I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
pu
t
er
E
ng
ineering
(
I
J
E
CE
)
Vo
l.
15
,
No
.
6
,
Decem
b
er
20
25
,
p
p
.
5
4
5
3
~
5
4
6
5
I
SS
N:
2088
-
8
7
0
8
,
DOI
: 1
0
.
1
1
5
9
1
/ijece.
v
15
i
6
.
pp
5
4
5
3
-
5
4
6
5
5453
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
ec
e.
ia
esco
r
e.
co
m
M
emo
ry
less
state
-
recov
ery cry
ptan
a
ly
sis
method for
lig
htweight
stream
ciphe
r
–
A
5
/1
K
hedk
a
r
Abo
li Audu
m
ba
r
1,
2
,
Uda
y
P
a
nd
it
K
ho
t
3
,
B
a
la
j
i
G
.
H
o
g
a
de
1
1
D
e
p
a
r
t
me
n
t
o
f
El
e
c
t
r
o
n
i
c
s E
n
g
i
n
e
e
r
i
n
g
,
T
e
r
n
a
E
n
g
i
n
e
e
r
i
n
g
C
o
l
l
e
g
e
,
N
a
v
i
M
u
mb
a
i
,
I
n
d
i
a
2
D
e
p
a
r
t
me
n
t
o
f
El
e
c
t
r
o
n
i
c
s a
n
d
T
e
l
e
c
o
mm
u
n
i
c
a
t
i
o
n
,
P
i
l
l
a
i
C
o
l
l
e
g
e
o
f
E
n
g
i
n
e
e
r
i
n
g
,
N
e
w
P
a
n
v
e
l
,
I
n
d
i
a
3
D
e
p
a
r
t
me
n
t
o
f
El
e
c
t
r
o
n
i
c
s a
n
d
T
e
l
e
c
o
mm
u
n
i
c
a
t
i
o
n
E
n
g
i
n
e
e
r
i
n
g
,
S
t
.
F
r
a
n
c
i
s I
n
st
i
t
u
t
e
o
f
T
e
c
h
n
o
l
o
g
y
,
M
u
mb
a
i
,
I
n
d
i
a
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
J
an
2
9
,
2
0
2
5
R
ev
is
ed
Au
g
2
2
,
2
0
2
5
Acc
ep
ted
Sep
1
6
,
2
0
2
5
Cry
p
t
o
lo
g
y
re
fe
rs
to
t
h
e
d
isc
ip
li
n
e
c
o
n
c
e
rn
e
d
with
se
c
u
ri
n
g
c
o
m
m
u
n
ica
ti
o
n
a
n
d
d
a
ta
i
n
tra
n
sit
b
y
tran
sf
o
rm
in
g
it
in
to
a
n
u
n
in
telli
g
i
b
le
f
o
r
m
,
th
e
re
b
y
p
re
v
e
n
ti
n
g
in
ter
p
re
tatio
n
b
y
u
n
a
u
th
o
rize
d
e
n
ti
ti
e
s.
Cry
p
tan
a
ly
sis
is
th
e
st
u
d
y
a
n
d
p
ra
c
ti
c
e
o
f
a
n
a
l
y
z
in
g
c
ry
p
t
o
g
ra
p
h
ic
sy
ste
m
s
wit
h
t
h
e
a
im
o
f
u
n
c
o
v
e
ri
n
g
th
e
ir
we
a
k
n
e
ss
e
s,
fin
d
in
g
v
u
l
n
e
ra
b
il
it
ies
a
n
d
o
b
tai
n
in
g
u
n
a
u
th
o
ri
z
e
d
a
c
c
e
ss
to
e
n
c
ry
p
ted
d
a
ta.
A5
/
1
is
a
li
g
h
t
we
ig
h
t
stre
a
m
c
ip
h
e
r
u
se
d
to
p
r
o
tec
t
G
S
M
c
o
m
m
u
n
ica
ti
o
n
s.
T
h
e
re
a
re
two
m
e
m
o
ry
les
s
c
ry
p
tan
a
ly
sis
te
c
h
n
i
q
u
e
s
u
se
d
fo
r
th
is
c
ip
h
e
r
wh
ich
a
re
G
o
li
c
’
s
G
u
e
ss
-
a
n
d
-
d
e
term
in
e
a
tt
a
c
k
a
n
d
Zh
a
n
g
’s
Ne
a
r
Co
ll
isio
n
a
tt
a
c
k
.
In
t
h
is
p
a
p
e
r
a
n
e
w
g
u
e
ss
in
g
tec
h
n
i
q
u
e
c
a
ll
e
d
m
o
v
e
g
u
e
ss
in
g
tec
h
n
iq
u
e
u
se
d
to
c
o
n
str
u
c
t
li
n
e
a
r
e
q
u
a
ti
o
n
fil
ter
a
l
o
n
g
wi
th
G
o
li
c
’s
g
u
e
ss
a
n
d
d
e
term
in
e
tec
h
n
iq
u
e
is
st
u
d
ied
.
Two
m
o
d
ifi
c
a
ti
o
n
s
in
m
o
v
e
g
u
e
ss
in
g
tec
h
n
iq
u
e
a
re
p
r
o
p
o
se
d
fo
r
re
c
o
v
e
ry
o
f
in
tern
a
l
sta
tes
S
0
a
n
d
S
1
.
F
u
rth
e
r,
a
n
o
v
e
l
a
lg
o
rit
h
m
is
p
ro
p
o
se
d
to
se
lec
t
th
e
m
o
d
ifi
c
a
ti
o
n
t
o
g
e
t
m
in
imu
m
ti
m
e
c
o
m
p
lex
i
ty
fo
r
r
e
c
o
v
e
ry
o
f
in
ter
n
a
l
sta
tes
S
0
a
n
d
S
1
.
T
h
e
p
ro
p
o
se
d
a
lg
o
rit
h
m
g
i
v
e
s
m
in
im
u
m
ti
m
e
c
o
m
p
lex
i
ty
o
f
2
29
.
3138
a
t
t
=
1
4
fo
r
re
c
o
v
e
ry
o
f
S
0
sta
te
a
n
d
2
43
.
246
fo
r
re
c
o
v
e
ry
o
f
S
1
a
t
t
=
22
.
K
ey
w
o
r
d
s
:
C
r
y
p
tan
aly
s
is
Gu
ess
-
an
d
-
d
eter
m
in
e
attac
k
T
im
e
-
co
m
p
lex
ity
GSM
A5
/1
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Kh
ed
k
ar
Ab
o
li Au
d
u
m
b
ar
Dep
ar
tm
en
t o
f
E
lectr
o
n
ics E
n
g
in
ee
r
in
g
,
T
er
n
a
E
n
g
in
ee
r
in
g
C
o
lleg
e
Plo
t N
o
1
2
,
Sec
-
2
2
Op
p
.
Ner
u
l Railway
Statio
n
W
,
Ph
ase
I
I
,
Ner
u
l,
Nav
i M
u
m
b
ai,
Ma
h
ar
a
s
h
tr
a
4
0
0
7
0
6
,
I
n
d
ia
E
m
ail: a
b
o
lik
h
ed
k
ar
@
g
m
ail.
c
o
m
1.
I
NT
RO
D
UCT
I
O
N
No
w
-
a
-
d
ay
s
m
an
y
ap
p
licatio
n
s
o
n
in
ter
n
et
o
f
th
in
g
s
(
I
o
T
)
an
d
e
m
b
ed
d
ed
s
y
s
tem
s
ar
e
d
ev
elo
p
e
d
.
Su
ch
ap
p
licatio
n
u
s
es
g
lo
b
al
s
y
s
tem
f
o
r
m
o
b
ile
co
m
m
u
n
icatio
n
s
(
GSM)
tech
n
o
lo
g
y
f
o
r
co
m
m
u
n
icatio
n
.
Als
o
,
m
o
b
ile
n
etwo
r
k
s
u
s
e
GSM
to
tr
an
s
m
it
p
er
s
o
n
al
in
f
o
r
m
atio
n
o
n
r
ad
io
lin
k
s
[
1
]
.
T
h
e
A5
/1
s
tr
ea
m
cip
h
e
r
ar
e
wid
ely
u
s
ed
in
GSM
co
m
m
u
n
icatio
n
,
h
as
b
ee
n
th
e
s
u
b
ject
o
f
ex
te
n
s
iv
e
cr
y
p
tan
aly
s
is
s
in
ce
its
in
ce
p
tio
n
.
E
v
er
s
in
ce
its
p
r
o
p
o
s
al,
A5
/1
h
as
b
ee
n
attac
k
ed
with
v
a
r
i
o
u
s
cr
y
p
tan
al
y
tic
m
eth
o
d
s
s
u
ch
as
Satis
f
iab
ilit
y
(
SAT)
-
b
ased
cr
y
p
tan
aly
s
is
,
tim
e/m
em
o
r
y
/d
ata
t
r
ad
e
-
o
f
f
att
ac
k
s
,
g
u
ess
‐
an
d
‐
d
ete
r
m
in
e
att
ac
k
s
,
n
ea
r
co
llis
io
n
attac
k
(
NC
A)
,
cip
h
er
tex
t
attac
k
,
q
u
an
tu
m
attac
k
o
n
r
ed
u
ce
d
v
er
s
io
n
o
f
C
ip
h
er
[
2
]
–
[
1
1
]
[
1
2
]
,
[
1
3
]
.
R
ain
b
o
w
T
ab
le
g
en
er
atio
n
m
eth
o
d
u
s
ed
to
p
er
f
o
r
m
tim
e/m
em
o
r
y
/
d
ata
tr
ad
e
-
o
f
f
(
T
MD
T
O)
attac
k
also
h
as h
ig
h
ch
an
ce
s
o
f
co
llis
io
n
[
1
4
]
.
M
o
s
t
o
f
th
e
p
r
ac
tical
attac
k
s
o
n
A5
/1
r
eq
u
ir
e
lar
g
e
,
p
r
ec
o
m
p
u
te
d
r
a
in
b
o
w
tab
le
wh
ich
s
ig
n
if
ican
tly
in
cr
ea
s
es th
e
m
e
m
o
r
y
c
o
m
p
lex
ities
[
1
5
]
–
[
1
8
]
.
Me
m
o
r
y
less
cr
y
p
tan
aly
s
is
te
ch
n
iq
u
es
o
n
th
e
A5
/
1
cip
h
e
r
in
v
o
lv
e
a
n
aly
zin
g
th
e
cip
h
e
r
with
o
u
t
r
ely
in
g
o
n
p
r
e
v
io
u
s
s
tates
o
r
m
em
o
r
y
.
T
h
ese
tech
n
iq
u
es
aim
to
b
r
ea
k
th
e
e
n
cr
y
p
tio
n
b
y
ex
am
in
i
n
g
t
h
e
alg
o
r
ith
m
'
s
s
tr
u
ctu
r
e
an
d
p
r
o
p
er
ties
with
o
u
t
co
n
s
id
er
in
g
h
is
to
r
ical
d
ata.
B
y
s
tu
d
y
in
g
th
e
A5
/1
alg
o
r
ith
m
,
cr
y
p
tan
aly
s
ts
ca
n
u
n
co
v
e
r
v
u
l
n
er
ab
ilit
ies an
d
wea
k
n
ess
es th
at
co
u
ld
p
o
ten
tially
co
m
p
r
o
m
i
s
e
its
s
ec
u
r
ity
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
5
3
-
5
4
6
5
5454
E
x
is
tin
g
Me
m
o
r
y
less
cr
y
p
tan
a
ly
s
is
tech
n
iq
u
es
n
am
el
y
g
u
ess
an
d
d
eter
m
i
n
e
attac
k
an
d
n
ea
r
co
llis
io
n
attac
k
th
at
d
eter
m
in
es
S1
r
ec
o
v
er
y
s
tate
an
d
S0
r
ec
o
v
er
y
s
tate,
r
esp
ec
tiv
ely
.
Gu
ess
an
d
d
eter
m
in
e
attac
k
h
ig
h
ly
r
elies
o
n
id
en
tif
y
in
g
p
a
tter
n
s
an
d
r
elatio
n
s
h
ip
s
b
etwe
en
k
ey
an
d
ci
p
h
er
tex
t
[
1
9
]
.
A
n
ew
lo
w
k
ey
s
tr
ea
m
g
u
ess
-
an
d
-
d
eter
m
in
e
(
GD)
att
ac
k
was
p
r
o
p
o
s
ed
in
[
2
0
]
g
i
v
es
th
e
tim
e
co
m
p
lex
ity
o
f
2
52
.
T
h
is
co
m
p
lex
ity
is
h
ig
h
er
t
h
an
th
e
Go
lic’
s
GD
attac
k
’
s
tim
e
co
m
p
lex
ity
o
f
2
43
.
15
[
2
1
]
.
Z
h
a
n
g
’
s
n
ea
r
co
llis
io
n
a
ttack
r
ec
o
v
e
r
’
s
S
0
s
tate
with
th
e
tim
e
co
m
p
lex
ity
o
f
2
53.
19
[
8
]
.
Xu
et
a
l.
f
u
r
th
e
r
wo
r
k
ed
o
n
Go
lic’
s
GD
attac
k
an
d
Z
h
a
n
g
’
s
n
ea
r
co
llis
io
n
attac
k
an
d
was
f
o
u
n
d
th
at
th
e
co
m
p
lex
ity
o
f
Go
li
c'
s
S1
r
ec
o
v
er
y
attac
k
is
n
o
lo
wer
th
an
2
46.
04
b
u
t
h
ig
h
er
t
h
an
th
e
p
r
e
v
io
u
s
ly
b
el
iev
ed
2
43
.
On
th
e
o
th
er
h
an
d
,
Z
h
an
g
'
s
n
ea
r
co
llis
io
n
attac
k
r
ec
o
v
er
s
S0
with
t
h
e
tim
e
co
m
p
lex
ity
2
53.
19
:
s
u
ch
a
co
m
p
lex
ity
ca
n
b
e
f
u
r
th
er
l
o
wer
ed
to
2
50.
78
with
th
e
h
elp
o
f
m
o
v
e
g
u
ess
in
g
tech
n
iq
u
e
[
2
2
]
.
T
h
e
2
-
b
it
m
o
v
e
g
u
ess
in
g
b
ased
g
u
ess
an
d
d
eter
m
in
e
attac
k
o
n
A5
/1
t
h
at
ca
n
r
ec
o
v
er
in
ter
n
al
S0
an
d
S1
s
tate
with
th
e
co
m
p
lex
ities
o
f
2
43.
92
at
t=2
1
an
d
2
4
3.
25
at
t=
2
2
,
r
esp
ec
tiv
ely
[
2
2
]
.
T
h
is
r
esear
ch
f
o
cu
s
es
o
n
e
n
h
a
n
cin
g
t
h
e
cr
y
p
tan
al
y
s
is
o
f
th
e
A5
/1
s
tr
ea
m
ci
p
h
er
b
y
r
ef
i
n
in
g
th
e
m
o
v
e
g
u
ess
in
g
tech
n
iq
u
e
u
s
ed
to
r
ec
o
v
er
in
ter
n
al
s
tates.
W
h
ile
p
r
io
r
wo
r
k
,
s
u
ch
as
Go
lic
[
2
1
]
an
d
th
e
m
o
v
e
g
u
ess
in
g
tech
n
iq
u
e
to
k
ee
p
[
8
]
as
f
ix
ed
cl
o
ck
b
it
m
et
h
o
d
[
2
2
]
,
e
x
p
lo
r
ed
p
a
r
tial
d
ep
en
d
en
cies
am
o
n
g
clo
c
k
b
its
,
th
e
r
esear
ch
g
ap
lies
in
th
e
lack
o
f
u
n
d
e
r
s
tan
d
in
g
a
n
d
s
y
s
tem
atic
tr
ea
tm
en
t
o
f
th
e
b
eh
av
io
r
wh
e
n
o
th
e
r
clo
ck
b
its
ar
e
h
eld
co
n
s
tan
t
in
th
e
s
to
p
-
an
d
-
g
o
m
ec
h
an
i
s
m
.
T
h
is
s
tu
d
y
s
y
s
tem
atica
ll
y
in
v
esti
g
ates
th
e
b
eh
av
io
r
o
f
th
e
cip
h
er
wh
e
n
an
y
o
n
e
o
u
t
o
f
t
h
r
ee
clo
ck
b
its
ar
e
h
eld
co
n
s
tan
t,
r
ev
ea
lin
g
h
o
w
s
u
c
h
co
n
f
ig
u
r
atio
n
s
af
f
e
ct
th
e
s
to
p
-
an
d
-
g
o
m
ec
h
an
is
m
.
T
wo
n
o
v
el
m
o
d
if
icatio
n
s
a
r
e
p
r
o
p
o
s
ed
in
th
e
m
o
v
e
g
u
ess
in
g
tech
n
iq
u
e
allo
win
g
f
o
r
ef
f
ec
tiv
e
r
ec
o
v
er
y
o
f
i
n
ter
n
al
s
tates
S0
an
d
S1
.
A
d
y
n
a
m
ic
d
ec
is
io
n
alg
o
r
ith
m
is
also
im
p
lem
en
te
d
in
th
ese
m
o
d
if
ie
d
tech
n
iq
u
es to
g
iv
e
m
i
n
im
u
m
tim
e
c
o
m
p
le
x
ity
.
Alth
o
u
g
h
th
ese
in
n
o
v
atio
n
s
i
m
p
r
o
v
e
u
p
o
n
ex
is
tin
g
cr
y
p
ta
n
aly
tic
s
tr
ateg
ies
f
o
r
A
5
/1
,
t
h
ey
ca
n
also
o
f
f
er
a
f
r
am
ewo
r
k
th
at
ca
n
b
e
g
en
er
alize
d
to
o
th
er
lin
ea
r
f
ee
d
b
ac
k
s
h
if
t r
e
g
is
ter
(
L
FS
R
)
-
b
ased
s
tr
ea
m
cip
h
er
s
.
T
h
e
s
tu
d
y
p
av
es
th
e
way
f
o
r
f
u
tu
r
e
r
esear
ch
in
to
ad
a
p
tiv
e
cr
y
p
tan
aly
s
is
an
d
lig
h
twe
ig
h
t
cip
h
er
d
esig
n
,
esp
ec
ially
in
ap
p
licatio
n
s
wh
e
r
e
co
m
p
u
tatio
n
al
ef
f
icien
c
y
is
cr
itical,
s
u
ch
as e
m
b
ed
d
e
d
an
d
I
o
T
s
y
s
tem
s
.
T
h
is
p
ap
er
is
o
r
g
an
ized
as
f
o
l
lo
ws.
Sectio
n
2
p
r
o
v
id
es
b
r
ie
f
in
f
o
r
m
atio
n
o
n
k
ey
s
tr
ea
m
g
en
er
atio
n
p
r
o
ce
d
u
r
es
in
A5
/1
ci
p
h
er
.
Sectio
n
3
d
is
cu
s
s
es
th
e
Go
lic’
s
Gu
ess
an
d
Dete
r
m
in
e
attac
k
[
2
1
]
a
n
d
g
iv
es
a
b
r
ief
r
ev
iew
o
f
2
-
b
it
m
o
v
e
g
u
ess
an
d
d
eter
m
in
e
attac
k
[
2
2
]
.
L
ater
,
th
e
m
o
d
if
icatio
n
s
in
th
e
m
o
v
e
g
u
ess
in
g
tech
n
iq
u
e
ar
e
i
n
tr
o
d
u
ce
d
in
s
ec
tio
n
4
an
d
p
r
o
p
o
s
ed
a
n
alg
o
r
ith
m
in
s
ec
tio
n
5
to
s
elec
t
th
e
p
r
o
p
er
m
o
d
if
ied
tech
n
iq
u
e
f
o
r
m
in
im
u
m
tim
e
co
m
p
lex
ity
.
R
esu
lts
an
d
Dis
cu
s
s
io
n
ar
e
d
o
n
e
i
n
s
ec
tio
n
6
.
Sectio
n
7
co
n
clu
d
es
th
e
r
esear
ch
wo
r
k
.
2.
T
H
E
K
E
Y
-
ST
R
E
A
M
G
E
N
E
RATI
O
N
P
RO
C
E
DUR
E
I
N
A5
/1
A5
/1
was
d
esig
n
ed
to
p
r
o
v
id
e
o
v
er
-
th
e
-
air
co
m
m
u
n
icatio
n
p
r
iv
ac
y
f
o
r
th
e
GSM
ce
llu
lar
telep
h
o
n
e
s
tan
d
ar
d
a
n
d
h
as
b
ee
n
wid
ely
u
s
ed
in
GSM
telep
h
o
n
y
in
E
u
r
o
p
e
a
n
d
th
e
USA.
I
n
its
d
esig
n
,
t
h
r
ee
c
o
m
b
in
e
d
L
FS
R
s
with
ir
r
eg
u
lar
clo
ck
in
g
ar
e
u
s
ed
to
en
cr
y
p
t
b
u
r
s
ts
o
f
tr
af
f
ic,
as
is
r
eq
u
ir
ed
in
G
SM
[
2
3
]
.
A5
/1
h
as
3
-
L
FS
R
r
eg
is
ter
s
R
1
,
R
2
an
d
R
3
with
s
ize
s
1
9
,
2
2
,
2
3
r
esp
e
ctiv
ely
m
ak
in
g
it
6
4
-
b
it
in
ter
n
al
s
tate
as
s
h
o
wn
in
Fig
u
r
e
1
.
T
h
ese
s
tates a
t tim
e
t
(
t =
0
,
1
,
2
….
)
is
r
e
p
r
esen
ted
as
[
2
2
]
:
S
t
=
(
R
1
t
,
R
2
t
,
R
3
t
)
=
(
S
t
[
0
,
…1
8
]
,
S
t
[
1
9
,
…4
0
]
,
S
t
[
4
1
,
…6
3
]
)
=
(
R
1
t
[
0
,
…1
8
]
,
R
2
t
[
0
,
…2
1
]
,
R
3
t
[
0
,
…2
2
]
)
(
1
)
B
ef
o
r
e
g
en
er
atin
g
th
e
o
u
tp
u
t
b
it
,
A5
/1
r
o
u
n
d
f
u
n
ctio
n
will u
p
d
ate
th
e
in
ter
n
al
s
tate
→
+
1
in
a
s
to
p
-
an
d
-
g
o
m
an
n
er
as f
o
llo
ws:
−
C
o
m
p
u
te
as
=
(
1
[
8
]
⋅
2
[
10
]
)
⨁
(
1
[
8
]
⋅
3
[
10
]
)
⨁
(
2
[
10
]
⋅
3
[
10
]
)
=
(
[
8
]
⋅
[
29
]
)
⨁
(
[
8
]
⋅
[
51
]
)
⨁
(
[
29
]
⋅
[
51
]
)
(
2
)
w
h
er
e
‘
.
’
is
th
e
AND
o
f
2
-
b
its
−
If
1
[
8
]
=
[
8
]
≠
,
1
+
1
←
1
,
o
th
er
wis
e,
ca
ll Up
d
at
eR1
as f
o
llo
ws
1
+
1
[
ⅈ
]
←
1
[
ⅈ
−
1
]
ⅈ
∈
[
1
,
18
]
1
[
18
]
⨁
1
[
17
]
⨁
1
[
16
]
⨁
1
[
13
]
(
3
)
−
If
2
[
10
]
=
[
29
]
≠
,
2
+
1
←
2
,
o
th
er
wis
e,
ca
ll Up
d
at
eR2
as f
o
llo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Memo
r
yle
s
s
s
ta
te
-
r
ec
o
ve
r
y
cryp
ta
n
a
lysi
s
meth
o
d
fo
r
lig
h
tw
eig
h
t
…
(
K
h
ed
ka
r
A
b
o
li A
u
d
u
mb
a
r
)
5455
2
+
1
[
ⅈ
]
←
2
[
ⅈ
−
1
]
ⅈ
∈
[
1
,
21
]
←
2
[
21
]
⨁
2
[
20
]
(
4
)
−
If
3
[
10
]
=
[
51
]
≠
,
3
+
1
←
3
,
o
th
er
wis
e,
ca
ll Up
d
at
eR1
as f
o
llo
ws:
3
+
1
[
ⅈ
]
←
3
[
ⅈ
−
1
]
ⅈ
∈
[
1
,
22
]
←
3
[
22
]
⨁
3
[
21
]
⨁
3
[
20
]
⨁
3
[
7
]
(
5
)
T
h
en
th
e
Ou
t
p
u
t k
e
y
s
tr
ea
m
b
it
is
g
en
er
ated
as
:
=
1
+
1
[
18
]
⨁
2
+
1
[
21
]
⨁
3
+
1
[
22
]
=
+
1
[
18
]
⨁
+
1
[
40
]
⨁
+
1
[
63
]
(
6
)
Fig
u
r
e
1
.
A5
/
1
s
tr
ea
m
cip
h
e
r
[
2
4
]
3.
E
XI
ST
I
NG
AT
T
ACK
F
O
R
A5
/1
Alth
o
u
g
h
t
h
er
e
ar
e
m
an
y
tech
n
iq
u
es
f
o
r
attac
k
s
f
o
r
A5
/1
,
th
is
p
ap
er
d
is
cu
s
s
es
o
n
ly
th
e
m
em
o
r
y
less
tech
n
iq
u
es
o
f
attac
k
s
s
u
ch
as
1
.
Gu
ess
an
d
Dete
r
m
in
e
atta
ck
an
d
2
.
Nea
r
C
o
llis
io
n
attac
k
.
Nea
r
co
llis
io
n
attac
k
is
ch
allen
g
ed
in
m
an
y
way
s
,
af
ter
th
e
th
eo
r
etica
l
an
aly
s
is
an
d
p
r
ac
tical
im
p
lem
en
tatio
n
s
it
i
s
o
b
s
er
v
ed
th
at
n
o
n
-
r
an
d
o
m
n
ess
claim
ed
in
[
8
]
ca
n
h
ar
d
l
y
b
e
ac
h
iev
e
d
s
o
co
n
clu
d
ed
th
at
Nea
r
c
o
llis
io
n
attac
k
ca
n
n
o
t
h
av
e
less
tim
e
co
m
p
lex
ity
co
m
p
ar
ed
to
Gu
ess
an
d
Dete
r
m
i
n
e
attac
k
[
2
5
]
.
An
d
h
e
n
ce
,
in
t
h
is
p
ap
er
d
is
cu
s
s
io
n
is
d
o
n
e
o
n
l
y
with
Gu
ess
-
an
d
-
Dete
r
m
in
e
attac
k
.
3
.
1
.
G
o
lic’
s
g
ues
s
-
a
nd
-
det
er
m
ine a
t
t
a
c
k
[
2
1
]
Fo
r
ea
ch
s
tep
i
=
0
,
1
,
2
…,
w
h
eth
er
t
h
e
r
e
g
is
ter
s
R
1
;
R
2
;
R
3
ar
e
u
p
d
ated
o
r
n
o
t
d
ep
e
n
d
s
o
n
th
e
th
r
ee
clo
ck
b
its
ⅈ
[
8
,
29
,
51
]
.
Su
ch
3
‐
b
it
clo
ck
s
ca
n
also
b
e
r
eg
ar
d
ed
as
a
3
‐
b
i
t
in
teg
er
ⅈ
∈
{
0
,
1
…
…
7
}
d
ef
in
ed
as
in
(
7
).
ⅈ
[
0
,
1
,
2
]
=
ⅈ
[
8
,
29
,
51
]
=
(
,
,
)
(
7
)
I
n
Go
lic'
s
g
u
ess
‐
an
d
‐
d
eter
m
in
e
m
o
d
el,
th
e
ad
v
e
r
s
ar
y
aim
s
at
r
ec
o
v
er
in
g
th
e
in
itial
s
tate
1
,
th
e
s
tate
r
ig
h
t
b
ef
o
r
e
t
h
e
g
en
e
r
atio
n
o
f
z
0
.
S
o
,
th
e
to
‐
be
‐
g
u
ess
ed
clo
ck
s
ar
e
ⅈ
f
o
r
i
=
1
,
2
,
….
W
ith
th
e
k
n
o
wled
g
e
o
f
c
i
,
ea
ch
b
it
o
f
s
i
+1
ca
n
b
e
r
e
p
r
esen
ted
as
a
lin
ea
r
co
m
b
in
atio
n
o
f
ⅈ
b
its
an
d
,
f
o
llo
win
g
(
7
)
,
t
h
e
ad
v
e
r
s
ar
y
ca
n
d
ed
u
ce
th
r
ee
lin
ea
r
e
q
u
atio
n
s
.
ⅈ
[
8
]
=
ⅈ
[
29
]
=
ⅈ
[
51
]
=
Fro
m
th
e
o
u
tp
u
t
ⅈ
,
th
e
ad
v
e
r
s
ar
y
ca
n
f
u
r
th
er
d
ed
u
ce
o
n
e
lin
ea
r
eq
u
atio
n
.
ⅈ
=
ⅈ
+
1
[
18
]
⊕
ⅈ
+
1
[
40
]
⊕
ⅈ
+
1
[
63
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
5
3
-
5
4
6
5
5456
I
n
o
th
er
w
o
r
d
s
,
b
y
g
u
ess
in
g
3
‐
b
it
ⅈ
,
th
e
ad
v
e
r
s
ar
y
ca
n
d
e
d
u
c
e
4
lin
ea
r
eq
u
atio
n
s
o
f
s
tate
b
its
.
Go
lic
p
r
o
p
o
s
e
a
b
asic
attac
k
th
at
g
u
ess
3
t
clo
ck
b
its
1
…
…
.
B
ased
o
n
th
e
t
+
1
o
u
tp
u
t
b
its
0
…
…
,
th
e
ad
v
er
s
ar
y
ca
n
d
e
d
u
ce
a
s
y
s
tem
o
f
a
v
er
ag
in
g
1
+
3
+
4
3
lin
ea
r
eq
u
atio
n
s
.
Fo
r
t
≥
1
4
.
3
8
,
th
e
s
y
s
tem
ca
n
in
v
o
lv
e
1
+
3
+
4
3
≥
63
.
3
2
eq
u
atio
n
s
wh
ich
is
s
u
f
f
icien
t
f
o
r
id
en
tif
y
in
g
th
e
co
r
r
ec
t
g
u
ess
u
n
iq
u
ely
with
“h
ig
h
p
r
o
b
a
b
ilit
y
”.
Alth
o
u
g
h
th
e
n
u
m
b
er
o
f
eq
u
atio
n
s
an
d
th
e
“h
ig
h
p
r
o
b
ab
ilit
y
”
h
a
v
e
n
ev
er
b
ee
n
v
er
if
ied
,
th
e
co
m
p
lex
ity
o
f
Go
lic'
s
att
ac
k
is
u
s
u
ally
b
eliev
ed
as
2
3
≥
2
43
⋅
15
s
tep
s
wh
er
e
ea
ch
s
tep
in
v
o
lv
es
th
e
s
o
lu
tio
n
o
f
a
lin
e
ar
s
y
s
tem
.
A
p
p
ar
en
tly
,
s
u
ch
a
co
m
p
lex
ity
ev
alu
atio
n
ass
u
m
es
th
at
th
e
w
r
o
n
g
‐
g
u
ess
o
r
ien
ted
lin
ea
r
eq
u
atio
n
s
y
s
tem
ac
ts
r
an
d
o
m
ly
,
an
d
its
r
an
k
g
r
o
ws
li
n
ea
r
ly
with
t
to
6
3
.
3
2
.
I
t
is
later
p
r
o
v
ed
th
at
s
u
ch
an
ass
u
m
p
tio
n
is
n
o
t tr
u
e
f
o
r
A5
/1
.
B
esid
es,
Go
lic
also
n
o
tices
t
h
at
n
o
t
all
3
t
clo
ck
b
its
1
…
…
ar
e
to
b
e
g
u
ess
ed
in
d
e
p
en
d
e
n
tly
.
Acc
o
r
d
in
g
to
t
h
e
s
to
p
‐
an
d
-
g
o
m
ec
h
an
is
m
,
th
er
e
ar
e
o
cc
asio
n
s
wh
er
e
o
n
ly
2
o
u
t
o
f
th
e
3
L
FS
R
s
ar
e
u
p
d
ated
(
ⅈ
∉
{0
,
7
})
an
d
1
o
u
t
o
f
th
e
3
ⅈ
+
1
b
its
ar
e
alr
ea
d
y
k
n
o
wn
i
n
ⅈ
.
T
o
av
o
id
s
u
ch
r
e
d
u
n
d
an
t
b
it
g
u
ess
es,
Go
lic
p
r
o
p
o
s
e
“b
r
an
c
h
in
g
tech
n
iq
u
e”
wh
er
e
a
tr
ee
s
tr
u
ctu
r
e
is
ap
p
lied
to
tr
ac
k
th
e
k
n
o
wn
b
its
to
f
u
r
th
er
lo
wer
th
e
co
m
p
lex
it
y
.
Ho
wev
er
,
s
in
ce
th
e
b
r
a
n
ch
in
g
tech
n
i
q
u
e
d
ep
en
d
s
o
n
t
h
e
clo
ck
d
y
n
am
ic
v
alu
es,
th
e
co
m
p
lex
ities
in
d
id
n
o
t
tak
e
th
e
ef
f
ec
t
o
f
th
e
b
r
an
ch
i
n
g
tech
n
iq
u
e
in
to
th
e
ev
al
u
atio
n
s
.
T
h
is
tech
n
iq
u
e
d
etec
ts
S
1
s
tate
wh
ich
m
ay
ca
u
s
e
d
ela
y
ed
p
r
o
ce
s
s
o
f
k
ey
d
etec
tio
n
.
3
.
2
.
A
brief
re
v
iew
o
f
m
o
v
e
g
ues
s
ing
t
ec
hn
i
qu
e
[
2
2
]
T
h
e
2
‐
b
it
mo
ve
p
a
tter
n
∈
{
0
,
⋯
3
}
ac
co
r
d
in
g
to
t
h
e
3
‐
b
it c
lo
ck
=
[
8
,
29
,
51
]
.
Su
ch
a
m
o
v
e
p
atter
n
ca
n
b
e
eq
u
i
v
alen
tly
r
e
g
ar
d
ed
as 2
-
d
im
en
tio
n
al
b
in
a
r
y
v
ec
to
r
d
ef
in
e
d
in
(
8
)
.
=
[
0
,
1
]
=
(
[
8
]
⊕
[
29
]
,
[
8
]
⊕
[
51
]
)
=
(
,
)
2
2
(
8
)
W
ith
th
e
k
n
o
wled
g
e
o
f
in
ab
o
v
e
e
q
u
atio
n
,
two
eq
u
atio
n
s
ar
e
d
ed
u
ce
d
in
(
9
)
.
[
8
]
⊕
[
29
]
=
[
8
]
⊕
[
51
]
=
(
9
)
T
h
e
f
o
u
r
p
o
s
s
ib
le
v
alu
es
o
f
,
r
ef
er
r
ed
as
Mo
v
e
0
–
3
,
c
o
r
r
esp
o
n
d
s
to
d
if
f
er
e
n
t
m
o
v
e
m
en
ts
in
A5
/1
L
FS
R
s
tr
an
s
f
o
r
m
in
g
to
+
1
.
−
Mo
v
e
0
: Fr
o
m
th
e
L
FS
R
ac
tio
n
asp
ec
t,
u
p
d
ateR1
,
u
p
d
ateR2
an
d
u
p
d
ateR3
ar
e
all
ca
lled
.
T
h
is
co
r
r
esp
o
n
d
s
to
clo
ck
v
al
u
es
ⅈ
∈
{
0
,
7
}
o
r
e
q
u
iv
alen
tl
y
s
t
[
8
,
29
,
51
]
∈
{
(
0
,
0
,
0
)
,
(
1
,
1
,
1
)
}
−
Mo
v
e
1
:
On
ly
u
p
d
ateR2
a
n
d
u
p
d
ateR3
ar
e
ca
lled
co
r
r
esp
o
n
d
i
n
g
to
ⅈ
∈
{
1
,
6
}
o
r
e
q
u
iv
al
en
tly
[
8
,
29
,
51
]
∈
{
(
0
,
1
,
1
)
,
(
1
,
0
,
0
)
}
−
Mo
v
e
2
:
On
ly
u
p
d
ateR1
an
d
u
p
d
ateR3
ar
e
ca
lled
co
r
r
esp
o
n
d
in
g
t
o
ⅈ
∈
{
2
,
5
}
OR
[
8
,
29
,
51
]
∈
{
(
1
,
0
,
1
)
,
(
0
,
1
,
0
)
}
.
−
Mo
v
e
3
:
On
ly
u
p
d
ateR1
an
d
u
p
d
ateR2
ar
e
ca
lled
c
o
r
r
esp
o
n
d
in
g
to
C
ⅈ
∈
{
3
,
4
}
OR
s
t
[
8
,
29
,
51
]
∈
{
(
1
,
1
,
0
)
,
(
0
,
0
,
1
)
}
Acc
o
r
d
in
g
to
th
e
d
ef
in
itio
n
,
t
h
e
L
FS
R
ac
tio
n
s
b
ef
o
r
e
g
en
er
at
in
g
th
e
o
u
tp
u
t
k
ey
s
tr
ea
m
b
its
0
…
…
ca
n
b
e
r
ep
r
esen
ted
as
0
…
…
.
I
n
th
is
g
u
ess
an
d
d
eter
m
in
e
attac
k
,
f
ir
s
t
g
u
ess
th
e
m
o
v
em
en
t
co
r
r
esp
o
n
d
in
g
to
th
e
tr
an
s
f
o
r
m
atio
n
→
+
1
an
d
m
ain
tain
s
a
li
n
ea
r
eq
u
atio
n
s
et
BC
b
y
ad
d
in
g
n
ew
eq
u
atio
n
s
ac
co
r
d
in
g
to
an
d
t
h
e
o
u
t
p
u
t
.
Fo
r
ea
c
h
s
tep
t
,
th
er
e
ar
e
th
r
ee
lin
ea
r
e
q
u
atio
n
s
:
two
ar
e
f
r
o
m
eq
u
atio
n
(
9
)
ac
c
o
r
d
in
g
to
th
e
m
o
v
e
g
u
ess
an
d
o
n
e
is
f
r
o
m
t
h
e
o
u
tp
u
t
as
=
+
1
[
18
]
⨁
+
1
[
40
]
⨁
+
1
[
63
]
(
1
0
)
So
,
ea
ch
2
‐
b
it m
o
v
e
g
u
ess
r
esu
lts
in
th
r
ee
eq
u
atio
n
s
.
3
.
2
.
1
.
M
o
v
e
g
ues
s
ing
ba
s
ed
r
ec
o
v
er
ing
S
0
s
t
a
t
e
T
h
e
m
o
v
e
eq
u
atio
n
s
in
(
9)
a
n
d
th
e
o
u
tp
u
t
eq
u
atio
n
in
(
10
)
co
r
r
esp
o
n
d
to
t
h
e
in
ter
n
al
s
tates
at
d
if
f
er
en
t
tim
e
i
n
s
tan
ce
s
.
B
u
t
th
is
attac
k
is
tar
g
eted
f
o
r
r
ec
o
v
er
in
g
t
h
e
in
itial
s
tate
0
.
T
h
er
e
f
o
r
e,
th
e
in
ter
n
a
l
s
tates
at
d
if
f
er
en
t
tim
e
in
s
tan
ce
t
s
h
o
u
ld
b
e
r
ep
r
esen
ted
b
y
0
b
its
f
o
r
d
ed
u
cin
g
0
r
elate
d
eq
u
atio
n
s
.
T
h
e
s
tate
is
d
ed
u
ce
d
f
r
o
m
0
b
y
tak
i
n
g
th
e
m
o
v
es
0
…
…
as
0
→
0
1
→
1
…
→
−
2
−
1
→
−
1
(
1
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Memo
r
yle
s
s
s
ta
te
-
r
ec
o
ve
r
y
cryp
ta
n
a
lysi
s
meth
o
d
fo
r
lig
h
tw
eig
h
t
…
(
K
h
ed
ka
r
A
b
o
li A
u
d
u
mb
a
r
)
5457
T
h
e
m
o
v
es
0
…
…
−
1
co
r
r
esp
o
n
d
s
to
th
e
lin
ea
r
tr
a
n
s
f
o
r
m
atio
n
s
in
L
FS
R
s
s
o
ea
ch
b
it
is
a
lin
ea
r
co
m
b
in
atio
n
o
f
0
b
its
:
s
u
ch
a
li
n
ea
r
c
o
m
b
in
atio
n
ca
n
b
e
r
eg
ar
d
ed
as
a
in
n
er
p
r
o
d
u
ct
o
f
0
an
d
a
6
4
‐
b
it
wo
r
d
∈
2
64
.
I
n
o
r
d
er
to
tr
ac
k
all
s
tate
b
its
in
0
…
.
.
b
its
,
it
is
d
ef
in
ed
b
y
6
4
x
6
4
b
in
ar
y
m
atr
ices
0
…
…
∈
(
2
64
)
64
s
.
t.
ⅈ
=
ⅈ
0
f
o
r
all
i
=
0
,
…,
t
.
T
h
e
r
o
w
v
ec
to
r
o
f
ⅈ
is
d
en
o
ted
as
ⅈ
[
j
]
f
o
r
j
=
0
,
…,
6
3
.
I
n
th
is
way
,
th
e
s
tate
b
it
ⅈ
[
j]
ca
n
b
e
co
m
p
u
ted
as
th
e
in
n
er
p
r
o
d
u
ce
o
f
in
itial
s
0
an
d
r
o
w
v
ec
t
o
r
ⅈ
[
j
]
.
E
ac
h
s
tate
b
it
o
f
ca
n
b
e
u
n
if
o
r
m
l
y
ex
p
r
ess
ed
as a
lin
ea
r
co
m
b
in
atio
n
o
f
0
b
its
as
[
ⅈ
]
=
[
ⅈ
]
⋅
0
,
ⅈ
=
0
…
63
,
=
0
,
1
,
2
(
1
2
)
Fo
r
t
co
n
s
ec
u
tiv
e
m
o
v
em
en
ts
0
…
…
−
1
an
d
th
e
co
r
r
esp
o
n
d
in
g
o
u
tp
u
t
0
⋯
⋯
−
1
,
th
e
co
r
r
esp
o
n
d
in
g
lin
ea
r
eq
u
atio
n
s
s
et
BC
ca
n
b
e
d
ed
u
ce
d
as
ⅇ
(
(
⋯
0
⋅
−
1
)
,
(
0
…
−
1
)
)
→
Su
ch
B
C
ca
n
b
e
r
eg
ar
d
ed
as a
lin
ea
r
eq
u
atio
n
s
y
s
tem
in
(
13
).
=
,
wh
er
e
∈
2
3
×
64
,
∈
2
64
,
∈
2
3
(
1
3
)
an
d
th
e
s
o
lu
tio
n
o
f
th
e
eq
u
atio
n
a
b
o
v
e
co
r
r
esp
o
n
d
s
to
all
c
an
d
id
ate
0
.
T
h
e
n
u
m
b
er
o
f
s
o
lu
tio
n
s
d
ep
en
d
s
o
n
th
e
r
an
k
o
f
th
e
m
atr
ix
A
a
n
d
it
s
ex
ten
d
ed
m
atr
i
x
as in
(
14
).
=
[
,
]
(
1
4
)
−
If
r
a
n
k
(
A
)
=
r
a
n
k
(
E
)
,
th
er
e
will b
e
2
64
−
s
o
lu
tio
n
s
wh
er
e
is
th
e
p
o
s
itiv
e
in
teg
er
d
ef
in
ed
in
(
15
)
a
s
th
e
r
an
k
o
f
th
e
m
atr
i
x
A
;
=
(
)
(
1
5
)
−
If
r
a
n
k(
A
)
≠
r
a
n
k
(
E
)
,
th
e
r
e
will n
o
s
o
lu
tio
n
at
all.
W
ith
th
e
g
u
ess
ed
m
o
v
es
0
…
…
−
1
an
d
th
e
o
b
s
er
v
ed
o
u
t
p
u
t
b
its
0
⋯
⋯
−
1
,
we
ar
e
n
o
w
ab
le
to
ac
q
u
ir
e
b
o
th
A
a
n
d
b
alo
n
g
with
th
e
ex
ten
d
ed
m
atr
ix
E
in
(
1
4
)
.
T
h
e
p
r
o
b
ab
ilit
y
o
f
r
a
n
k
(
A
)
=
r
a
n
k
(
E
):
−
Fo
r
th
e
co
r
r
ec
t g
u
ess
o
f
0
…
…
−
1
,
r
a
n
k
(
A
)
=
r
a
n
k
(
E
)
is
co
n
s
tan
tly
tr
u
e.
−
If
m
0
,
…,
mt
−
1
,
th
e
p
r
o
b
a
b
ilit
y
o
f
r
a
n
k
(
A
)
=
r
a
n
k
(
E
)
is
d
ef
in
ed
as
(
0
≤
≤
1
)
.
Acc
o
r
d
in
g
to
o
u
r
an
aly
s
is
,
s
u
ch
'
s
g
r
o
ws g
r
ad
u
a
lly
with
t
an
d
s
h
o
u
ld
b
e
m
ea
s
u
r
ed
p
r
ac
tically
.
So
,
th
e
p
r
o
b
ab
ilit
y
o
f
r
a
n
k
(
A
)
=
r
a
n
k
(
E
)
ca
n
b
e
f
o
r
m
ally
r
ep
r
esen
ted
as (
16)
.
[
(
)
=
(
)
]
=
1
0
…
…
−
1
is
co
r
r
ec
tly
g
u
ess
ed
=
∈
[
0
,
1
]
0
…
…
−
1
is
wr
o
n
g
ly
g
u
es
s
ed
(
1
6
)
T
h
e
g
en
e
r
al
p
r
o
ce
s
s
o
f
s
u
ch
a
n
attac
k
ca
n
b
e
s
u
m
m
ar
ized
as f
o
llo
ws:
S1
: G
u
ess
0
…
…
−
1
,
o
b
s
er
v
e
0
⋯
⋯
−
1
an
d
d
ed
u
ce
th
e
lin
ea
r
s
y
s
tem
r
ep
r
esen
te
d
as
=
S2
: D
o
th
e
r
an
k
test
ch
ec
k
,
wh
eth
er
r
a
n
k
(
A
)
≠
r
a
n
k
(
E
)
S3
:
T
r
av
er
s
in
g
th
e
r
em
ain
in
g
s
0
ca
n
d
id
ates a
n
d
i
d
en
tify
th
e
co
r
r
ec
t
0
with
ad
d
itio
n
al
o
u
tp
u
t
b
its
⋯
⋯
−
1
g
en
er
ated
b
y
th
e
en
cr
y
p
tio
n
o
r
ac
le.
C
o
m
p
lex
ity
a
n
aly
s
is
:
in
s
tep
3
,
th
er
e
ar
e
2
2
ca
n
d
id
ate
(
0
…
…
−
1
)
'
s
.
Acc
o
r
d
in
g
to
(
16
)
,
a
v
er
ag
i
n
g
(
⋅
2
2
)
m
o
v
e
p
atter
n
ca
n
d
id
ates
ca
n
p
ass
th
e
test
.
Ad
d
in
g
β
t
in
(
1
5
)
,
th
e
a
v
er
ag
in
g
tim
e
co
m
p
l
ex
ity
ca
n
b
e
co
m
p
u
ted
as in
(
17
).
C
o
m
p
=
2
2
+
⋅
2
2
+
64
−
=
2
2
+
2
2
+
64
−
+
(
1
7
)
B
y
r
an
d
o
m
ly
s
elec
tin
g
2
30
((
0
…
…
−
1
)
,
(
0
⋯
⋯
−
1
)
)
p
air
s
a
n
d
p
er
f
o
r
m
in
g
t
h
e
attac
k
.
T
h
e
v
alu
e
o
b
tain
ed
f
o
r
an
d
f
o
r
d
if
f
e
r
en
t
v
alu
es
o
f
t’
s
ar
e
s
h
o
wn
in
T
ab
le
1
.
T
h
e
lo
west
tim
e
co
m
p
lex
ity
is
2
36
.
56
co
r
r
esp
o
n
d
in
g
t
o
t
=
1
6
.
Fo
r
test
in
g
p
u
r
p
o
s
es,
th
e
alg
o
r
ith
m
g
iv
e
n
in
[
2
1
]
is
ex
ec
u
te
d
an
d
th
e
v
al
u
es
o
f
t
h
e
tim
e
c
o
m
p
lex
ity
o
b
tain
ed
a
r
e
s
h
o
wn
in
T
a
b
le
1
.
Alth
o
u
g
h
th
e
v
al
u
es
o
b
ta
in
ed
ar
e
s
lig
h
tly
d
if
f
e
r
en
t
g
i
v
en
in
[
2
1
]
m
ay
b
e
b
ec
au
s
e
o
f
r
a
n
d
o
m
k
ey
g
en
e
r
a
tio
n
,
b
u
t t
h
e
p
atter
n
o
b
tain
e
d
i
s
s
am
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
5
3
-
5
4
6
5
5458
T
ab
le
1
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
7
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
0
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
6
3
1
.
9
5
4
-
0
.
1
0
4
3
.
9
3
17
5
9
.
4
8
2
7
-
1
.
9
6
3
6
.
7
8
7
3
4
.
6
8
1
6
-
0
.
1
1
4
3
.
2
0
18
6
0
.
4
0
0
8
-
2
.
9
5
3
7
.
3
6
8
3
7
.
5
8
4
5
-
0
.
1
1
4
2
.
2
9
19
6
1
.
1
4
5
9
-
4
.
1
4
3
8
.
4
9
9
4
0
.
5
5
0
1
-
0
.
1
1
4
1
.
3
3
20
6
1
.
7
8
7
5
-
5
.
4
9
4
0
.
1
4
10
4
3
.
5
1
5
7
-
0
.
1
2
4
0
.
3
5
21
6
2
.
3
6
4
1
-
7
.
0
1
4
2
.
0
3
11
4
6
.
4
6
0
1
-
0
.
1
5
3
9
.
3
8
22
6
2
.
8
7
3
9
-
8
.
6
9
4
4
.
0
0
12
4
9
.
3
6
7
5
-
0
.
1
8
3
8
.
4
4
23
6
3
.
2
9
5
2
-
1
0
.
5
2
4
6
.
0
0
13
5
2
.
1
6
1
9
-
0
.
2
4
3
7
.
5
9
24
6
3
.
6
0
8
2
-
1
2
.
4
9
4
8
.
0
0
14
5
4
.
6
7
6
6
-
0
.
3
6
3
6
.
9
6
25
6
3
.
8
1
1
-
1
4
.
5
5
5
0
.
0
0
15
5
6
.
7
3
7
6
-
0
.
6
5
3
6
.
6
2
26
6
3
.
9
2
2
5
-
1
6
.
6
3
5
2
.
0
0
16
5
8
.
3
0
4
-
1
.
1
9
3
6
.
5
6
27
6
3
.
9
7
3
4
-
1
8
.
6
4
5
4
.
0
0
3
.
2
.
2
.
M
o
v
e
g
ues
s
ing
ba
s
ed
r
ec
o
v
er
in
g
S
1
s
t
a
t
e
Fo
r
r
ec
o
v
e
r
in
g
s
1
ac
co
r
d
i
n
g
t
o
0
⋯
⋯
−
1
,
we
d
o
n
o
t
n
ee
d
to
g
u
ess
0
.
W
e
g
u
ess
d
ir
ec
tly
th
e
t
−
1
m
o
v
e
p
atter
n
s
1
…
…
−
1
an
d
ac
q
u
ir
e
th
e
lin
ea
r
eq
u
atio
n
s
y
s
tem
=
o
f
s
izes
∈
2
(
2
−
1
)
×
64
,
∈
2
64
,
∈
2
2
−
1
.
T
h
er
e
f
o
r
e,
th
e
g
e
n
er
al
p
r
o
c
ess
h
as n
o
w
b
ec
o
m
e:
S1
: G
u
ess
m
o
v
es
1
…
…
−
1
an
d
m
ain
tai
n
a
lin
ea
r
s
y
s
tem
=
S2
: D
o
th
e
m
atr
ix
r
an
k
test
an
d
d
is
ca
r
d
th
e
wr
o
n
g
g
u
ess
es satis
f
y
in
g
r
a
n
k
(
A
)
≠
r
a
n
k
(
E
)
S3
:
T
r
av
er
s
e
th
e
r
em
ai
n
in
g
1
ca
n
d
id
ates a
n
d
id
e
n
tify
th
e
co
r
r
e
ct
1
with
ad
d
itio
n
al
o
u
tp
u
t
b
its
⋯
⋯
−
1
.
I
n
S1
,
s
tar
t f
r
o
m
1
=
I
an
d
ac
q
u
ir
e
th
e
b
it c
o
n
d
itio
n
s
o
n
(
1
…
…
−
1
)
an
d
(
1
⋯
⋯
−
1
)
.
B
esid
es,
lettin
g
1
= x
= (
0
…
…
63
)
,
th
er
e
is
also
an
e
q
u
at
io
n
d
ed
u
ce
d
f
r
o
m
0
ac
co
r
d
i
n
g
t
o
(
1
0
)
as
18
⊕
40
⊕
63
=
0
(
1
8
)
C
o
m
p
lex
ity
a
n
aly
s
is
:
Am
o
n
g
th
e
2
2
(
−
1
)
m
o
v
es
1
…
…
−
1
,
th
er
e
is
a
p
o
r
tio
n
o
f
p
ass
in
g
th
e
r
an
k
test
an
d
th
e
a
v
er
ag
in
g
r
a
n
k
(
A
)
is
β
t
as g
iv
en
in
(
15
)
.
So
,
th
e
c
o
m
p
lex
ity
ca
n
b
e
ev
alu
ated
as:
C
o
m
p
=
2
2
(
−
1
)
+
⋅
2
2
(
−
1
)
+
64
−
=
2
2
(
−
1
)
+
2
2
(
−
1
)
+
64
−
+
(
1
9
)
T
h
e
an
d
p
ar
am
eter
s
ar
e
p
r
ac
tically
ev
alu
ated
,
an
d
th
e
v
al
u
e
o
f
co
m
p
lex
ity
ar
e
s
h
o
wn
in
T
ab
le
2
.
T
h
e
lo
west c
o
m
p
lex
ity
ac
h
iev
e
d
is
2
43
.
251
at
t
=
22.
T
ab
le
2
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
9
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
1
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
7
19
0
57
18
5
1
.
3
7
7
8
-
0
.
4
6
9
0
2
4
6
.
1
5
3
8
22
0
56
19
5
3
.
9
4
3
9
-
0
.
8
1
6
9
7
4
5
.
2
4
1
9
25
0
55
20
5
6
.
3
7
8
2
-
1
.
2
8
6
7
5
4
4
.
3
5
2
10
28
0
54
21
5
8
.
6
4
6
2
-
1
.
9
3
7
7
7
4
3
.
5
4
5
11
31
0
53
22
6
0
.
6
1
7
9
-
2
.
9
1
6
4
3
4
3
.
2
5
1
12
34
0
52
23
6
2
.
0
8
8
3
-
4
.
5
1
6
2
6
4
4
.
2
1
9
13
3
6
.
9
9
9
3
-
0
.
0
0
0
2
5
5
1
.
0
0
0
24
6
3
.
0
0
3
6
-
6
.
7
3
2
5
6
4
6
.
0
2
6
14
3
9
.
9
9
1
9
-
0
.
0
0
5
1
4
5
0
.
0
0
2
25
6
3
.
5
1
2
6
-
9
.
3
1
3
5
0
4
8
.
0
0
3
15
4
2
.
9
5
8
6
-
0
.
0
3
2
0
6
4
9
.
0
0
9
26
6
3
.
7
8
1
-
1
2
.
0
6
4
8
3
5
0
.
0
0
0
16
4
5
.
8
6
7
6
-
0
.
0
9
9
2
4
4
8
.
0
3
3
27
6
3
.
9
1
2
9
-
1
4
.
8
4
8
2
3
5
2
.
0
0
0
17
4
8
.
6
8
2
6
-
0
.
2
2
9
6
6
4
7
.
0
8
7
28
6
3
.
9
7
0
1
-
1
7
.
5
7
6
0
9
5
4
.
0
0
0
4.
P
RO
P
O
SE
D
M
O
D
I
F
I
E
D
M
O
VE
G
UE
S
SI
NG
T
E
CH
N
I
Q
UE
Go
lic
h
as
n
o
tice
t
h
at
n
o
t
all
3
t
clo
ck
b
its
1
…
…
ar
e
t
o
b
e
g
u
ess
ed
in
d
ep
e
n
d
en
tly
.
Acc
o
r
d
in
g
to
th
e
s
to
p
‐
an
d
-
g
o
m
ec
h
a
n
is
m
,
t
h
er
e
ar
e
o
cc
asio
n
s
wh
e
r
e
o
n
ly
2
o
u
t
o
f
th
e
3
L
FS
R
s
ar
e
u
p
d
ated
(
ⅈ
∉
{0
,
7
})
an
d
1
o
u
t
o
f
th
e
3
ⅈ
+
1
b
its
ar
e
alr
ea
d
y
k
n
o
wn
in
ⅈ
[
1
2
]
.
B
u
t
wh
i
ch
v
alu
e
o
f
ⅈ
+
1
s
h
o
u
ld
b
e
c
o
n
s
id
e
r
ed
th
at
r
em
ain
ed
co
n
s
tan
t.
2
-
b
it m
o
v
e
p
atter
n
in
[
2
2
]
ass
u
m
ed
o
n
l
y
[
8
]
an
d
ca
lcu
lated
th
e
tim
e
co
m
p
l
ex
ity
.
I
n
th
is
p
ap
er
e
q
u
atio
n
(
8
)
o
f
e
x
is
tin
g
tech
n
iq
u
e
h
as
b
ee
n
m
o
d
if
ied
b
y
ta
k
in
g
eith
e
r
[
29
]
o
r
[
51
]
co
n
s
tan
t.
B
ec
au
s
e
o
f
t
h
ese
p
r
o
p
o
s
ed
ass
u
m
p
tio
n
s
,
m
o
v
e
p
a
tter
n
ar
e
c
h
an
g
e
d
wh
ich
ca
u
s
es
th
e
ch
an
g
e
in
th
e
v
alu
es o
f
u
p
d
ate
r
e
g
is
ter
an
d
h
ad
a
wid
e
ef
f
ec
t
o
n
ca
lcu
latio
n
o
f
tim
e
co
m
p
lex
ity
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Memo
r
yle
s
s
s
ta
te
-
r
ec
o
ve
r
y
cryp
ta
n
a
lysi
s
meth
o
d
fo
r
lig
h
tw
eig
h
t
…
(
K
h
ed
ka
r
A
b
o
li A
u
d
u
mb
a
r
)
5459
4
.
1
.
M
o
difica
t
io
n 1
-
wit
h
[
]
co
ns
t
a
nt
E
q
u
iv
alen
tly
b
in
ar
y
v
ec
to
r
o
f
d
im
en
s
io
n
2
d
ef
in
e
d
f
o
r
[
29
]
is
as
f
o
llo
ws:
=
[
0
,
1
]
=
(
[
8
]
⊕
[
29
]
,
[
29
]
⊕
[
51
]
)
=
(
1
,
1
)
2
2
(
2
0
)
W
ith
th
e
k
n
o
wled
g
e
o
f
in
ab
o
v
e
eq
u
atio
n
,
2
e
q
u
atio
n
s
ar
e
d
ed
u
ce
d
as f
o
llo
ws:
[
8
]
⊕
[
29
]
=
1
[
29
]
⊕
[
51
]
=
1
(
2
1
)
T
h
e
4
p
o
s
s
ib
le
v
alu
es
o
f
,
r
ef
er
r
ed
as
Mo
v
e
0
–
3
,
co
r
r
esp
o
n
d
s
to
d
if
f
e
r
en
t
m
o
v
em
en
ts
i
n
A5
/1
L
FS
R
s
tr
an
s
f
o
r
m
in
g
to
+
1
.
−
Mo
v
e
0
: Fr
o
m
th
e
L
FS
R
ac
tio
n
asp
ec
t,
u
p
d
ateR1
,
u
p
d
ateR2
an
d
u
p
d
ateR3
ar
e
all
ca
lled
.
T
h
is
co
r
r
esp
o
n
d
s
to
clo
ck
v
al
u
es
ⅈ
∈
{
0
,
7
}
o
r
e
q
u
iv
alen
tl
y
s
t
[
8
,
29
,
51
]
∈
{
(
0
,
0
,
0
)
,
(
1
,
1
,
1
)
}
−
Mo
v
e
1
:
On
ly
u
p
d
ateR1
a
n
d
u
p
d
ateR3
ar
e
ca
lled
co
r
r
esp
o
n
d
i
n
g
to
ⅈ
∈
{
1
,
6
}
o
r
e
q
u
iv
al
en
tly
[
8
,
29
,
51
]
∈
{
(
0
,
1
,
0
)
,
(
1
,
0
,
1
)
}
−
Mo
v
e
2
:
On
ly
u
p
d
ateR2
an
d
u
p
d
ateR3
ar
e
ca
lled
co
r
r
esp
o
n
d
in
g
t
o
ⅈ
∈
{
2
,
5
}
OR
[
8
,
29
,
51
]
∈
{
(
1
,
0
,
0
)
,
(
0
,
1
,
1
)
}
.
−
Mo
v
e
3
:
On
ly
u
p
d
ateR1
an
d
u
p
d
ateR2
ar
e
ca
lled
c
o
r
r
esp
o
n
d
in
g
to
C
ⅈ
∈
{
3
,
4
}
OR
s
t
[
8
,
29
,
51
]
∈
{
(
1
,
1
,
0
)
,
(
0
,
0
,
1
)
}
As co
n
s
id
er
ed
in
s
ec
tio
n
3
,
3
rd
eq
u
atio
n
is
co
n
s
id
er
ed
as
=
+
1
[
18
]
⨁
+
1
[
40
]
⨁
+
1
[
63
]
(
2
2
)
4
.
1
.
1
.
Rec
o
v
er
y
o
f
S
0
s
t
a
t
e
wi
t
h
[
]
co
ns
t
a
nt
B
y
m
o
d
if
icatio
n
as
d
is
cu
s
s
ed
in
s
ec
tio
n
4
.
1
a
n
ew
c
o
d
e
is
im
p
lem
en
ted
,
a
n
d
th
e
tim
e
co
m
p
lex
ity
is
ca
lcu
lated
b
y
(
17
)
.
T
h
e
r
esu
lt
o
b
tain
ed
is
s
h
o
wn
in
T
a
b
le
3
.
T
h
e
lo
west
tim
e
co
m
p
lex
ity
ac
h
iev
ed
is
2
31
.
66
co
r
r
esp
o
n
d
in
g
t
o
t
=
1
5
.
4
.
1
.
2
.
Rec
o
v
er
y
o
f
S
1
s
t
a
t
e
wit
h
[
]
co
ns
t
a
nt
Fo
r
r
ec
o
v
er
y
o
f
S1
s
tate
ac
c
o
r
d
in
g
to
m
o
d
if
icatio
n
m
a
d
e
in
4
.
1
t
h
e
lo
west
tim
e
c
o
m
p
lex
ity
is
ca
lcu
lated
b
ased
o
n
(
19
)
.
T
h
e
r
esu
lt
o
b
tain
ed
is
s
h
o
wn
in
T
ab
le
4
.
T
h
e
lo
west
tim
e
co
m
p
lex
ity
ac
h
iev
ed
is
2
43
.
246
co
r
r
esp
o
n
d
in
g
t
o
t
=
2
2
.
T
ab
le
3
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
7
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
0
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
6
3
2
.
2
8
6
-
3
.
9
8
3
9
.
7
2
17
5
9
.
5
8
5
1
-
8
.
8
9
3
4
.
0
6
7
3
4
.
9
5
8
9
-
4
.
1
9
3
8
.
8
4
18
6
0
.
4
8
3
5
-
1
0
.
9
7
3
6
.
0
0
8
3
7
.
8
2
8
5
-
4
.
2
8
3
7
.
8
8
19
6
1
.
2
0
5
8
-
1
3
.
2
5
3
8
.
0
0
9
4
0
.
7
7
7
4
-
4
.
3
1
3
6
.
9
0
20
6
1
.
8
2
4
4
-
1
5
.
6
2
4
0
.
0
0
10
4
3
.
7
3
3
3
-
4
.
3
6
3
5
.
9
0
21
6
2
.
3
8
3
3
-
1
8
.
0
2
4
2
.
0
0
11
4
6
.
6
7
0
3
-
4
.
4
2
3
4
.
9
0
22
6
2
.
8
8
2
5
-
2
0
.
6
5
4
4
.
0
0
12
4
9
.
5
6
7
4
-
4
.
5
3
3
3
.
9
0
23
6
3
.
2
9
8
5
-
2
3
.
3
2
4
6
.
0
0
13
5
2
.
3
4
6
2
-
4
.
7
8
3
2
.
8
8
24
6
3
.
6
0
9
3
-
2
6
.
6
7
4
8
.
0
0
14
5
4
.
8
3
9
1
-
5
.
2
4
3
2
.
0
0
25
6
3
.
8
1
1
3
-
30
50
15
5
6
.
8
7
7
2
-
6
.
0
0
3
1
.
6
6
26
6
3
.
9
2
2
5
-
29
52
16
5
8
.
4
2
4
4
-
7
.
2
0
3
2
.
4
0
27
6
3
.
9
7
3
4
-
30
54
T
ab
le
4
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
9
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
1
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
7
19
0
57
18
5
1
.
3
7
7
8
-
0
.
4
6
8
1
3
4
6
.
1
5
4
8
22
0
56
19
5
3
.
9
4
4
-
0
.
8
1
7
7
5
4
5
.
2
4
0
9
25
0
55
20
5
6
.
3
7
8
2
-
1
.
2
9
0
2
3
4
4
.
3
4
9
10
28
0
54
21
5
8
.
6
4
6
1
-
1
.
9
2
7
9
3
4
3
.
5
5
4
11
31
0
53
22
6
0
.
6
1
8
-
2
.
9
2
5
5
7
4
3
.
2
4
6
12
34
0
52
23
6
2
.
0
8
8
3
-
4
.
5
3
3
4
7
4
4
.
2
1
7
13
3
6
.
9
9
9
3
-
0
.
0
0
0
3
2
5
1
.
0
0
0
24
6
3
.
0
0
3
6
-
6
.
7
4
7
2
0
4
6
.
0
2
6
14
3
9
.
9
9
1
9
-
0
.
0
0
6
6
4
5
0
.
0
0
1
25
6
3
.
5
1
2
6
-
9
.
3
3
7
9
1
4
8
.
0
0
3
15
4
2
.
9
5
8
6
-
0
.
0
2
9
6
5
4
9
.
0
1
1
26
6
3
.
7
8
1
-
1
2
.
1
4
0
7
7
5
0
.
0
0
0
16
4
5
.
8
6
7
6
-
0
.
0
9
0
6
3
4
8
.
0
4
1
27
6
3
.
9
1
2
9
-
1
5
.
0
7
6
9
5
5
2
.
0
0
0
17
4
8
.
6
8
2
7
-
0
.
2
3
2
7
0
4
7
.
0
8
4
28
6
3
.
9
7
0
2
-
1
8
.
0
1
1
3
1
5
4
.
0
0
0
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
5
3
-
5
4
6
5
5460
4
.
2
.
M
o
difica
t
io
n 2
-
wit
h
[
]
co
ns
t
a
nt
E
q
u
iv
alen
tly
b
in
ar
y
v
ec
to
r
o
f
d
im
en
s
io
n
2
d
ef
in
e
d
f
o
r
[
51
]
is
as
f
o
llo
ws:
=
[
0
,
1
]
=
(
[
8
]
⊕
[
51
]
,
[
29
]
⊕
[
51
]
)
=
(
2
,
2
)
2
2
(
2
3
)
W
ith
th
e
k
n
o
wled
g
e
o
f
in
ab
o
v
e
e
q
u
atio
n
,
2
e
q
u
atio
n
s
ar
e
d
ed
u
ce
d
as f
o
llo
ws:
[
8
]
⊕
[
51
]
=
2
[
29
]
⊕
[
51
]
=
2
(
2
4
)
T
h
e
4
p
o
s
s
ib
le
v
alu
es
o
f
,
r
ef
er
r
ed
as
Mo
v
e
0
–
3
,
co
r
r
esp
o
n
d
s
to
d
if
f
e
r
en
t
m
o
v
em
en
ts
i
n
A5
/1
L
FS
R
s
tr
an
s
f
o
r
m
in
g
to
+
1
.
−
Mo
v
e
0
: Fr
o
m
th
e
L
FS
R
ac
tio
n
asp
ec
t,
u
p
d
ateR1
,
u
p
d
ateR2
an
d
u
p
d
ateR3
ar
e
all
ca
lled
.
T
h
is
co
r
r
esp
o
n
d
s
to
clo
ck
v
al
u
es
ⅈ
∈
{
0
,
7
}
o
r
e
q
u
iv
alen
tl
y
s
t
[
8
,
29
,
51
]
∈
{
(
0
,
0
,
0
)
,
(
1
,
1
,
1
)
}
−
Mo
v
e
1
:
On
ly
u
p
d
ateR1
a
n
d
u
p
d
ateR2
ar
e
ca
lled
co
r
r
esp
o
n
d
i
n
g
to
ⅈ
∈
{
1
,
6
}
o
r
e
q
u
iv
al
en
tly
[
8
,
29
,
51
]
∈
{
(
0
,
0
,
1
)
,
(
1
,
1
,
0
)
}
−
Mo
v
e
2
:
On
ly
u
p
d
ateR2
an
d
u
p
d
ateR3
ar
e
ca
lled
co
r
r
esp
o
n
d
in
g
t
o
ⅈ
∈
{
2
,
5
}
OR
[
8
,
29
,
51
]
∈
{
(
1
,
0
,
0
)
,
(
0
,
1
,
1
)
}
.
−
Mo
v
e
3
:
On
ly
u
p
d
ateR1
an
d
u
p
d
ateR3
ar
e
ca
lled
c
o
r
r
esp
o
n
d
in
g
to
C
ⅈ
∈
{
3
,
4
}
OR
s
t
[
8
,
29
,
51
]
∈
{
(
1
,
0
,
1
)
,
(
0
,
1
,
0
)
}
As co
n
s
id
er
ed
in
s
ec
tio
n
3
,
3
rd
eq
u
atio
n
is
co
n
s
id
er
ed
as
=
+
1
[
18
]
⨁
+
1
[
40
]
⨁
+
1
[
63
]
(
2
5
)
4
.
2
.
1
.
Rec
o
v
er
y
o
f
S
0
s
t
a
t
e
wit
h
[
]
co
ns
t
a
nt
B
y
m
o
d
if
icatio
n
as
d
is
cu
s
s
ed
in
s
ec
tio
n
4
.
2
a
n
ew
c
o
d
e
is
im
p
lem
en
ted
,
a
n
d
th
e
tim
e
co
m
p
lex
ity
is
ca
lcu
lated
b
y
(
17
)
.
T
h
e
r
esu
lt
o
b
tain
ed
is
s
h
o
wn
in
T
a
b
le
5
.
T
h
e
lo
west
tim
e
co
m
p
lex
ity
ac
h
iev
ed
is
2
29
.
313
co
r
r
esp
o
n
d
in
g
t
o
t
=
1
4
.
T
ab
le
5
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
7
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
0
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
6
3
2
.
4
5
1
9
-
6
.
9
7
7
3
6
.
5
7
0
17
5
9
.
6
5
5
8
-
1
3
.
4
2
7
3
4
.
0
0
2
7
3
5
.
0
9
7
5
-
7
.
1
2
8
3
5
.
7
7
4
18
6
0
.
5
4
0
7
-
1
5
.
8
0
3
3
6
.
0
0
0
8
3
7
.
9
5
0
6
-
7
.
1
8
7
3
4
.
8
6
2
19
6
1
.
2
4
7
2
-
1
8
.
3
5
2
3
8
.
0
0
0
9
4
0
.
8
9
1
1
-
7
.
2
0
8
3
3
.
9
0
0
20
6
1
.
8
5
0
1
-
2
0
.
8
2
0
4
0
.
0
0
0
10
4
3
.
8
4
2
3
-
7
.
2
4
3
3
2
.
9
1
3
21
6
2
.
3
9
6
4
-
2
3
.
5
7
3
4
2
.
0
0
0
11
4
6
.
7
7
6
8
-
7
.
2
8
7
3
1
.
9
3
7
22
6
2
.
8
8
8
-
2
5
.
9
1
2
4
4
.
0
0
0
12
4
9
.
6
7
0
1
-
7
.
4
0
5
3
0
.
9
3
5
23
6
3
.
3
0
0
4
-
29
46
13
5
2
.
4
4
5
4
-
7
.
7
7
5
2
9
.
8
8
0
24
6
3
.
6
0
9
8
-
30
48
14
5
4
.
9
3
4
3
-
8
.
4
9
4
2
9
.
3
1
3
25
6
3
.
8
1
1
4
-
30
50
15
5
6
.
9
6
6
6
-
9
.
6
2
6
3
0
.
2
2
1
26
6
3
.
9
2
2
5
-
30
52
16
5
8
.
5
0
5
6
-
1
1
.
2
9
8
3
2
.
0
2
5
5
27
6
3
.
9
7
3
4
-
30
54
4
.
2
.
2
.
Rec
o
v
er
y
o
f
S
1
s
t
a
t
e
wit
h
[
]
co
ns
t
a
nt
Fo
r
r
ec
o
v
er
y
o
f
S
1
s
tate
ac
c
o
r
d
in
g
t
o
m
o
d
i
f
icatio
n
m
ad
e
in
4
.
2
th
e
lo
west
tim
e
co
m
p
lex
ity
is
ca
lcu
lated
b
ased
o
n
eq
u
atio
n
1
9
.
T
h
e
r
esu
lt
o
b
tain
e
d
is
s
h
o
wn
i
n
T
a
b
le
6
.
T
h
e
l
o
west
tim
e
co
m
p
lex
ity
ac
h
iev
ed
is
2
43
.
251
co
r
r
esp
o
n
d
i
n
g
to
t
=
22.
R
em
ar
k
:
Mo
v
e
g
u
ess
an
d
d
eter
m
in
e
tech
n
i
q
u
e
d
is
cu
s
s
ed
in
s
ec
tio
n
3
an
d
s
ec
tio
n
4
is
b
a
s
ed
o
n
th
e
Go
lic’
s
o
b
s
er
v
atio
n
s
s
tatin
g
o
n
ly
2
o
u
t
o
f
3
L
FS
R
s
ar
e
u
p
d
ated
an
d
1
o
u
t
o
f
3
L
FS
R
s
alwa
y
s
r
etain
s
its
p
r
ev
io
u
s
s
tate.
So
in
[
1
2
]
,
[
8
]
clo
ck
b
it
is
co
n
s
id
er
ed
c
o
m
m
o
n
,
an
d
th
e
L
o
west
tim
e
co
m
p
le
x
ity
r
esu
lt
th
at
we
o
b
tain
to
r
ec
o
v
er
S0
an
d
S1
s
tate
is
2
36
.
56
co
r
r
esp
o
n
d
in
g
to
t
=
1
6
an
d
2
43
.
251
at
t
=
2
2
.
W
e
h
av
e
also
o
b
tain
ed
th
e
ti
m
e
co
m
p
lex
ity
r
esu
lt
b
y
k
ee
p
i
n
g
th
e
[
29
]
an
d
[
51
]
clo
ck
b
it
s
tate
as
p
r
ev
io
u
s
s
tate
an
d
o
b
s
er
v
e
d
th
at
tim
e
co
m
p
lex
ity
ca
n
ag
ain
b
e
lo
wer
ed
f
u
r
th
er
.
T
o
r
ec
o
v
e
r
S
0
an
d
S
1
s
tate
th
e
o
b
tain
ed
lo
west
tim
e
co
m
p
lex
ity
is
2
31
.
66
co
r
r
esp
o
n
d
in
g
to
t
=
1
5
,
2
43
.
246
at
t
=
2
2
f
o
r
[
29
]
an
d
2
29
.
313
co
r
r
esp
o
n
d
in
g
t
o
t
=
1
4
,
2
43
.
251
co
r
r
e
s
p
o
n
d
in
g
to
t
=
2
2
f
o
r
[
51
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Memo
r
yle
s
s
s
ta
te
-
r
ec
o
ve
r
y
cryp
ta
n
a
lysi
s
meth
o
d
fo
r
lig
h
tw
eig
h
t
…
(
K
h
ed
ka
r
A
b
o
li A
u
d
u
mb
a
r
)
5461
T
ab
le
6
.
T
h
e
v
alu
es o
f
an
d
in
eq
u
atio
n
1
9
with
2
30
r
an
d
o
m
t
ests
f
o
r
S
1
r
ec
o
v
er
y
u
s
in
g
m
o
v
e
g
u
ess
-
an
d
-
d
eter
m
in
e
att
ac
k
t
l
o
g
l
o
g
C
o
mp.
t
l
o
g
l
o
g
C
o
mp.
7
19
0
57
18
5
1
.
3
7
7
8
-
0
.
4
6
8
3
5
4
6
.
1
5
4
1
5
8
22
0
56
19
5
3
.
9
4
4
-
0
.
8
1
7
1
6
4
5
.
2
4
1
2
2
9
25
0
55
20
5
6
.
3
7
8
2
-
1
.
2
8
7
8
2
4
4
.
3
5
1
7
4
10
28
0
54
21
5
8
.
6
4
6
2
-
1
.
9
2
9
1
4
4
3
.
5
5
3
1
2
11
31
0
53
22
6
0
.
6
1
8
-
2
.
9
1
6
2
8
4
3
.
2
5
1
5
6
12
34
0
52
23
6
2
.
0
8
8
3
-
4
.
5
3
0
7
3
4
4
.
2
1
7
5
7
13
3
6
.
9
9
9
3
-
0
.
0
0
0
4
8
5
1
.
0
0
0
2
1
24
6
3
.
0
0
3
7
-
6
.
7
4
7
6
2
4
6
.
0
2
6
5
3
14
3
9
.
9
9
1
9
-
0
.
0
0
5
8
1
5
0
.
0
0
2
2
8
25
6
3
.
5
1
2
6
-
9
.
3
5
9
8
0
4
8
.
0
0
3
0
7
15
4
2
.
9
5
8
6
-
0
.
0
2
9
5
1
4
9
.
0
1
1
8
8
26
6
3
.
7
8
1
-
1
2
.
1
6
1
5
0
5
0
.
0
0
0
3
6
16
4
5
.
8
6
7
6
-
0
.
0
9
4
6
5
4
8
.
0
3
7
7
5
27
6
3
.
9
1
2
9
-
1
5
.
0
6
5
8
9
5
2
.
0
0
0
0
4
17
4
8
.
6
8
2
6
-
0
.
2
3
2
4
3
4
7
.
0
8
5
0
0
28
6
3
.
9
7
0
1
-
1
8
.
0
0
1
7
6
5
4
.
0
0
0
0
0
5.
A
L
G
O
R
I
T
H
M
T
O
S
E
L
E
C
T
A
C
O
N
S
T
A
N
T
C
L
O
C
K
B
I
T
U
S
E
D
F
O
R
G
U
E
S
S
A
N
D
D
E
T
E
R
M
I
NE
T
E
C
H
N
I
Q
U
E
Acc
o
r
d
in
g
to
s
to
p
-
an
d
-
g
o
m
ec
h
an
is
m
in
s
ec
tio
n
2
th
e
o
cc
asi
o
n
wh
en
1
o
u
t o
f
3
clo
ck
b
its
r
em
ain
s
a
s
p
r
ev
io
u
s
,
d
ep
en
d
s
o
n
(
2
)
.
U
s
in
g
th
is
(
2
)
it
h
as
b
ee
n
d
e
r
i
v
ed
th
at
wh
ich
o
f
th
e
clo
c
k
b
it
will
r
em
ain
s
am
e
as
th
e
p
r
ev
io
u
s
clo
ck
b
it
an
d
th
is
clo
ck
b
it
is
co
n
s
id
er
ed
co
n
s
tan
t.
Usi
n
g
th
is
co
n
s
tan
t
clo
ck
b
it
m
o
v
e
eq
u
atio
n
o
f
(
9
)
,
o
r
(
2
1
)
,
o
r
(
2
4
)
will
b
e
c
alcu
lated
.
Usi
n
g
th
is
,
f
u
r
th
e
r
r
ec
o
v
er
y
p
r
o
ce
s
s
o
f
S
0
an
d
S
1
will b
e
d
o
n
e
as d
is
cu
s
s
ed
in
s
e
ctio
n
3
an
d
4
.
Attack
p
r
o
ce
d
u
r
e:
T
h
e
s
tep
s
in
v
o
lv
ed
i
n
attac
k
p
r
o
ce
d
u
r
e
ar
e
as f
o
llo
ws.
The general process of such an attack can be summarised as follows:
Step 1: Compute
from equation (2)
Step 2: Check which clock bit
≠
Step 3: That clock bit is considered common bit in 2
-
bit Move equation.
If
(s[8] !=
)
then
(a)
Acquire
ℓ
keystream bits
z
0
, …,
z
ℓ
−
1
(b)
Initialise S ←
ϕ
for collecting
s
0
candidates
(c)
Guess (
m
0
, …,
m
t
−
1
) and do the following sub steps:
a.
Acquire
the
equations
BC
getBC((
m
0
,
…,
m
t
−
1
),
(
z
0
,….,
z
t
−
1
))
by
ca
ll
in
g
Algorithm 1.
b.
De
du
ce
th
e
A
and
b
in
(
13
)
ac
co
rd
in
g
to
BC
an
d
co
mp
ut
e
th
e
ex
te
nd
ed
matrix
E
in (
14
).
c.
Compute
rank
(
A
)
an
d
rank
(
E
),
if
rank
(
A
)
≠
rank
(
E
)
,
su
ch
a
mo
ve
me
nt
gu
es
s
is wrong, go back to Step 3 for the next movement guess.
d.
Fo
r
al
l
2
64
−
rank
(
A
)
solutions
to
A
x
T
=b
T
,
se
t
̂
0
←
x
an
d
ge
ne
ra
te
th
e
ke
ys
tr
ea
m
bits
̂
0
,
….,
̂
−
1
,
̂
,
…
;
̂
ℓ
−
1
.
e.
If (
̂
,…..,
̂
ℓ
−
1
) =
(
,…..,
ℓ
−
1
), add such
̂
0
into S.
(d)
Return S.
Else If
(s[29] !=
)
then
(a)
Acquire
ℓ
keystream bits
z
0
, …,
z
ℓ
−
1
(b)
Initialise S ←
ϕ
for collecting
s
0
candidates
(c)
Guess (
m
0
, …,
m
t
−
1
) and do the following sub steps:
a.
Acquire
the
equations
BC
getBC((
m
0
,
…,
m
t
−
1
),
(
z
0
,….,
z
t
−
1
))
by
ca
ll
in
g
Algorithm 2.
b.
De
du
ce
th
e
A
and
b
in
(
13
)
ac
co
rd
in
g
to
BC
an
d
co
mp
ut
e
th
e
ex
te
nd
ed
matrix
E
in (
14
).
c.
Compute
rank
(
A
)
an
d
rank
(
E
),
if
rank
(
A
)
≠
rank
(
E
)
,
su
ch
a
mo
ve
me
nt
gu
es
s
is wrong, go back to Step 3 for the next movement guess.
d.
Fo
r
al
l
2
64
−
rank
(
A
)
solutions
to
A
x
T
=b
T
,
se
t
̂
0
←
x
an
d
ge
ne
ra
te
th
e
ke
ys
tr
ea
m
bits
̂
0
,
….,
̂
−
1
,
̂
,
…
;
̂
ℓ
−
1
.
e.
If (
̂
,…..,
̂
ℓ
−
1
) =
(
,…..,
ℓ
−
1
), add such
̂
0
into S.
(d)
Return S.
Else If
(s[51] !=
)
then
(a)
Acquire
ℓ
keystream bits
z
0
, …,
z
ℓ
−
1
(b)
Initialise S ←
ϕ
for collecting
s
0
candidates
(c)
Guess (
m
0
, …,
m
t
−
1
) and do the following sub steps:
a.
Acquire
the
equations
BC
getBC((
m
0
,
…,
m
t
−
1
),
(
z
0
,….,
z
t
−
1
))
by
ca
ll
in
g
Algorithm 3.
b.
De
du
ce
th
e
A
and
b
in
(
13
)
ac
co
rd
in
g
to
BC
an
d
co
mp
ut
e
th
e
ex
te
nd
ed
matrix
E
in (
14
).
c.
Compute
rank
(
A
)
an
d
rank
(
E
),
if
rank
(
A
)
≠
rank
(
E
)
,
su
ch
a
mo
ve
me
nt
gu
es
s
is wrong, go back to Step 3 for the next movement guess.
d.
Fo
r
al
l
2
64
−
rank
(
A
)
solutions
to
A
x
T
=b
T
,
se
t
̂
0
←
x
an
d
ge
ne
ra
te
th
e
ke
ys
tr
ea
m
bits
̂
0
,
….,
̂
−
1
,
̂
,
…
;
̂
ℓ
−
1
.
e.
If (
̂
,…..,
̂
ℓ
−
1
) =
(
,…..,
ℓ
−
1
), add such
̂
0
into S.
(d)
Return S.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
5
3
-
5
4
6
5
5462
Alg
o
r
ith
m
1
.
Ded
u
ce
th
e
s
et
o
f
eq
u
atio
n
s
ac
c
o
r
d
in
g
to
th
e
g
i
v
en
m
o
v
es a
n
d
o
u
tp
u
t
b
its
1.
procedure
getBC (movements (
m
0
, …,
m
t
−
1
)
∈
{0,3}
t
, output bits (
0
,
…
.
.
,
−
1
)
∈
2
2.
Initialise the words
W
0
←
I
3.
Initialise the linear equations set BC
←
ϕ
4.
Initialise
x
=
(
x
0
,
….
,
x
63
)
as
ve
ct
or
of
63
un
kn
ow
n
Bo
ol
ea
n
va
ri
ab
le
s
co
rr
e
sp
on
di
ng
to the 64 state bits of
s
0
5.
for
i
= 0, 1, …,
t
−
1
do
a.
Represent
m
i
= (
μ
,
ν
)
∈
{0, …, 3} as Equation (8)
b.
Update BC by adding the following equations
i.
(
ⅈ
[
8
]
⊕
ⅈ
[
29
]
)
⋅
=
ii.
(
ⅈ
[
8
]
⊕
ⅈ
[
51
]
)
⋅
=
c.
Deduce
w
i
+1
according
to
w
i
by
ca
ll
in
g
w
i
+1
←
UpdW(
m
i
,
w
i
)
de
fi
ne
d
in
Al
go
ri
th
m
3 ref. [12].
d.
Up
da
te
BC
by
ad
di
ng
th
e
f
ol
lo
wi
ng
li
ne
ar
eq
ua
ti
on
s
co
rr
es
po
nd
in
g
to
Eq
ua
ti
o
n
(
10
)
i.
(
ⅈ
+
1
[
18
]
⊕
ⅈ
+
1
[
40
]
⊕
ⅈ
+
1
[
63
]
) .
=
ⅈ
6.
End for
7.
Return
BC
8.
End Procedure
Alg
o
r
ith
m
2
.
Ded
u
ce
th
e
s
et
o
f
eq
u
atio
n
s
ac
c
o
r
d
in
g
to
th
e
g
i
v
en
m
o
v
es a
n
d
o
u
tp
u
t
b
its
1.
procedure
getBC (movements (
m
0
, …,
m
t
−
1
)
∈
{0,3}
t
, output bits (
0
,
…
.
.
,
−
1
)
∈
2
2.
Initialise the words
W
0
←
I
3.
Initialise the linear equations set BC
←
ϕ
4.
Initialise
x
=
(
x
0
,
….
,
x
63
)
as
ve
ct
or
of
63
un
kn
ow
n
Bo
ol
ea
n
va
ri
ab
le
s
co
rr
e
sp
on
di
ng
to the 64 state bits of
s
0
5.
for
i
= 0, 1, …,
t
−
1
do
a.
Represent
mi
= (
1
,
1)
∈
{0, …, 3} as Equation (20)
b.
Update BC by adding the following equations
i.
(
ⅈ
[
8
]
⊕
ⅈ
[
29
]
)
⋅
=
1
ii.
(
ⅈ
[
29
]
⊕
ⅈ
[
51
]
)
⋅
=
1
c.
Deduce
w
i
+1
according
to
w
i
by
ca
ll
in
g
w
i
+1
←
UpdW(
m
i
,
w
i
)
de
fi
ne
d
in
Al
go
ri
th
m
3 ref. [12].
d.
Up
da
te
BC
by
ad
di
ng
th
e
f
ol
lo
wi
ng
li
ne
ar
eq
ua
ti
on
s
co
rr
es
po
nd
in
g
to
Eq
ua
ti
o
n
(
10
)
i.
(
ⅈ
+
1
[
18
]
⊕
ⅈ
+
1
[
40
]
⊕
ⅈ
+
1
[
63
]
) .
=
ⅈ
6.
End for
7.
Return
BC
8.
End Procedure
Alg
o
r
ith
m
3
.
Ded
u
ce
th
e
s
et
o
f
eq
u
atio
n
s
ac
c
o
r
d
in
g
to
th
e
g
i
v
en
m
o
v
es a
n
d
o
u
tp
u
t
b
its
1.
procedure
getBC (movements (
m
0
, …,
m
t
−
1
)
∈
{0,3}
t
, output bits (
0
,
…
.
.
,
−
1
)
∈
2
2.
Initialise the words
W
0
←
I
3.
Initialise the linear equations set BC
←
ϕ
4.
Initialise
x
=
(
x
0
,
….
,
x
63
)
as
ve
ct
or
of
63
un
kn
ow
n
Bo
ol
ea
n
va
ri
ab
le
s
co
rr
e
sp
on
di
ng
to the 64 state bits of
s
0
5.
for
i
= 0, 1, …,
t
−
1
do
a.
Represent
mi
= (
2
,
2
)
∈
{0, …, 3} as Equation (23)
b.
Update BC by adding the following equations
i.
(
ⅈ
[
8
]
⊕
ⅈ
[
51
]
)
⋅
=
2
ii.
(
ⅈ
[
29
]
⊕
ⅈ
[
51
]
)
⋅
=
2
c.
Deduce
w
i
+1
according
to
w
i
by
ca
ll
in
g
w
i
+1
←
UpdW(
m
i
,
w
i
)
de
fi
ne
d
in
Al
go
ri
th
m
3 ref. [12].
d.
Up
da
te
BC
by
ad
di
ng
th
e
f
ol
lo
wi
ng
li
ne
ar
eq
ua
ti
on
s
co
rr
es
po
nd
in
g
to
Eq
ua
ti
o
n
(
10
)
i.
(
ⅈ
+
1
[
18
]
⊕
ⅈ
+
1
[
40
]
⊕
ⅈ
+
1
[
63
]
) .
=
ⅈ
6.
End for
7.
Return
BC
8.
End Procedure
Fo
llo
w
th
e
s
am
e
attac
k
p
r
o
ce
d
u
r
e
as g
iv
en
ab
o
v
e
to
R
ec
o
v
er
S
1
s
tate,
b
u
t n
o
n
ee
d
to
g
u
ess
m
0
m
o
v
e
.
6.
RE
SU
L
T
AND
DI
SCUS
SI
O
N
T
h
e
s
elec
tiv
e
m
o
d
if
ied
Gu
es
s
an
d
Dete
r
m
in
e
attac
k
alg
o
r
ith
m
g
iv
es
th
e
m
eth
o
d
to
s
elec
t
wh
ich
clo
ck
b
it
is
to
b
e
co
n
s
id
er
e
d
co
n
s
tan
t
an
d
at
e
v
er
y
tim
e
o
f
ex
ec
u
tin
g
t
h
e
attac
k
s
o
th
at
m
in
im
u
m
tim
e
co
m
p
lex
ity
is
ac
h
iev
ed
t
o
r
ec
o
v
er
S
0
a
n
d
S
1
s
tate.
T
h
e
p
r
a
ctica
l
r
esu
lts
o
b
tain
ed
f
o
r
r
ec
o
v
er
y
o
f
S
0
an
d
S
1
s
tate
o
f
p
r
o
p
o
s
ed
m
eth
o
d
o
lo
g
y
ar
e
d
is
cu
s
s
ed
h
er
e.
Evaluation Warning : The document was created with Spire.PDF for Python.