I
nte
rna
t
io
na
l J
o
urna
l o
f
Adv
a
nces in Applie
d Science
s
(
I
J
AAS)
Vo
l.
15
,
No
.
1
,
Ma
r
ch
20
26
,
p
p
.
2
9
3
~
3
0
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I
SS
N:
2252
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8
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:
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v15.
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tern
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)
a
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stil
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m
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s’
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with
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iate
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o
m
m
u
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m
s.
K
ey
w
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s
:
C
ir
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atr
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ased
cr
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in
cr
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b
l
em
.
I
n
c
o
d
e
-
b
ased
cr
y
p
t
o
g
r
ap
h
y
,
th
e
co
m
m
o
n
en
cr
y
p
tio
n
s
ch
em
es
ar
e
Mc
E
liece
en
cr
y
p
tio
n
s
ch
em
e
,
Nied
er
r
eiter
en
cr
y
p
tio
n
s
ch
em
e.
R
ec
en
tly
,
th
e
h
y
b
r
i
d
Mc
E
liece
en
cr
y
p
tio
n
s
ch
em
e
(
Hy
ME
S
)
was
ex
p
lain
ed
[
7
]
.
L
ik
e
th
e
R
SA
en
cr
y
p
tio
n
s
ch
em
e,
th
e
co
n
v
en
tio
n
al
Mc
E
liece
s
y
s
te
m
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
f
o
r
s
ec
u
r
e
in
f
o
r
m
atio
n
co
m
m
u
n
icatio
n
f
ails
to
ca
p
tu
r
e
th
e
atten
ti
on
o
f
r
esear
ch
er
s
,
u
s
er
s
,
an
d
in
d
u
s
tr
ies
,
as
th
e
m
ag
n
itu
d
e
o
f
th
e
g
en
er
a
tio
n
m
atr
ix
u
s
ed
in
th
e
p
u
b
lic
k
ey
o
f
th
e
ap
p
r
o
ac
h
is
r
elativ
ely
en
o
r
m
o
u
s
.
Ho
wev
er
,
th
e
v
u
l
n
er
ab
ilit
y
a
s
s
o
ciate
d
with
th
e
co
n
v
en
tio
n
al
en
cr
y
p
tio
n
s
ch
e
m
es lik
e
R
SA a
n
d
D
i
f
f
ie
-
He
l
lm
an
alg
o
r
ith
m
s
wh
en
u
s
ed
o
n
q
u
an
tu
m
c
o
m
p
u
tin
g
h
as m
ad
e
it o
n
e
o
f
th
e
f
o
cu
s
es
o
f
r
esear
ch
in
q
u
an
tu
m
s
ec
u
r
ity
.
On
e
o
f
th
e
m
ajo
r
f
o
c
u
s
es
o
n
th
e
co
n
v
en
tio
n
al
Mc
E
liece
s
y
s
tem
o
f
en
cr
y
p
tio
n
is
r
ed
u
ct
io
n
in
th
e
m
ag
n
itu
d
e
o
f
th
e
s
y
s
tem
k
ey
.
Fo
r
ex
am
p
le,
Fath
alla
an
d
Aza
b
[
8
]
in
v
esti
g
ated
th
e
n
o
tio
n
o
f
em
p
lo
y
in
g
co
m
p
ac
t
r
e
p
r
esen
tatio
n
o
f
th
e
s
h
ar
ed
m
atr
ix
o
f
th
e
co
n
v
en
ti
o
n
al
Mc
E
liece
s
y
s
tem
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
.
T
h
e
r
esear
ch
in
[
9
]
,
[
1
0
]
p
r
o
p
o
s
ed
q
u
asi
-
cy
clic
(
QC
)
alter
n
a
n
t a
lo
n
g
with
th
e
q
u
asi
-
d
y
a
d
ic
(
QD
)
Go
p
p
a
c
o
d
es
to
r
ed
u
c
e
th
e
m
ag
n
itu
d
e
o
f
th
e
co
n
v
en
tio
n
al
Mc
E
liece
s
ch
em
e
o
f
in
f
o
r
m
atio
n
e
n
c
r
y
p
tio
n
,
wh
ich
th
e
y
s
u
cc
ee
d
ed
in
r
ed
u
cin
g
it
f
r
o
m
s
ev
er
al
h
u
n
d
r
ed
th
o
u
s
an
d
b
its
to
2
0
k
ilo
b
its
.
T
h
e
p
u
r
p
o
s
e
o
f
th
o
s
e
co
n
s
tr
u
cts
is
to
u
s
e
f
ir
s
t
r
o
w
p
er
m
u
tatio
n
s
to
p
r
o
d
u
ce
th
e
e
n
tire
m
atr
ix
.
Ad
d
itio
n
ally
,
th
o
s
e
co
n
s
tr
u
cti
o
n
s
en
ab
le
m
ess
ag
e
en
cr
y
p
tio
n
u
s
i
n
g
o
n
ly
th
e
in
iti
al
r
o
w
o
f
th
e
m
atr
i
x
r
ath
er
t
h
a
n
th
e
en
tire
m
atr
ix
.
I
n
b
o
th
s
c
en
ar
io
s
,
th
e
b
in
a
r
y
p
ar
am
eter
s
r
em
ain
s
ec
u
r
e
ev
e
n
af
ter
m
u
ltip
le
attac
k
s
.
Oth
er
m
eth
o
d
s
p
r
o
p
o
s
ed
to
d
ep
lete
th
e
m
ag
n
itu
d
e
o
f
th
e
s
h
ar
ed
k
ey
o
f
t
h
e
tr
ad
itio
n
a
l
Mc
E
liece
in
f
o
r
m
ati
o
n
en
cr
y
p
tio
n
s
ch
em
e
in
clu
d
e
“a
lg
e
b
r
aic
g
e
o
m
etr
i
c
(
AG)
co
d
es”
[
1
1
]
,
“
g
en
er
ali
ze
d
R
ee
d
-
So
lo
m
o
n
(
GR
S)
co
d
es”
[
1
2
]
,
“lo
w
-
d
e
n
s
ity
p
ar
ity
ch
ec
k
(
L
DPC
)
co
d
es”
[
1
3
]
,
“
R
ee
d
-
Mu
ller
(
R
M)
co
d
es”
[
1
4
]
,
“lo
w
-
r
an
k
p
ar
ity
ch
ec
k
(
L
R
PC
)
co
d
es”
[
1
5
]
,
a
n
d
m
an
y
m
o
r
e
th
at
allo
w
f
o
r
s
h
o
r
ter
s
h
a
r
ed
k
ey
s
,
h
a
v
e
b
ee
n
d
em
o
n
s
tr
ated
t
h
r
o
u
g
h
v
ar
io
u
s
co
d
es.
T
h
e
m
aj
o
r
ity
o
f
t
h
ese
v
ar
iatio
n
s
h
a
v
e
b
ee
n
s
u
cc
ess
f
u
lly
cr
y
p
tan
aly
ze
d
,
ev
en
th
o
u
g
h
t
h
e
o
r
ig
i
n
al
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
s
till
s
ec
u
r
e
[
1
6
]
,
[
1
7
]
.
T
h
e
alter
n
at
iv
e
co
d
es
m
u
s
t
b
e
tr
ea
ted
with
ca
u
tio
n
b
ec
au
s
e
o
f
th
eir
ex
ce
s
s
iv
e
s
tr
u
ctu
r
e,
e
v
en
with
th
eir
p
r
o
m
is
in
g
f
ea
t
u
r
es.
Po
lar
c
o
d
in
g
was
in
itially
in
ten
d
ed
to
b
e
a
m
eth
o
d
,
a
k
in
to
p
r
ev
i
o
u
s
Pin
s
k
er
an
d
Ma
s
s
ey
s
ch
em
es,
f
o
r
i
n
cr
ea
s
in
g
th
e
c
u
to
f
f
r
ate
o
f
s
eq
u
en
tial
d
ec
o
d
i
n
g
.
T
h
e
s
ec
r
et
to
r
aisi
n
g
th
e
cu
to
f
f
r
ate
is
to
tak
e
a
v
ec
to
r
c
h
an
n
el
(
wh
eth
e
r
it’s
n
atu
r
ally
o
cc
u
r
r
in
g
o
r
p
u
r
p
o
s
ef
u
lly
p
r
o
d
u
ce
d
)
,
s
p
lit
it
u
p
in
to
m
u
ltip
le
in
ter
r
elate
d
s
u
b
s
id
iar
y
ch
a
n
n
els,
an
d
th
en
ap
p
ly
a
s
ep
ar
ate
o
r
d
er
e
d
d
ec
o
d
er
o
n
ev
e
r
y
s
u
b
ch
a
n
n
el.
Po
lar
co
d
i
n
g
was
in
itially
in
ten
d
ed
to
b
e
a
lo
w
-
co
m
p
lex
ity
r
ec
u
r
s
iv
e
ch
a
n
n
el
co
m
b
in
in
g
an
d
s
p
litt
in
g
o
p
er
atio
n
o
f
th
is
k
i
n
d
,
with
th
e
g
o
al
o
f
b
ein
g
em
p
lo
y
ed
as in
ter
n
al
co
d
e
in
a
co
m
p
o
s
ite
s
ch
em
e
with
ex
ter
n
al
co
n
v
o
lu
tio
n
al
c
o
d
in
g
an
d
o
r
d
er
e
d
d
ec
o
d
in
g
.
Nev
er
th
eless
,
th
e
in
itial
g
o
al
o
f
in
cr
ea
s
in
g
th
e
cu
to
f
f
r
ate
to
ch
an
n
el
ca
p
ac
ity
was
ac
tu
al
ly
ac
h
iev
ed
with
o
u
t
th
e
n
ee
d
f
o
r
a
n
o
u
ter
co
d
e
b
ec
a
u
s
e
th
e
p
o
lar
in
n
e
r
co
d
e
p
r
o
v
ed
to
b
e
s
o
s
u
cc
es
s
f
u
l
[
1
8
]
.
W
ith
t
h
e
co
n
tin
u
o
u
s
g
r
o
wth
in
th
e
f
iel
d
o
f
q
u
an
t
u
m
co
m
p
u
tin
g
,
th
e
r
e
is
n
ee
d
f
o
r
m
o
r
e
r
o
b
u
s
t
s
ch
em
es
th
at
will
b
e
d
if
f
icu
lt
f
o
r
th
e
q
u
a
n
tu
m
s
y
s
t
em
s
to
b
r
ea
k
.
I
n
th
is
p
ap
e
r
,
a
m
o
d
if
ied
Mc
E
liece
p
u
b
lic
k
e
y
en
cr
y
p
tio
n
s
y
s
tem
with
h
ig
h
lev
el
o
f
s
ec
u
r
ity
t
o
p
r
o
d
u
ce
a
s
ec
u
r
e
co
m
m
u
n
i
ca
tio
n
s
ch
em
e
b
ased
o
n
p
o
la
r
co
d
e,
wh
ich
ca
n
p
r
ev
en
t Bric
k
el’
s
attac
k
d
u
r
in
g
co
m
m
u
n
icatio
n
.
T
h
e
o
b
jectiv
es o
f
th
is
p
ap
er
ar
e
s
u
m
m
a
r
ized
as f
o
llo
ws:
−
C
r
ea
te
a
m
o
d
if
ied
f
o
r
m
o
f
M
cE
liece
cr
y
p
to
s
y
s
tem
th
at
u
s
e
s
cir
cu
lan
t
m
atr
ices
to
m
ak
e
i
ts
p
u
b
lic
k
ey
s
m
aller
.
T
h
is
will
r
ed
u
ce
a
m
ajo
r
wea
k
n
ess
o
f
Mc
E
liece
c
r
y
p
to
s
y
s
tem
,
its
lar
g
e
p
u
b
lic
k
ey
s
ize.
T
h
e
id
ea
will
h
elp
in
im
p
r
o
v
in
g
th
e
u
s
ag
e
o
f
Mc
E
liece
cr
y
p
to
s
y
s
tem
in
s
itu
atio
n
s
wh
er
e
li
m
ited
s
to
r
ag
e
ca
p
ac
ity
wo
u
ld
n
o
t a
llo
w
all
it
s
cr
y
p
to
g
r
a
m
s
to
b
e
tr
a
n
s
m
itted
.
−
T
o
ex
a
m
in
e
th
e
s
ec
u
r
ity
o
f
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
b
as
ed
o
n
cir
c
u
lan
t
m
at
r
ices
ag
ain
s
t
b
o
th
class
ic
an
d
q
u
a
n
tu
m
attac
k
s
.
T
h
is
i
n
v
o
lv
es
s
cr
u
tin
izin
g
t
h
e
ab
ilit
y
o
f
cir
cu
lan
t
m
atr
ices,
co
m
p
lem
en
ted
b
y
s
cr
am
b
lin
g
an
d
p
er
m
u
tatio
n
m
atr
ices,
to
en
s
u
r
e
th
e
s
ec
u
r
ity
o
f
co
n
f
id
en
tial m
ess
ag
e
tr
an
s
m
is
s
io
n
.
−
T
o
an
aly
ze
an
d
c
o
m
p
ar
e
th
e
e
f
f
ec
tiv
en
ess
o
f
th
e
Mc
E
liece
s
y
s
tem
b
ased
o
n
a
cir
c
u
lan
t
m
atr
ix
with
th
e
Mc
E
liece
s
y
s
tem
.
I
n
t
h
is
ar
ea
,
we
p
lan
to
test
an
d
an
aly
ze
th
e
s
p
ee
d
o
f
e
n
cr
y
p
tio
n
an
d
d
ec
r
y
p
tio
n
,
th
e
tim
e
r
eq
u
ir
ed
f
o
r
g
en
er
atin
g
a
k
ey
,
an
d
,
m
o
s
t
im
p
o
r
tan
tly
,
t
h
e
ef
f
ec
tiv
en
ess
an
d
ef
f
icien
c
y
ad
d
ed
b
y
th
e
u
s
e
o
f
a
cir
cu
la
n
t m
atr
ix
.
T
h
e
r
est
o
f
th
e
d
o
cu
m
e
n
t
is
s
t
r
u
ctu
r
ed
as
f
o
llo
ws:
s
ec
tio
n
2
d
escr
ib
es
s
ev
er
al
r
esear
ch
p
ap
er
s
wh
ich
ar
e
clo
s
ely
r
elate
d
to
c
o
d
e
-
b
a
s
ed
en
cr
y
p
tio
n
s
ch
em
es
,
s
ec
ti
o
n
3
p
r
o
v
id
es
in
f
o
r
m
atio
n
a
b
o
u
t
th
e
tr
ad
itio
n
al
Mc
E
liece
alg
o
r
ith
m
an
d
b
asic
p
r
in
cip
les
ab
o
u
t
c
ir
cu
lan
t
m
a
tr
ices.
Sectio
n
4
p
r
esen
ts
th
e
p
r
o
p
o
s
ed
cir
cu
lan
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
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8
8
1
4
A
n
o
ve
l c
ir
cu
la
n
t m
a
tr
ix
-
b
a
s
ed
McE
liece
fr
a
mewo
r
k
fo
r
s
ec
u
r
e
d
ig
ita
l c
o
mmu
n
ica
tio
n
(
R
a
viku
ma
r
I
n
a
k
o
ti
)
295
m
atr
ix
-
b
ased
Mc
E
liece
cr
y
p
to
s
y
s
tem
in
th
is
p
ap
er
.
S
ec
tio
n
5
o
f
th
is
p
ap
er
p
r
esen
ts
th
e
s
ec
u
r
ity
an
aly
s
is
an
d
co
m
p
ar
is
o
n
o
f
th
e
p
r
o
p
o
s
ed
cir
cu
lan
t
m
atr
ix
-
b
ased
Mc
E
liec
e
cr
y
p
to
s
y
s
tem
,
an
d
last
ly
,
s
ec
tio
n
6
o
f
th
is
p
ap
er
d
r
aws
a
co
n
clu
s
io
n
o
n
th
e
cir
c
u
lan
t m
atr
ix
-
b
ased
Mc
E
liece
c
r
y
p
to
s
y
s
tem
.
2.
RE
L
AT
E
D
WO
RK
On
e
o
f
th
e
f
ir
s
t
an
d
lo
n
g
est
-
last
in
g
p
u
b
lic
-
k
ey
en
c
r
y
p
tio
n
s
ch
em
es
is
th
e
Mc
E
liece
s
y
s
tem
f
o
r
in
f
o
r
m
atio
n
en
co
d
in
g
,
c
r
ea
ted
in
th
e
y
ea
r
1
9
7
8
b
y
R
o
b
er
t
J
.
Mc
E
liece
.
I
ts
d
ef
en
s
e
ag
ain
s
t
q
u
an
tu
m
attac
k
s
,
wh
ich
co
m
e
f
r
o
m
its
u
s
e
o
f
e
r
r
o
r
-
c
o
r
r
ec
tin
g
co
d
es,
h
as
en
ti
ce
d
a
s
ig
n
if
ican
t
in
ter
est
b
y
n
u
m
er
o
u
s
en
titi
es
in
th
e
cr
y
p
t
o
g
r
a
p
h
y
wo
r
ld
.
T
h
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
th
o
r
o
u
g
h
ly
co
v
er
ed
in
t
h
is
liter
atu
r
e
r
e
v
iew,
wh
ich
also
ex
p
lo
r
es
its
r
ec
en
t
d
e
v
elo
p
m
en
ts
,
s
ec
u
r
ity
f
ea
tu
r
e
s
,
im
p
lem
en
tatio
n
d
if
f
icu
lties
,
an
d
th
eo
r
etica
l
u
n
d
er
p
in
n
in
g
s
.
T
h
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
wa
s
in
tr
o
d
u
ce
d
b
y
R
o
b
er
t
J
.
Mc
E
liece
,
wh
o
al
s
o
s
u
g
g
est
ed
u
s
in
g
alg
eb
r
aic
co
d
in
g
t
h
eo
r
y
f
o
r
p
u
b
lic
-
k
ey
en
cr
y
p
tio
n
.
I
t
d
escr
i
b
es
th
e
f
u
n
d
am
en
tal
i
d
ea
s
an
d
d
r
aws
atten
tio
n
to
its
p
o
s
s
ib
le
b
en
ef
its
o
v
e
r
al
ter
n
ativ
e
en
c
r
y
p
tio
n
tech
n
iq
u
es.
Ma
d
u
n
i
et
a
l.
[
1
9
]
ex
am
in
e
a
Mc
E
liece
cr
y
p
to
s
y
s
tem
v
a
r
iatio
n
th
at
c
an
g
u
a
r
an
tee
t
h
at
en
c
o
d
in
g
u
tili
ze
d
as
th
e
s
h
ar
ed
g
en
e
r
ate
d
k
ey
is
n
o
m
o
r
e
s
im
ilar
to
th
e
p
er
m
u
tatio
n
o
f
t
h
e
s
ec
r
et/u
n
s
h
ar
e
d
c
o
d
e.
As
a
r
esu
lt,
th
e
ad
o
p
tio
n
o
f
tr
a
d
itio
n
al
co
d
e
f
am
ilies
,
s
u
ch
as
R
ee
d
-
So
lo
m
o
n
co
d
es,
wh
ich
h
av
e
b
ee
n
lo
n
g
-
s
tan
d
in
g
ex
clu
s
io
n
s
f
r
o
m
th
e
co
n
v
en
tio
n
al
Mc
E
liece
s
y
s
tem
o
f
in
f
o
r
m
atio
n
en
c
o
d
i
n
g
d
u
e
t
o
s
af
ety
co
n
ce
r
n
s
,
m
a
y
b
e
g
iv
e
n
an
o
t
h
er
lo
o
k
.
T
h
is
el
ev
ated
th
e
p
u
b
lic
k
ey
’
s
s
ec
u
r
ity
lev
el.
T
h
e
p
r
im
ar
y
b
en
ef
its
o
f
th
e
s
u
g
g
ested
ap
p
r
o
ac
h
ar
e
th
ese
b
ec
au
s
e
it i
s
wid
ely
r
ec
o
g
n
ized
th
at
th
ese
ca
teg
o
r
ies
o
f
en
c
o
d
i
n
g
s
ca
n
r
esu
lt
in
a
d
ec
r
ea
s
e
in
m
ag
n
itu
d
e
o
f
t
h
e
s
h
ar
ed
k
ey
s
o
r
,
co
m
p
ar
a
b
ly
,
a
n
in
cr
ea
s
e
in
in
f
o
r
m
atio
n
d
ec
o
d
i
n
g
r
esis
tiv
ity
.
New
p
ar
am
eter
s
f
o
r
t
h
e
co
n
v
en
tio
n
al
Mc
E
liece
an
d
Nied
e
r
r
eiter
cr
y
p
to
s
y
s
tem
s
ar
e
p
r
o
p
o
s
ed
[
2
0
]
,
g
u
ar
an
teein
g
b
aselin
e
s
ec
u
r
ity
ac
r
o
s
s
all
in
v
esti
g
ated
th
r
ea
ts
.
T
h
e
m
o
d
if
ied
s
ettin
g
s
tak
e
in
to
c
o
n
s
i
d
er
atio
n
th
e
im
p
r
o
v
e
d
th
r
ea
t,
th
e
r
ec
e
n
tly
ad
d
ed
b
in
a
r
y
Go
p
p
a
co
d
e
l
is
t
d
ec
o
d
in
g
ap
p
r
o
ac
h
,
a
n
d
th
e
o
p
tio
n
to
em
p
lo
y
co
d
e
s
eq
u
en
ce
s
th
at
ar
en
’
t
m
u
ltip
le
o
f
two
.
Fo
r
th
e
s
am
e
lev
el
o
f
s
ec
u
r
ity
,
th
e
r
esu
ltin
g
s
h
ar
ed
-
k
ey
len
g
th
s
ar
e
s
ig
n
if
ican
tly
s
h
o
r
ter
co
m
p
a
r
ed
to
p
r
io
r
p
a
r
am
eter
s
ettin
g
s
s
elec
tio
n
.
T
o
f
u
r
th
er
clar
if
y
o
n
th
e
n
at
u
r
e
o
f
th
e
co
n
v
en
tio
n
al
Mc
E
liece
cr
y
p
t
o
s
y
s
tem
’
s
q
u
an
tu
m
r
esis
tan
ce
,
th
e
au
th
o
r
s
ex
am
in
e
p
o
te
n
tial
q
u
an
tu
m
attac
k
s
an
d
s
u
g
g
est
d
ef
en
s
e
s
tr
ateg
ies
.
Pu
n
ey
a
n
i
an
d
B
h
at
[
2
1
]
d
e
m
o
n
s
tr
ated
ex
ec
u
tio
n
o
f
a
co
n
v
en
tio
n
al
Mc
E
liece
s
y
s
tem
o
f
in
f
o
r
m
atio
n
e
n
cr
y
p
tio
n
o
n
a
f
ield
-
p
r
o
g
r
am
m
a
b
le
g
ate
ar
r
ay
(
FP
GA)
,
g
u
ar
a
n
tee
in
g
th
e
s
ec
u
r
ity
g
u
id
e
lin
es p
r
o
v
id
ed
b
y
th
e
E
u
r
o
p
ea
n
T
elec
o
m
m
u
n
icatio
n
s
Stan
d
ar
d
s
I
n
s
titu
te
f
o
r
th
e
n
ex
t
wav
e
o
f
en
cr
y
p
tio
n
s
y
s
tem
s
th
at
ar
e
r
o
b
u
s
t
to
q
u
a
n
tu
m
r
esis
t
an
ce
.
E
m
p
lo
y
in
g
a
s
h
ar
ed
k
ey
with
a
b
y
te
len
g
t
h
o
f
2
,
0
9
7
,
1
5
2
,
t
h
e
s
u
g
g
ested
im
p
lem
e
n
tatio
n
b
y
th
e
a
u
th
o
r
s
o
f
f
er
s
q
u
an
tu
m
s
ec
u
r
ity
with
b
its
b
e
y
o
n
d
1
2
8
.
T
h
e
s
u
g
g
ested
s
y
s
tem
is
b
u
ilt
ar
o
u
n
d
a
h
ar
d
war
e
an
d
s
o
f
twar
e
s
ettin
g
s
th
at
m
ak
es
u
s
e
o
f
an
AX1
4
lite
in
ter
f
ac
e
to
lin
k
an
AR
M
C
o
r
tex
-
A5
3
co
r
e
t
o
a
co
p
r
o
ce
s
s
o
r
.
T
h
e
s
tate
-
of
-
th
e
-
ar
t
co
m
p
r
e
h
en
s
iv
e
o
v
e
r
v
ie
w,
co
m
p
o
n
e
n
t
-
by
-
co
m
p
o
n
en
t
alg
o
r
ith
m
ic
d
escr
ip
tio
n
,
an
d
im
p
lem
en
tatio
n
o
f
th
is
cr
y
p
to
s
y
s
tem
ar
e
p
r
esen
ted
[
2
2
]
.
Dif
f
er
en
t
Mc
E
liec
e
cr
y
p
to
s
y
s
tem
attac
k
s
ar
e
co
v
er
ed
in
s
ep
a
r
ate
s
ec
tio
n
s
.
Asi
d
e
f
r
o
m
s
im
u
latio
n
o
f
th
e
cr
y
p
to
s
y
s
tem
o
n
d
if
f
e
r
en
t
ex
te
n
s
io
n
d
eg
r
ee
s
,
th
e
au
th
o
r
s
also
p
r
esen
t
ex
p
e
r
im
en
tal
r
esu
lts
u
s
in
g
Go
p
p
a
co
d
es.
T
h
e
au
t
h
o
r
s
co
n
clu
d
e
d
th
e
r
esu
lts
an
d
t
h
e
d
if
f
er
en
t im
p
lem
en
tatio
n
-
r
elate
d
is
s
u
es b
ased
o
n
th
e
s
im
u
lat
io
n
s
th
at
wer
e
r
u
n
.
T
h
e
m
o
d
er
n
v
a
r
ian
ts
o
f
th
e
class
ical
cr
y
p
to
s
y
s
tem
s
p
u
t
f
o
r
th
b
y
Har
o
ld
Nied
er
r
eiter
(
1
9
8
6
)
a
n
d
R
o
b
er
t
J
.
Mc
E
liece
(
1
9
7
8
)
ar
e
ex
am
in
ed
[
2
3
]
.
Fiv
e
d
if
f
er
e
n
t
co
d
e
-
b
ased
s
h
ar
ed
k
ey
s
y
s
tem
s
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
h
av
e
b
e
en
th
o
r
o
u
g
h
ly
r
ev
iewe
d
.
I
t
is
d
em
o
n
s
tr
a
ted
th
at
th
er
e
ar
e
s
er
io
u
s
p
r
o
b
lem
s
with
s
ev
er
al
co
n
tem
p
o
r
ar
y
e
x
p
o
s
itio
n
s
o
f
t
r
ad
itio
n
al
Mc
E
liece
an
d
Nied
er
r
eiter
s
y
s
tem
s
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
.
I
t
h
as
b
ee
n
d
e
m
o
n
s
tr
ated
,
in
p
a
r
ticu
lar
,
th
at
XGRS
en
cr
y
p
tio
n
s
y
s
te
m
s
,
wh
ich
b
ases
its
elf
o
n
th
e
b
r
o
a
d
en
ed
R
ee
d
-
So
lo
m
o
n
co
d
e,
co
n
tain
m
u
ltip
le
f
laws
an
d
is
n
o
t
as
s
ec
u
r
e
a
g
ain
s
t
th
e
in
f
o
r
m
atio
n
s
et
d
ec
o
d
in
g
attac
k
as
it
is
s
u
p
p
o
s
ed
to
b
e
.
I
t
is
d
em
o
n
s
t
r
ated
th
at
b
o
th
t
h
e
s
h
ar
ed
an
d
u
n
s
h
ar
e
d
en
cr
y
p
tio
n
k
e
y
s
o
cc
u
p
y
a
s
ig
n
if
ican
t
q
u
o
ta
o
f
s
to
r
ag
e
a
n
d
th
at
k
ey
g
en
er
atio
n
a
n
d
d
ec
r
y
p
tio
n
in
c
o
n
tem
p
o
r
ar
y
c
r
y
p
to
s
y
s
tem
s
ta
k
e
a
lo
n
g
tim
e.
A
n
o
v
el
c
o
d
e
-
b
ased
d
ig
ital
s
ig
n
atu
r
e
b
u
ilt
o
n
th
e
Mc
E
lie
ce
s
y
s
tem
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
is
p
r
o
p
o
s
ed
[
2
4
]
.
Alg
o
r
ith
m
s
f
o
r
th
e
co
n
s
tr
u
ctio
n
o
f
s
h
ar
ed
k
ey
,
s
ig
n
in
g
,
a
n
d
au
th
e
n
ticatio
n
ar
e
s
h
o
wn
.
T
h
e
p
u
b
lic
k
ey
is
cr
ea
te
d
b
y
th
e
k
ey
g
en
er
atio
n
alg
o
r
ith
m
u
s
in
g
r
an
d
o
m
in
v
er
s
e
m
atr
ices.
C
o
m
p
ar
ed
to
t
h
e
C
FS
s
ch
em
e,
th
e
s
ig
n
in
g
alg
o
r
ith
m
is
less
co
m
p
lex
an
d
tak
es
less
co
m
p
u
tin
g
tim
e
to
s
ig
n
a
d
o
cu
m
en
t.
Fo
r
g
er
ies
ca
n
b
e
r
ec
o
g
n
ize
d
b
y
th
e
v
e
r
if
icatio
n
alg
o
r
ith
m
.
I
t
is
d
em
o
n
s
tr
at
ed
th
at
th
e
s
u
g
g
ested
s
ch
em
e
is
r
esis
tan
t
to
s
tr
u
ctu
r
al
attac
k
s
u
s
in
g
p
u
b
lic
k
ey
s
.
B
ir
h
an
u
et
a
l.
[
2
5
]
em
p
lo
y
an
ir
r
eg
u
lar
co
d
e
v
er
s
io
n
o
f
th
e
QC
-
L
DPC
an
d
th
e
q
u
asi
-
cy
clic
m
o
d
er
ate
-
d
en
s
ity
p
ar
ity
-
ch
ec
k
(
QC
-
MD
PC
)
in
p
lace
o
f
G
o
p
p
a
c
o
d
e,
wh
ich
is
u
s
ed
in
tan
d
em
t
o
ad
d
r
ess
p
r
ev
io
u
s
b
o
ttlen
ec
k
s
in
th
e
s
y
s
tem
.
R
esu
lt
s
o
b
tain
ed
b
y
th
e
p
r
o
p
o
s
ed
m
et
h
o
d
also
co
n
f
ir
m
ed
th
at
th
e
len
g
th
o
f
th
e
s
h
ar
ed
k
ey
was
ap
p
r
o
p
r
iately
s
h
o
r
ten
ed
.
T
h
e
f
ac
t
t
h
at
th
is
r
elea
s
e
o
f
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
m
o
r
e
r
esis
tan
t
to
m
ess
ag
e
-
r
es
en
d
th
r
ea
ts
is
an
o
t
h
er
b
en
ef
it
o
v
er
th
e
p
r
ev
io
u
s
iter
atio
n
.
Su
tr
ad
h
ar
[
2
6
]
f
o
cu
s
es
o
n
th
e
d
ev
elo
p
m
en
t
o
f
in
d
is
tin
g
u
is
h
ab
ilit
y
u
n
d
e
r
ad
ap
tiv
e
ch
o
s
en
cip
h
er
tex
t
attac
k
2
(
I
ND
-
C
C
A2
)
s
ec
u
r
e
v
er
s
io
n
o
f
t
h
e
c
o
n
v
e
n
tio
n
al
M
cE
liece
s
y
s
tem
o
f
in
f
o
r
m
atio
n
en
cr
y
p
tio
n
.
T
h
e
au
t
h
o
r
s
e
m
p
l
o
y
th
e
S
-
r
e
p
etitio
n
en
cr
y
p
tio
n
o
f
S/2
d
if
f
e
r
en
t
in
f
o
r
m
atio
n
with
a
s
in
g
le
ty
p
ic
al
p
er
m
u
tatio
n
,
wh
ic
h
co
n
tr
a
d
ict
th
e
S
-
r
ep
etitio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
1
4
I
n
t J Ad
v
Ap
p
l Sci
,
Vo
l.
1
5
,
No
.
1
,
Ma
r
c
h
2
0
2
6
:
293
-
3
0
2
296
en
cr
y
p
tio
n
o
f
s
in
g
le
in
f
o
r
m
at
io
n
in
o
th
e
r
m
o
d
if
icatio
n
.
N
ew
Mc
E
liece
s
y
s
tem
o
f
in
f
o
r
m
a
tio
n
en
cr
y
p
tio
n
,
with
its
f
o
u
n
d
atio
n
o
n
p
u
n
c
tu
r
ed
R
M
co
d
es
ar
e
[
2
7
]
.
T
h
ey
ef
f
ec
tiv
ely
d
em
o
n
s
tr
ate
th
e
in
e
f
f
icac
y
o
f
well
-
k
n
o
wn
s
ec
u
r
ity
th
r
ea
ts
o
n
th
e
s
u
g
g
ested
R
M
co
d
e
-
b
ased
Mc
E
liece
cr
y
p
to
s
y
s
tem
,
in
clu
d
in
g
th
e
Min
d
er
-
Sh
o
k
r
o
llah
i,
C
h
izh
o
v
-
B
o
r
o
d
in
,
an
d
s
q
u
a
r
e
co
d
e
atta
ck
s
.
I
n
o
r
d
er
to
g
u
ar
d
ag
ai
n
s
t
th
e
af
o
r
em
en
tio
n
ed
attac
k
s
o
n
th
e
s
u
g
g
ested
R
M
co
d
e
-
b
ased
c
r
y
p
t
o
s
y
s
tem
s
,
th
e
au
th
o
r
s
d
ev
is
ed
an
id
ea
l
p
u
n
ctu
r
i
n
g
s
ch
em
e.
Sp
ec
if
ically
,
th
ey
d
eter
m
in
ed
th
e
p
r
ec
is
e
ar
ea
s
o
f
p
u
n
ctu
r
i
n
g
p
o
s
itio
n
s
wh
er
e
th
e
g
en
e
r
ato
r
m
atr
i
x
’
s
least
am
o
u
n
t
o
f
p
u
n
ctu
r
e
d
co
lu
m
n
s
co
u
ld
b
e
f
o
u
n
d
.
3.
P
RE
L
I
M
I
NAR
I
E
S
T
h
e
wo
r
k
in
g
o
f
co
n
v
en
tio
n
Mc
E
liece
cr
y
p
to
s
y
s
tem
an
d
cir
cu
lan
t
m
atr
ix
ar
e
d
es
cr
ib
ed
i
n
th
is
s
ec
tio
n
.
3
.
1
.
T
he
M
cEl
iece
cr
y
pto
s
y
s
t
em
R
o
b
er
t
J
.
Mc
E
liece
cr
ea
ted
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
,
a
p
u
b
lic
-
k
ey
e
n
cr
y
p
tio
n
alg
o
r
ith
m
,
in
1
9
7
8
.
I
n
co
n
tr
ast
to
n
u
m
e
r
o
u
s
o
th
e
r
p
u
b
lic
-
k
e
y
en
cr
y
p
tio
n
alg
o
r
ith
m
s
th
at
d
ep
en
d
o
n
d
is
cr
ete
lo
g
ar
ith
m
p
r
o
b
lem
s
o
r
f
ac
to
r
izatio
n
d
if
f
icu
lties
,
M
cE
liece
is
b
ased
o
n
th
e
c
h
allen
g
e
o
f
d
ec
o
d
in
g
er
r
o
r
-
c
o
r
r
ec
t
i
n
g
co
d
es,
wh
ich
ar
e
f
r
eq
u
e
n
tly
u
tili
ze
d
in
d
ig
ital
c
o
m
m
u
n
icatio
n
s
.
T
h
e
Mc
E
liec
e
cr
y
p
to
s
y
s
tem
r
el
ies
o
n
Go
p
p
a
co
d
e,
p
er
m
u
tatio
n
m
atr
ix
,
a
n
d
an
in
v
er
tib
le
m
a
tr
ix
to
s
cr
am
b
le
th
e
p
lain
tex
t
an
d
co
n
ce
al
a
s
ec
r
et
k
e
y
.
T
h
e
co
r
r
esp
o
n
d
in
g
co
d
ewo
r
d
g
en
er
ate
d
is
f
in
ally
p
er
m
u
ted
b
ef
o
r
e
tr
an
s
m
itted
th
r
o
u
g
h
th
e
tr
an
s
m
is
s
io
n
ch
an
n
el.
T
h
e
p
lain
tex
t
f
r
o
m
th
e
s
en
d
er
is
f
ir
s
t
s
cr
am
b
led
,
an
d
th
e
g
e
n
er
ated
co
d
e
wo
r
d
is
p
er
m
u
ted
.
A
s
et
o
f
b
its
u
p
to
t
f
r
o
m
th
e
co
d
ewo
r
d
a
r
e
f
lip
p
e
d
,
with
t
r
ep
r
esen
tin
g
th
e
e
r
r
o
r
c
o
r
r
ec
ti
o
n
co
d
e
o
f
th
e
g
e
n
er
ated
co
d
ewo
r
d
.
T
h
e
p
u
b
lic
k
ey
o
f
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
a
co
m
b
in
atio
n
o
f
n
o
n
-
s
i
n
g
u
lar
‘
k
’
b
y
‘
k
’
s
cr
am
b
le
m
at
r
ix
,
a
‘
k
’
b
y
‘
n
’
g
en
er
ato
r
m
atr
ix
,
an
d
‘
n
b
y
n
’
p
er
m
u
tatio
n
m
atr
ix
.
T
h
e
en
cr
y
p
tio
n
a
n
d
d
ec
r
y
p
tio
n
p
r
o
ce
s
s
o
f
th
e
Mc
E
liec
e
cr
y
p
to
s
y
s
tem
is
g
iv
en
as f
o
llo
ws
.
3
.
1
.
1
.
P
riv
a
t
e
k
e
y
g
ener
a
t
i
o
n
T
h
e
p
r
i
v
ate
k
e
y
o
f
th
e
Mc
E
l
iece
cr
y
p
to
s
y
s
tem
is
a
c
o
m
b
i
n
atio
n
o
f
th
e
g
en
e
r
at
o
r
m
atr
i
x
‘
G
’
,
th
e
s
cr
am
b
lin
g
m
atr
ix
‘
S
’
,
an
d
th
e
r
an
d
o
m
p
er
m
u
tatio
n
m
atr
ix
‘
P
’
.
T
h
e
g
en
e
r
ato
r
m
atr
i
x
is
g
iv
en
as sh
o
wn
in
(
1
)
.
=
(
|
(
−
)
′
)
(
1
)
W
h
er
e
‘
I
k
’
is
an
id
en
tity
m
atr
i
x
o
f
d
im
e
n
s
io
n
k
by
k
,
an
d
′
is
a
r
an
d
o
m
p
er
m
u
tatio
n
m
atr
ix
o
f
d
im
en
s
io
n
‘
(
−
)
’
.
T
h
e
p
ar
ity
ch
ec
k
m
atr
ix
o
f
th
e
lin
ea
r
c
o
d
e
is
o
b
tain
ed
as
s
h
o
wn
in
(
2
)
.
T
h
e
p
r
i
v
ate
k
ey
i
s
‘
S
k
’
is
th
e
co
m
b
in
atio
n
o
f
th
e
m
atr
ice
s
‘
G
’
,
‘
S
’
,
an
d
‘
P
’
.
−
(
)
=
(
′
|
(
−
)
)
(
2
)
3
.
1
.
2
.
P
ub
lic
k
ey
g
ener
a
t
io
n
T
h
e
p
u
b
lic
k
e
y
co
n
s
is
ts
o
f
‘
k
’
b
y
‘
n
’
m
atr
ix
G
′
d
ef
in
ed
b
y
‘
G
.
S
.
P
’
an
d
er
r
o
r
c
o
r
r
ec
tin
g
ca
p
ab
ilit
y
t.
T
h
e
en
cr
y
p
ti
o
n
p
r
o
ce
s
s
o
f
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
as
f
o
llo
ws:
g
iv
en
a
m
ess
ag
e
m
∈
2
,
a
r
an
d
o
m
er
r
o
r
v
ec
to
r
e
∈
2
is
ch
o
s
en
with
a
we
ig
h
t w
≤
t a
n
d
c
o
m
p
u
te
co
r
r
esp
o
n
d
in
g
en
cr
y
p
ted
m
ess
ag
e
as (
3
)
.
=
′
+
(
3
)
T
o
d
ec
r
y
p
t th
e
e
n
cr
y
p
te
d
m
ess
ag
e,
th
e
f
o
llo
win
g
is
co
m
p
u
te
d
as sh
o
wn
in
(
4
)
.
−
1
=
+
−
1
(
4
)
Sin
ce
‘
P
’
is
a
p
er
m
u
tatio
n
m
a
tr
ix
,
‘
−
1
=
’
is
eq
u
ally
a
p
er
m
u
tatio
n
m
atr
ix
as
s
u
ch
,
th
e
v
ec
to
r
‘
−
1
’
h
as
th
e
s
am
e
weig
h
t
as
e
.
T
h
er
ef
o
r
e,
‘
’
ca
n
b
e
o
b
tain
e
d
b
y
d
ec
o
d
in
g
‘
−
1
’
.
Fin
ally
,
‘
’
ca
n
b
e
m
u
l
tip
lied
by
−
1
as (
)
−
1
to
o
b
tain
.
3
.
2
.
Circ
ula
nt
ma
t
rix
C
ir
cu
lan
t
m
atr
ix
is
a
m
atr
ix
i
n
wh
ich
ea
ch
r
o
w
r
elativ
e
to
th
e
p
r
ev
i
o
u
s
r
o
w
v
ec
t
o
r
is
r
o
tated
o
n
e
elem
en
t
to
th
e
r
ig
h
t
[
2
8
]
.
T
h
e
p
r
o
d
u
ct
o
f
cir
cu
la
n
t
m
atr
ices
is
also
a
cir
cu
lan
t
m
atr
ix
a
n
d
co
m
m
u
tativ
e
[
2
9
]
.
Ma
tr
ix
A
is
a
cir
cu
lan
t
m
atr
ix
with
en
tr
ies
g
en
er
ated
f
r
o
m
th
e
n
-
v
ec
to
r
{
1
,
2
,
.
.
.
,
}
b
y
cy
clic
ally
p
er
m
u
tin
g
its
en
tr
ies,
an
d
is
o
f
th
e
f
o
r
m
as sh
o
wn
in
(
5
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
A
n
o
ve
l c
ir
cu
la
n
t m
a
tr
ix
-
b
a
s
ed
McE
liece
fr
a
mewo
r
k
fo
r
s
ec
u
r
e
d
ig
ita
l c
o
mmu
n
ica
tio
n
(
R
a
viku
ma
r
I
n
a
k
o
ti
)
297
=
[
1
2
…
1
…
−
1
.
.
.
2
.
.
.
3
.
.
.
…
.
.
.
1
]
(
5
)
Fo
r
ex
am
p
le,
we
d
ef
in
e
t
h
e
cir
cu
lan
t m
atr
ix
g
e
n
er
ated
b
y
th
r
ee
elem
en
ts
as sh
o
wn
in
(
6
)
.
3
(
,
,
)
=
[
]
(
6
)
T
h
e
b
lo
ck
cir
c
u
lan
t
o
f
cir
c
u
lan
t
m
atr
ix
(
)
is
an
M
×
M
m
atr
ix
f
o
r
ev
e
r
y
=
1
,
2
,
.
.
.
,
.
T
h
en
{
1
,
2
,
.
.
.
,
}
g
en
er
ates
an
N
M
×
N
M
b
lo
ck
s
cir
cu
lan
t
m
atr
ices.
An
ex
am
p
le
o
f
a
b
lo
ck
cir
cu
lan
t m
atr
ix
(
C
ir
c
B
)
f
o
r
‘
A
’
d
ef
in
ed
as sh
o
wn
in
(
7
)
to
(
9
)
.
=
[
1
2
3
4
5
6
7
8
8
]
;
ℎ
(
7
)
=
(
,
,
)
=
[
]
(
8
)
W
e
h
av
e
,
=
(
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
AT
E
R
I
AL
S AN
D
M
E
T
H
O
D
I
n
th
is
s
ec
tio
n
,
th
e
ci
r
cu
lan
t
m
atr
ix
-
b
ased
Mc
E
liece
cr
y
p
to
s
y
s
tem
is
p
r
esen
ted
in
d
etail.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
u
s
es
th
e
s
tr
u
ctu
r
e
o
f
th
e
tr
ad
itio
n
al
Mc
E
l
iece
s
ch
em
e
b
u
t
in
tr
o
d
u
ce
s
th
e
cir
cu
lan
t
m
atr
ices
to
im
p
r
o
v
e
th
e
ef
f
icac
y
o
f
s
to
r
ag
e
an
d
co
m
p
u
tatio
n
.
T
h
e
M
cE
liece
cr
y
p
to
s
y
s
tem
is
b
ased
o
n
t
h
e
d
if
f
icu
lty
o
f
d
ec
o
d
in
g
r
an
d
o
m
lin
ea
r
c
o
d
es,
o
f
ten
u
s
in
g
Go
p
p
a
co
d
es,
wh
ich
m
ak
es
it
r
esis
tan
t
to
q
u
an
tu
m
-
b
ased
attac
k
s
.
Usi
n
g
cir
cu
lan
t
m
atr
ices
allo
ws
f
o
r
a
r
ed
u
ctio
n
in
k
ey
s
ize,
en
h
an
cin
g
th
e
s
y
s
tem
'
s
p
r
ac
ti
ca
lity
.
T
h
e
p
r
o
p
o
s
ed
s
y
s
tem
co
n
s
is
ts
o
f
th
r
ee
p
h
ases
,
wh
i
ch
in
clu
d
e
k
ey
g
e
n
er
atio
n
p
h
ase,
e
n
c
r
y
p
tio
n
p
h
ase
,
an
d
d
ec
r
y
p
tio
n
p
h
ase.
4
.
1
.
K
ey
g
ener
a
t
io
n pha
s
e
I
n
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
,
k
ey
g
en
e
r
atio
n
i
n
v
o
lv
es
cr
ea
tin
g
a
p
u
b
lic
a
n
d
p
r
iv
at
e
k
ey
p
air
th
r
o
u
g
h
a
Go
p
p
a
co
d
e
(
o
r
a
co
m
p
ar
ab
le
er
r
o
r
-
c
o
r
r
ec
tin
g
co
d
e)
an
d
ap
p
ly
in
g
r
a
n
d
o
m
tr
an
s
f
o
r
m
atio
n
s
to
co
n
ce
al
its
s
tr
u
ctu
r
e.
B
y
u
s
in
g
cir
cu
lan
t
m
at
r
ices,
s
to
r
ag
e
r
eq
u
ir
e
m
en
ts
f
o
r
th
e
g
e
n
er
ato
r
m
atr
i
x
ar
e
m
in
im
ized
,
as
th
e
en
tire
m
atr
ix
ca
n
b
e
g
en
e
r
ated
f
r
o
m
ju
s
t
o
n
e
r
o
w.
T
o
g
e
n
er
ate
th
e
p
r
i
v
ate
an
d
p
u
b
lic
k
e
y
,
a
co
d
e
p
ar
a
m
eter
n
r
e
p
r
esen
tin
g
co
d
e
len
g
th
,
k
r
ep
r
esen
tin
g
th
e
d
im
en
s
io
n
o
f
th
e
co
d
e
p
ar
am
eter
,
an
d
er
r
o
r
co
r
r
ec
tio
n
ca
p
ab
ilit
y
t
is
ch
o
s
en
.
Af
ter
ch
o
o
s
in
g
th
e
p
a
r
am
eter
s
,
th
e
b
in
ar
y
G
o
p
p
a
co
d
e
c
,
with
a
g
en
er
ato
r
m
atr
ix
‘
G
’
o
f
s
ize
k
x
n
is
ch
o
s
en
.
i)
Prim
ar
y
k
e
y
g
e
n
er
atio
n
:
t
o
c
o
n
s
tr
u
ct
th
e
p
r
iv
ate
k
ey
,
a
g
e
n
er
ato
r
m
atr
ix
‘
G
’
is
cr
ea
ted
f
o
r
th
e
ch
o
s
en
G
o
p
p
a
co
d
e,
s
tr
u
ctu
r
ed
as
a
cir
cu
lan
t
m
atr
ix
.
Fo
r
ex
am
p
le,
a
v
ec
to
r
v
=
{
v
0
,
v
1
,
v
2
,
…
,
v
n
−
1
}
is
ch
o
s
en
to
g
en
er
ate
a
cir
cu
la
n
t
m
atr
ix
‘
G
’
b
y
r
o
tatin
g
v
r
o
w
b
y
r
o
w.
I
n
ad
d
itio
n
to
G
,
two
s
ec
r
et
m
atr
ices
‘
S
’
an
d
‘
P
’
r
ep
r
esen
tin
g
a
r
an
d
o
m
in
v
er
tib
le
k
×
k
m
atr
ix
u
s
ed
to
p
er
m
u
tate
t
h
e
p
lain
tex
t
a
n
d
a
r
an
d
o
m
p
er
m
u
tatio
n
m
at
r
ix
o
f
s
ize
n
×
n
th
at
s
cr
am
b
led
th
e
o
r
d
er
o
f
t
h
e
co
d
e
b
its
ar
e
d
ef
i
n
ed
.
T
h
e
p
r
iv
ate
k
ey
is
th
en
g
en
e
r
ated
to
c
o
m
p
o
s
e
th
e
tr
ip
le
(
G
,
S
,
P
)
as sh
o
wn
in
(
1
0
)
.
=
(
,
,
)
(
1
0
)
W
h
er
e
‘
G
’
is
th
e
g
en
er
at
o
r
m
at
r
ix
o
f
Go
p
p
a
co
d
e
in
cir
cu
lan
t
m
atr
ix
f
o
r
m
,
‘
S
’
is
th
e
s
cr
am
b
lin
g
m
atr
i
x
to
m
ask
th
e
s
tr
u
ctu
r
e
o
f
‘
G
’
an
d
‘
P
’
is
th
e
p
er
m
u
tatio
n
m
atr
i
x
to
d
is
g
u
is
e
th
e
ar
r
an
g
em
en
t
o
f
t
h
e
co
d
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
1
4
I
n
t J Ad
v
Ap
p
l Sci
,
Vo
l.
1
5
,
No
.
1
,
Ma
r
c
h
2
0
2
6
:
293
-
3
0
2
298
ii)
Pu
b
lic
k
ey
g
e
n
er
atio
n
:
t
h
e
p
u
b
lic
k
ey
G
′
is
co
m
p
u
ted
as sh
o
wn
in
(
1
1
)
.
′
=
(
1
1
)
B
ec
au
s
e
‘
G
’
is
cir
cu
lan
t,
it
r
eq
u
ir
es
m
u
ch
less
s
to
r
ag
e
s
p
a
ce
.
G
′
is
p
u
b
lis
h
ed
as
a
p
u
b
lic
k
ey
,
wh
ile
(
G
,
S
,
P
)
ar
e
k
ep
t
p
r
iv
ate.
4
.
2
.
E
ncry
ptio
n
ph
a
s
e
T
o
en
cr
y
p
t
a
m
ess
ag
e
m
u
s
in
g
th
e
p
u
b
lic
k
ey
G
′
g
en
er
ated
in
th
e
k
ey
g
e
n
er
atio
n
p
h
ase,
‘
’
is
r
ep
r
esen
ted
as
a
b
i
n
ar
y
v
ec
to
r
o
f
len
g
th
‘
k
’
.
T
h
e
en
co
d
e
d
m
e
s
s
ag
e
‘
c
’
is
co
m
p
u
ted
b
y
m
u
ltip
ly
in
g
th
e
p
lai
n
m
ess
ag
e
m
with
th
e
p
u
b
lic
k
e
y
G
′
an
d
ad
d
in
g
a
r
an
d
o
m
e
r
r
o
r
v
ec
to
r
‘
e
’
o
f
weig
h
t
‘
t
’
with
ex
a
ctly
‘
t
’
r
an
d
o
m
b
its
s
et
to
1
as sh
o
wn
in
(
1
2
)
.
=
′
+
(
1
2
)
T
h
e
er
r
o
r
v
ec
to
r
e
c
o
m
p
licate
s
th
e
d
ec
o
d
in
g
p
r
o
ce
s
s
with
o
u
t
ac
ce
s
s
to
th
e
p
r
iv
ate
k
ey
,
th
u
s
s
af
eg
u
ar
d
i
n
g
t
h
e
cip
h
er
tex
t'
s
s
ec
u
r
ity
.
4
.
3
.
T
ra
ns
m
it
t
he
ciphert
ex
t
T
h
e
f
in
al
en
cr
y
p
ted
o
u
tp
u
t
,
d
en
o
te
d
as
'
c'
is
s
en
t
f
r
o
m
th
e
s
en
d
e
r
to
th
e
r
ec
eiv
er
o
v
e
r
th
e
co
m
m
u
n
icatio
n
ch
an
n
el.
T
h
is
cip
h
er
tex
t
'
c'
n
o
w
co
n
tain
s
th
e
o
r
ig
in
al
m
ess
ag
e
f
u
lly
en
co
d
ed
th
r
o
u
g
h
th
e
en
cr
y
p
tio
n
alg
o
r
ith
m
,
c
o
m
b
in
ed
with
a
d
elib
er
ately
a
d
d
ed
r
an
d
o
m
er
r
o
r
v
ec
to
r
f
o
r
en
h
a
n
ce
d
s
ec
u
r
ity
.
T
h
is
tr
an
s
m
is
s
io
n
s
tep
co
m
p
letes
th
e
en
cr
y
p
tio
n
p
h
ase,
en
s
u
r
in
g
th
e
m
ess
ag
e
r
em
ain
s
p
r
o
tecte
d
u
n
til
p
r
o
p
e
r
d
ec
r
y
p
tio
n
with
th
e
s
h
ar
e
d
k
e
y
o
cc
u
r
s
at
th
e
r
ec
eiv
er
'
s
en
d
.
4
.
4
.
Dec
ry
pti
o
n
ph
a
s
e
I
n
th
is
p
h
ase,
th
e
p
r
im
ar
y
k
e
y
(
,
,
)
is
u
s
ed
to
d
ec
r
y
p
t
t
h
e
cip
h
er
tex
t
‘
’
.
T
h
e
f
ir
s
t
s
tep
in
th
is
p
h
ase
is
to
u
n
d
o
th
e
p
er
m
u
tati
o
n
ap
p
lie
d
d
u
r
in
g
th
e
en
cr
y
p
ti
o
n
b
y
co
m
p
u
tin
g
‘
c
P
−
1
’
as sh
o
wn
in
(
1
3
)
.
′
=
−
1
=
(
′
+
)
−
1
=
+
−
1
(
1
3
)
Af
ter
r
em
o
v
i
n
g
th
e
p
er
m
u
tatio
n
,
th
e
s
cr
am
b
lin
g
m
atr
ix
is
in
v
er
ted
b
y
m
u
ltip
ly
in
g
c
′
by
S
−
1
as sh
o
wn
in
(
1
4
)
.
−
1
′
=
−
1
(
+
−
1
)
=
+
−
1
−
1
(
1
4
)
T
h
e
s
tep
y
ield
s
a
s
cr
am
b
led
c
o
d
ewo
r
d
with
m
in
o
r
er
r
o
r
,
wh
ich
ca
n
b
e
r
eso
lv
ed
d
u
r
in
g
er
r
o
r
co
r
r
ec
tio
n
p
h
ase.
4
.
5
.
E
rr
o
r
c
o
rr
ec
t
io
n
T
o
c
o
r
r
e
c
t
e
r
r
o
r
s
i
n
t
h
e
s
c
r
a
m
b
l
e
d
c
o
d
e
w
o
r
d
,
t
h
e
e
r
r
o
r
-
c
o
r
r
e
c
ti
n
g
a
l
g
o
r
i
t
h
m
f
o
r
G
o
p
p
a
c
o
d
e
a
s
s
o
ci
a
t
e
d
w
i
t
h
‘
G
’
i
s
u
s
e
d
t
o
d
e
c
o
d
e
‘
mG
’
a
n
d
r
e
c
t
i
f
y
a
n
y
e
r
r
o
r
i
n
t
r
o
d
u
c
e
d
b
y
‘
e
’
.
T
h
e
r
e
t
r
i
e
v
e
d
‘
m
’
i
s
d
e
c
o
d
e
d
r
e
s
i
d
u
a
l
e
r
r
o
r
s
a
r
e
r
e
m
o
v
e
d
f
r
o
m
t
h
e
c
o
d
e
w
o
r
d
.
T
h
e
e
r
r
o
r
-
c
o
r
r
e
c
t
i
n
g
a
l
g
o
r
i
t
h
m
c
o
n
s
is
ts
o
f
t
h
e
f
o
l
l
o
win
g
s
t
e
p
s
.
i)
Sy
n
d
r
o
m
e
ca
lcu
latio
n
:
i
n
th
is
s
tag
e,
th
e
s
y
n
d
r
o
m
e
is
d
eter
m
in
ed
to
d
etec
t
th
e
p
r
esen
ce
an
d
lo
ca
tio
n
o
f
er
r
o
r
s
.
T
h
e
s
y
n
d
r
o
m
e
s
o
f
f
e
r
s
ess
en
tial
in
f
o
r
m
atio
n
r
e
g
a
r
d
in
g
t
h
e
p
o
s
itio
n
s
o
f
th
e
e
r
r
o
r
s
an
d
is
ca
lcu
lated
as sh
o
wn
in
(
1
5
)
.
=
(
1
5
)
H
e
r
e
,
H
r
e
p
r
e
s
e
n
ts
t
h
e
p
a
r
i
t
y
-
ch
e
c
k
m
a
t
r
i
x
r
el
a
t
e
d
t
o
t
h
e
G
o
p
p
a
c
o
d
e
,
a
n
d
H
T
i
s
i
ts
t
r
a
n
s
p
o
s
e
.
s
=
e
H
T
b
e
c
a
u
s
e
mG
′
H
T
=
0
i
f
s
=
0
,
t
h
e
n
n
o
e
r
r
o
r
s
a
r
e
p
r
e
s
e
n
t
a
n
d
i
f
s
≠
0
,
i
t
i
n
d
i
c
a
te
s
t
h
e
p
r
e
s
e
n
c
e
o
f
e
r
r
o
r
.
ii)
E
r
r
o
r
lo
ca
to
r
p
o
ly
n
o
m
ial
ca
lcu
lato
r
:
in
th
is
p
h
ase
,
th
e
er
r
o
r
lo
ca
tio
n
is
d
eter
m
in
ed
b
y
c
alcu
latin
g
th
e
er
r
o
r
lo
ca
t
o
r
p
o
ly
n
o
m
ial
σ
(
x
)
u
s
in
g
th
e
s
y
n
d
r
o
m
e
co
m
p
u
ted
i
n
th
e
s
y
n
d
r
o
m
e
ca
lc
u
latio
n
p
h
ase.
T
h
e
er
r
o
r
lo
ca
t
o
r
p
o
ly
n
o
m
ial
h
elp
s
id
en
tify
th
e
e
r
r
o
r
lo
ca
tio
n
s
,
w
h
ich
is
d
ef
in
ed
as sh
o
wn
in
(
1
6
)
.
(
)
=
∏
(
1
−
)
=
1
(
1
6
)
W
h
er
e
X
i
ar
e
th
e
lo
ca
tio
n
s
o
f
th
e
er
r
o
r
s
in
th
e
r
ec
eiv
e
d
v
ec
to
r
.
T
o
d
eter
m
in
e
th
e
er
r
o
r
lo
ca
to
r
p
o
ly
n
o
m
ial
σ
(
x
)
,
we
u
s
e
a
P
atter
s
o
n
s
y
n
d
r
o
m
e
-
b
ased
d
ec
o
d
i
n
g
alg
o
r
ith
m
s
p
ec
if
ically
d
esig
n
ed
f
o
r
Go
p
p
a
co
d
es.
Fo
r
ex
am
p
le,
co
n
s
id
er
a
s
im
p
le
s
ce
n
ar
io
with
t
wo
er
r
o
r
s
lo
ca
ted
at
p
o
s
itio
n
X
1
an
d
X
2
,
th
e
er
r
o
r
lo
ca
t
o
r
p
o
ly
n
o
m
ial
wo
u
l
d
b
e
r
e
p
r
esen
ted
as sh
o
wn
i
n
(
1
7
)
.
(
)
=
(
1
−
1
)
(
1
−
2
)
=
1
−
(
1
+
2
)
+
1
2
2
(
1
7
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
A
n
o
ve
l c
ir
cu
la
n
t m
a
tr
ix
-
b
a
s
ed
McE
liece
fr
a
mewo
r
k
fo
r
s
ec
u
r
e
d
ig
ita
l c
o
mmu
n
ica
tio
n
(
R
a
viku
ma
r
I
n
a
k
o
ti
)
299
T
h
e
co
ef
f
icien
ts
o
f
σ
(
x
)
ca
n
b
e
d
et
er
m
in
ed
f
r
o
m
th
e
s
y
n
d
r
o
m
e
v
ec
to
r
‘
s
’
.
iii)
So
lv
in
g
th
e
e
r
r
o
r
lo
ca
to
r
p
o
l
y
n
o
m
ial:
o
n
ce
th
e
er
r
o
r
lo
ca
to
r
p
o
ly
n
o
m
ial
(
)
is
o
b
tain
ed
,
th
e
n
ex
t
s
tep
is
to
s
o
lv
e
it
to
id
en
tif
y
th
e
er
r
o
r
p
o
s
itio
n
s
.
Fi
n
d
in
g
th
e
r
o
o
ts
o
f
(
)
=
0
r
ev
ea
ls
th
e
lo
ca
tio
n
s
X
i
o
f
th
e
er
r
o
r
s
.
T
h
ese
r
o
o
ts
ca
n
b
e
f
o
u
n
d
th
r
o
u
g
h
d
if
f
e
r
en
t
alg
e
b
r
aic
m
eth
o
d
s
tailo
r
ed
to
th
e
f
in
ite
f
ield
u
tili
ze
d
b
y
th
e
Go
p
p
a
co
d
e
.
Fo
r
e
x
am
p
le,
(
)
=
1
−
1
f
o
r
a
s
in
g
le
er
r
o
r
at
p
o
s
itio
n
X
1
,
s
o
lv
in
g
(
)
=
0
im
m
ed
iately
g
iv
es
=
1
.
iv
)
C
alcu
latin
g
th
e
er
r
o
r
m
ag
n
itu
d
e:
af
ter
p
in
p
o
in
tin
g
er
r
o
r
p
o
s
itio
n
s
,
er
r
o
r
m
a
g
n
itu
d
es
ca
n
b
e
d
eter
m
in
e
d
.
I
n
Go
p
p
a
c
o
d
es,
e
r
r
o
r
s
ar
e
g
en
er
ally
b
i
n
ar
y
(
0
o
r
1
)
,
m
ea
n
in
g
t
h
e
er
r
o
r
v
ec
to
r
e
h
as
n
o
n
-
ze
r
o
v
al
u
es
o
n
ly
at
th
e
s
p
ec
if
ie
d
er
r
o
r
lo
ca
tio
n
s
.
T
h
e
er
r
o
r
v
ec
to
r
e
is
co
n
s
tr
u
cted
b
y
s
ettin
g
th
e
i
d
en
tifie
d
er
r
o
r
p
o
s
itio
n
as
1
.
W
h
en
m
u
ltip
le
er
r
o
r
s
ar
e
p
r
esen
t,
ea
c
h
id
en
tif
ied
p
o
s
itio
n
in
th
e
v
ec
to
r
is
ass
ig
n
ed
a
v
alu
e
o
f
1
,
with
all
o
th
er
p
o
s
itio
n
s
lef
t a
s
0
.
v)
C
o
r
r
ec
tin
g
th
e
e
r
r
o
r
s
an
d
r
etr
iev
in
g
th
e
o
r
i
g
in
al
m
ess
ag
e:
in
th
is
p
h
ase,
t
h
e
er
r
o
r
s
in
th
e
r
ec
eiv
e
d
cip
h
er
tex
t ‘
c
’
ar
e
co
r
r
ec
ted
u
s
in
g
th
e
id
en
tifie
d
er
r
o
r
v
ec
to
r
‘
e
’
as sh
o
wn
in
(
1
8
)
.
−
=
′
(
1
8
)
T
h
is
p
r
o
d
u
ce
s
th
e
co
d
ewo
r
d
‘
mG
’
,
wh
ic
h
allo
ws
u
s
to
r
ec
o
v
e
r
th
e
o
r
ig
i
n
al
m
ess
ag
e
‘
m
’
b
y
ap
p
ly
in
g
s
tan
d
ar
d
d
ec
o
d
in
g
m
eth
o
d
s
u
s
ed
f
o
r
li
n
ea
r
co
d
es.
4.
6
.
Securit
y
a
na
ly
s
is
o
f
t
he
M
cEl
iece
cr
y
pto
s
y
s
t
em
An
an
aly
s
is
o
f
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
u
s
in
g
cir
c
u
lan
t
m
atr
ices
ev
alu
ates
its
r
esil
ien
ce
ag
ain
s
t
v
ar
io
u
s
attac
k
s
,
an
d
m
o
r
e
s
p
e
cif
ically
,
th
o
s
e
wh
ich
a
r
e
en
a
b
led
b
y
p
r
o
p
er
ties
o
f
cir
cu
lan
t
m
atr
ices.
Alth
o
u
g
h
im
p
r
o
v
in
g
s
to
r
ag
e
ef
f
icien
c
y
,
th
er
e
ar
e
p
ar
ticu
lar
d
if
f
ic
u
lties
in
tr
o
d
u
ce
d
b
y
th
e
u
s
e
o
f
cir
c
u
lan
t
m
atr
ices.
T
h
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
,
b
ein
g
a
s
t
an
d
ar
d
a
p
p
r
o
ac
h
,
ta
k
es
ad
v
an
tag
e
o
f
th
e
d
if
f
icu
lty
o
f
r
an
d
o
m
ly
d
ec
o
d
in
g
lin
ea
r
co
d
es.
Mo
r
e
s
p
ec
if
ically
,
f
o
r
a
r
an
d
o
m
ly
s
elec
ted
g
en
er
ato
r
m
atr
i
x
‘
G
’
,
th
e
task
o
f
r
an
d
o
m
ly
lin
ea
r
co
d
e
d
ec
o
d
in
g
(
d
ec
o
d
in
g
a
r
an
d
o
m
ly
lin
ea
r
c
o
d
e,
n
am
ely
f
in
d
i
n
g
th
e
in
itial
m
ess
ag
e
an
d
th
e
er
r
o
r
a
d
d
ed
)
with
o
u
t
th
e
p
r
iv
ate
k
e
y
is
d
if
f
icu
lt.
G
o
p
p
a
co
d
es
ar
e
ch
o
s
en
d
u
e
t
o
th
eir
im
p
r
o
v
ed
er
r
o
r
c
o
r
r
ec
ti
o
n
an
d
r
esil
ien
ce
to
ef
f
icien
t
d
ec
o
d
in
g
alg
o
r
ith
m
s
an
d
,
m
o
r
e
s
p
ec
if
ically
,
t
o
th
o
s
e
in
tr
o
d
u
ce
d
b
y
q
u
an
tu
m
th
r
ea
ts
.
I
n
t
h
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
,
s
ec
u
r
ity
ag
ain
s
t
p
o
ten
tial
attac
k
s
is
ac
h
iev
e
d
b
y
u
tili
zin
g
th
e
ar
r
an
g
em
e
n
t
o
f
Go
p
p
a
co
d
e
an
d
two
p
r
iv
ate
tr
an
s
f
o
r
m
atio
n
s
:
f
ir
s
t
ly
,
b
y
ap
p
ly
in
g
an
in
v
er
t
ib
le
r
an
d
o
m
m
atr
ix
S
f
o
r
d
is
tu
r
b
in
g
th
e
in
itial
m
ess
ag
e
,
an
d
s
ec
o
n
d
ly
b
y
u
tili
zin
g
a
p
er
m
u
tatio
n
m
atr
ix
P
f
o
r
r
an
d
o
m
ly
r
ea
r
r
an
g
i
n
g
co
d
e
p
o
s
itio
n
s
.
Ad
o
p
tin
g
cir
cu
lan
t
m
atr
ices
in
th
e
Mc
E
liece
cr
y
p
to
s
y
s
tem
alter
s
th
e
p
u
b
lic
k
e
y
G′
,
r
e
q
u
ir
in
g
a
ca
r
ef
u
l
ass
es
s
m
en
t o
f
an
y
v
u
ln
er
a
b
ilit
ies r
elate
d
to
cir
cu
lan
t m
atr
ix
ch
ar
ac
ter
is
tics
.
4.
7
.
Securit
y
o
f
t
he
circ
ula
nt
m
a
t
rix
ba
s
ed
Mc
E
liece
cr
y
pto
s
y
s
t
em
First
o
f
all,
th
e
u
s
e
o
f
cir
cu
la
n
t
m
atr
ices
h
elp
s
to
m
in
im
iz
e
th
e
s
ize
o
f
th
e
k
ey
.
I
t
also
im
p
o
s
es
a
p
atter
n
o
n
th
e
g
en
er
ato
r
m
atr
ix
th
at
m
ig
h
t
af
f
ec
t
s
ec
u
r
ity
.
I
n
th
is
s
ec
tio
n
,
we
e
x
am
in
e
th
e
p
r
im
ar
y
t
y
p
es
o
f
attac
k
s
in
d
etail,
f
o
cu
s
in
g
p
ar
t
icu
lar
ly
o
n
h
o
w
th
e
u
tili
za
tio
n
o
f
a
cir
c
u
lan
t m
atr
ix
i
n
f
lu
en
c
es th
em
.
4
.
7
.
1
.
Str
uct
ura
l
a
t
t
a
ck
T
h
is
ty
p
e
o
f
attac
k
tr
ies
to
f
i
n
d
p
atter
n
s
in
t
h
e
p
u
b
lic
k
ey
G
′
to
f
ig
u
r
e
o
u
t
th
e
p
r
iv
ate
k
ey
.
Sin
ce
a
cir
cu
lan
t
m
atr
ix
G
h
as
a
r
ep
ea
tin
g
,
cir
cu
lar
s
tr
u
ctu
r
e,
an
attac
k
er
m
ig
h
t
u
s
e
th
is
to
tr
y
an
d
r
eb
u
ild
th
e
m
atr
ix
.
T
o
s
to
p
th
is
,
a
r
an
d
o
m
p
er
m
u
tatio
n
m
atr
ix
is
u
s
ed
to
m
ix
u
p
th
e
cir
cu
lar
p
atter
n
o
f
G
,
m
ak
in
g
it
h
ar
d
to
s
ee
th
e
s
tr
u
ctu
r
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t m
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ased
Mc
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M
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Tr
a
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El
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e
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mp
r
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e
n
t
f
a
c
t
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r
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u
b
l
i
c
k
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y
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i
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e
~
2
5
6
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B
~
2
5
6
b
y
t
e
s
~1
,
0
0
0
x
r
e
d
u
c
t
i
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t
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(
k
×
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(
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a
st
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r
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p
t
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r
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t
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6.
CO
NCLU
SI
O
N
T
h
is
is
a
p
ap
er
d
escr
ib
in
g
an
alter
n
ativ
e
im
p
lem
en
tatio
n
o
f
th
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Mc
E
liece
cr
y
p
to
g
r
ap
h
ic
alg
o
r
ith
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th
at
u
tili
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s
cir
cu
lan
t
m
atr
ix
r
ep
r
esen
tatio
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s
.
On
e
o
f
th
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we
ak
n
ess
es
ass
o
ciate
d
with
u
s
in
g
th
is
ty
p
e
o
f
s
y
s
tem
is
th
at
it
o
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ten
r
eq
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i
r
es
v
er
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l
ar
g
e,
co
m
p
lex
k
e
y
s
izes
to
p
r
o
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e
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i
g
h
lev
els
o
f
s
ec
u
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ity
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h
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th
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s
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ap
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s
s
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m
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m
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ller
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ize
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ile
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ate
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ess
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t
v
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u
r
e
c
h
an
n
els.
B
y
d
ec
r
e
asin
g
th
e
k
ey
s
izes
ass
o
ciate
d
with
th
e
Mc
E
liece
cr
y
p
to
g
r
ap
h
ic
alg
o
r
ith
m
,
th
e
a
u
th
o
r
s
b
eliev
e
th
ei
r
wo
r
k
will a
llo
w
f
o
r
im
p
r
o
v
ed
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tio
n
o
f
th
e
alg
o
r
ith
m
in
co
n
tem
p
o
r
ar
y
d
ig
ital
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ev
ices
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clu
d
in
g
th
o
s
e
co
n
n
ec
ted
to
th
e
I
o
T
,
as
well
as
m
o
b
ile
d
e
v
ices
u
s
in
g
wir
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etwo
r
k
s
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Alth
o
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g
h
co
n
ce
r
n
s
r
eg
ar
d
in
g
wea
k
n
ess
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t
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tr
u
ctu
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p
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t,
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s
e
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p
r
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p
er
m
u
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d
s
cr
am
b
lin
g
m
atr
ices
m
in
i
m
izes
th
ese
r
is
k
s
.
T
h
is
o
f
f
er
s
th
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s
ec
u
r
ity
o
f
th
e
s
y
s
tem
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g
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s
t
b
o
th
r
eg
u
lar
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n
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q
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n
tu
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att
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Fu
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r
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r
esear
ch
c
o
u
ld
i
m
p
r
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v
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th
is
m
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d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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t J Ad
v
Ap
p
l Sci
I
SS
N:
2252
-
8
8
1
4
A
n
o
ve
l c
ir
cu
la
n
t m
a
tr
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b
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s
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fr
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R
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Giv
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is
in
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p
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q
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F
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Au
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RE
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NC
E
S
[
1
]
M
.
L
.
-
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a
r
c
í
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C
.
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a
v
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o
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c
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v
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p
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[
2
]
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B
i
n
d
a
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d
A
.
K
.
S
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g
h
,
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[
3
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K
.
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ss
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.
[
4
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H
.
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.
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h
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r
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t
h
,
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V
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d
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a
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a
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d
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.
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.
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r
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sa
d
,
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u
a
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m
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e
s
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l
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t
c
r
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p
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:
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q
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ms,
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EE
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A
c
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ss
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l
.
1
3
,
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p
.
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[
5
]
O
.
A
l
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b
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m
,
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s wit
h
q
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d
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m c
r
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,
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.
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p
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[
6
]
Z.
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.
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u
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t
a
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,
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u
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.
[
7
]
J.
O
.
D.
M
o
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a
l
,
A
.
D.
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O
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s
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8
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[
2
9
]
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.
S
.
C
.
P
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t
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,
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.
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S O
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v
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h
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in
th
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p
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rtme
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ter
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ste
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s E
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ra
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k
h
a
p
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h
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h
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c
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R.
Am
b
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d
k
a
r
C
h
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d
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c
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