
nne
ls
,
a
nd
th
e
n
a
ppl
y
a
s
e
pa
r
a
te
or
de
r
e
d
de
c
ode
r
on
e
ve
r
y
s
ubc
ha
nne
l.
P
ol
a
r
c
odi
ng
w
a
s
in
it
ia
ll
y
in
te
nde
d
to
be
a
lo
w
-
c
om
pl
e
xi
ty
r
e
c
ur
s
iv
e
c
ha
nne
l
c
om
bi
ni
ng
a
nd
s
pl
it
ti
ng
ope
r
a
ti
on
of
th
is
ki
nd,
w
it
h
th
e
goa
l
of
be
in
g
e
m
pl
oye
d a
s
i
nt
e
r
na
l
c
ode
i
n a
c
om
pos
it
e
s
c
he
m
e
w
it
h e
xt
e
r
na
l
c
onvolut
io
na
l
c
odi
ng a
nd or
de
r
e
d de
c
odi
ng.
N
e
ve
r
th
e
le
s
s
,
th
e
in
it
ia
l
goa
l
of
in
c
r
e
a
s
in
g
th
e
c
ut
of
f
r
a
te
to
c
h
a
nne
l
c
a
pa
c
it
y
w
a
s
a
c
tu
a
ll
y
a
c
hi
e
v
e
d
w
it
hout
th
e
ne
e
d
f
or
a
n
out
e
r
c
ode
be
c
a
us
e
th
e
pol
a
r
in
ne
r
c
ode
pr
ove
d
to
be
s
o
s
uc
c
e
s
s
f
ul
[
18]
.
W
it
h
th
e
c
ont
in
uous
gr
ow
th
in
th
e
f
ie
ld
of
qua
nt
um
c
om
put
in
g,
th
e
r
e
is
ne
e
d
f
or
m
or
e
r
obus
t
s
c
he
m
e
s
th
a
t
w
il
l
be
di
f
f
ic
ul
t
f
or
th
e
qua
nt
um
s
ys
te
m
s
to
br
e
a
k.
I
n
th
is
p
a
pe
r
,
a
m
o
di
f
ie
d
M
c
E
li
e
c
e
publ
ic
ke
y
e
nc
r
ypt
io
n
s
y
s
te
m
w
it
h
hi
gh
le
ve
l
of
s
e
c
ur
it
y
to
pr
oduc
e
a
s
e
c
ur
e
c
om
m
uni
c
a
ti
on
s
c
he
m
e
ba
s
e
d
on
pol
a
r
c
ode
,
w
hi
c
h
c
a
n
pr
e
ve
nt
B
r
ic
ke
l’
s
a
tt
a
c
k dur
in
g c
om
m
uni
c
a
ti
on. T
he
obj
e
c
ti
ve
s
of
t
hi
s
pa
pe
r
a
r
e
s
um
m
a
r
iz
e
d a
s
f
ol
lo
w
s
:
−
C
r
e
a
te
a
m
odi
f
ie
d
f
or
m
of
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
th
a
t
us
e
s
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
to
m
a
ke
it
s
publ
ic
ke
y
s
m
a
ll
e
r
.
T
hi
s
w
il
l
r
e
duc
e
a
m
a
jo
r
w
e
a
kne
s
s
of
M
c
E
li
e
c
e
c
r
yp
to
s
ys
te
m
,
it
s
la
r
ge
publ
ic
ke
y
s
iz
e
.
T
h
e
id
e
a
w
il
l
he
lp
in
im
pr
ovi
ng
th
e
us
a
ge
of
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
in
s
it
ua
ti
ons
w
he
r
e
li
m
it
e
d
s
to
r
a
ge
c
a
pa
c
it
y w
oul
d not a
ll
ow
a
ll
i
ts
c
r
ypt
ogr
a
m
s
t
o be
t
r
a
ns
m
it
te
d.
−
T
o
e
xa
m
in
e
th
e
s
e
c
ur
it
y
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
ba
s
e
d
on
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
a
ga
in
s
t
bot
h
c
la
s
s
i
c
a
nd
qua
nt
um
a
tt
a
c
k
s
.
T
hi
s
in
vol
ve
s
s
c
r
ut
in
iz
in
g
th
e
a
bi
li
ty
of
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
,
c
om
pl
e
m
e
nt
e
d
by
s
c
r
a
m
bl
in
g a
nd pe
r
m
ut
a
ti
on ma
tr
ic
e
s
, t
o e
ns
ur
e
t
he
s
e
c
ur
it
y of
c
onf
id
e
nt
ia
l
m
e
s
s
a
ge
t
r
a
n
s
m
is
s
io
n.
−
T
o
a
na
ly
z
e
a
nd
c
om
pa
r
e
th
e
e
f
f
e
c
ti
ve
ne
s
s
of
th
e
M
c
E
li
e
c
e
s
ys
te
m
ba
s
e
d
on
a
c
ir
c
ul
a
nt
m
a
tr
ix
w
it
h
th
e
M
c
E
li
e
c
e
s
y
s
te
m
.
I
n
th
is
a
r
e
a
,
we
pl
a
n
to
te
s
t
a
nd
a
na
ly
z
e
th
e
s
pe
e
d
of
e
nc
r
ypt
io
n
a
nd
de
c
r
ypt
io
n,
th
e
ti
m
e
r
e
qui
r
e
d
f
or
ge
ne
r
a
ti
ng
a
ke
y,
a
nd,
m
os
t
im
por
ta
nt
ly
,
th
e
e
f
f
e
c
ti
ve
ne
s
s
a
nd e
f
f
ic
ie
nc
y
a
dde
d
by
th
e
us
e
of
a
c
ir
c
ul
a
nt
m
a
tr
ix
.
T
he
r
e
s
t
of
th
e
do
c
um
e
nt
is
s
tr
uc
tu
r
e
d
a
s
f
ol
lo
w
s
:
s
e
c
ti
on
2
de
s
c
r
ib
e
s
s
e
ve
r
a
l
r
e
s
e
a
r
c
h
pa
pe
r
s
w
hi
c
h
a
r
e
c
lo
s
e
ly
r
e
la
te
d
to
c
ode
-
ba
s
e
d
e
nc
r
ypt
io
n
s
c
he
m
e
s
,
s
e
c
ti
on
3
pr
ovi
de
s
in
f
or
m
a
ti
on
a
bout
th
e
t
r
a
di
ti
ona
l
M
c
E
li
e
c
e
a
lg
or
it
hm
a
nd
ba
s
ic
pr
in
c
ip
le
s
a
bout
c
ir
c
ul
a
nt
m
a
tr
i
c
e
s
.
S
e
c
ti
on
4
pr
e
s
e
nt
s
th
e
pr
opos
e
d
c
ir
c
ul
a
nt
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2252
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8814
A
nov
e
l
c
ir
c
ul
ant
m
at
r
ix
-
bas
e
d M
c
E
li
e
c
e
f
r
am
e
w
o
r
k
f
or
s
e
c
u
r
e
di
gi
ta
l
c
om
m
uni
c
at
io
n
(
R
av
ik
um
ar
I
nak
ot
i
)
295
m
a
tr
ix
-
ba
s
e
d
M
c
E
li
e
c
e
c
r
ypt
os
ys
t
e
m
in
th
is
pa
pe
r
.
S
e
c
ti
on
5
of
th
is
pa
pe
r
pr
e
s
e
nt
s
th
e
s
e
c
ur
it
y
a
na
ly
s
is
a
nd
c
om
pa
r
is
on
of
th
e
pr
opos
e
d
c
ir
c
ul
a
nt
m
a
tr
ix
-
ba
s
e
d
M
c
E
li
e
c
e
c
r
ypt
os
ys
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m
,
a
nd
la
s
tl
y, s
e
c
ti
on
6
of
th
is
p
a
pe
r
dr
a
w
s
a
c
onc
lu
s
io
n on the
c
ir
c
ul
a
nt
m
a
tr
ix
-
ba
s
e
d
M
c
E
li
e
c
e
c
r
y
pt
os
ys
te
m
.
2.
R
E
L
A
T
E
D
WORK
O
ne
of
th
e
f
ir
s
t
a
nd
lo
nge
s
t
-
la
s
ti
ng
publ
ic
-
ke
y
e
nc
r
ypt
io
n
s
c
he
m
e
s
i
s
th
e
M
c
E
li
e
c
e
s
y
s
te
m
f
or
in
f
or
m
a
ti
on
e
nc
odi
ng,
c
r
e
a
te
d
in
th
e
ye
a
r
1978
by
R
ob
e
r
t
J
.
M
c
E
li
e
c
e
.
I
ts
de
f
e
n
s
e
a
ga
in
s
t
qua
nt
um
a
tt
a
c
k
s
,
w
hi
c
h
c
om
e
f
r
om
it
s
us
e
of
e
r
r
or
-
c
or
r
e
c
ti
ng
c
ode
s
,
ha
s
e
nt
ic
e
d
a
s
ig
ni
f
ic
a
nt
in
te
r
e
s
t
by
num
e
r
ous
e
nt
it
ie
s
in
th
e
c
r
ypt
ogr
a
phy
w
or
ld
.
T
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
is
th
or
oughly
c
ove
r
e
d
in
th
is
li
te
r
a
tu
r
e
r
e
vi
e
w
,
w
hi
c
h
a
ls
o
e
xpl
or
e
s
it
s
r
e
c
e
nt
de
v
e
lo
pm
e
nt
s
,
s
e
c
ur
it
y
f
e
a
tu
r
e
s
,
im
pl
e
m
e
nt
a
ti
on
di
f
f
ic
ul
ti
e
s
,
a
nd
th
e
or
e
ti
c
a
l
unde
r
pi
nni
ngs
.
T
he
M
c
E
li
e
c
e
c
r
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os
y
s
te
m
wa
s
in
tr
oduc
e
d
by
R
obe
r
t
J
.
M
c
E
li
e
c
e
,
w
ho
a
l
s
o
s
ugge
s
t
ed
us
in
g
a
lg
e
br
a
ic
c
odi
ng
th
e
or
y
f
or
publ
ic
-
ke
y
e
nc
r
ypt
io
n.
I
t
de
s
c
r
ib
e
s
th
e
f
unda
m
e
nt
a
l
id
e
a
s
a
nd
dr
a
w
s
a
tt
e
nt
io
n
to
it
s
pos
s
ib
le
be
ne
f
it
s
ove
r
a
lt
e
r
na
ti
ve
e
nc
r
ypt
io
n
te
c
hni
que
s
.
M
a
duni
e
t
al
.
[
19]
e
xa
m
in
e
a
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
va
r
ia
ti
on
th
a
t
c
a
n
gua
r
a
nt
e
e
th
a
t
e
n
c
odi
ng
ut
il
iz
e
d
a
s
th
e
s
ha
r
e
d
ge
ne
r
a
te
d
ke
y
i
s
no
m
or
e
s
im
il
a
r
to
th
e
pe
r
m
ut
a
ti
on
of
th
e
s
e
c
r
e
t/
uns
ha
r
e
d
c
ode
.
A
s
a
r
e
s
ul
t,
th
e
a
dopt
io
n
of
tr
a
di
ti
ona
l
c
ode
f
a
m
il
ie
s
,
s
uc
h
a
s
R
e
e
d
-
S
ol
om
on
c
ode
s
,
w
hi
c
h
ha
ve
be
e
n
lo
ng
-
s
ta
ndi
ng
e
xc
lu
s
io
ns
f
r
om
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
te
m
of
in
f
or
m
a
ti
on
e
nc
odi
ng
due
to
s
a
f
e
ty
c
onc
e
r
ns
,
m
a
y
b
e
gi
ve
n
a
not
he
r
lo
ok.
T
hi
s
e
le
va
te
d
th
e
publ
ic
ke
y’
s
s
e
c
ur
it
y l
e
ve
l.
T
he
pr
im
a
r
y be
ne
f
it
s
of
t
he
s
ugge
s
te
d a
ppr
oa
c
h a
r
e
t
he
s
e
be
c
a
u
s
e
i
t
is
w
id
e
ly
r
e
c
ogni
z
e
d
th
a
t
th
e
s
e
c
a
te
gor
ie
s
of
e
nc
odi
ngs
c
a
n r
e
s
ul
t
in
a
de
c
r
e
a
s
e
i
n m
a
gni
tu
de
of
t
he
s
ha
r
e
d ke
ys
or
, c
om
pa
r
a
bl
y,
a
n
in
c
r
e
a
s
e
i
n i
nf
or
m
a
ti
on de
c
odi
ng r
e
s
is
ti
vi
ty
.
N
e
w
pa
r
a
m
e
te
r
s
f
or
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
a
nd
N
ie
de
r
r
e
it
e
r
c
r
ypt
os
ys
te
m
s
a
r
e
pr
opos
e
d
[
20]
,
gua
r
a
nt
e
e
in
g
ba
s
e
li
ne
s
e
c
ur
it
y
a
c
r
os
s
a
ll
in
ve
s
ti
ga
t
e
d
th
r
e
a
ts
.
T
he
m
odi
f
ie
d
s
e
tt
in
gs
ta
ke
in
to
c
ons
id
e
r
a
ti
on
th
e
im
pr
ove
d
th
r
e
a
t,
th
e
r
e
c
e
nt
ly
a
dde
d
bi
na
r
y
G
oppa
c
ode
li
s
t
de
c
odi
ng
a
ppr
oa
c
h,
a
nd
th
e
opt
io
n
to
e
m
pl
oy
c
ode
s
e
que
nc
e
s
th
a
t
a
r
e
n’
t
m
ul
ti
pl
e
of
two.
F
o
r
th
e
s
a
m
e
le
ve
l
of
s
e
c
ur
it
y,
th
e
r
e
s
ul
ti
ng
s
ha
r
e
d
-
ke
y
le
ngt
hs
a
r
e
s
ig
ni
f
ic
a
nt
ly
s
hor
te
r
c
om
pa
r
e
d
to
pr
io
r
pa
r
a
m
e
te
r
s
e
tt
in
gs
s
e
le
c
ti
on.
T
o
f
ur
th
e
r
c
la
r
if
y
on
th
e
na
tu
r
e
of
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
’
s
qua
nt
um
r
e
s
is
ta
nc
e
,
th
e
a
ut
hor
s
e
xa
m
in
e
pot
e
nt
ia
l
qua
nt
um
a
tt
a
c
ks
a
nd
s
ugge
s
t
de
f
e
ns
e
s
tr
a
te
gi
e
s
.
P
une
ya
ni
a
nd
B
ha
t
[
21]
de
m
ons
tr
a
te
d
e
xe
c
ut
io
n
of
a
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
te
m
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n
on
a
f
ie
ld
-
pr
ogr
a
m
m
a
bl
e
g
a
te
a
r
r
a
y
(
F
P
G
A
)
,
gua
r
a
nt
e
e
in
g
th
e
s
e
c
ur
it
y
gui
de
li
ne
s
pr
ovi
de
d by the
E
ur
ope
a
n T
e
le
c
om
m
uni
c
a
ti
ons
S
ta
n
da
r
ds
I
ns
ti
tu
te
f
or
t
he
ne
xt
w
a
ve
of
e
nc
r
ypt
io
n
s
ys
te
m
s
th
a
t
a
r
e
r
obu
s
t
to
qua
nt
um
r
e
s
is
t
a
nc
e
.
E
m
pl
oyi
ng
a
s
h
a
r
e
d
ke
y
w
it
h
a
byt
e
le
ngt
h
of
2,097,152,
th
e
s
ugge
s
te
d
im
pl
e
m
e
nt
a
ti
on
by
th
e
a
ut
hor
s
of
f
e
r
s
qua
nt
um
s
e
c
ur
it
y
w
it
h
bi
ts
be
yond
128.
T
he
s
ugge
s
te
d
s
ys
te
m
is
bui
lt
a
r
ound
a
ha
r
dw
a
r
e
a
nd
s
of
twa
r
e
s
e
tt
in
gs
th
a
t
m
a
ke
s
us
e
of
a
n
A
X
14
li
te
in
te
r
f
a
c
e
to
li
nk
a
n
A
R
M
C
or
te
x
-
A
53
c
or
e
to
a
c
opr
oc
e
s
s
or
.
T
he
s
ta
t
e
-
of
-
th
e
-
a
r
t
c
om
pr
e
he
ns
iv
e
ove
r
vi
e
w
,
c
om
pone
nt
-
by
-
c
om
pone
nt
a
lg
or
it
hm
ic
de
s
c
r
ip
ti
on,
a
nd
im
pl
e
m
e
nt
a
ti
on
of
th
is
c
r
ypt
os
ys
te
m
a
r
e
pr
e
s
e
nt
e
d
[
22]
.
D
if
f
e
r
e
n
t
M
c
E
li
e
c
e
c
r
ypt
os
y
s
te
m
a
tt
a
c
k
s
a
r
e
c
ove
r
e
d
in
s
e
pa
r
a
te
s
e
c
ti
on
s
.
A
s
id
e
f
r
om
s
im
ul
a
ti
on
of
th
e
c
r
ypt
os
ys
te
m
on
di
f
f
e
r
e
nt
e
xt
e
ns
io
n
de
gr
e
e
s
,
th
e
a
ut
hor
s
a
ls
o
pr
e
s
e
nt
e
xpe
r
im
e
nt
a
l
r
e
s
ul
ts
us
in
g
G
oppa
c
ode
s
.
T
he
a
ut
hor
s
c
onc
lu
de
d t
he
r
e
s
ul
ts
a
nd t
he
di
f
f
e
r
e
nt
i
m
pl
e
m
e
nt
a
ti
on
-
r
e
la
te
d i
s
s
ue
s
ba
s
e
d on th
e
s
im
ul
a
ti
ons
t
ha
t
w
e
r
e
r
un.
T
he
m
ode
r
n
va
r
ia
nt
s
of
th
e
c
la
s
s
ic
a
l
c
r
ypt
os
ys
te
m
s
put
f
or
th
by
H
a
r
ol
d
N
ie
de
r
r
e
it
e
r
(
1986)
a
nd
R
obe
r
t
J
.
M
c
E
li
e
c
e
(
1978)
a
r
e
e
xa
m
in
e
d
[
23]
.
F
iv
e
di
f
f
e
r
e
nt
c
ode
-
ba
s
e
d
s
ha
r
e
d
k
e
y
s
ys
t
e
m
s
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n
ha
ve
be
e
n
th
or
oughly
r
e
vi
e
w
e
d.
I
t
is
de
m
ons
tr
a
te
d
th
a
t
th
e
r
e
a
r
e
s
e
r
io
us
pr
obl
e
m
s
w
it
h
s
e
ve
r
a
l
c
ont
e
m
por
a
r
y
e
xpos
it
io
n
s
of
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
a
nd
N
ie
d
e
r
r
e
it
e
r
s
ys
te
m
s
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n.
I
t
ha
s
be
e
n
de
m
ons
tr
a
te
d,
in
pa
r
ti
c
ul
a
r
,
th
a
t
X
G
R
S
e
nc
r
ypt
io
n
s
ys
te
m
s
,
w
hi
c
h
ba
s
e
s
it
s
e
lf
on
th
e
br
oa
de
ne
d
R
e
e
d
-
S
ol
om
on
c
ode
,
c
ont
a
in
m
ul
ti
pl
e
f
la
w
s
a
nd
is
not
a
s
s
e
c
ur
e
a
ga
i
ns
t
th
e
in
f
or
m
a
ti
on
s
e
t
de
c
odi
ng
a
tt
a
c
k
a
s
it
is
s
uppos
e
d
to
be
.
I
t
is
de
m
ons
tr
a
te
d
th
a
t
bot
h
th
e
s
ha
r
e
d
a
nd
u
ns
ha
r
e
d
e
nc
r
ypt
io
n
ke
ys
oc
c
upy
a
s
ig
ni
f
ic
a
nt
quot
a
of
s
to
r
a
ge
a
nd t
ha
t
ke
y ge
ne
r
a
ti
on a
nd de
c
r
ypt
io
n i
n c
ont
e
m
por
a
r
y c
r
ypt
os
ys
te
m
s
t
a
ke
a
l
ong ti
m
e
.
A
nove
l
c
ode
-
ba
s
e
d
di
gi
ta
l
s
ig
na
tu
r
e
bui
lt
on
th
e
M
c
E
li
e
c
e
s
ys
te
m
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n
is
pr
opos
e
d
[
24]
.
A
lg
or
it
hm
s
f
or
th
e
c
ons
tr
uc
ti
on
o
f
s
ha
r
e
d
ke
y
,
s
ig
ni
ng,
a
nd
a
ut
he
nt
ic
a
ti
on
a
r
e
s
how
n.
T
he
publ
ic
ke
y
is
c
r
e
a
te
d
by
th
e
ke
y
ge
ne
r
a
ti
on
a
lg
or
it
hm
us
in
g
r
a
ndom
in
ve
r
s
e
m
a
tr
ic
e
s
.
C
om
pa
r
e
d
to
th
e
C
F
S
s
c
he
m
e
,
th
e
s
ig
ni
ng
a
lg
or
it
hm
is
le
s
s
c
om
pl
e
x
a
nd
t
a
ke
s
l
e
s
s
c
om
put
in
g
ti
m
e
to
s
ig
n
a
doc
um
e
nt
.
F
or
ge
r
ie
s
c
a
n
be
r
e
c
ogni
z
e
d
by
th
e
ve
r
if
ic
a
ti
on
a
lg
or
it
hm
.
I
t
is
de
m
ons
tr
a
te
d
th
a
t
th
e
s
ugg
e
s
te
d
s
c
he
m
e
i
s
r
e
s
i
s
ta
nt
to
s
tr
uc
tu
r
a
l
a
tt
a
c
ks
us
in
g publi
c
ke
y
s
.
B
ir
ha
nu
e
t
al
.
[
25]
e
m
pl
oy
a
n
ir
r
e
gul
a
r
c
ode
ve
r
s
io
n
of
t
he
Q
C
-
L
D
P
C
a
nd
th
e
qua
s
i
-
c
yc
li
c
m
ode
r
a
te
-
de
ns
it
y
pa
r
it
y
-
c
he
c
k
(
Q
C
-
M
D
P
C
)
in
pl
a
c
e
of
G
oppa
c
ode
,
w
hi
c
h
is
us
e
d
in
ta
nd
e
m
to
a
ddr
e
s
s
pr
e
vi
ous
bot
tl
e
ne
c
k
s
in
th
e
s
y
s
te
m
.
R
e
s
ul
t
s
obt
a
in
e
d
by
th
e
pr
opos
e
d
m
e
th
od
a
ls
o
c
onf
ir
m
e
d
th
a
t
th
e
le
ngt
h
of
th
e
s
ha
r
e
d
ke
y
w
a
s
a
ppr
opr
ia
te
ly
s
hor
te
ne
d.
T
he
f
a
c
t
th
a
t
th
is
r
e
le
a
s
e
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
is
m
or
e
r
e
s
is
ta
nt
to
m
e
s
s
a
g
e
-
r
e
s
e
nd
th
r
e
a
t
s
is
a
not
he
r
be
ne
f
it
ove
r
th
e
pr
e
vi
ous
it
e
r
a
ti
on.
S
ut
r
a
dha
r
[
26]
f
oc
us
e
s
on
th
e
de
ve
lo
pm
e
nt
of
in
di
s
ti
ngui
s
ha
bi
li
ty
unde
r
a
da
pt
iv
e
c
h
os
e
n
c
ip
he
r
te
xt
a
tt
a
c
k
2
(
I
N
D
-
C
C
A
2
)
s
e
c
ur
e
ve
r
s
io
n
of
th
e
c
onve
nt
io
na
l
M
c
E
li
e
c
e
s
ys
te
m
of
in
f
or
m
a
ti
on
e
n
c
r
ypt
io
n.
T
he
a
ut
hor
s
e
m
pl
oy
th
e
S
-
r
e
pe
ti
ti
on
e
nc
r
ypt
io
n
of
S
/2
di
f
f
e
r
e
nt
in
f
or
m
a
ti
on
w
it
h
a
s
in
gl
e
ty
pi
c
a
l
pe
r
m
ut
a
ti
on,
w
hi
c
h
c
ont
r
a
di
c
t
th
e
S
-
r
e
pe
ti
ti
on
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
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8814
I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
. 15, No. 1, M
a
r
c
h 2026
:
293
-
302
296
e
nc
r
ypt
io
n
of
s
in
gl
e
in
f
or
m
a
ti
on
in
ot
he
r
m
odi
f
ic
a
ti
on.
N
e
w
M
c
E
li
e
c
e
s
y
s
te
m
of
in
f
or
m
a
ti
on
e
nc
r
ypt
io
n
,
w
it
h
it
s
f
ounda
ti
on
on
punc
tu
r
e
d
R
M
c
ode
s
a
r
e
[
27]
.
T
he
y
e
f
f
e
c
ti
ve
ly
de
m
ons
tr
a
te
th
e
in
e
f
f
ic
a
c
y
of
w
e
ll
-
known
s
e
c
ur
it
y
th
r
e
a
ts
on
th
e
s
ugge
s
te
d
R
M
c
ode
-
ba
s
e
d
M
c
E
li
e
c
e
c
r
ypt
os
y
s
te
m
,
in
c
lu
di
ng
th
e
M
in
de
r
-
S
hokr
ol
la
hi
,
C
hi
z
hov
-
B
or
odi
n,
a
nd
s
qua
r
e
c
ode
a
tt
a
c
ks
.
I
n
or
de
r
to
gua
r
d
a
ga
in
s
t
th
e
a
f
or
e
m
e
nt
io
ne
d
a
tt
a
c
ks
on
th
e
s
ugge
s
t
e
d
R
M
c
ode
-
ba
s
e
d
c
r
ypt
os
ys
te
m
s
,
th
e
a
ut
hor
s
de
vi
s
e
d
a
n
id
e
a
l
punc
tu
r
in
g
s
c
he
m
e
.
S
pe
c
if
ic
a
ll
y,
th
e
y
de
te
r
m
in
e
d
th
e
pr
e
c
is
e
a
r
e
a
s
of
punc
tu
r
in
g
pos
it
io
ns
w
he
r
e
th
e
ge
ne
r
a
to
r
m
a
tr
ix
’
s
le
a
s
t
a
m
ount
of
punc
tu
r
e
d c
ol
um
ns
c
oul
d be
f
ound.
3.
P
R
E
L
I
M
I
N
A
R
I
E
S
T
he
w
or
ki
ng
of
c
onve
nt
io
n
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
a
nd
c
ir
c
ul
a
nt
m
a
tr
ix
a
r
e
de
s
c
r
ib
e
d
i
n
th
is
s
e
c
ti
on
.
3.1.
T
h
e
M
c
E
li
e
c
e
c
r
yp
t
os
ys
t
e
m
R
obe
r
t
J
.
M
c
E
li
e
c
e
c
r
e
a
te
d
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
,
a
publ
i
c
-
ke
y
e
nc
r
ypt
io
n
a
lg
or
it
hm
,
in
1978.
I
n
c
ont
r
a
s
t
to
num
e
r
ous
ot
he
r
publ
ic
-
ke
y
e
nc
r
ypt
io
n
a
lg
or
it
hm
s
th
a
t
de
pe
nd
on
di
s
c
r
e
t
e
lo
ga
r
it
hm
pr
obl
e
m
s
or
f
a
c
to
r
iz
a
ti
on di
f
f
ic
ul
ti
e
s
, M
c
E
li
e
c
e
i
s
ba
s
e
d on the
c
ha
ll
e
nge
of
de
c
odi
ng e
r
r
o
r
-
c
or
r
e
c
ti
ng
c
ode
s
, w
hi
c
h a
r
e
f
r
e
que
nt
ly
ut
il
iz
e
d
in
di
gi
ta
l
c
om
m
uni
c
a
ti
ons
. T
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
r
e
l
ie
s
on G
oppa
c
ode
, pe
r
m
ut
a
ti
o
n
m
a
tr
ix
,
a
nd
a
n
in
ve
r
t
ib
le
m
a
tr
ix
to
s
c
r
a
m
bl
e
th
e
pl
a
in
te
xt
a
nd
c
onc
e
a
l
a
s
e
c
r
e
t
ke
y.
T
he
c
or
r
e
s
ponding
c
ode
w
or
d
ge
ne
r
a
te
d
is
f
in
a
ll
y
pe
r
m
ut
e
d
be
f
or
e
tr
a
ns
m
it
te
d
th
r
ough
th
e
tr
a
ns
m
is
s
io
n
c
h
a
nne
l.
T
h
e
pl
a
in
te
xt
f
r
om
th
e
s
e
nde
r
is
f
ir
s
t
s
c
r
a
m
bl
e
d
,
a
nd
th
e
ge
ne
r
a
te
d
c
ode
w
or
d
is
pe
r
m
ut
e
d.
A
s
e
t
of
bi
ts
up
to
t
f
r
om
th
e
c
ode
w
or
d
a
r
e
f
li
ppe
d,
w
it
h
t
r
e
pr
e
s
e
nt
in
g
th
e
e
r
r
or
c
or
r
e
c
ti
on
c
ode
of
th
e
ge
ne
r
a
te
d
c
ode
w
or
d.
T
he
publ
ic
ke
y
of
M
c
E
li
e
c
e
c
r
ypt
os
ys
t
e
m
is
a
c
om
bi
na
ti
on
of
non
-
s
in
gu
la
r
‘
k
’
by
‘
k
’
s
c
r
a
m
bl
e
m
a
tr
ix
,
a
‘
k
’
by
‘
n
’
ge
ne
r
a
to
r
m
a
tr
ix
,
a
nd
‘
n
by
n
’
pe
r
m
ut
a
ti
on
m
a
tr
ix
.
T
he
e
nc
r
ypt
i
on
a
nd
de
c
r
ypt
io
n
pr
oc
e
s
s
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
i
s
gi
ve
n a
s
f
ol
lo
w
s
.
3.1.1.
P
r
iv
at
e
k
e
y ge
n
e
r
at
io
n
T
he
pr
iv
a
te
ke
y
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
t
e
m
is
a
c
om
bi
na
t
io
n
of
th
e
ge
ne
r
a
t
or
m
a
tr
ix
‘
G
’
,
th
e
s
c
r
a
m
bl
in
g m
a
tr
ix
‘
S
’
,
a
nd t
he
r
a
ndom pe
r
m
ut
a
ti
on ma
tr
ix
‘
P
’
. T
h
e
ge
ne
r
a
to
r
m
a
tr
ix
i
s
gi
ve
n a
s
s
how
n i
n (
1)
.
=
(
|
(
−
)
′
)
(
1)
W
he
r
e
‘
I
k
’
is
a
n
id
e
nt
i
ty
m
a
tr
ix
of
di
m
e
ns
io
n
k
by
k
,
a
nd
′
is
a
r
a
n
dom
pe
r
m
ut
a
ti
on
m
a
t
r
ix
of
di
m
e
ns
io
n
‘
(
−
)
’
. T
he
pa
r
it
y c
he
c
k m
a
tr
ix
of
th
e
l
in
e
a
r
c
ode
i
s
obt
a
in
e
d a
s
s
ho
w
n i
n (
2)
.
T
he
pr
iv
a
te
ke
y i
s
‘
S
k
’
i
s
th
e
c
om
bi
na
ti
on of
t
he
m
a
tr
ic
e
s
‘
G
’
, ‘
S
’
,
a
nd ‘
P
’
.
−
(
)
=
(
′
|
(
−
)
)
(
2)
3.1.2.
P
u
b
li
c
k
e
y ge
n
e
r
at
io
n
T
he
publ
ic
ke
y
c
ons
is
t
s
of
‘
k
’
by
‘
n
’
m
a
tr
ix
G
′
de
f
in
e
d
by
‘
G
.
S
.
P
’
a
nd
e
r
r
or
c
or
r
e
c
ti
ng
c
a
pa
bi
li
ty
t.
T
he
e
nc
r
ypt
io
n
pr
oc
e
s
s
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
is
a
s
f
ol
l
ow
s
:
g
iv
e
n
a
m
e
s
s
a
ge
m
∈
2
,
a
r
a
ndom
e
r
r
or
ve
c
to
r
e
∈
2
is
c
hos
e
n w
it
h a
w
e
ig
ht
w
≤t
a
nd c
om
put
e
c
or
r
e
s
pond
in
g e
nc
r
ypt
e
d m
e
s
s
a
g
e
a
s
(
3)
.
=
′
+
(
3)
T
o de
c
r
ypt
t
he
e
nc
r
ypt
e
d m
e
s
s
a
g
e
, t
he
f
ol
lo
w
in
g i
s
c
om
put
e
d a
s
s
how
n i
n (
4)
.
−
1
=
+
−
1
(
4)
S
in
c
e
‘
P
’
is
a
pe
r
m
ut
a
ti
on
m
a
tr
ix
,
‘
−
1
=
’
is
e
qua
ll
y
a
pe
r
m
ut
a
ti
on
m
a
tr
ix
a
s
s
uc
h,
th
e
ve
c
to
r
‘
−
1
’
ha
s
th
e
s
a
m
e
w
e
ig
ht
a
s
e
.
T
he
r
e
f
or
e
,
‘
’
c
a
n
be
obt
a
in
e
d
by
de
c
odi
n
g
‘
−
1
’
.
F
in
a
ll
y,
‘
’
c
a
n
be
m
ul
ti
pl
ie
d
by
−
1
a
s
(
)
−
1
to
obt
a
in
.
3.2.
C
ir
c
u
la
n
t
m
at
r
ix
C
ir
c
ul
a
nt
m
a
tr
ix
is
a
m
a
tr
ix
in
w
hi
c
h
e
a
c
h
r
ow
r
e
la
ti
ve
to
th
e
pr
e
vi
ous
r
ow
ve
c
to
r
is
r
ot
a
te
d
one
e
le
m
e
nt
to
th
e
r
ig
ht
[
28]
.
T
he
pr
oduc
t
o
f
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
is
a
ls
o
a
c
ir
c
ul
a
nt
m
a
tr
ix
a
nd
c
om
m
ut
a
ti
ve
[
29
]
.
M
a
tr
ix
A
is
a
c
ir
c
ul
a
nt
m
a
tr
ix
w
it
h
e
nt
r
ie
s
ge
ne
r
a
te
d
f
r
o
m
th
e
n
-
ve
c
to
r
{
1
,
2
,
.
.
.
,
}
by
c
yc
li
c
a
ll
y
pe
r
m
ut
in
g i
ts
e
nt
r
ie
s
, a
nd i
s
of
t
he
f
or
m
a
s
s
how
n i
n (
5)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
A
dv A
ppl
S
c
i
I
S
S
N
:
2252
-
8814
A
nov
e
l
c
ir
c
ul
ant
m
at
r
ix
-
bas
e
d M
c
E
li
e
c
e
f
r
am
e
w
o
r
k
f
or
s
e
c
u
r
e
di
gi
ta
l
c
om
m
uni
c
at
io
n
(
R
av
ik
um
ar
I
nak
ot
i
)
297
=
[
1
2
…
1
…
−
1
.
.
.
2
.
.
.
3
.
.
.
…
.
.
.
1
]
(
5)
F
or
e
xa
m
pl
e
, w
e
de
f
in
e
t
he
c
ir
c
ul
a
nt
m
a
tr
ix
ge
ne
r
a
te
d by thr
e
e
e
le
m
e
nt
s
a
s
s
how
n i
n (
6)
.
3
(
,
,
)
=
[
]
(
6)
T
he
bl
oc
k
c
ir
c
ul
a
nt
of
c
ir
c
ul
a
nt
m
a
tr
ix
(
)
is
a
n
M
×
M
m
a
tr
ix
f
or
e
ve
r
y
=
1
,
2
,
.
.
.
,
.
T
he
n
{
1
,
2
,
.
.
.
,
}
ge
ne
r
a
te
s
a
n
N
M
×
N
M
bl
oc
ks
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
.
A
n
e
xa
m
pl
e
of
a
bl
oc
k
c
ir
c
ul
a
nt
m
a
tr
ix
(
C
ir
c
B
)
f
or
‘
A
’
de
f
in
e
d a
s
s
how
n i
n (
7)
to
(
9)
.
=
[
1
2
3
4
5
6
7
8
8
]
;
ℎ
(
7)
=
(
,
,
)
=
[
]
(
8)
W
e
ha
ve
,
=
(
1
,
2
,
3
)
=
[
1
2
3
3
1
2
2
3
1
]
;
=
(
4
,
5
,
6
)
=
[
4
5
6
6
4
5
5
6
4
]
;
=
(
7
,
8
,
9
)
=
[
7
8
9
9
7
8
8
9
7
]
(9
)
4.
M
A
T
E
R
I
A
L
S
A
N
D
M
E
T
H
O
D
I
n
th
is
s
e
c
ti
on,
th
e
c
ir
c
ul
a
nt
m
a
tr
ix
-
ba
s
e
d
M
c
E
li
e
c
e
c
r
ypt
o
s
ys
te
m
is
pr
e
s
e
nt
e
d
in
de
ta
il
.
T
he
pr
opos
e
d
m
e
th
od
us
e
s
th
e
s
tr
uc
tu
r
e
of
th
e
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
s
c
he
m
e
but
in
tr
oduc
e
s
th
e
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
to
im
pr
ove
th
e
e
f
f
ic
a
c
y
of
s
to
r
a
ge
a
nd
c
om
put
a
ti
on.
T
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
is
ba
s
e
d
on
th
e
di
f
f
ic
ul
ty
o
f
de
c
odi
ng
r
a
ndom
li
ne
a
r
c
ode
s
,
of
te
n
us
in
g
G
oppa
c
ode
s
,
w
hi
c
h
m
a
ke
s
it
r
e
s
is
ta
nt
to
qua
nt
um
-
ba
s
e
d
a
tt
a
c
k
s
.
U
s
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
a
ll
ow
s
f
or
a
r
e
duc
ti
on
in
ke
y s
iz
e
,
e
nh
a
nc
in
g
th
e
s
ys
te
m
'
s
pr
a
c
ti
c
a
li
ty
. T
he
pr
opos
e
d
s
ys
te
m
c
on
s
is
ts
of
t
hr
e
e
pha
s
e
s
, w
hi
c
h i
nc
lu
de
ke
y ge
n
e
r
a
ti
on pha
s
e
,
e
nc
r
ypt
io
n pha
s
e
,
a
nd de
c
r
ypt
io
n pha
s
e
.
4.1.
K
e
y
ge
n
e
r
at
io
n
p
h
as
e
I
n
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
,
k
e
y
ge
ne
r
a
ti
on
in
vol
ve
s
c
r
e
a
ti
ng
a
publ
ic
a
nd
pr
iv
a
te
ke
y
pa
ir
th
r
ough
a
G
oppa
c
ode
(
or
a
c
om
pa
r
a
bl
e
e
r
r
or
-
c
or
r
e
c
ti
ng
c
o
de
)
a
nd
a
ppl
yi
ng
r
a
ndom
t
r
a
ns
f
or
m
a
ti
ons
to
c
onc
e
a
l
it
s
s
tr
uc
tu
r
e
.
B
y
us
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
,
s
to
r
a
ge
r
e
qui
r
e
m
e
nt
s
f
or
th
e
ge
ne
r
a
to
r
m
a
tr
ix
a
r
e
m
in
im
iz
e
d,
a
s
th
e
e
nt
ir
e
m
a
tr
ix
c
a
n
be
ge
ne
r
a
te
d
f
r
om
ju
s
t
one
r
ow
.
T
o
ge
n
e
r
a
te
th
e
pr
iv
a
te
a
nd
publ
ic
ke
y,
a
c
ode
pa
r
a
m
e
te
r
n
r
e
pr
e
s
e
nt
in
g
c
ode
le
ngt
h,
k
r
e
pr
e
s
e
nt
in
g
th
e
di
m
e
ns
io
n
of
th
e
c
ode
pa
r
a
m
e
te
r
,
a
nd
e
r
r
or
c
or
r
e
c
ti
on
c
a
pa
bi
li
ty
t
is
c
hos
e
n.
A
f
te
r
c
hoos
in
g
th
e
pa
r
a
m
e
te
r
s
,
th
e
bi
na
r
y
G
oppa
c
ode
c
,
w
it
h
a
ge
ne
r
a
to
r
m
a
tr
ix
‘
G
’
of
s
iz
e
k
x
n
is
c
hos
e
n.
i)
P
r
im
a
r
y
ke
y
ge
ne
r
a
ti
on
:
t
o
c
ons
tr
uc
t
th
e
pr
iv
a
te
ke
y,
a
ge
ne
r
a
to
r
m
a
tr
ix
‘
G
’
is
c
r
e
a
te
d
f
or
th
e
c
hos
e
n
G
oppa
c
ode
,
s
tr
uc
tu
r
e
d
a
s
a
c
ir
c
ul
a
nt
m
a
tr
ix
.
F
or
e
xa
m
pl
e
,
a
ve
c
to
r
v
=
{
v
0
,
v
1
,
v
2
,
…
,
v
n
−
1
}
is
c
hos
e
n
to
ge
ne
r
a
te
a
c
ir
c
ul
a
nt
m
a
tr
ix
‘
G
’
by
r
ot
a
ti
ng
v
r
ow
by
r
ow
.
I
n a
d
di
ti
on
to
G
,
two s
e
c
r
e
t
m
a
tr
ic
e
s
‘
S
’
a
nd
‘
P
’
r
e
pr
e
s
e
nt
in
g
a
r
a
ndom
in
ve
r
ti
bl
e
k
×
k
m
a
tr
ix
us
e
d
to
pe
r
m
ut
a
te
th
e
pl
a
in
te
xt
a
nd
a
r
a
ndom
pe
r
m
ut
a
ti
on
m
a
tr
ix
of
s
iz
e
n
×
n
th
a
t
s
c
r
a
m
bl
e
d
th
e
or
de
r
of
th
e
c
ode
bi
ts
a
r
e
d
e
f
in
e
d.
T
h
e
pr
iv
a
te
ke
y
i
s
th
e
n ge
ne
r
a
te
d t
o c
om
pos
e
t
h
e
t
r
ip
le
(
G
,
S
,
P
)
a
s
s
how
n i
n (
10)
.
=
(
,
,
)
(
10)
W
he
r
e
‘
G
’
is
th
e
ge
ne
r
a
to
r
m
a
tr
ix
of
G
oppa
c
ode
in
c
ir
c
ul
a
nt
m
a
tr
ix
f
or
m
,
‘
S
’
is
th
e
s
c
r
a
m
bl
in
g
m
a
tr
ix
to
m
a
s
k t
he
s
tr
uc
tu
r
e
of
‘
G
’
a
nd ‘
P
’
i
s
th
e
pe
r
m
ut
a
ti
on ma
tr
ix
t
o di
s
gui
s
e
t
he
a
r
r
a
nge
m
e
nt
of
t
he
c
od
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
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nt
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A
dv A
ppl
S
c
i
,
V
ol
. 15, No. 1, M
a
r
c
h 2026
:
293
-
302
298
ii)
P
ubl
ic
ke
y
ge
ne
r
a
ti
on
:
t
he
publi
c
ke
y
G
′
is
c
om
put
e
d a
s
s
how
n i
n
(
11)
.
′
=
(
11)
B
e
c
a
us
e
‘
G
’
is
c
ir
c
ul
a
nt
,
it
r
e
qui
r
e
s
m
uc
h
le
s
s
s
to
r
a
ge
s
p
a
c
e
.
G
′
is
publ
is
he
d
a
s
a
publ
ic
ke
y,
w
hi
le
(
G
,
S
,
P
)
a
r
e
ke
pt
pr
iv
a
te
.
4.2.
E
n
c
r
yp
t
io
n
p
h
as
e
T
o
e
nc
r
ypt
a
m
e
s
s
a
ge
m
us
in
g
th
e
publ
ic
ke
y
G
′
ge
ne
r
a
te
d
in
th
e
ke
y
ge
ne
r
a
ti
on
pha
s
e
,
‘
’
is
r
e
pr
e
s
e
nt
e
d
a
s
a
bi
na
r
y
ve
c
to
r
of
le
ngt
h
‘
k
’
.
T
he
e
n
c
ode
d
m
e
s
s
a
ge
‘
c
’
is
c
om
put
e
d
by
m
ul
ti
pl
yi
ng
th
e
pl
a
in
m
e
s
s
a
ge
m
w
it
h
th
e
publ
ic
ke
y
G
′
a
nd
a
ddi
ng a
r
a
ndom
e
r
r
or
ve
c
to
r
‘
e
’
of
w
e
ig
ht
‘
t
’
w
it
h
e
xa
c
tl
y
‘
t
’
r
a
ndom
bi
ts
s
e
t
to
1
a
s
s
how
n i
n (
12)
.
=
′
+
(
12)
T
he
e
r
r
or
ve
c
to
r
e
c
om
pl
ic
a
te
s
th
e
de
c
odi
ng
pr
oc
e
s
s
w
it
hout
a
c
c
e
s
s
to
th
e
pr
iv
a
te
ke
y,
th
us
s
a
f
e
gua
r
di
ng
th
e
c
ip
he
r
te
xt
'
s
s
e
c
ur
it
y.
4.3.
T
r
an
s
m
it
t
h
e
c
ip
h
e
r
t
e
xt
T
he
f
in
a
l
e
nc
r
ypt
e
d
out
put
,
d
e
not
e
d
a
s
'
c
'
is
s
e
nt
f
r
om
th
e
s
e
nde
r
to
th
e
r
e
c
e
iv
e
r
ove
r
th
e
c
om
m
uni
c
a
ti
on
c
ha
nne
l.
T
hi
s
c
ip
he
r
te
xt
'
c
'
now
c
ont
a
in
s
th
e
or
ig
in
a
l
m
e
s
s
a
g
e
f
ul
ly
e
nc
ode
d
th
r
ough
th
e
e
nc
r
ypt
io
n
a
lg
or
it
hm
,
c
om
bi
ne
d
w
it
h
a
de
li
be
r
a
te
ly
a
dde
d
r
a
n
dom
e
r
r
or
ve
c
to
r
f
or
e
nha
nc
e
d
s
e
c
ur
it
y.
T
hi
s
tr
a
ns
m
is
s
io
n
s
te
p
c
om
pl
e
te
s
th
e
e
nc
r
ypt
io
n
pha
s
e
,
e
n
s
ur
in
g
th
e
m
e
s
s
a
ge
r
e
m
a
in
s
pr
ot
e
c
te
d
unt
il
pr
ope
r
de
c
r
ypt
io
n w
it
h t
he
s
ha
r
e
d ke
y oc
c
ur
s
a
t
th
e
r
e
c
e
iv
e
r
'
s
e
nd.
4.4.
D
e
c
r
yp
t
io
n
p
h
as
e
I
n
th
is
pha
s
e
,
th
e
pr
im
a
r
y
ke
y
(
,
,
)
is
u
s
e
d
to
d
e
c
r
ypt
th
e
c
ip
h
e
r
te
xt
‘
’
.
T
h
e
f
ir
s
t
s
te
p
in
th
is
pha
s
e
i
s
t
o undo the p
e
r
m
ut
a
ti
on a
ppl
ie
d dur
in
g t
he
e
nc
r
ypt
io
n by c
om
put
in
g ‘
c
P
−
1
’
a
s
s
how
n i
n (
13)
.
′
=
−
1
=
(
′
+
)
−
1
=
+
−
1
(
13)
A
f
te
r
r
e
m
ovi
ng t
he
pe
r
m
ut
a
ti
on, t
he
s
c
r
a
m
bl
in
g m
a
tr
ix
i
s
i
nve
r
t
e
d by mul
ti
pl
yi
ng
c
′
by
S
−
1
a
s
s
how
n i
n (
14)
.
−
1
′
=
−
1
(
+
−
1
)
=
+
−
1
−
1
(
14)
T
he
s
te
p yi
e
ld
s
a
s
c
r
a
m
bl
e
d c
ode
w
or
d w
it
h m
in
or
e
r
r
or
, w
hi
c
h c
a
n be
r
e
s
ol
ve
d dur
in
g e
r
r
or
c
or
r
e
c
ti
on pha
s
e
.
4.5.
E
r
r
or
c
o
r
r
e
c
t
io
n
T
o
c
or
r
e
c
t
e
r
r
or
s
in
th
e
s
c
r
a
m
b
le
d c
o
de
w
or
d, t
h
e
e
r
r
o
r
-
c
or
r
e
c
t
in
g
a
lg
or
it
hm
f
or
G
o
pp
a
c
o
de
a
s
s
o
c
i
a
t
e
d
w
it
h
‘
G
’
is
us
e
d
to
de
c
o
d
e
‘
mG
’
a
nd
r
e
c
ti
f
y
a
n
y
e
r
r
or
i
nt
r
odu
c
e
d
by
‘
e
’
.
T
h
e
r
e
tr
i
e
v
e
d
‘
m
’
is
d
e
c
od
e
d
r
e
s
id
u
a
l
e
r
r
or
s
a
r
e
r
e
m
ov
e
d
f
r
om
t
h
e
c
od
e
w
or
d.
T
h
e
e
r
r
o
r
-
c
or
r
e
c
t
in
g
a
lg
o
r
it
hm
c
on
s
i
s
t
s
of
t
h
e
f
ol
l
ow
in
g
s
t
e
p
s
.
i)
S
yndr
om
e
c
a
lc
ul
a
ti
on:
in
th
is
s
ta
g
e
,
th
e
s
yndr
om
e
is
de
t
e
r
m
in
e
d
to
de
te
c
t
th
e
pr
e
s
e
nc
e
a
nd
lo
c
a
ti
on
of
e
r
r
or
s
.
T
he
s
yndr
om
e
s
of
f
e
r
s
e
s
s
e
nt
i
a
l
in
f
or
m
a
ti
on
r
e
ga
r
d
in
g
th
e
pos
it
io
ns
of
th
e
e
r
r
or
s
a
nd
is
c
a
lc
ul
a
te
d a
s
s
how
n i
n (
15)
.
=
(
15)
H
e
r
e
,
H
r
e
pr
e
s
e
nt
s
t
he
pa
r
it
y
-
c
he
c
k
m
a
tr
ix
r
e
l
a
t
e
d
to
t
he
G
o
pp
a
c
od
e
,
a
nd
H
T
is
i
ts
tr
a
n
s
po
s
e
.
s
=
e
H
T
be
c
a
u
s
e
mG
′
H
T
=
0
if
s
=
0
, t
h
e
n
n
o
e
r
r
or
s
a
r
e
pr
e
s
e
n
t
a
nd
i
f
s
≠
0
, i
t
in
d
ic
a
t
e
s
t
he
pr
e
s
e
n
c
e
of
e
r
r
or
.
ii)
E
r
r
or
lo
c
a
to
r
po
ly
nom
ia
l
c
a
lc
ul
a
to
r
:
in
th
is
pha
s
e
,
th
e
e
r
r
or
lo
c
a
ti
on
is
de
te
r
m
in
e
d
by
c
a
lc
ul
a
ti
ng
th
e
e
r
r
or
lo
c
a
to
r
pol
yno
m
ia
l
σ
(
x
)
us
in
g
th
e
s
yndr
om
e
c
om
put
e
d
in
th
e
s
yndr
om
e
c
a
lc
ul
a
ti
on
pha
s
e
.
T
h
e
e
r
r
or
l
oc
a
to
r
pol
ynomi
a
l
he
lp
s
i
de
nt
if
y t
he
e
r
r
or
lo
c
a
ti
ons
, w
hi
c
h i
s
de
f
in
e
d a
s
s
ho
w
n i
n (
16)
.
(
)
=
∏
(
1
−
)
=
1
(
16)
W
he
r
e
X
i
a
r
e
th
e
lo
c
a
ti
ons
of
th
e
e
r
r
or
s
in
th
e
r
e
c
e
iv
e
d
v
e
c
to
r
.
T
o
de
te
r
m
in
e
th
e
e
r
r
or
lo
c
a
to
r
pol
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l
σ
(
x
)
,
w
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us
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a
P
a
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r
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on
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a
lg
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s
pe
c
if
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a
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de
s
ig
ne
d
f
or
G
oppa
c
ode
s
. F
or
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xa
m
pl
e
, c
ons
id
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r
a
s
im
pl
e
s
c
e
na
r
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h t
w
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r
r
or
s
l
oc
a
te
d a
t
pos
it
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n
X
1
a
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X
2
, t
he
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oc
a
to
r
pol
ynomi
a
l
w
oul
d be
r
e
pr
e
s
e
nt
e
d a
s
s
how
n i
n (
17)
.
(
)
=
(
1
−
1
)
(
1
−
2
)
=
1
−
(
1
+
2
)
+
1
2
2
(
17)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2252
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8814
A
nov
e
l
c
ir
c
ul
ant
m
at
r
ix
-
bas
e
d M
c
E
li
e
c
e
f
r
am
e
w
o
r
k
f
or
s
e
c
u
r
e
di
gi
ta
l
c
om
m
uni
c
at
io
n
(
R
av
ik
um
ar
I
nak
ot
i
)
299
T
he
c
oe
f
f
ic
ie
nt
s
of
σ
(
x
)
c
a
n be
de
te
r
m
in
e
d f
r
om
t
he
s
yndr
om
e
ve
c
t
or
‘
s
’
.
iii)
S
ol
vi
ng
th
e
e
r
r
o
r
lo
c
a
to
r
po
ly
nom
ia
l:
onc
e
th
e
e
r
r
or
lo
c
a
to
r
po
ly
nom
ia
l
(
)
is
obt
a
in
e
d,
th
e
ne
xt
s
te
p
is
to
s
ol
ve
it
to
id
e
nt
if
y
th
e
e
r
r
or
pos
it
io
ns
.
F
in
di
ng
th
e
r
oot
s
of
(
)
=
0
r
e
ve
a
ls
th
e
lo
c
a
ti
ons
X
i
of
th
e
e
r
r
or
s
.
T
he
s
e
r
oot
s
c
a
n
be
f
ound
th
r
ough
di
f
f
e
r
e
nt
a
lg
e
br
a
ic
m
e
th
ods
ta
il
or
e
d
to
th
e
f
in
it
e
f
ie
ld
u
ti
li
z
e
d
by t
he
G
oppa
c
ode
. F
or
e
xa
m
pl
e
,
(
)
=
1
−
1
f
or
a
s
in
gl
e
e
r
r
o
r
a
t
pos
it
io
n
X
1
, s
ol
vi
ng
(
)
=
0
im
m
e
di
a
te
ly
gi
ve
s
=
1
.
iv
)
C
a
lc
ul
a
ti
ng
th
e
e
r
r
or
m
a
gni
tu
de
:
a
f
te
r
pi
npoi
nt
in
g
e
r
r
or
pos
it
io
ns
,
e
r
r
or
m
a
gni
tu
de
s
c
a
n
be
de
te
r
m
in
e
d.
I
n
G
oppa
c
ode
s
,
e
r
r
or
s
a
r
e
ge
ne
r
a
ll
y
bi
na
r
y
(
0
or
1)
,
m
e
a
ni
n
g
th
e
e
r
r
or
ve
c
to
r
e
ha
s
non
-
z
e
r
o
va
lu
e
s
onl
y
a
t
th
e
s
p
e
c
if
ie
d
e
r
r
or
lo
c
a
ti
ons
.
T
he
e
r
r
or
ve
c
to
r
e
is
c
o
ns
tr
uc
te
d
by
s
e
tt
in
g
th
e
id
e
nt
if
ie
d
e
r
r
or
pos
it
io
n
a
s
1.
W
h
e
n
m
ul
ti
pl
e
e
r
r
or
s
a
r
e
pr
e
s
e
nt
,
e
a
c
h
id
e
nt
if
ie
d pos
it
io
n
in
th
e
v
e
c
to
r
is
a
s
s
ig
ne
d a
va
lu
e
of
1, w
it
h a
ll
ot
he
r
pos
it
io
ns
l
e
f
t
a
s
0.
v)
C
or
r
e
c
ti
ng
th
e
e
r
r
or
s
a
nd
r
e
tr
ie
vi
ng
th
e
or
ig
in
a
l
m
e
s
s
a
ge
:
i
n
th
is
pha
s
e
,
th
e
e
r
r
or
s
in
th
e
r
e
c
e
iv
e
d
c
ip
he
r
te
xt
‘
c
’
a
r
e
c
or
r
e
c
te
d us
in
g t
he
i
de
nt
if
ie
d e
r
r
or
ve
c
to
r
‘
e
’
a
s
s
how
n i
n (
18)
.
−
=
′
(
18)
T
hi
s
pr
oduc
e
s
th
e
c
ode
w
or
d
‘
mG
’
,
w
hi
c
h
a
ll
ow
s
u
s
to
r
e
c
ov
e
r
t
he
or
ig
in
a
l
m
e
s
s
a
ge
‘
m
’
by
a
ppl
yi
ng
s
ta
nda
r
d de
c
odi
ng me
th
ods
u
s
e
d f
or
l
in
e
a
r
c
ode
s
.
4.
6.
S
e
c
u
r
it
y
an
al
ys
is
of
t
h
e
M
c
E
li
e
c
e
c
r
yp
t
os
y
s
t
e
m
A
n
a
na
ly
s
is
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
us
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
e
va
lu
a
te
s
it
s
r
e
s
il
ie
nc
e
a
ga
in
s
t
va
r
io
us
a
tt
a
c
ks
,
a
nd
m
or
e
s
pe
c
if
ic
a
ll
y
,
th
os
e
w
hi
c
h
a
r
e
e
na
bl
e
d
by
pr
ope
r
ti
e
s
of
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
.
A
lt
hough
im
pr
ovi
ng
s
to
r
a
ge
e
f
f
ic
ie
nc
y,
th
e
r
e
a
r
e
pa
r
ti
c
ul
a
r
di
f
f
ic
ul
ti
e
s
in
tr
oduc
e
d
by
th
e
us
e
of
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
.
T
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
,
be
in
g
a
s
ta
nda
r
d
a
ppr
oa
c
h,
ta
ke
s
a
dv
a
nt
a
ge
of
th
e
di
f
f
ic
ul
ty
of
r
a
ndoml
y
de
c
odi
ng
li
ne
a
r
c
ode
s
.
M
or
e
s
p
e
c
if
ic
a
ll
y,
f
or
a
r
a
ndoml
y
s
e
l
e
c
te
d
g
e
ne
r
a
to
r
m
a
tr
ix
‘
G
’
,
th
e
ta
s
k
of
r
a
ndoml
y
li
ne
a
r
c
ode
de
c
odi
ng
(
de
c
odi
ng
a
r
a
ndoml
y
li
ne
a
r
c
ode
,
na
m
e
ly
f
in
di
ng
th
e
in
it
ia
l
m
e
s
s
a
ge
a
nd
th
e
e
r
r
or
a
dde
d)
w
it
hout
th
e
pr
iv
a
te
ke
y
is
di
f
f
ic
ul
t.
G
oppa
c
ode
s
a
r
e
c
ho
s
e
n
due
to
th
e
ir
im
pr
ove
d
e
r
r
or
c
or
r
e
c
ti
on
a
nd
r
e
s
il
ie
nc
e
to
e
f
f
ic
ie
nt
de
c
odi
ng a
lg
or
it
hm
s
a
nd, mor
e
s
pe
c
if
ic
a
ll
y, t
o t
hos
e
i
nt
r
oduc
e
d by qua
nt
um
t
hr
e
a
ts
. I
n t
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
,
s
e
c
ur
it
y
a
ga
in
s
t
pot
e
nt
i
a
l
a
tt
a
c
k
s
is
a
c
hi
e
ve
d
by
ut
il
iz
in
g
th
e
a
r
r
a
nge
m
e
nt
of
G
oppa
c
ode
a
nd
two
pr
iv
a
te
tr
a
ns
f
or
m
a
ti
ons
:
f
ir
s
t
ly
,
by
a
ppl
yi
ng
a
n
in
ve
r
ti
bl
e
r
a
ndom
m
a
tr
ix
S
f
or
di
s
tu
r
bi
ng
th
e
in
it
ia
l
m
e
s
s
a
ge
,
a
nd
s
e
c
ondl
y
by
ut
il
iz
in
g
a
pe
r
m
ut
a
ti
on
m
a
tr
ix
P
f
or
r
a
ndoml
y
r
e
a
r
r
a
ngi
ng
c
ode
pos
it
io
ns
.
A
dopt
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
in
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
a
lt
e
r
s
th
e
publ
ic
ke
y
G′
,
r
e
qui
r
in
g
a
c
a
r
e
f
ul
a
s
s
e
s
s
m
e
nt
of
a
ny vulne
r
a
bi
li
ti
e
s
r
e
la
t
e
d t
o c
ir
c
ul
a
nt
m
a
tr
ix
c
ha
r
a
c
te
r
is
ti
c
s
.
4.
7
.
S
e
c
u
r
it
y of
t
h
e
c
ir
c
u
la
n
t
m
at
r
ix
b
as
e
d
Mc
E
li
e
c
e
c
r
yp
t
o
s
ys
t
e
m
F
ir
s
t
of
a
ll
,
th
e
us
e
of
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
he
lp
s
to
m
in
im
iz
e
t
he
s
iz
e
of
th
e
ke
y.
I
t
a
l
s
o
im
pos
e
s
a
pa
tt
e
r
n
on
th
e
ge
n
e
r
a
to
r
m
a
tr
ix
th
a
t
m
ig
ht
a
f
f
e
c
t
s
e
c
ur
it
y.
I
n
t
hi
s
s
e
c
ti
on,
w
e
e
xa
m
in
e
th
e
pr
im
a
r
y
ty
pe
s
of
a
tt
a
c
ks
i
n de
ta
il
, f
oc
us
in
g pa
r
ti
c
ul
a
r
ly
on how the
ut
il
iz
a
ti
on of
a
c
ir
c
ul
a
nt
m
a
tr
ix
i
nf
lu
e
nc
e
s
t
he
m
.
4.7.1.
S
t
r
u
c
t
u
r
al
at
t
ac
k
T
hi
s
ty
pe
of
a
tt
a
c
k
tr
ie
s
to
f
in
d
pa
tt
e
r
ns
in
th
e
pub
li
c
ke
y
G
′
to
f
ig
ur
e
out
th
e
pr
iv
a
te
ke
y.
S
in
c
e
a
c
ir
c
ul
a
nt
m
a
tr
ix
G
ha
s
a
r
e
pe
a
ti
ng,
c
ir
c
ul
a
r
s
tr
uc
tu
r
e
,
a
n
a
tt
a
c
k
e
r
m
ig
ht
us
e
th
is
to
tr
y
a
nd
r
e
bui
ld
th
e
m
a
tr
ix
.
T
o
s
to
p
th
is
,
a
r
a
ndom
pe
r
m
ut
a
ti
on
m
a
tr
ix
is
us
e
d
to
m
ix
up
t
he
c
ir
c
ul
a
r
pa
tt
e
r
n
o
f
G
,
m
a
ki
ng
it
ha
r
d
to
s
e
e
th
e
s
tr
uc
tu
r
e
.
A
ls
o,
a
not
he
r
r
a
ndom
m
a
tr
ix
‘
S
’
is
us
e
d
to
m
ix
u
p
e
a
c
h
r
ow
of
th
e
ge
ne
r
a
to
r
m
a
tr
ix
G
,
hi
di
n
g
th
e
c
ir
c
ul
a
nt
pa
tt
e
r
n
e
ve
n
m
or
e
.
A
s
a
r
e
s
ul
t,
th
e
publ
i
c
ke
y
G
′
=
S
G
P
doe
s
n'
t
s
how
th
e
c
ir
c
ul
a
nt
na
tu
r
e
of
G
,
m
a
ki
ng t
hi
s
t
ype
of
a
tt
a
c
k l
e
s
s
e
f
f
e
c
ti
ve
.
4.7.2.
K
n
ow
n
p
la
in
t
e
xt
at
t
ac
k
I
n
a
known
-
pl
a
in
te
xt
a
tt
a
c
k,
a
n
a
dve
r
s
a
r
y
m
a
y
obt
a
in
pa
ir
s
of
pl
a
in
te
xt
s
a
nd
c
ip
he
r
te
xt
m
e
s
s
a
ge
s
a
nd
tr
y
to
ut
il
iz
e
th
is
da
ta
to
r
e
tr
ie
ve
th
e
pr
iv
a
te
ke
y.
F
or
e
xa
m
pl
e
,
le
t’
s
a
s
s
um
e
th
e
a
tt
a
c
ke
r
pos
s
e
s
s
e
s
th
e
m
e
s
s
a
ge
m
a
nd i
ts
c
or
r
e
s
ponding c
ip
he
r
te
xt
‘
c
’
i
s
a
s
s
how
n i
n (
1
9)
.
=
′
+
(
19)
W
he
r
e
‘
e
’
is
a
n
e
r
r
or
ve
c
to
r
of
a
known
w
e
ig
ht
‘
’
.
I
f
a
n
a
tt
a
c
ke
r
ga
th
e
r
s
e
nough
pa
ir
s
(
m
,
c
)
,
th
e
y
c
oul
d
a
tt
e
m
pt
t
o s
ol
ve
f
o
r
G
′
.
H
ow
e
ve
r
, s
in
c
e
G
pr
im
e
e
qua
ls
S
G
P
, t
he
y s
ti
ll
ne
e
d t
o s
e
pa
r
a
te
out
‘
S
’
a
nd
‘
P
’
, w
hi
c
h
is
ha
r
d
to
do
be
c
a
us
e
m
a
tr
ix
ope
r
a
ti
ons
a
r
e
c
om
pl
e
x a
nd
th
e
r
e
a
r
e
e
xt
r
a
e
r
r
or
s
.
T
he
s
e
c
ur
it
y
c
om
e
s
f
r
om
how
ha
r
d
it
is
to
de
c
ode
a
li
ne
a
r
c
ode
a
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f
in
d
th
e
e
xa
c
t
e
r
r
o
r
ve
c
to
r
'
e
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w
it
hout
knowing
th
e
p
r
iv
a
te
ke
y
m
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ic
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‘
S
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P
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300
4.7.3.
D
e
c
od
in
g
at
t
ac
k
A
de
c
o
di
ng
a
t
ta
c
k
tr
ie
s
to
ge
t
th
e
or
ig
in
a
l
m
e
s
s
a
g
e
m
by
s
ol
vi
ng
th
e
e
qu
a
ti
on
c
=
m
G
′
+
e
di
r
e
c
tl
y
.
I
n
th
is
pr
oc
e
s
s
,
bot
h
m
a
nd
e
n
e
e
d
to
be
f
o
und
w
he
n
G
′
a
nd
c
a
r
e
k
now
n.
T
he
pr
obl
e
m
i
s
h
a
r
d
b
e
c
a
us
e
de
c
od
in
g
a
r
a
ndom
li
ne
a
r
c
ode
is
a
n
N
P
-
h
a
r
d
t
a
s
k
,
w
hi
c
h
is
w
hy
M
c
E
li
e
c
e
c
r
ypt
o
s
y
s
te
m
is
s
e
c
ur
e
.
E
v
e
n
th
ough
us
i
ng
a
c
ir
c
ul
a
nt
m
a
tr
ix
f
or
G
m
i
ght
m
a
ke
it
s
s
tr
uc
tu
r
e
e
a
s
ie
r
to
g
ue
s
s
,
th
e
m
a
tr
ic
e
s
‘
S
’
a
nd
‘
P
’
m
ix
up
th
e
s
tr
uc
tu
r
e
,
hi
di
ng
it
f
r
om
a
tt
a
c
k
e
r
s
.
T
he
S
te
r
n
a
lg
or
it
hm
a
n
d
ot
h
e
r
in
f
or
m
a
ti
on
-
s
e
t
de
c
odi
ng
m
e
th
o
ds
a
r
e
a
m
ong
th
e
be
s
t
w
a
ys
to
tr
y
a
nd
s
o
lv
e
th
i
s
pr
obl
e
m
.
H
o
w
e
v
e
r
,
th
e
s
e
m
e
th
ods
a
r
e
not
pr
a
c
ti
c
a
l
f
or
la
r
g
e
c
od
e
s
iz
e
s
li
ke
n
=2
,
048 without
knowi
ng
pr
iv
a
te
ke
y
m
a
tr
ic
e
s
‘
S
’
a
nd
‘
P
’
.
T
h
e
c
od
e
p
a
r
a
m
e
te
r
s
n
a
nd
t
a
r
e
c
ho
s
e
n
s
o t
h
a
t
e
ve
n
t
he
be
s
t
d
e
c
o
di
ng m
e
th
od
s
w
oul
d
ne
e
d t
oo
m
uc
h c
o
m
put
in
g p
ow
e
r
, e
v
e
n f
or
qua
nt
um
c
om
put
e
r
s
.
4.7
.4.
Q
u
an
t
u
m
at
t
a
c
k
Q
u
a
nt
um
c
om
p
ut
e
r
s
c
a
n
s
o
lv
e
c
e
r
t
a
in
m
a
t
h
pr
obl
e
m
s
m
uc
h
f
a
s
t
e
r
th
a
n
r
e
g
ul
a
r
c
o
m
p
ut
e
r
s
,
s
om
e
ti
m
e
s
th
ou
s
a
n
d
s
or
m
il
li
on
s
o
f
ti
m
e
s
q
ui
c
k
e
r
.
A
pr
o
m
in
e
n
t
e
xa
m
pl
e
i
s
S
h
or
'
s
a
lg
or
it
hm
,
w
hi
c
h
e
f
f
ic
i
e
nt
l
y
a
ddr
e
s
s
e
s
th
e
in
te
ge
r
f
a
c
to
r
i
z
a
ti
on
pr
o
bl
e
m
,
r
e
n
de
r
in
g
R
S
A
a
nd
E
C
C
s
u
s
c
e
pt
ib
l
e
to
qu
a
n
tu
m
a
tt
a
c
k
s
.
H
o
w
e
v
e
r
,
d
e
c
odi
ng
r
a
n
dom
li
ne
a
r
c
ode
s
(
t
h
e
f
oun
da
ti
o
n
of
th
e
M
c
E
li
e
c
e
c
r
ypt
o
s
ys
te
m
)
r
e
m
a
in
s
c
ha
ll
e
n
gi
n
g
e
v
e
n
f
or
q
ua
nt
u
m
c
om
put
e
r
s
.
G
r
ov
e
r
'
s
a
l
gor
it
h
m
,
kn
ow
n
f
or
a
c
c
e
le
r
a
t
in
g
br
ut
e
-
f
o
r
c
e
s
e
a
r
c
h,
i
s
no
t
p
a
r
t
ic
ul
a
r
ly
e
f
f
e
c
t
iv
e
i
n
th
i
s
c
on
te
xt
,
a
s
th
e
de
c
o
di
ng
pr
o
bl
e
m
d
o
e
s
n
ot
e
a
s
il
y
a
ll
o
w
f
or
br
ut
e
-
f
or
c
e
a
ppr
oa
c
he
s
.
I
n
th
e
M
c
E
li
e
c
e
c
r
y
pt
o
s
y
s
te
m
,
t
he
d
e
c
odi
ng
pr
o
bl
e
m
r
e
m
a
i
ns
di
f
f
ic
ul
t
b
e
c
a
u
s
e
of
th
e
va
s
t
nu
m
b
e
r
of
p
os
s
ib
l
e
e
r
r
or
v
e
c
to
r
s
e
,
w
hi
c
h
G
r
ov
e
r
'
s
a
l
gor
i
th
m
a
lo
n
e
c
a
n
not
e
f
f
ic
ie
nt
l
y r
e
d
uc
e
.
4.7.5.
D
u
al
-
c
od
e
at
t
ac
k
I
n
dua
l
-
c
ode
a
tt
a
c
ks
,
a
n
a
dve
r
s
a
r
y
tr
ie
s
to
le
ve
r
a
ge
in
f
o
r
m
a
ti
on
f
r
om
th
e
dua
l
c
ode
of
th
e
publ
ic
c
ode
,
w
hi
c
h
c
ons
i
s
ts
of
v
e
c
to
r
s
or
th
ogona
l
to
a
ll
c
ode
w
or
ds
i
n
′
.
W
it
h
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
,
th
e
s
tr
uc
tu
r
e
of
th
e
dua
l
c
ode
m
a
y be
m
or
e
p
r
e
di
c
ta
bl
e
. T
he
s
c
r
a
m
bl
in
g pr
ovi
d
e
d by
a
nd
pr
e
s
e
r
ve
s
t
he
r
a
ndomne
s
s
of
t
he
publ
ic
c
ode
’
a
nd
it
s
dua
l,
e
ns
ur
in
g
th
a
t
du
a
l
-
c
ode
a
tt
a
c
k
s
a
r
e
no
m
or
e
e
f
f
e
c
ti
ve
th
a
n
th
e
y
a
r
e
a
ga
in
s
t
th
e
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
.
5.
P
E
R
F
O
R
M
A
N
C
E
C
O
M
P
A
R
I
S
O
N
O
F
T
R
A
D
I
T
I
O
N
A
L
M
C
E
L
I
E
C
E
A
N
D
T
H
E
C
I
R
C
U
L
A
N
T
M
A
T
R
I
X
-
B
A
S
E
D
M
C
E
L
I
E
C
E
T
he
pe
r
f
or
m
a
nc
e
e
va
lu
a
ti
on
of
th
e
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
ut
il
iz
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
ve
r
s
us
th
e
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
c
e
nt
e
r
s
on
th
r
e
e
ke
y
f
a
c
to
r
s
:
ke
y
s
iz
e
,
e
nc
r
ypt
io
n
a
nd
de
c
r
ypt
io
n
s
pe
e
d,
a
nd
s
e
c
ur
it
y
le
ve
l
is
s
how
n
in
T
a
bl
e
1.
T
he
M
c
E
li
e
c
e
c
r
ypt
os
ys
te
m
a
c
hi
e
ve
s
s
ig
ni
f
ic
a
nt
a
dva
nt
a
ge
s
ove
r
c
onve
nt
io
na
l
publ
ic
ke
y
c
r
ypt
ogr
a
phy
in
bot
h
ke
y
s
iz
e
a
nd
e
nc
r
ypt
io
n
s
pe
e
d
by
im
pl
e
m
e
nt
in
g
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
,
m
a
ki
ng
it
ve
r
y
pr
a
c
ti
c
a
l.
T
he
c
ir
c
ul
a
nt
m
a
tr
ix
s
tr
uc
tu
r
e
a
ls
o
in
tr
oduc
e
s
m
in
or
vul
ne
r
a
bi
li
ty
to
s
om
e
a
tt
a
c
k
ty
pe
s
;
how
e
ve
r
,
us
in
g
s
uf
f
ic
ie
nt
s
c
r
a
m
bl
in
g
m
e
th
ods
a
lo
ng
w
it
h
pr
ope
r
pa
r
a
m
e
te
r
s
e
le
c
ti
on
a
ll
ow
s
f
or
th
e
s
e
c
ur
it
y
of
th
e
s
ys
te
m
to
r
e
m
a
in
in
ta
c
t.
T
hus
,
it
pr
ovi
de
s
a
good
b
a
la
nc
e
be
twe
e
n
s
e
c
ur
it
y
a
nd
pe
r
f
or
m
a
nc
e
,
us
e
f
ul
f
or
a
ppl
ic
a
ti
ons
w
he
r
e
s
to
r
a
ge
a
nd
c
om
put
a
ti
ona
l
r
e
s
our
c
e
s
ne
e
d
to
be
m
in
im
a
l,
f
or
e
xa
m
pl
e
,
in
te
r
ne
t
of
t
hi
ngs
(
I
oT
)
de
vi
c
e
s
a
nd mobi
le
A
pp
s
.
T
a
bl
e
1. C
om
pa
r
is
on of
tr
a
di
ti
ona
l
M
c
E
li
e
c
e
a
nd
th
e
c
ir
c
ul
a
nt
m
a
tr
ix
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ba
s
e
d
M
c
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li
e
c
e
M
e
t
r
i
c
T
r
a
di
t
i
ona
l
M
c
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l
i
e
c
e
C
i
r
c
ul
a
nt
m
a
t
r
i
x
-
ba
s
e
d
M
c
E
l
i
e
c
e
I
m
pr
ove
m
e
nt
f
a
c
t
or
P
ubl
i
c
ke
y s
i
z
e
~
256 K
B
~
256 byt
e
s
~1
,
000x r
e
duc
t
i
on
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nc
r
ypt
i
on
s
pe
e
d
O
(
k×n)
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(
nl
ogn)
F
a
s
t
e
r
e
nc
r
ypt
i
on w
i
t
h F
F
T
D
e
c
r
ypt
i
on
s
pe
e
d
C
om
pa
r
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bl
e
C
om
pa
r
a
bl
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m
i
l
a
r
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e
c
ur
i
t
y
l
e
ve
l
V
e
r
y
st
r
ong
S
t
r
ong, s
l
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ght
t
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t
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c
a
l
r
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r
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m
bl
e
d
P
r
a
c
t
i
c
a
l
l
y c
om
pa
r
a
bl
e
6.
C
O
N
C
L
U
S
I
O
N
T
hi
s
i
s
a
pa
pe
r
de
s
c
r
ib
in
g
a
n
a
lt
e
r
na
ti
ve
im
pl
e
m
e
nt
a
ti
on
of
t
he
M
c
E
li
e
c
e
c
r
ypt
ogr
a
phi
c
a
lg
or
it
hm
th
a
t
ut
il
iz
e
s
c
ir
c
ul
a
nt
m
a
tr
ix
r
e
pr
e
s
e
nt
a
ti
ons
. O
ne
of
t
he
w
e
a
kne
s
s
e
s
a
s
s
o
c
ia
te
d w
it
h us
in
g t
hi
s
t
ype
of
s
ys
te
m
is
th
a
t
it
of
te
n
r
e
qui
r
e
s
ve
r
y
la
r
ge
,
c
om
pl
e
x
k
e
y
s
iz
e
s
to
pr
ovi
de
hi
gh
le
ve
ls
of
s
e
c
ur
it
y.
T
he
a
ut
hor
s
of
th
is
pa
pe
r
di
s
c
us
s
how
va
r
io
us
pr
ope
r
ti
e
s
of
c
ir
c
ul
a
nt
m
a
tr
ic
e
s
a
l
lo
w
f
or
th
e
de
ve
lo
pm
e
nt
of
s
m
a
ll
e
r
ke
y
s
i
z
e
w
hi
le
s
ti
ll
pr
ovi
di
ng
a
de
qua
te
pr
ot
e
c
ti
on
f
o
r
m
e
s
s
a
ge
s
s
e
nt
vi
a
s
e
c
ur
e
c
ha
nne
ls
.
B
y
de
c
r
e
a
s
in
g
th
e
ke
y
s
iz
e
s
a
s
s
oc
ia
t
e
d w
it
h t
he
M
c
E
li
e
c
e
c
r
ypt
ogr
a
phi
c
a
lg
or
it
hm
, t
he
a
ut
h
or
s
be
li
e
ve
t
he
ir
w
or
k w
il
l
a
ll
ow
f
or
im
pr
ove
d
a
dopt
io
n
of
th
e
a
lg
or
it
hm
in
c
ont
e
m
por
a
r
y
di
gi
ta
l
de
vi
c
e
s
,
in
c
lu
di
ng
th
os
e
c
onne
c
te
d
to
th
e
I
oT
,
a
s
w
e
ll
a
s
m
obi
le
de
vi
c
e
s
us
in
g
w
ir
e
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C
E
S
[
1]
M
.
L
.
-
G
a
r
c
í
a
a
nd
E
.
C
.
-
N
a
va
r
r
o,
“
P
os
t
-
qua
nt
um
a
ut
he
nt
i
c
a
t
i
on
f
r
a
m
e
w
or
k
ba
s
e
d
on
i
r
i
s
r
e
c
ogni
t
i
on
a
nd
hom
om
or
phi
c
e
nc
r
ypt
i
on,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 155015
–
155030, 2025, doi
:
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C
C
E
S
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[
2]
E
.
B
i
nda
l
a
nd
A
.
K
.
S
i
ngh,
“
S
e
c
ur
e
a
nd
c
om
pa
c
t
:
a
ne
w
va
r
i
a
nt
o
f
M
c
E
l
i
e
c
e
c
r
ypt
os
ys
t
e
m
,”
I
E
E
E
A
c
c
e
s
s
,
vol
.
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pp. 35586
–
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:
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C
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[
3]
K
.
I
w
a
m
ur
a
a
nd
A
.
A
.
A
.
M
.
K
a
m
a
l
,
“
S
e
c
ur
e
us
e
r
a
ut
he
nt
i
c
a
t
i
on
w
i
t
h
i
nf
or
m
a
t
i
on
t
he
or
e
t
i
c
s
e
c
ur
i
t
y
us
i
ng
s
e
c
r
e
t
s
ha
r
i
ng
-
ba
s
e
d
s
e
c
ur
e
c
om
put
a
t
i
on,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 9015
–
9031, 2025, doi
:
10.1109/
A
C
C
E
S
S
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[
4]
H
.
A
.
S
ha
r
a
t
h,
J
.
V
r
i
nda
va
na
m
,
S
.
D
a
na
,
a
nd
S
.
N
.
P
r
a
s
a
d,
“
Q
ua
nt
um
-
r
e
s
i
l
i
e
nt
c
r
ypt
ogr
a
phy:
a
s
ur
ve
y
on
c
l
a
s
s
i
c
a
l
a
nd
qua
nt
u
m
a
l
gor
i
t
hm
s
,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 172854
–
172877, 2025, doi
:
10.1109/
A
C
C
E
S
S
.2025.3612982.
[
5]
O
. A
l
i
br
a
hi
m
, “
U
nve
i
l
i
ng S
a
m
s
ung
qua
nt
um
G
a
l
a
xy:
s
e
c
u
r
i
ng s
m
a
r
t
phone
s
w
i
t
h qua
nt
um
a
nd pos
t
-
qua
nt
um
c
r
ypt
ogr
a
phy,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 73202
–
73218, 2025, doi
:
10.1109/
A
C
C
E
S
S
.2025.3563826.
[
6]
Z
.
Z
.
S
un
e
t
a
l
.
,
“
Q
ua
nt
um
bl
oc
kc
ha
i
n
r
e
l
yi
ng
on
qua
nt
um
s
e
c
ur
e
di
r
e
c
t
c
om
m
uni
c
a
t
i
on
ne
t
w
or
k,”
I
E
E
E
I
nt
e
r
ne
t
of
T
hi
ngs
J
our
nal
, vol
. 12, no. 10, pp. 14375
–
14385, 2025, doi
:
10.1109/
J
I
O
T
.2025.3526
443.
[
7]
J
. O
.
D.
M
or
a
l
, A
.
D.
i
O
l
i
us
,
G
. V
i
da
l
,
P
. M
.
C
r
e
s
po,
a
nd J
. E
. M
a
r
t
i
ne
z
,
“
C
ybe
r
s
e
c
ur
i
t
y i
n c
r
i
t
i
c
a
l
i
nf
r
a
s
t
r
uc
t
ur
e
s
:
a
po
s
t
-
qua
nt
um
c
r
ypt
ogr
a
phy
pe
r
s
pe
c
t
i
ve
,”
I
E
E
E
I
nt
e
r
ne
t
of
T
hi
ngs
J
our
nal
,
vol
.
11,
no.
18,
pp.
30217
–
30244,
2024,
doi
:
10.1109/
J
I
O
T
.2024.3410702.
[
8]
E
.
F
a
t
ha
l
l
a
a
nd
M
.
A
z
a
b,
“
B
e
yond
c
l
a
s
s
i
c
a
l
c
r
ypt
ogr
a
phy:
a
s
y
s
t
e
m
a
t
i
c
r
e
vi
e
w
of
pos
t
-
qua
nt
um
ha
s
h
-
ba
s
e
d
s
i
gna
t
ur
e
s
c
he
m
e
s
,
s
e
c
ur
i
t
y, a
nd opt
i
m
i
z
a
t
i
ons
,”
I
E
E
E
A
c
c
e
s
s
, vol
. 12, pp. 175969
–
175987, 2024, doi
:
10.1109/
A
C
C
E
S
S
.2024.3485602.
[
9]
K
.
W
a
ng,
J
.
D
ong,
S
.
W
a
ng,
Z
.
Y
ua
n,
L
.
S
ha
,
a
nd
F
.
X
i
a
o,
“
R
S
A
K
A
-
V
D
T
:
de
s
i
gni
ng
r
e
l
i
a
bl
e
a
nd
pr
ova
bl
y
s
e
c
ur
e
a
ut
he
nt
i
c
a
t
e
d
ke
y
a
gr
e
e
m
e
nt
s
c
h
e
m
e
f
or
ve
hi
c
ul
a
r
di
gi
t
a
l
t
w
i
n
ne
t
w
or
ks
,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
V
e
hi
c
ul
ar
T
e
c
hnol
ogy
,
vol
.
74,
no.
8,
pp. 12330
–
12346, 2025, doi
:
10.1109/
T
V
T
.2025.3552481.
[
10]
X
.
R
e
n
e
t
al
.
,
“
B
ui
l
di
ng
r
e
s
i
l
i
e
nt
W
e
b
3.0
i
nf
r
a
s
t
r
uc
t
ur
e
w
i
t
h
qua
nt
um
i
nf
o
r
m
a
t
i
on
t
e
c
hnol
ogi
e
s
a
nd
bl
oc
kc
ha
i
n:
a
n
a
m
bi
l
a
t
e
r
a
l
vi
e
w
,”
P
r
oc
e
e
di
ngs
of
t
he
I
E
E
E
, vol
. 112, no. 11, pp. 1686
–
1715, 2024, doi
:
10
.1109/
J
P
R
O
C
.2024.3520803.
[
11]
K
.
S
.
S
hi
m
,
B
.
K
i
m
,
a
nd
W
.
L
e
e
,
“
R
e
s
e
a
r
c
h
on
qua
nt
um
ke
y,
di
s
t
r
i
but
i
on
ke
y
a
nd
pos
t
-
qua
nt
um
c
r
ypt
ogr
a
phy
ke
y
a
ppl
i
e
d
pr
ot
oc
ol
s
f
or
da
t
a
s
c
i
e
n
c
e
a
nd
w
e
b
s
e
c
ur
i
t
y,”
J
ou
r
nal
of
W
e
b
E
ngi
ne
e
r
i
ng
,
vol
.
23,
no.
6,
pp.
813
–
830,
2024,
doi
:
10.13052/
j
w
e
1540
-
9589.2365.
[
12]
B
.
C
hou
dh
ur
y,
A
.
H
ot
a
,
M
.
K
a
r
m
a
ka
r
,
S
.
S
a
ha
,
A
. N
a
g
, a
nd
S
.
N
a
ndi
,
“
A
c
o
m
p
r
e
he
ns
i
ve
s
u
r
v
e
y
on
pr
e
ve
r
s
us
pos
t
qua
nt
um
s
e
c
ur
i
t
y
s
c
he
m
e
s
f
o
r
5
G
-
e
na
bl
e
d
I
o
T
a
p
pl
i
c
a
t
i
o
ns
,”
I
E
E
E
A
c
c
e
s
s
,
vo
l
.
1
3,
pp
. 1
59
305
–
1
59
333
, 2
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5,
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i
:
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109
/
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C
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S
S
.2
025
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608
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[
13]
S
.
B
a
j
r
i
ć
,
“
E
na
bl
i
ng
s
e
c
ur
e
a
nd
t
r
us
t
w
or
t
hy
qua
nt
um
ne
t
w
or
ks
:
c
ur
r
e
nt
s
t
a
t
e
-
of
-
t
he
-
a
r
t
,
ke
y
c
ha
l
l
e
nge
s
,
a
nd
pot
e
nt
i
a
l
s
ol
ut
i
ons
,
”
I
E
E
E
A
c
c
e
s
s
, vol
. 11, pp. 128801
–
128809, 2023, doi
:
10.1109/
A
C
C
E
S
S
.2023.3333020.
[
14]
J
.
Z
ha
ng,
F
.
Z
ha
ng,
a
nd
X
.
H
ua
ng,
“
T
he
or
y
a
nd
a
ppl
i
c
a
t
i
ons
of
s
e
que
nt
i
a
l
l
y
t
hr
e
s
hol
d
publ
i
c
-
ke
y
c
r
ypt
ogr
a
phy:
pr
a
c
t
i
c
a
l
pr
i
va
t
e
ke
y
s
a
f
e
gua
r
di
ng
a
nd
s
e
c
ur
e
us
e
f
or
i
ndi
vi
dua
l
us
e
r
s
,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
I
nf
or
m
at
i
on
F
or
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ns
i
c
s
and
Se
c
ur
i
t
y
,
vol
.
20
,
pp. 3220
–
3233, 2025, doi
:
10.1109/
T
I
F
S
.2025.3552202.
[
15]
H
.
W
e
n
e
t
al
.
,
“
S
e
c
ur
e
opt
i
c
a
l
i
m
a
ge
c
om
m
uni
c
a
t
i
on
us
i
ng
doubl
e
r
a
ndo
m
t
r
a
ns
f
or
m
a
t
i
on
a
nd
m
e
m
r
i
s
t
i
ve
c
ha
os
,”
I
E
E
E
P
hot
oni
c
s
J
our
nal
, vol
. 15, no. 1, pp. 1
–
11, 2023, doi
:
10.1109/
J
P
H
O
T
.2022.32
33129.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
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I
nt
J
A
dv A
ppl
S
c
i
,
V
ol
. 15, No. 1, M
a
r
c
h 2026
:
293
-
302
302
[
16]
M
. E
l
-
H
a
de
dy, A
. A
be
l
i
a
n, K
.
L
e
e
,
B
. N
.
C
he
ng, a
nd W
.
-
M
. H
w
u, “
A
N
U
B
I
S
:
hybr
i
d F
P
A
A
-
F
P
G
A
a
r
c
hi
t
e
c
t
ur
e
f
or
e
nt
r
opy
-
ba
s
e
d
t
r
ue
r
a
ndom
num
be
r
ge
ne
r
a
t
i
on
i
n
s
e
c
ur
e
U
A
V
c
om
m
uni
c
a
t
i
on,”
I
E
E
E
E
m
be
dde
d
Sy
s
t
e
m
s
L
e
t
t
e
r
s
,
vol
.
17,
no.
3,
pp.
164
–
167,
2025, doi
:
10.1109/
L
E
S
.2024.3510365.
[
17]
M
.
A
.
K
ha
n
e
t
al
.
,
“
A
n
i
m
pr
ovi
s
e
d
c
e
r
t
i
f
i
c
a
t
e
-
ba
s
e
d
pr
oxy
s
i
gna
t
ur
e
us
i
ng
hype
r
e
l
l
i
pt
i
c
c
ur
ve
c
r
ypt
ogr
a
phy
f
or
s
e
c
ur
e
U
A
V
c
om
m
uni
c
a
t
i
ons
,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
I
nt
e
l
l
i
ge
nt
T
r
ans
por
t
at
i
on
Sy
s
t
e
m
s
,
vol
.
26,
no.
4,
pp.
5264
–
5275,
2025,
doi
:
10.1109/
T
I
T
S
.2024.3524575.
[
18]
A
.
G
i
or
ge
t
t
i
e
t
al
.
,
“
G
e
ne
r
a
l
i
z
e
d
qua
nt
um
-
a
s
s
i
s
t
e
d
di
gi
t
a
l
s
i
gna
t
ur
e
s
e
r
vi
c
e
i
n
a
n
S
D
N
-
c
ont
r
ol
l
e
d
qua
nt
um
-
i
nt
e
gr
a
t
e
d
opt
i
c
a
l
ne
t
w
or
k,”
J
our
nal
of
O
pt
i
c
al
C
om
m
uni
c
at
i
ons
and
N
e
t
w
o
r
k
i
ng
,
vol
.
17,
no.
2,
pp.
A
155
--
A
164,
2025,
doi
:
10.1364/
J
O
C
N
.534089.
[
19]
P
.
K
.
M
a
duni
,
I
.
B
yun,
J
.
S
e
o,
a
nd
K
.
K
o,
“
H
ybr
i
d
qua
nt
um
-
s
a
f
e
c
r
ypt
ogr
a
phi
c
s
c
he
m
e
w
i
t
h
s
e
c
ur
e
ke
y
e
xc
ha
ng
e
a
nd
s
i
gna
t
ur
e
s
c
he
m
e
,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 147650
–
147665, 2025, doi
:
10.1109/
A
C
C
E
S
S
.2025.3600068.
[
20]
Y
.
H
a
r
i
pr
a
s
a
d,
S
.
S
.
I
ye
nga
r
,
a
nd
N
.
K
.
C
ha
udha
r
y,
“
S
e
c
ur
i
ng
t
he
f
ut
ur
e
:
a
dva
nc
e
d
e
nc
r
ypt
i
on
f
or
qua
nt
um
-
s
a
f
e
vi
de
o
t
r
a
ns
m
i
s
s
i
on,”
I
E
E
E
T
r
ans
ac
t
i
ons
on C
ons
um
e
r
E
l
e
c
t
r
oni
c
s
, vol
. 71, no. 1, pp.
140
–
153, 2025, doi
:
10.1109/
T
C
E
.2024.3473542.
[
21]
V
.
P
une
ya
ni
a
nd
K
.
V
B
ha
t
,
“
Q
ua
nt
um
-
r
e
s
i
s
t
a
nt
bl
oc
kc
ha
i
n
pr
ot
oc
ol
s
f
or
s
e
c
ur
e
t
r
a
ns
a
c
t
i
ons
,”
I
E
E
E
A
c
c
e
s
s
,
vol
.
13,
pp. 108984
–
108991, 2025, doi
:
10.1109/
A
C
C
E
S
S
.2025.3581955.
[
22]
K
.
B
.
A
.
K
um
a
r
,
L
.
S
.
M
ohi
t
h,
K
.
J
a
i
n,
P
.
K
r
i
s
hna
n,
N
.
V
e
nka
t
a
c
ha
l
a
m
,
a
nd
R
.
B
uyya
,
“
P
os
t
-
qua
nt
um
c
r
ypt
ogr
a
phy
-
ba
s
e
d
m
ul
t
i
m
e
di
a
e
nc
r
ypt
i
on
c
om
m
uni
c
a
t
i
on
s
c
he
m
e
i
n
I
oT
c
on
s
um
e
r
e
l
e
c
t
r
oni
c
s
,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
C
ons
um
e
r
E
l
e
c
t
r
oni
c
s
,
vol
. 71, no. 2, pp. 4995
–
5006, 2025, doi
:
10.1109/
T
C
E
.2025.3572949.
[
23]
S
.
H
us
s
a
i
n,
A
.
T
uf
a
i
l
,
H
.
A
.
A
.
G
.
N
a
i
m
,
M
.
A
.
K
ha
n,
a
nd
G
.
B
a
r
b,
“
E
va
l
ua
t
i
o
n
of
c
om
put
a
t
i
ona
l
l
y
e
f
f
i
c
i
e
nt
i
de
nt
i
t
y
-
ba
s
e
d
pr
oxy
s
i
gna
t
ur
e
s
,”
I
E
E
E
O
pe
n J
our
nal
of
t
he
C
om
put
e
r
Soc
i
e
t
y
, vol
. 6, pp. 846
–
861,
2025, doi
:
10.1109/
O
J
C
S
.2025.3573638.
[
24]
A
.
S
ha
r
m
a
a
nd
S
.
R
a
ni
,
“
P
os
t
-
qua
nt
um
c
r
ypt
og
r
a
phy
(
P
Q
C
)
f
or
I
oT
-
c
ons
um
e
r
e
l
e
c
t
r
oni
c
s
de
vi
c
e
s
i
nt
e
gr
a
t
e
d
w
i
t
h
de
e
p
l
e
a
r
ni
ng,”
I
E
E
E
T
r
ans
ac
t
i
ons
on C
ons
um
e
r
E
l
e
c
t
r
oni
c
s
, vol
. 71, no. 2, pp. 4925
–
4933, 20
25, doi
:
10.1109/
T
C
E
.2025.3569904.
[
25]
S
.
L
.
B
i
r
ha
nu,
M
.
G
ha
di
m
i
,
Y
.
H
a
i
,
P
.
S
e
e
l
i
ng,
R
.
B
a
s
s
ol
i
,
a
nd
F
.
H
.
P
.
F
i
t
z
e
k,
“
A
s
ur
ve
y
of
c
ont
i
nuous
va
r
i
a
bl
e
qua
nt
um
ke
y
di
s
t
r
i
but
i
on i
n qua
nt
um
c
om
m
uni
c
a
t
i
on,”
I
E
E
E
A
c
c
e
s
s
, vol
. 13, pp. 166027
–
166061, 2025, doi
:
10.1109/
A
C
C
E
S
S
.2025.3610519.
[
26]
K
.
S
ut
r
a
dha
r
,
“
A
qua
nt
um
c
r
ypt
ogr
a
phi
c
p
r
ot
oc
ol
f
or
s
e
c
ur
e
ve
hi
c
ul
a
r
c
o
m
m
uni
c
a
t
i
on,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
I
nt
e
l
l
i
ge
nt
T
r
ans
por
t
at
i
on Sy
s
t
e
m
s
, vol
. 25, no. 5, pp. 3513
–
3522, 2024, doi
:
10.1109/
T
I
T
S
.2023.3322728.
[
27]
R
.
Z
ha
ng,
L
.
Z
ha
ng,
K
.
-
K
.
R
.
C
hoo,
a
nd
T
.
C
h
e
n,
“
D
yna
m
i
c
a
ut
he
nt
i
c
a
t
e
d
a
s
ym
m
e
t
r
i
c
gr
oup
ke
y
a
gr
e
e
m
e
nt
w
i
t
h
s
e
nde
r
non
-
r
e
pudi
a
t
i
on
a
nd
pr
i
va
c
y
f
or
gr
oup
-
o
r
i
e
nt
e
d
a
ppl
i
c
a
t
i
ons
,”
I
E
E
E
T
r
ans
ac
t
i
ons
on
D
e
pe
ndabl
e
and
Se
c
u
r
e
C
om
put
i
ng
,
vol
.
20
,
no. 1, pp. 492
–
505, 2023, doi
:
10.1109/
T
D
S
C
.2021.3138445.
[
28]
V
. K
um
a
r
e
t
al
.
, “
D
e
s
i
gn of
s
e
c
ur
e
a
nd
e
f
f
i
c
i
e
nt
f
r
a
m
e
w
or
k f
or
ve
hi
c
ul
a
r
di
gi
t
a
l
t
w
i
n ne
t
w
or
ks
us
i
ng E
C
C
,”
I
E
E
E
A
c
c
e
s
s
, vol
. 12,
pp. 194352
–
194366, 2024, doi
:
10.1109/
A
C
C
E
S
S
.2024.3511654.
[
29]
D
.
S
.
C
.
P
ut
r
a
nt
o,
R
.
W
.
W
a
r
dha
ni
,
H
.
T
.
L
a
r
a
s
a
t
i
,
a
nd
H
.
K
i
m
,
“
S
pa
c
e
a
nd
t
i
m
e
-
e
f
f
i
c
i
e
nt
qua
nt
um
m
ul
t
i
pl
i
e
r
i
n
pos
t
qua
nt
um
c
r
ypt
ogr
a
phy e
r
a
,”
I
E
E
E
A
c
c
e
s
s
, vol
. 11, pp. 21848
–
21862, 2023, doi
:
10.1109/
A
C
C
E
S
S
.2023.3252504.
B
I
O
G
R
A
P
H
I
E
S
O
F
A
U
T
H
O
R
S
Ravikumar In
akoti
is pursuing his Ph.
D. in the De
partment of
Co
mputer Scie
nce
and Sys
tems En
gineering
at Andh
ra Univers
ity, Vi
sakhapatn
am
,
and c
ompleted his M.Tech
.
in
2015
from
Pydah
College
of
Engineering.
He
worked
as
an
as
sistant
professor
in
the
Department
of
Computer
Science
and
Engineering
at
Welfare
Insti
tute
of
Technology
and
Management,
Visakhapatnam,
India.
His
researc
h
interests
include
w
ireless
sensor
network
s
,
big
data
analytics,
computer
networks,
network
security
,
and
MANE
Ts.
He
can
be
contacted
at email
: ravirk1
228@
gmail.co
m.
James
Stephen
Meka
is
a
respected
academician,
currently
serving
as
the
national
chair
professor
at
the
Dr.
B.R.
Ambedkar
Chair,
Andhr
a
University,
under
the
Ministry
of
Social
Justice
and
Empowerment,
Government
of
India.
Academically,
he
holds
a
Ph.D.
in
Computer
Science
and
Systems
Engine
ering
from
Andhra
Univer
sity,
along
with
multiple
Master’s
degrees in MCA
and
M.Phil.
(CS),
M.Div., M.B.A.,
and M.Tech.
(CST).
He
can be cont
acted at em
ail: j
amessteph
enm@
gmail.co
m.
Padala
Venkata
Gopala
Durga
Prasad
Reddy
is
a
senior
pr
ofessor
in
the
Department
of
Computer
Science
and
Systems
Engineering,
Andhra
University,
Visakhapatnam,
India
,
where
he
previously
worked
as
a
Vice
-
Chanc
ellor.
He
produced
more
than
60
Ph.D’s
and
published
more
than
250
quality
r
esearch
articl
es.
He
owns
more
than
15
patents
and
3
copyrights
.
His
research
interests
include
mac
hine
learning,
artificial
intelligenc
e,
IoT
,
and
wireless
networks
.
He
can
be
contacted
at
email:
prasadreddy.vizag@gmail.com.
Evaluation Warning : The document was created with Spire.PDF for Python.