I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
p
ute
r
E
ng
in
ee
ring
(
I
J
E
CE
)
Vo
l.
8
,
No
.
3
,
J
u
n
e
201
8
,
p
p
.
1814
~
1
8
2
1
I
SS
N:
2
0
8
8
-
8708
,
DOI
: 1
0
.
1
1
5
9
1
/
i
j
ec
e
.
v8
i
3
.
p
p
1
8
1
4
-
1821
1814
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ia
e
s
co
r
e
.
co
m/
jo
u
r
n
a
ls
/in
d
ex
.
p
h
p
/
I
JE
C
E
Desig
n and
I
m
ple
m
en
tatio
n of
a Se
cure Co
mm
un
ica
tion
Protoco
l
M.
K
.
Vis
wa
na
t
h
1
,
M
.
Ra
njit
h K
u
m
a
r
2
1
De
p
a
rte
m
e
n
t
o
f
M
a
th
e
m
a
ti
c
s,
Ra
jala
k
sh
m
i
En
g
in
e
e
rin
g
Co
ll
e
g
e
,
T
h
a
n
d
a
lam
,
Ch
e
n
n
a
i
–
6
0
2
1
0
5
,
T
a
m
il
Na
d
u
,
In
d
ia
2
Re
se
a
rc
h
S
c
h
o
lar,
Re
se
a
rc
h
a
n
d
De
v
e
lo
p
m
e
n
t
Ce
n
tre,
Bh
a
ra
th
iar
Un
iv
e
rsit
y
,
Co
im
b
a
to
re
–
6
4
1
0
4
6
,
T
a
m
il
Na
d
u
,
In
d
ia
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Ma
y
22
,
2
0
1
7
R
ev
i
s
ed
No
v
3
0
,
2
0
1
7
A
cc
ep
ted
Dec
7
,
2
0
1
7
T
h
e
m
a
in
o
b
jec
t
o
f
th
is
p
a
p
e
r
is
to
p
re
se
n
t
a
m
u
tu
a
l
a
u
th
e
n
ti
c
a
ti
o
n
p
ro
t
o
c
o
l
th
a
t
g
u
a
ra
n
tee
s
se
c
u
rit
y
,
in
teg
rit
y
a
n
d
a
u
th
e
n
ti
c
it
y
o
f
m
e
ss
a
g
e
s,
tran
sf
e
rre
d
o
v
e
r
a
n
e
t
w
o
rk
s
y
ste
m
.
In
th
is
p
a
p
e
r
a
s
y
m
m
e
tri
c
k
e
y
c
r
y
p
to
s
y
ste
m
,
th
a
t
sa
ti
sf
ie
s
a
ll
th
e
a
b
o
v
e
re
q
u
irem
e
n
ts,
is
d
e
v
e
lo
p
e
d
u
sin
g
t
h
e
o
re
m
s
o
f
J.R.
Ch
e
n
,
I.
M
.
V
i
n
o
g
ra
d
o
v
a
n
d
F
e
r
m
a
t
a
n
d
th
e
d
e
c
ima
l
e
x
p
a
n
sio
n
o
f
a
n
irratio
n
a
l
n
u
m
b
e
r.
K
ey
w
o
r
d
:
C
h
e
n
’
s
t
h
eo
r
e
m
Fer
m
at
’
s
t
w
o
s
q
u
ar
es t
h
eo
r
e
m
P
s
eu
d
o
in
v
er
s
e
R
ab
in
cr
y
p
to
s
y
s
te
m
Vin
o
g
r
ad
o
v
’
s
t
h
eo
r
e
m
Co
p
y
rig
h
t
©
2
0
1
8
In
stit
u
te o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
rin
g
a
n
d
S
c
ien
c
e
.
Al
l
rig
h
ts
re
se
rv
e
d
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
M.
R
an
j
ith
K
u
m
ar
,
Dep
ar
t
m
en
t o
f
Ma
th
e
m
at
ics,
B
h
ar
ath
iar
Un
iv
er
s
it
y
,
C
o
i
m
b
ato
r
e
–
6
4
1
0
4
6
,
T
am
il
Nad
u
,
I
n
d
ia.
E
m
ail: a
n
n
a
m
.
r
an
j
it
h
@
g
m
ail.
co
m
1.
I
NT
RO
D
UCT
I
O
N
T
h
e
cr
y
p
to
g
r
ap
h
ic
co
m
m
u
n
i
t
y
h
a
s
b
ee
n
p
er
tin
en
tl
y
m
o
r
e
s
u
cc
e
s
s
f
u
l
in
t
h
e
r
elate
d
f
ield
o
f
id
en
ti
f
icatio
n
an
d
i
n
te
g
r
it
y
,
w
h
er
e
t
h
e
au
th
e
n
tic
u
s
er
s
tr
y
to
co
n
v
i
n
ce
ea
c
h
o
t
h
er
o
f
t
h
eir
id
en
tit
y
a
n
d
th
e
in
te
g
r
it
y
o
f
th
e
s
ec
r
et
m
e
s
s
a
g
e
ex
ch
a
n
g
ed
o
v
er
an
elec
tr
o
n
ic
ch
an
n
el
[
1
]
,
[
2
]
.
I
n
o
r
d
in
ar
y
c
o
m
m
u
n
icatio
n
s
a
n
in
tr
u
d
er
ca
n
s
ee
al
l
t
h
e
e
x
c
h
a
n
g
ed
m
e
s
s
a
g
es,
ca
n
d
elete
,
ad
d
o
r
alter
an
d
r
ed
ir
ec
t
m
es
s
ag
es,
ca
n
i
n
itiate
th
e
p
r
o
to
co
l
w
it
h
a
n
o
th
er
p
ar
t
y
a
n
d
r
e
-
u
s
e
m
e
s
s
a
g
es
f
r
o
m
p
ar
t
o
f
co
m
m
u
n
ica
tio
n
s
[
3
]
,
[
4
]
.
H
en
ce
cr
y
p
to
g
r
ap
h
ic
to
o
ls
ar
e
v
er
y
cr
u
cial
i
n
s
ec
r
e
t
co
m
m
u
n
icatio
n
s
,
as
it
p
r
ev
e
n
ts
u
n
a
u
t
h
o
r
ized
p
er
s
o
n
s
f
r
o
m
ac
q
u
ir
i
n
g
,
s
to
r
ed
d
ata
b
et
w
ee
n
co
m
p
u
ter
s
o
r
m
e
s
s
a
g
es tr
an
s
f
er
r
ed
b
et
w
ee
n
t
wo
m
u
t
u
all
y
au
t
h
e
n
ticated
p
ar
ties
.
W
e
d
escr
ib
e
in
th
is
p
ap
er
h
o
w
t
h
e
ab
o
v
e
ca
p
ab
ilit
ies
ar
e
i
n
co
r
p
o
r
ated
in
th
e
co
m
m
u
n
ic
atio
n
s
y
s
te
m
d
ev
elo
p
ed
h
er
e
u
s
i
n
g
t
h
e
b
r
o
ad
id
ea
p
r
o
p
o
s
ed
in
[
5
]
.
Ho
w
ev
er
th
e
tec
h
n
iq
u
es
u
s
ed
h
er
e
ar
e
q
u
ite
d
if
f
er
e
n
t
f
r
o
m
t
h
e
o
n
e
u
s
ed
i
n
[
5
]
,
b
u
t
is
clo
s
e
to
th
e
o
n
e
u
s
ed
in
[
6
]
.
W
e
m
a
k
e
u
s
e
o
f
[
7
]
-
[
9
]
an
d
th
e
Fer
m
a
t’
s
t
w
o
s
q
u
ar
es
t
h
eo
r
e
m
[
1
0
]
in
cr
ea
ti
n
g
t
h
e
k
e
y
s
f
o
r
en
cr
y
p
t
in
g
t
h
e
p
lain
te
x
t
a
n
d
al
s
o
th
e
R
ab
i
n
cr
y
p
to
s
y
s
te
m
[
1
1
]
,
w
it
h
o
u
t
t
h
e
m
o
d
u
l
u
s
b
ei
n
g
m
ad
e
p
u
b
lic
f
o
r
e
n
cr
y
p
ti
n
g
th
e
m
es
s
ag
e
d
ig
es
t.
I
n
th
i
s
p
r
o
to
co
l
b
o
th
th
e
s
e
n
d
er
an
d
r
ec
eiv
er
o
f
a
m
e
s
s
a
g
e
ca
n
co
n
s
tr
u
ct
ea
c
h
o
th
er
’
s
k
e
y
i
n
a
d
d
itio
n
to
th
eir
o
w
n
k
e
y
as i
n
t
h
e
ca
s
e
o
f
[
6
]
.
T
h
e
r
est
o
f
t
h
e
p
ap
er
is
o
r
g
a
n
ized
as
f
o
llo
w
s
.
I
n
Sectio
n
2
w
e
d
escr
ib
e
t
h
e
b
asic
id
ea
o
f
R
ab
i
n
cr
y
p
to
s
y
s
te
m
s
.
I
n
Sectio
n
3
we
g
i
v
e
s
o
m
e
b
ac
k
g
r
o
u
n
d
ab
o
u
t
th
e
p
s
e
u
d
o
in
v
er
s
e
o
f
a
r
ec
ta
n
g
u
lar
m
atr
i
x
[
1
2
]
,
[
5
]
.
I
n
Sectio
n
4
w
e
e
x
p
lai
n
th
e
Go
ld
b
ac
h
co
n
j
ec
tu
r
e
an
d
Fer
m
at
’
s
t
w
o
s
q
u
ar
es
t
h
eo
r
em
.
R
ea
d
er
s
f
a
m
iliar
w
it
h
Sectio
n
1
to
4
,
m
a
y
p
r
o
ce
ed
d
ir
ec
tly
to
Sectio
n
5
o
f
th
i
s
p
ap
er
.
T
h
e
w
o
r
k
i
n
g
o
f
th
e
al
g
o
r
it
h
m
i
s
illu
s
tr
ated
w
it
h
a
n
e
x
a
m
p
le
i
n
Sectio
n
6
an
d
t
h
e
p
ap
er
co
n
cl
u
d
es
w
i
th
a
Sectio
n
o
n
th
e
s
ec
u
r
it
y
asp
ec
t
s
o
f
t
h
e
s
y
s
te
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Desig
n
a
n
d
I
m
p
leme
n
ta
io
n
o
f
a
S
ec
u
r
e
C
o
mmu
n
ica
tio
n
P
r
o
to
c
o
l
(
M.
K
.
V
is
w
a
n
a
th
)
1815
2.
R
AB
I
N
CRYP
T
O
SYS
T
E
M
T
h
e
aim
o
f
th
is
ch
ap
ter
is
to
d
is
cu
s
s
th
e
R
ab
i
n
cr
y
p
to
s
y
s
te
m
s
w
h
o
s
e
s
ec
u
r
it
y
i
s
b
ased
o
n
co
m
p
u
tatio
n
al
as
s
u
m
p
tio
n
s
r
el
at
ed
to
th
e
in
te
g
er
f
ac
to
r
izatio
n
[
1
3
]
.
T
h
e
R
ab
in
p
u
b
lic
-
k
e
y
en
cr
y
p
tio
n
s
c
h
e
m
e
[1
]
,
[
1
4
]
w
a
s
th
e
f
ir
s
t
e
x
a
m
p
l
e
o
f
a
p
r
o
v
ab
ly
s
ec
u
r
e
p
u
b
lic
-
k
e
y
en
cr
y
p
tio
n
s
c
h
e
m
e
-
th
e
p
r
o
b
lem
f
ac
ed
b
y
a
p
ass
iv
e
ad
v
er
s
ar
y
o
f
r
ec
o
v
er
in
g
p
lai
n
tex
t
f
r
o
m
s
o
m
e
g
i
v
en
cip
h
er
tex
t
is
co
m
p
u
tatio
n
all
y
eq
u
i
v
alen
t
to
f
ac
to
r
in
g
.
T
h
e
s
ec
u
r
it
y
o
f
R
ab
in
i
s
m
o
r
e
clo
s
el
y
r
ela
t
ed
to
f
ac
to
r
in
g
th
a
n
R
S
A
.
I
t
d
ea
ls
w
i
th
t
h
e
p
r
o
b
lem
t
h
at
if
.
n
p
q
w
h
er
e
p
an
d
q
ar
e
d
is
tin
ct
p
r
im
es
t
h
en
s
q
u
ar
in
g
is
a
f
o
u
r
-
to
-
o
n
e
m
ap
,
s
o
it
is
n
ec
ess
ar
y
to
h
av
e
a
r
u
le
to
ch
o
o
s
e
th
e
c
o
r
r
ec
t so
lu
tio
n
w
h
ile
d
ec
r
y
p
ti
n
g
th
e
cr
y
p
to
tex
t.
1)
C
h
o
o
s
e
t
w
o
r
an
d
o
m
p
r
i
m
es
p
an
d
q
s
u
c
h
th
a
t
3
(
m
o
d
4
)
pq
an
d
s
et
.
n
p
q
.
2)
n
is
m
ad
e
p
u
b
lic
an
d
,
pq
is
k
ep
t a
s
s
ec
r
et.
T
o
en
cr
y
p
t a
m
es
s
ag
e
m
,
co
m
p
u
te
2
m
o
d
C
m
n
3)
T
o
r
ec
o
v
er
p
lain
tex
t
m
f
r
o
m
C
,
o
n
e
d
o
es th
e
f
o
llo
w
i
n
g
:
a.
Use
th
e
e
x
te
n
d
ed
E
u
clid
ea
n
al
g
o
r
ith
m
to
f
i
n
d
t
h
e
in
te
g
er
s
a
an
d
b
s
atis
f
y
i
n
g
.
.
1
a
p
b
q
.
No
te
th
a
t
a
an
d
b
ca
n
b
e
co
m
p
u
ted
o
n
ce
an
d
f
o
r
all
d
u
r
in
g
t
h
e
k
e
y
g
e
n
er
atio
n
s
ta
g
e.
b.
C
o
m
p
u
te
(
1
)
4
m
o
d
p
r
C
p
an
d
(
1
)
4
m
o
d
q
s
C
q
.
c.
Fin
d
t
h
e
f
o
u
r
s
q
u
ar
e
r
o
o
ts
o
f
C
m
o
d
u
lo
n
.
T
h
e
y
ar
e
1
.
.
.
.
(
m
o
d
)
m
a
p
s
b
q
r
n
2
.
.
.
.
(
m
o
d
)
m
a
p
s
b
q
r
n
31
m
n
m
42
m
n
m
an
d
d
ec
id
es
w
h
ic
h
o
f
th
e
s
e
is
m
.
A
d
r
a
w
b
ac
k
o
f
R
ab
in
’
s
p
u
b
lic
-
k
e
y
s
c
h
e
m
e
is
t
h
at
t
h
e
r
ec
eiv
er
is
f
ac
ed
w
i
th
t
h
e
tas
k
o
f
s
e
l
ec
tin
g
th
e
co
r
r
ec
t
p
lain
tex
t
f
r
o
m
a
m
o
n
g
th
e
f
o
u
r
p
o
s
s
ib
ilit
ies.
T
h
is
a
m
b
ig
u
it
y
i
n
d
ec
r
y
p
tio
n
ca
n
ea
s
i
l
y
b
e
o
v
er
co
m
e
in
p
r
ac
tice
b
y
ad
d
in
g
p
r
e
-
s
p
ec
if
ied
r
ed
u
n
d
an
c
y
to
t
h
e
o
r
ig
i
n
a
l
p
lain
te
x
t
p
r
io
r
to
en
cr
y
p
tio
n
.
T
h
en
,
w
it
h
h
i
g
h
p
r
o
b
a
b
ilit
y
,
ex
ac
t
l
y
o
n
e
o
f
t
h
e
f
o
u
r
s
q
u
ar
e
r
o
o
ts
1
m
,
2
m
,
3
m
,
4
m
o
f
a
le
g
iti
m
ate
cip
h
er
tex
t
C
w
ill
p
o
s
s
e
s
s
th
is
r
ed
u
n
d
a
n
c
y
,
a
n
d
t
h
e
r
ec
ei
v
er
w
ill
s
elec
t
t
h
i
s
as
th
e
in
te
n
d
ed
p
lain
te
x
t.
I
f
n
o
n
e
o
f
t
h
e
s
q
u
ar
e
r
o
o
ts
o
f
C
pos
s
ess
es
th
is
r
ed
u
n
d
an
c
y
,
th
en
t
h
e
r
ec
eiv
er
s
h
o
u
ld
r
ej
ec
t
C
as
a
f
r
a
u
d
u
le
n
t
m
es
s
ag
e.
T
h
i
s
ca
s
e
d
o
es
n
o
t
ar
is
e
w
i
th
t
h
e
p
r
o
b
le
m
in
h
an
d
.
3.
M
O
O
RE
-
P
E
NRO
SE
I
NV
E
RSE
(
P
S
E
UDO
I
NV
E
R
SE
)
3
.
1
.
Def
ini
t
io
n
L
et
m
×
n
AR
an
d
n
×
m
XR
,
th
en
t
h
e
f
o
llo
w
i
n
g
e
q
u
atio
n
s
ar
e
u
s
ed
to
d
ef
in
e
th
e
p
s
eu
d
o
in
v
er
s
e
o
f
a
r
ec
tan
g
u
lar
m
atr
i
x
A
[
1
2
]
,
[
1
4
]
.
A
X
A
A
(
1
)
X
A
X
X
(
2
)
T
A
X
A
X
(
3
)
T
X
A
X
A
(
4
)
E
q
u
atio
n
s
(
1
)
th
r
o
u
g
h
(
4
)
ar
e
ca
lled
th
e
P
en
r
o
s
e
co
n
d
itio
n
s
[
1
5
]
.
3
.
2
.
Def
ini
t
io
n
A
p
s
e
u
d
o
in
v
er
s
e
o
f
r
ec
ta
n
g
u
lar
m
atr
ix
m
×
n
AR
is
also
a
r
ec
ta
n
g
u
lar
m
atr
i
x
#
n
×
m
X
A
R
s
atis
f
y
in
g
E
q
u
atio
n
s
(
1
)
t
h
r
o
u
g
h
(
4
)
.
A
p
s
e
u
d
o
in
v
er
s
e
is
s
o
m
eti
m
es
ca
lled
t
h
e
Mo
o
r
e
–
P
en
r
o
s
e
i
n
v
er
s
e
a
f
ter
th
e
p
io
n
ee
r
in
g
w
o
r
k
d
o
n
e
b
y
Mo
o
r
e
(
1
9
2
0
,
1
9
3
5
)
an
d
P
en
r
o
s
e
(
1
9
5
5
)
.
3
.
3
.
Co
ns
t
ruct
io
n o
f
ps
eudo
i
nv
e
rse
Fo
r
a
g
iv
e
n
m
×
n
AR
,
th
e
p
s
e
u
d
o
in
v
er
s
e
#
n
×
m
AR
is
u
n
iq
u
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
8
,
No
.
3
,
J
u
n
e
2
0
1
8
:
1
8
1
4
–
1821
1816
a.
I
f
mn
an
d
r
a
n
k
A
m
th
e
n
#1
AA
.
b.
I
f
mn
an
d
r
a
n
k
A
m
th
e
n
T
AA
is
n
o
n
-
s
i
n
g
u
lar
a
n
d
1
#
TT
A
A
A
A
(
5
)
c.
I
f
mn
an
d
r
a
n
k
A
n
th
en
T
AA
is
n
o
n
-
s
i
n
g
u
lar
a
n
d
1
#
TT
A
A
A
A
(
6
)
3
.
4
.
Co
nje
ct
ure
a.
I
f
A
is
a
r
ec
ta
n
g
u
lar
m
a
tr
ix
in
m
×
n
R
f
o
r
m
ed
b
y
t
h
e
mn
co
n
s
ec
u
t
iv
e
d
ec
i
m
al
p
lace
s
o
f
a
n
y
ir
r
atio
n
al
n
u
m
b
er
,
w
ith
mn
,
th
en
r
a
n
k
A
m
an
d
A
is
al
w
a
y
s
r
i
g
h
t in
v
er
tib
le.
b.
I
f
A
is
a
r
ec
tan
g
u
lar
m
atr
i
x
in
m
×
n
R
f
o
r
m
ed
b
y
t
h
e
mn
co
n
s
ec
u
t
iv
e
d
ec
i
m
al
p
lace
s
o
f
a
n
y
ir
r
atio
n
a
l
n
u
m
b
er
,
w
ith
mn
,
th
en
r
a
n
k
A
n
an
d
A
is
alw
a
y
s
le
f
t i
n
v
er
t
ib
le.
4.
T
H
E
G
O
L
DB
ACH
CO
NJ
E
CT
UR
E
I
n
1
7
4
2
,
C
.
Go
ld
b
ac
h
co
n
j
ec
t
u
r
ed
th
at,
“eve
r
y
o
d
d
n
u
mb
er
g
r
ea
ter
th
a
n
n
in
e
is
ex
p
r
ess
i
b
le
a
s
th
e
s
u
m
o
f
th
r
ee
p
r
imes”
an
d
“
ev
ery
ev
en
n
u
mb
er
g
r
ea
ter
t
h
a
n
fo
u
r
is
ex
p
r
ess
ib
le
a
s
th
e
s
u
m
o
f
tw
o
o
d
d
p
r
imes”
.
T
h
e
f
ir
s
t
o
n
e
is
ca
l
l
ed
th
e
o
d
d
Go
ld
b
ac
h
co
n
j
ec
tu
r
e
an
d
th
e
s
ec
o
n
d
o
n
e
i
s
ca
lle
d
th
e
ev
e
n
Go
ld
b
ac
h
co
n
j
ec
tu
r
e
[
1
7
]
.
I
n
1
9
3
7
,
I
.
M.
Vin
o
g
r
ad
o
v
e
s
tab
lis
h
ed
th
e
o
d
d
Go
ld
b
ac
h
co
n
j
ec
tu
r
e.
B
u
t
th
e
ev
e
n
Go
ld
b
ac
h
co
n
j
ec
tu
r
e
is
s
till
a
n
o
p
en
q
u
e
s
tio
n
a
n
d
th
e
b
est r
es
u
lt o
b
tain
ed
s
o
f
ar
is
g
i
v
en
b
y
J
i
n
R
u
n
C
h
e
n
i
n
1
9
6
6
.
4
.
1
.
Vino
g
ra
do
v
’
s
t
heo
re
m
I
t
w
as
s
h
ow
n
i
n
1
9
3
7
b
y
I
.
M.
Vin
o
g
r
ad
o
v
[
9
]
th
at,
“
A
ll
s
u
fficien
tly
la
r
g
e
o
d
d
in
teg
ers
a
r
e
ex
p
r
ess
ib
le
a
s
a
s
u
m
o
f
th
r
ee
p
r
imes”
.
Vin
o
g
r
ad
o
v
p
r
o
v
ed
th
e
th
r
ee
-
p
r
i
m
es
th
eo
r
e
m
b
y
a
n
al
y
tical
m
ea
n
s
,
u
s
i
n
g
m
aj
o
r
ar
c/m
in
o
r
ar
c
d
ec
o
m
p
o
s
i
tio
n
.
4
.
2
.
Chen’
s
t
heo
re
m
I
n
1
9
6
6
J
in
R
u
n
C
h
en
[
7
]
m
a
d
e
co
n
s
id
er
ab
le
p
r
o
g
r
ess
in
s
ettin
g
t
h
e
ev
e
n
Go
ld
b
ac
h
co
n
j
ec
tu
r
e;
in
[8
]
C
h
en
p
r
o
v
ed
th
e
f
o
llo
w
i
n
g
th
eo
r
e
m
.
“A
la
r
g
e
ev
en
in
teg
er
ca
n
b
e
ex
p
r
ess
ed
a
s
th
e
s
u
m
o
f
a
p
r
ime
a
n
d
th
e
p
r
o
d
u
ct
o
f
a
tmo
s
t
tw
o
p
r
imes”
.
C
h
en
’
s
th
eo
r
e
m
is
a
g
ia
n
t
s
tep
to
w
ar
d
s
s
o
l
v
i
n
g
t
h
e
Go
ld
b
ac
h
co
n
j
ec
tu
r
e,
an
d
is
a
r
e
m
ar
k
ab
le
r
es
u
lt
u
s
i
n
g
th
e
Siev
e
m
et
h
o
d
s
.
5.
T
H
E
N
E
W
SCH
E
M
E
T
h
e
m
ai
n
id
ea
o
f
t
h
is
p
ap
er
is
,
to
d
ev
elo
p
a
n
e
w
cr
y
p
to
s
y
s
t
e
m
u
s
i
n
g
C
h
e
n
’
s
t
h
eo
r
e
m
,
Vi
n
o
g
r
ad
o
v
’
s
th
eo
r
e
m
a
n
d
th
e
Fer
m
at
’
s
t
w
o
s
q
u
ar
es
t
h
eo
r
e
m
,
w
h
ic
h
p
r
o
v
id
es
co
n
f
id
en
tiali
t
y
,
a
u
t
h
en
t
ici
t
y
a
n
d
in
te
g
r
it
y
o
f
th
e
s
ec
r
et
m
ess
a
g
e
s
h
ar
ed
o
v
er
a
p
u
b
lic
c
h
a
n
n
el.
T
h
is
w
o
r
k
i
s
a
n
o
v
e
l
m
et
h
o
d
o
f
d
ev
elo
p
i
n
g
a
co
m
m
u
n
icatio
n
p
r
o
to
co
l
w
h
ic
h
is
s
ec
u
r
e
ag
ai
n
s
t a
ll th
e
k
n
o
w
n
p
o
s
s
ib
le
attac
k
s
.
T
h
e
p
r
o
to
co
l is as f
o
llo
w
s
:
W
e
ar
e
lo
o
k
in
g
f
o
r
n
u
m
b
er
s
w
h
ic
h
s
at
is
f
y
t
h
e
f
o
llo
w
in
g
d
e
co
m
p
o
s
i
tio
n
(
a)
a
n
d
(
b
)
g
i
v
en
b
elo
w
a
n
d
ca
ll th
e
s
e
n
u
m
b
er
s
a
s
f
ea
s
ib
le
n
u
m
b
er
s
.
No
t a
l
l t
h
e
o
d
d
an
d
ev
en
in
teg
er
s
ar
e
f
ea
s
ib
le.
Fo
r
ex
a
m
p
le
1
1
an
d
1
4
ar
e
n
o
t
f
ea
s
ib
le.
A
M
A
T
L
AB
p
r
o
g
r
am
m
e
is
d
ev
elo
p
ed
t
o
ch
ec
k
w
h
eth
er
a
g
iv
e
n
ev
e
n
o
r
o
d
d
n
u
m
b
er
i
s
f
ea
s
ib
le.
U
s
in
g
M
A
T
L
A
B
t
h
e
f
o
llo
w
in
g
n
u
m
b
er
s
ar
e
f
o
u
n
d
to
b
e
f
e
asib
le:
1
0
0
,
1
0
1
,
1
0
0
2
,
9
9
9
,
1
5
0
,
1
5
1
,
1
0
2
9
,
1
5
7
8
an
d
th
eir
d
ec
o
m
p
o
s
itio
n
s
ar
e
g
iv
e
n
b
y
1
0
0
7
9
7
3
,
1
0
1
8
9
7
5
,
1
0
0
2
9
6
7
5
7
,
9
9
9
9
9
1
3
5
,
1
5
0
7
3
7
1
1
,
1
5
1
1
3
9
5
7
,
1
0
2
9
1
0
2
1
5
3
,
1
5
7
8
1
5
4
3
5
7
.
B
o
b
an
d
Alice
ch
o
o
s
e
o
n
l
y
f
ea
s
ib
le
n
u
m
b
er
s
f
o
r
t
h
is
p
r
o
to
co
l.
a.
Su
p
p
o
s
e
N
is
a
lar
g
e
e
v
e
n
i
n
te
g
er
,
th
en
N
s
atis
f
ie
s
t
h
e
d
ec
o
m
p
o
s
itio
n
11
N
P
r
s
,
w
h
er
e
1
r
an
d
1
s
ar
e
d
is
tin
ct
p
r
i
m
e
s
an
d
P
is
th
e
l
ar
g
est p
r
i
m
e
s
a
tis
f
y
in
g
t
h
is
r
el
atio
n
.
b.
I
f
M
is
a
lar
g
e
o
d
d
in
teg
er
,
t
h
e
n
M
s
atis
f
ie
s
t
h
e
d
ec
o
m
p
o
s
it
io
n
22
M
Q
r
s
,
w
h
er
e
2
r
an
d
2
s
a
r
e
ap
p
r
o
p
r
iate
d
is
tin
ct
p
r
i
m
es
a
n
d
Q
is
th
e
lar
g
es
t p
r
i
m
es sa
tis
f
y
i
n
g
t
h
i
s
r
elatio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Desig
n
a
n
d
I
m
p
leme
n
ta
io
n
o
f
a
S
ec
u
r
e
C
o
mmu
n
ica
tio
n
P
r
o
to
c
o
l
(
M.
K
.
V
is
w
a
n
a
th
)
1817
C
h
e
n
’
s
an
d
Vi
n
o
g
r
ad
o
v
’
s
th
e
o
r
em
s
g
u
ar
an
tee
t
h
e
ex
is
ten
c
e
o
f
t
w
o
p
r
i
m
es
P
an
d
Q
f
r
o
m
t
h
e
s
u
f
f
icie
n
tl
y
lar
g
e
f
ea
s
ib
le
n
u
m
b
er
s
N
an
d
M
.
5
.
1
.
I
nitia
l set
up
As
b
ef
o
r
e,
ass
u
m
e
t
w
o
p
r
o
t
ag
o
n
is
t
s
,
A
lice
a
n
d
B
o
b
.
A
n
a
u
t
h
en
t
icatio
n
p
r
o
to
co
l
is
ex
ec
u
t
ed
b
y
B
o
b
to
m
a
k
e
s
u
r
e
th
a
t
A
lice
w
a
n
ts
to
co
m
m
u
n
ica
te
w
it
h
h
i
m
.
A
lice
a
n
d
B
o
b
ch
o
o
s
e
t
w
o
lar
g
e
n
u
m
b
er
s
N
an
d
M
r
esp
ec
tiv
el
y
a
n
d
af
ter
ascer
tain
i
n
g
t
h
eir
id
en
tit
y
,
e
x
ch
a
n
g
e
it
o
v
er
a
s
ec
u
r
e
ch
an
n
el.
A
lice
th
e
n
ch
o
o
s
es
th
e
lar
g
e
s
t
p
r
i
m
es
1
N
o
f
th
e
f
o
r
m
41
t
,
2
N
o
f
th
e
f
o
r
m
43
t
less
th
a
n
N
.
Si
m
ilar
l
y
,
B
o
b
ch
o
o
s
es
th
e
lar
g
e
s
t
p
r
i
m
es
1
M
o
f
th
e
f
o
r
m
41
t
,
2
M
o
f
th
e
f
o
r
m
43
t
,
less
th
a
n
M
.
W
e
r
ec
all
th
e
Fer
m
at
’
s
t
w
o
s
q
u
ar
es th
eo
r
e
m
,
“
I
f
p
is
a
p
r
ime
n
u
mb
er o
f th
e
f
o
r
m
41
n
,
th
en
22
p
a
b
fo
r
s
o
me
in
teg
ers
,
ab
”.
W
e
ex
p
lo
it
th
is
t
h
eo
r
e
m
o
f
Fe
r
m
at’
s
,
to
o
b
tain
th
e
p
air
o
f
n
u
m
b
er
s
11
,
AB
an
d
22
,
AB
w
h
en
t
h
e
p
r
i
m
e
s
1
N
an
d
1
M
o
f
th
e
f
o
r
m
41
t
ar
e
k
n
o
w
n
.
22
1
1
1
N
A
B
an
d
22
1
2
2
M
A
B
.
No
w
B
o
b
an
d
Alice,
b
o
th
pos
s
ess
1
A
,
1
B
,
2
A
an
d
2
B
o
n
ce
t
h
e
y
ar
e
a
w
ar
e
o
f
N
an
d
M
.
Fo
r
ex
am
p
le,
i
f
1
104681
N
,
th
en
22
1
0
4
6
8
1
1
5
5
2
8
4
an
d
if
1
100957
M
th
en
22
1
0
0
9
5
7
3
0
9
7
4
.
T
h
u
s
b
o
th
t
h
e
u
s
er
s
B
o
b
an
d
A
lice
h
a
v
e
th
e
n
u
m
b
er
s
N
a
n
d
M
an
d
b
o
th
ca
n
co
m
p
u
te
1
2
1
1
,
,
,
N
N
A
B
an
d
1
2
2
2
,
,
,
M
M
A
B
.
T
h
ey
k
ee
p
th
e
p
air
o
f
f
o
u
r
tu
p
les
s
a
f
el
y
w
i
th
t
h
e
m
.
B
o
b
an
d
A
lice
ag
r
ee
f
o
r
an
ir
r
atio
n
al
n
u
m
b
er
I
w
h
ic
h
h
a
s
a
d
ec
im
a
l
ex
p
an
s
io
n
u
p
to
m
o
r
e
th
an
m
illi
o
n
p
lace
s
o
f
d
ec
i
m
als
an
d
I
is
k
ep
t a
s
s
ec
r
et.
5
.
2
.
P
la
inte
x
t
encr
y
ptio
n pro
t
o
co
l
W
h
en
A
lice
w
a
n
t
s
to
s
en
d
a
s
e
cr
et
m
ess
a
g
e
P
to
B
o
b
,
th
en
Ali
ce
h
as t
h
e
k
e
y
tu
p
le
s
1
2
1
1
,
,
,
N
N
A
B
an
d
1
2
2
2
,
,
,
M
M
A
B
w
it
h
h
er
,
co
m
p
u
ted
f
r
o
m
th
e
n
u
m
b
er
s
N
an
d
M
ex
ch
an
g
ed
o
v
er
a
s
ec
u
r
e
ch
a
n
n
el.
a.
I
f
1
B
is
a
f
ea
s
ib
le
n
u
m
b
er
,
th
e
n
s
h
e
ap
p
lies
C
h
e
n
’
s
t
h
eo
r
e
m
t
o
1
B
an
d
co
m
p
u
tes
12
,,
p
p
p
s
u
c
h
t
h
at
1
1
2
B
p
p
p
,
w
h
er
e
p
is
th
e
lar
g
e
s
t
p
r
i
m
e
an
d
12
pp
,
1
p
,
2
p
ar
e
d
is
tin
ct
p
r
i
m
es
s
atis
f
y
in
g
t
h
i
s
r
elatio
n
.
Si
m
ilar
l
y
i
f
2
A
is
f
ea
s
i
b
le,
s
h
e
co
m
p
u
te
s
12
,,
q
q
q
f
r
o
m
th
e
o
d
d
f
ea
s
ib
le
n
u
m
b
er
2
A
u
s
i
n
g
Vin
o
g
r
ad
o
v
’
s
t
h
eo
r
e
m
,
s
u
c
h
th
at
2
1
2
A
q
q
q
,
w
h
er
e
q
is
th
e
lar
g
es
t
p
r
im
e
an
d
12
,
qq
s
u
itab
le
d
is
tin
ct
p
r
i
m
es
12
qq
.
b.
No
w
,
A
lice
co
m
p
u
te
s
th
e
f
ir
s
t
en
cr
y
p
t
io
n
k
e
y
1
1
2
3
K
k
k
k
,
a
s
eq
u
en
ce
o
f
d
ec
i
m
al
p
lace
s
f
r
o
m
t
h
e
p
o
s
itio
n
q
in
th
e
e
x
p
an
s
io
n
o
f
th
e
ir
r
atio
n
al
n
u
m
b
er
I
,
w
h
ic
h
is
u
s
ed
to
b
eg
in
th
e
en
cr
y
p
tio
n
.
T
h
e
n
u
m
b
er
at
th
q
p
lace
,
s
ay
1
k
is
u
s
ed
to
s
u
b
s
tit
u
te
t
h
e
b
eg
i
n
n
in
g
le
tter
o
f
th
e
p
lai
n
tex
t
P
b
y
s
h
i
f
ti
n
g
t
h
e
alp
h
ab
et
b
y
1
k
u
n
it
s
.
A
f
ter
w
ar
d
s
th
e
p
r
o
ce
s
s
is
co
n
tin
u
ed
w
it
h
th
e
n
e
x
t
in
te
g
er
2
k
an
d
th
e
n
e
x
t
alp
h
ab
et
in
t
h
e
p
lai
n
te
x
t a
n
d
s
o
o
n
,
ti
ll t
h
e
e
n
tire
m
ess
a
g
e
i
s
e
n
cr
y
p
ted
.
T
h
is
en
cr
y
p
ted
m
es
s
a
g
e
s
a
y
'
C
is
o
b
tain
ed
b
y
u
s
i
n
g
th
e
k
e
y
q
o
f
B
o
b
.
c.
Nex
t,
A
lice
co
m
p
u
tes
h
er
e
n
cr
y
p
tio
n
k
e
y
m
atr
i
x
A
K
u
s
i
n
g
th
e
n
u
m
b
er
p
,
w
h
er
e
A
K
is
a
12
pp
r
ec
tan
g
u
lar
m
atr
i
x
a
n
d
t
h
e
en
tr
ies
o
f
A
K
ar
e
th
e
12
pp
co
n
s
ec
u
ti
v
e
d
ec
im
al
p
lace
s
p
ic
k
ed
f
r
o
m
th
e
p
o
s
itio
n
p
in
t
h
e
ex
p
a
n
s
io
n
o
f
I
.
d.
Sh
e
ar
r
an
g
es
t
h
e
cr
y
p
to
tex
t
'
C
in
b
lo
ck
s
o
f
le
n
g
th
2
p
w
i
th
it
s
n
u
m
er
ical
eq
u
i
v
ale
n
t
s
an
d
o
b
tain
s
t
h
e
f
i
n
al
cip
h
er
te
x
t
C
b
y
'
A
C
K
C
.
5
.
3
.
M
ess
a
g
e
inte
g
ri
t
y
encr
y
ptio
n pro
t
o
co
l
A
lice
co
m
p
u
tes
t
h
e
p
r
o
d
u
ct
22
n
N
M
.
T
h
e
in
teg
r
it
y
o
f
t
h
e
m
e
s
s
a
g
e
is
o
b
tain
ed
b
y
co
n
s
id
er
i
n
g
t
h
e
letter
s
1
2
3
4
,
,
,
m
m
m
m
m
(
s
a
y
)
o
cc
u
r
r
in
g
in
t
h
e
1
2
1
2
,
,
,
p
p
q
q
th
p
lace
s
o
f
th
e
f
ir
s
t
s
e
n
t
en
ce
in
P
.
T
h
e
co
m
p
ila
tio
n
o
f
w
o
r
d
i
n
t
h
e
e
x
ac
t
o
r
d
er
is
tak
e
n
a
s
m
es
s
ag
e
d
ig
e
s
t.
S
h
e
e
n
cr
y
p
ts
t
h
e
w
o
r
d
m
as
2
(
m
o
d
)
w
m
n
.
No
w
th
e
cip
h
er
te
x
t
C
an
d
th
e
en
cr
y
p
ted
m
es
s
ag
e
d
ig
e
s
t
w
ar
e
s
en
t
to
B
o
b
th
r
o
u
g
h
an
o
p
en
ch
an
n
el,
f
o
r
d
ec
r
y
p
tio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
8
,
No
.
3
,
J
u
n
e
2
0
1
8
:
1
8
1
4
–
1821
1818
5
.
4
.
Ciph
er
t
ex
t
decr
y
ptio
n pro
t
o
co
l
On
ce
B
o
b
r
ec
eiv
es th
e
cip
h
er
t
ex
t p
air
,
Cw
,
h
e
d
o
es th
e
f
o
llo
w
i
n
g
f
o
r
d
ec
r
y
p
tio
n
.
He
k
n
o
w
s
,
p
is
th
e
p
o
s
itio
n
o
f
th
e
d
ec
i
m
al
p
lace
to
s
tar
t,
in
t
h
e
ex
p
an
s
io
n
o
f
t
h
e
ir
r
atio
n
al
n
u
m
b
er
I
.
Fro
m
t
h
is
p
o
s
itio
n
o
f
p
,
h
e
c
o
llects
t
h
e
12
pp
co
n
s
ec
u
ti
v
e
d
ig
its
f
r
o
m
th
e
d
ec
i
m
al
ex
p
a
n
s
io
n
o
f
I
an
d
o
b
tain
s
th
e
r
ec
tan
g
u
lar
m
atr
i
x
A
K
o
f
o
r
d
e
r
12
pp
.
He
th
en
co
m
p
u
tes
th
e
p
s
eu
d
o
in
v
er
s
e
#
A
K
o
f
A
K
an
d
ap
p
lies
th
is
d
ec
r
y
p
tio
n
k
e
y
to
th
e
cip
h
er
tex
t
C
an
d
o
b
tain
s
'
C
,
#
'
A
C
K
C
,
w
h
er
e
C
is
ar
r
an
g
ed
in
b
lo
ck
s
o
f
1
p
-
t
u
p
les
w
i
th
i
ts
n
u
m
er
ical
eq
u
iv
ale
n
t.
No
w
h
e
k
n
o
w
s
h
i
s
k
e
y
v
al
u
e
q
an
d
o
b
tain
s
t
h
e
d
ec
i
m
al
p
lace
s
f
r
o
m
t
h
e
q
th
p
o
s
itio
n
o
f
th
e
d
ec
i
m
al
e
x
p
an
s
io
n
o
f
I
w
h
er
e
th
e
f
ir
s
t
en
cr
y
p
t
io
n
p
r
o
ce
s
s
h
a
s
b
eg
u
n
.
T
h
en
h
e
ca
n
ea
s
il
y
o
b
tain
s
t
h
e
p
lain
te
x
t
P
b
y
d
ec
r
y
p
tin
g
'
C
u
s
i
n
g
t
h
e
in
v
er
s
e
s
u
b
s
tit
u
tio
n
cip
h
er
o
f
B
o
b
.
T
h
is
p
r
o
ce
s
s
estab
lis
h
t
h
e
a
u
t
h
en
tic
it
y
o
f
t
h
e
m
es
s
ag
e
r
ec
eiv
ed
f
r
o
m
A
lice
as
t
h
e
m
e
s
s
a
g
e
i
s
lo
ck
ed
w
it
h
t
h
e
k
e
y
s
o
f
B
o
b
an
d
A
lice,
w
ith
o
u
t
f
o
r
m
all
y
ex
c
h
a
n
g
i
n
g
th
e
m
e
s
s
a
g
e
P
b
et
w
ee
n
B
o
b
an
d
A
l
ice.
5
.
5
.
Dec
ry
ptio
n P
ro
t
o
co
l
f
o
r
I
nte
g
rit
y
:
B
o
b
w
a
n
ts
to
co
m
p
u
te
m
o
d
wn
an
d
h
e
d
o
es it b
y
t
h
e
f
o
llo
w
i
n
g
m
et
h
o
d
.
a.
He
co
m
p
u
tes
2
2
1
4
2
m
o
d
N
N
m
w
N
an
d
2
2
1
4
2
m
o
d
M
M
m
w
M
.
b.
B
y
e
x
te
n
d
ed
E
u
clid
ea
n
al
g
o
r
ith
m
,
h
e
f
in
d
s
2
N
y
an
d
2
M
y
s
u
c
h
th
a
t
22
22
1
NM
y
N
y
M
.
c.
T
h
en
h
e
co
m
p
u
tes t
h
e
f
o
u
r
p
o
s
s
ib
ilit
ie
s
f
o
r
m
,
s
u
ch
t
h
at
2
2
2
2
1
2
2
m
o
d
N
M
M
N
r
y
N
m
y
M
m
n
21
r
n
r
2
2
2
2
3
2
2
m
o
d
N
M
M
N
r
y
N
m
y
M
m
n
43
r
n
r
.
I
f
B
o
b
w
an
t
s
to
r
ep
ly
to
th
e
m
ess
a
g
e
o
f
A
lice,
h
e
o
b
tain
s
th
e
n
e
w
k
e
y
s
2
K
an
d
B
K
u
s
in
g
th
e
v
al
u
es
o
f
2
B
an
d
1
A
an
d
co
n
tin
u
e
s
th
e
al
g
o
r
ith
m
e
x
ec
u
ted
b
y
A
l
ice.
He
co
m
p
u
tes
B
K
w
it
h
h
is
k
e
y
v
alu
e
q
an
d
co
m
p
u
tes
2
K
w
ith
th
e
h
elp
o
f
p
.
I
f
A
lice
w
a
n
t
s
to
co
n
ti
n
u
e
t
h
e
en
cr
y
p
ti
o
n
p
r
o
ce
s
s
,
A
l
ice
s
ele
cts
3
N
,
4
N
,
3
41
Nt
,
4
43
Nt
,
w
h
er
e
3
N
,
4
N
ar
e
th
e
f
ir
s
t
p
r
i
m
e
n
u
m
b
er
s
o
cc
u
r
r
in
g
j
u
s
t
af
ter
N
an
d
B
o
b
s
elec
ts
3
M
an
d
4
M
,
w
h
er
e
3
M
,
4
M
ar
e
t
h
e
f
ir
s
t
p
r
i
m
es
o
f
th
e
f
o
r
m
41
t
an
d
43
t
o
cc
u
r
r
in
g
j
u
s
t
af
ter
M
.
T
h
e
k
e
y
s
i
K
,
A
K
,
B
K
ar
e
co
m
p
u
ted
as b
ef
o
r
e
an
d
th
u
s
th
e
s
e
k
e
y
s
ar
e
d
y
n
a
m
ic.
6.
WO
RK
I
N
G
O
F
T
H
E
SY
ST
E
M
Ass
u
m
e
th
at
t
h
e
s
y
s
te
m
u
s
e
s
a
2
9
-
letter
alp
h
a
b
et
_
.
?
0
0
0
1
0
2
2
3
2
4
2
5
2
6
2
7
2
8
a
b
c
x
y
z
C
o
n
s
id
er
th
e
ca
s
e,
th
e
ir
r
atio
n
al
n
u
m
b
er
I
an
d
let
28816
N
an
d
47635
M
.
T
h
en
1
2
1
1
,
,
,
2
8
8
1
3
,
2
8
8
0
7
,
9
3
,
1
4
2
N
N
A
B
1
2
2
2
,
,
,
4
7
6
2
9
,
4
7
6
2
3
,
1
9
5
,
9
8
M
M
A
B
s
u
c
h
th
a
t
22
1
1
1
N
A
B
an
d
22
1
2
2
M
A
B
.
6
.
1
.
E
ncry
ptio
n
Ass
u
m
e
A
lice
co
n
tact
s
B
o
b
f
o
r
th
e
f
ir
s
t
ti
m
e.
Sh
e
p
ick
s
t
h
e
ev
e
n
n
u
m
b
er
1
B
f
r
o
m
1
N
an
d
t
h
e
o
d
d
n
u
m
b
er
2
A
f
r
o
m
1
M
.
I
f
12
,
BA
ar
e
f
ea
s
ib
le
n
u
m
b
er
s
,
th
e
n
s
h
e
co
m
p
u
tes
th
e
d
ec
o
m
p
o
s
itio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Desig
n
a
n
d
I
m
p
leme
n
ta
io
n
o
f
a
S
ec
u
r
e
C
o
mmu
n
ica
tio
n
P
r
o
to
c
o
l
(
M.
K
.
V
is
w
a
n
a
th
)
1819
1
1
2
1
4
2
1
2
7
5
3
B
p
p
p
f
o
r
th
e
ev
en
n
u
m
b
er
1
4
2
an
d
f
in
d
s
th
e
d
ec
o
m
p
o
s
itio
n
,
2
1
2
1
9
5
1
8
1
1
1
3
A
q
q
q
w
h
ich
e
x
is
t
f
o
r
f
ea
s
ib
le
n
u
m
b
er
s
b
y
d
ef
i
n
itio
n
.
Her
e
B
o
b
’
s
k
e
y
i
s
1
8
1
an
d
th
e
k
e
y
o
f
Alice
is
1
2
7
.
First
A
lice
f
i
n
d
s
th
e
d
ec
i
m
a
l
p
lace
s
f
r
o
m
t
h
e
p
o
s
itio
n
181
q
in
t
h
e
ex
p
an
s
io
n
o
f
.
No
w
,
1
K
=6
4
4
0
2
2
9
4
8
9
5
4
9
3
0
3
8
1
9
6
4
4
2
8
8
1
0
9
7
5
6
6
5
9
.
.
.
.
A
lice
e
n
c
r
y
p
t
s
t
h
e
co
n
f
id
e
n
tial
m
e
s
s
a
g
e,
n
a
m
el
y
th
e
P
lain
tex
t
P=
“
mee
t
a
t
th
e
litt
le
s
ch
o
o
lh
o
u
s
e
”
u
s
i
n
g
1
K
as,
ea
ch
ch
ar
ac
ter
in
t
h
e
p
lain
te
x
t
is
s
h
i
f
ted
w
it
h
th
e
co
r
r
esp
o
n
d
in
g
n
u
m
b
er
s
i
n
1
K
u
s
i
n
g
(
m
o
d
2
9
)
.
T
h
en
s
h
e
co
m
p
u
tes
t
h
e
in
itial
cr
y
p
to
tex
t
'
C
w
it
h
it
s
n
u
m
er
ical
eq
u
iv
ale
n
t a
n
d
ar
r
an
g
e
s
th
is
i
n
co
lu
m
n
s
o
f
le
n
g
th
th
r
ee
,
as a
m
atr
i
x
.
T
h
is
m
atr
i
x
'
C
is
g
i
v
en
b
y
,
1
8
2
5
2
8
1
6
2
0
2
2
0
6
1
1
1
9
0
0
'
0
8
2
8
0
1
0
9
1
1
1
9
1
9
1
6
0
8
2
5
0
8
0
2
2
7
0
1
1
9
1
5
1
5
2
2
1
4
0
9
C
A
lice
f
i
n
d
s
th
e
s
eq
u
e
n
ce
o
f
d
ec
i
m
al
p
lace
s
f
r
o
m
t
h
e
p
o
s
itio
n
127
p
an
d
ch
o
o
s
es
12
15
pp
co
n
s
ec
u
ti
v
e
d
ec
i
m
als
f
r
o
m
t
h
i
s
p
o
s
itio
n
i
n
th
e
ex
p
a
n
s
io
n
o
f
.
T
h
is
d
ec
im
al
s
e
q
u
en
ce
“
6
0
9
5
5
0
5
8
2
2
3
1
7
2
5
”
is
ar
r
an
g
ed
in
th
e
f
o
r
m
o
f
a
13
53
pp
r
ec
tan
g
u
lar
m
atr
ix
A
K
.
T
h
is
is
g
i
v
e
n
b
y
,
6
0
3
0
5
1
9
8
7
522
5
2
5
A
K
T
h
en
'
C
is
co
n
v
er
ted
i
n
to
th
e
f
in
al
cr
y
p
to
te
x
t
'
m
o
d
2
9
A
C
K
C
6
0
3
1
8
2
5
2
8
1
6
2
0
2
2
0
6
1
1
1
9
0
0
0
5
1
0
8
2
8
0
1
0
9
1
1
1
9
1
9
1
6
0
8
2
5
9
8
7
0
8
0
2
2
7
0
1
1
9
1
5
1
5
2
2
1
4
0
9
522
5
2
5
1
6
1
1
1
7
1
2
0
3
0
3
2
5
1
6
1
1
2
7
1
9
2
6
0
3
1
7
1
6
2
3
1
0
1
5
2
5
1
8
(
m
o
d
2
9
)
2
1
2
8
1
4
2
0
2
4
2
0
2
7
0
4
1
4
0
2
0
6
1
1
2
2
1
3
1
5
0
4
0
3
1
5
2
3
1
0
0
1
1
7
1
6
1
6
1
4
2
0
2
1
2
3
0
7
0
8
T
h
u
s
t
h
e
cip
h
er
te
x
t
C
is
“q
tvg
b
l_
?
lr
r
d
o
w
q
mru
n
q
d
q
yp
o
d
xu
e
u
z
k.
d
vq
p
ep
xlzo
x
h
.
s
ck
i”
.
No
te
th
at
30
P
an
d
50
C
.
Fo
r
m
es
s
ag
e
in
teg
r
it
y
,
A
lice
c
h
o
o
s
es
t
h
e
1
th
p
2
th
p
1
th
q
an
d
2
th
q
ch
ar
ac
ter
s
i
n
th
e
p
lai
n
tex
t
n
a
m
el
y
,
“
_
ee
e”
.
T
h
is
m
es
s
ag
e
d
i
g
est
w
it
h
its
n
u
m
er
ical
eq
u
iv
ale
n
t
:
2
6
0
4
0
4
0
4
m
is
en
cip
h
er
ed
as
w
b
y
u
s
i
n
g
22
1371875761
n
N
M
.
T
h
at
is
2
m
o
d
w
m
n
2
26040404
m
od
1371875761
n
914330048
m
od
1371875761
n
No
w
t
h
e
cip
h
er
te
x
t
C
an
d
th
e
en
cr
y
p
ted
m
ess
a
g
e
d
ig
e
s
t
w
ar
e
s
en
t to
B
o
b
th
r
o
u
g
h
a
n
o
p
en
ch
a
n
n
el.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
8
,
No
.
3
,
J
u
n
e
2
0
1
8
:
1
8
1
4
–
1821
1820
6
.
2
.
Dec
ry
ptio
n
B
o
b
ca
n
co
m
p
u
te
t
h
e
r
ec
tan
g
u
lar
m
atr
i
x
A
K
b
y
ap
p
l
y
in
g
t
h
e
k
e
y
p
o
f
A
lice
to
th
e
d
ec
i
m
al
ex
p
an
s
io
n
o
f
.
T
h
en
h
e
o
b
tain
s
th
e
p
s
eu
d
o
in
v
e
r
s
e
o
f
A
K
,
1
#
m
o
d
2
9
T
T
A
A
A
A
K
K
K
K
2
5
2
5
2
6
0
5
1
7
1
3
0
6
1
5
0
8
1
9
m
o
d
2
9
1
4
1
5
1
3
1
1
0
1
He
d
iv
id
es
th
e
cip
h
er
te
x
t
C
in
to
clo
ck
s
o
f
le
n
g
t
h
f
iv
e
an
d
d
ec
r
y
p
ts
it
b
y
ap
p
l
y
i
n
g
#
A
K
to
C
,
#
'
m
o
d
2
9
A
C
K
C
.
He
co
m
p
u
tes
t
h
e
d
ec
i
m
al
s
e
q
u
en
ce
1
K
,
s
tar
tin
g
f
r
o
m
t
h
e
p
o
s
i
tio
n
q
in
th
e
d
ec
i
m
al
ex
p
an
s
io
n
o
f
.
T
h
ese
d
ec
i
m
al
p
lace
s
ar
e
u
s
ed
to
d
ec
r
y
p
t
'
C
b
y
th
e
i
n
v
er
s
e
s
u
b
s
t
itu
tio
n
ci
p
h
er
an
d
B
o
b
o
b
tain
s
th
e
o
r
ig
in
al
s
ec
r
et
m
e
s
s
ag
e
""
P
m
e
e
t
a
t
t
h
e
s
c
h
o
o
l
h
o
u
s
e
.
Fo
r
d
ec
r
y
p
tio
n
o
f
t
h
e
m
ess
a
g
e
d
ig
est,
B
o
b
f
in
d
s
2
2
1
4
2
m
o
d
1
1
2
4
m
o
d
2
8
8
0
7
N
N
m
w
N
2
2
1
4
2
m
o
d
3
8
2
4
6
m
o
d
4
7
6
2
3
M
M
m
w
M
22
2
0
8
3
,
1
2
6
0
NM
yy
su
c
h
th
at
22
22
1
NM
y
N
y
M
an
d
it r
etu
r
n
s
th
e
f
o
u
r
p
o
s
s
ib
le
r
o
o
ts
,
2
2
2
2
1
2
2
m
o
d
9
5
0
5
4
5
7
0
3
N
M
M
N
r
y
N
m
y
M
m
n
21
421330058
r
n
r
2
2
2
2
3
2
2
m
o
d
2
6
0
4
0
4
0
4
N
M
M
N
r
y
N
m
y
M
m
n
43
1345835357
r
n
r
Am
o
n
g
t
h
ese
f
o
u
r
,
3
r
g
iv
es
h
i
m
th
e
o
r
ig
in
al
m
e
s
s
ag
e
d
ig
e
s
t.
B
o
b
ca
n
co
n
f
ir
m
it
b
y
co
n
s
id
e
r
in
g
t
h
e
letter
s
in
th
e
1
2
1
2
,
,
,
th
p
p
q
q
p
lace
o
f
th
e
p
lai
n
te
x
t
P
.
B
o
b
ca
n
r
ep
ly
to
A
lice
b
y
u
s
in
g
t
h
e
,
o
d
d
e
v
e
n
p
air
k
e
y
12
,
AB
as
b
ef
o
r
e.
T
h
is
p
r
o
ce
s
s
is
th
en
co
n
tin
u
ed
b
y
A
lice
u
s
i
n
g
t
h
e
n
e
w
p
r
i
m
e
p
air
s
34
,
NN
an
d
34
,
MM
an
d
it c
an
b
e
r
ep
ea
ted
an
y
n
u
m
b
er
o
f
t
i
m
e
s
as lo
n
g
as t
h
e
i
n
itial n
u
m
b
er
s
,
NM
ar
e
k
ep
t secr
et.
7.
CO
NCLU
SI
O
N
Th
e
cr
y
p
to
s
y
s
te
m
p
r
o
p
o
s
ed
h
er
e
is
q
u
ite
s
ec
u
r
e
as
i
t
is
d
i
f
f
icu
lt
to
o
b
tain
t
h
e
k
e
y
s
i
K
,
A
K
an
d
B
K
w
it
h
o
u
t
k
n
o
w
led
g
e
o
f
N
an
d
M
.
As t
h
e
p
r
i
m
e
p
air
s
12
,
NN
an
d
12
,
MM
ch
a
n
g
e
s
f
o
r
ea
ch
en
cr
y
p
tio
n
,
th
e
k
e
y
s
i
K
,
A
K
an
d
B
K
ar
e
d
y
n
a
m
ic
a
n
d
h
en
ce
t
h
e
s
y
s
te
m
is
s
ec
u
r
e
ag
ain
s
t
c
h
o
s
e
n
p
lain
tex
t
a
ttac
k
.
I
t
als
o
en
s
u
r
es
t
h
e
a
u
t
h
en
ticit
y
o
f
t
h
e
m
e
s
s
a
g
es
tr
a
n
s
f
er
r
ed
b
et
w
e
en
t
h
e
s
e
n
d
er
a
n
d
th
e
r
ec
eiv
e
r
as
t
i
s
lo
c
k
ed
w
it
h
th
e
k
e
y
s
o
f
B
o
b
an
d
A
lice.
T
h
e
R
ab
in
’
s
cr
y
p
to
s
y
s
te
m
w
it
h
o
u
t
th
e
m
o
d
u
lu
s
b
ei
n
g
m
ad
e
p
u
b
lic,
is
u
s
ed
i
n
en
cr
y
p
ti
n
g
t
h
e
m
es
s
ag
e
d
i
g
es
t
an
d
it e
n
s
u
r
es t
h
e
i
n
teg
r
it
y
o
f
t
h
e
m
e
s
s
a
g
e
tr
an
s
f
er
r
ed
.
T
h
e
u
s
e
o
f
t
h
e
i
n
te
g
er
s
a
p
p
ea
r
in
g
in
th
e
d
ec
i
m
al
e
x
p
an
s
io
n
o
f
(
n
o
t
m
ad
e
p
u
b
lic)
in
en
cr
y
p
tio
n
/d
ec
r
y
p
tio
n
,
e
n
ab
le
s
it to
b
e
s
a
f
e
ag
a
in
s
t t
h
e
u
s
u
al
m
et
h
o
d
s
o
f
cr
y
p
to
g
r
ap
h
ic
atta
ck
s
.
As lo
n
g
as
N
an
d
M
ar
e
n
o
t
k
n
o
w
n
it
is
i
m
p
o
s
s
ib
le
f
o
r
an
in
tr
u
d
er
to
b
r
ea
k
th
is
s
y
s
te
m
.
I
f
an
in
tr
u
d
er
p
r
eten
d
s
as
A
lice
an
d
s
e
n
d
s
B
o
b
a
m
e
s
s
a
g
e,
B
o
b
ca
n
s
e
n
d
a
s
ta
n
d
ar
d
tex
t
f
o
r
en
cr
y
p
tio
n
.
T
h
e
cip
h
er
te
x
t
o
f
t
h
i
s
s
ta
n
d
ar
d
m
es
s
ag
e
f
r
o
m
t
h
e
in
tr
u
d
er
,
en
ab
les B
o
b
to
ass
er
t th
e
au
th
e
n
t
icit
y
o
f
t
h
e
i
n
tr
u
d
er
.
T
h
e
p
r
o
p
o
s
ed
d
ata
en
cr
y
p
tio
n
s
ch
e
m
e
g
iv
e
n
ab
o
v
e
h
a
s
ad
v
an
ta
g
es
o
f
lar
g
e
k
e
y
s
p
ac
e,
h
ig
h
lev
e
l
s
ec
u
r
it
y
a
n
d
is
m
ath
e
m
atica
ll
y
a
n
d
co
m
p
u
tat
io
n
all
y
s
i
m
p
le
lik
e
[
5
]
,
[
1
8
]
.
T
h
e
s
y
s
te
m
is
s
ec
u
r
e
ag
ai
n
s
t
b
r
u
te
f
o
r
ce
attac
k
s
i
n
ce
t
h
e
k
e
y
s
ar
e
d
y
n
a
m
ic
an
d
t
h
e
len
g
t
h
o
f
t
h
e
p
lain
tex
t
a
n
d
th
e
cip
h
er
te
x
t
ar
e
n
o
t
eq
u
al.
T
h
u
s
th
e
s
y
s
te
m
i
s
s
ec
u
r
e
a
g
ai
n
s
t a
l
l p
o
s
s
ib
le
k
n
o
w
n
attac
k
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Desig
n
a
n
d
I
m
p
leme
n
ta
io
n
o
f
a
S
ec
u
r
e
C
o
mmu
n
ica
tio
n
P
r
o
to
c
o
l
(
M.
K
.
V
is
w
a
n
a
th
)
1821
RE
F
E
R
E
NC
E
S
[1
]
A
.
J.
M
e
n
e
z
e
s
,
P
.
C.
V
a
n
Oo
rc
h
o
t
a
n
d
S
.
A
.
V
a
n
sto
n
e
,
“
H
a
n
d
b
o
o
k
o
f
Ap
p
li
e
d
Cry
p
to
g
ra
p
h
y
”
,
CRC
P
re
ss
,
2
0
0
0
.
[2
]
Jo
h
n
M
a
rk
B.
Esp
a
lm
a
d
o
a
n
d
E
d
w
in
R.
A
rb
o
led
a
,
“
Da
re
A
lg
o
rit
h
m
:
A
Ne
w
S
e
c
u
rit
y
P
ro
t
o
c
o
l
b
y
In
teg
r
a
ti
o
n
o
f
Diff
e
r
e
n
t
Cry
p
to
g
ra
p
h
ic
T
e
c
h
n
iq
u
e
s,”
In
ter
n
a
t
io
n
a
l
J
o
u
rn
l
o
f
E
le
c
trica
l
a
n
d
Co
m
p
u
ter
E
n
g
in
e
e
rin
g
,
v
o
l.
7
,
n
o
.
2
,
p
p
.
1
0
3
2
-
1
0
4
1
,
2
0
1
7
.
[3
]
Ne
a
l
Ko
b
li
tz,
“
A
c
o
u
rs
e
in
N
u
mb
e
r T
h
e
o
ry
a
n
d
Cry
p
t
o
g
r
a
p
h
y
”
,
S
p
rin
g
e
r,
2
nd
e
d
it
io
n
,
1
9
9
4
.
[4
]
Rh
e
e
a
n
d
M
a
n
Yo
u
n
g
,
“
Cry
p
to
g
ra
p
h
y
a
n
d
S
e
c
u
re
Co
mm
u
n
ica
ti
o
n
s
”
,
M
c
G
ra
w
-
Hill
c
o
.
,
1
9
9
4
.
[5
]
M
.
K.
V
is
w
a
n
a
th
a
n
d
M
.
Ra
n
j
it
h
k
u
m
a
r,
“
A
se
c
u
re
c
r
y
p
to
s
y
ste
m
u
sin
g
th
e
d
e
c
ima
l
e
x
p
a
n
sio
n
o
f
a
n
Irra
ti
o
n
a
l
n
u
m
b
e
r,
”
Ap
p
l
ied
M
a
t
h
e
ma
ti
c
a
l
S
c
ien
c
e
s
, v
o
l.
9
,
p
p
.
5
2
9
3
-
5
3
0
3
,
2
0
1
5
.
[6
]
M
.
K.
V
isw
a
n
a
th
a
n
d
M
.
Ra
n
ji
t
h
k
u
m
a
r,
“
G
o
ld
b
a
c
h
Co
n
jec
tu
re
a
n
d
Cry
p
to
g
ra
p
h
y
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
P
u
re
a
n
d
Ap
p
li
e
d
M
a
th
e
ma
ti
c
s,
v
o
l.
1
1
6
,
n
o
.
2
,
p
p
.
4
0
3
-
4
1
3
,
2
0
1
7
.
[7
]
J.R.
Ch
e
n
,
“
On
th
e
re
p
re
se
n
tatio
n
o
f
a
larg
e
e
v
e
n
in
teg
e
r
a
s
th
e
su
m
o
f
a
p
rim
e
a
n
d
th
e
p
r
o
d
u
c
t
o
f
a
tm
o
st
tw
o
p
rim
e
s,”
Ke
x
u
e
T
o
n
g
b
a
o
(
Ch
i
n
e
se
)
, v
o
l.
1
7
,
p
p
.
3
6
5
-
3
8
6
,
1
9
6
6
.
[8
]
J.R.
Ch
e
n
,
“
On
th
e
re
p
re
se
n
tatio
n
o
f
a
larg
e
e
v
e
n
in
teg
e
r
a
s
th
e
su
m
o
f
a
p
rim
e
a
n
d
th
e
p
r
o
d
u
c
t
o
f
a
tm
o
st
tw
o
p
rim
e
s,”
S
c
i.
S
in
ica
,
v
o
l.
1
6
,
1
9
7
3
,
p
p
.
1
5
7
-
1
7
6
.
Ib
i
d
,
2
1
,
1
9
7
8
,
p
p
.
4
7
7
-
4
9
4
(C
h
in
e
se
).
[9
]
I.
M
.
V
i
n
o
g
ra
d
o
v
,
“
T
h
e
re
p
re
se
n
t
a
ti
o
n
o
f
a
n
o
d
d
n
u
m
b
e
r
a
s
a
su
m
o
f
t
h
re
e
p
rim
e
s,”
Do
k
l.
Aka
d
.
Na
u
k
,
S
S
S
R
1
5
,
1
9
3
7
,
p
p
.
1
6
9
-
1
7
2
,
R
u
ss
ia.
[1
0
]
I.
N.
He
rste
in
,
“
T
o
p
ic i
n
Al
g
e
b
ra
”
, 2
nd
Ed
it
i
o
n
,
W
il
y
Eas
tern
L
i
m
it
e
d
.
[1
1
]
S
.
L
e
ste
r
Hill
,
“
Cry
p
to
g
ra
p
h
y
in
a
n
a
lg
e
b
ra
ic alp
h
a
b
e
t,
”
Ame
r.
M
a
t
h
.
,
p
p
.
3
0
6
-
3
1
2
,
1
9
2
9
.
[1
2
]
R.
P
e
n
ro
se
,
“
A
g
e
n
e
ra
li
z
e
d
In
v
e
rs
e
f
o
r
m
a
tri
c
e
s,” Co
m
m
u
n
ica
ted
b
y
J.
A
.
T
o
d
d
Re
c
e
iv
e
d
2
6
Ju
ly
1
9
5
4
.
[1
3
]
R.
L
.
Riv
e
st,
A
.
S
h
a
m
ir
a
n
d
L
.
A
d
lem
a
n
,
“
A
m
e
th
o
d
f
o
r
o
b
tain
in
g
d
ig
it
a
l
sig
n
a
tu
re
s
a
n
d
p
u
b
li
c
k
e
y
c
r
y
p
to
sy
ste
m
s
”
Co
mm
u
n
ica
ti
o
n
s o
f
th
e
ACM
,
v
o
l
.
2
1
,
n
o.
2
p
p
.
1
2
0
-
1
2
6
,
1
9
7
8
.
[1
4
]
S
u
sh
m
a
P
ra
d
h
a
n
a
n
d
Biren
d
ra
Ku
m
a
r
S
h
a
rm
a
,
“
A
n
E
ff
icie
n
t
RS
A
Cr
y
p
to
s
y
ste
m
w
it
h
BM
-
P
RIM
E
M
e
th
o
d
,
”
In
ter
n
a
t
io
n
a
l
J
o
u
rn
a
l
o
f
I
n
fo
rm
a
t
io
n
&
S
e
c
u
rity,
v
o
l.
2
,
n
o
.
1
,
p
p
.
1
0
3
-
1
0
8
,
2
0
1
3
.
[1
5
]
P
re
d
ra
g
S
tan
im
iro
v
ic
a
n
d
M
io
m
i
r
S
tan
k
o
v
ic,
“
De
ter
m
in
a
n
ts
o
f
r
e
c
tan
g
u
lar
m
a
tri
c
e
s
a
n
d
M
o
o
re
-
Pe
n
ro
se
in
v
e
rse
,
”
No
v
i
sa
d
J
.
M
a
t
h
.
, v
ol
.
2
7
,
n
o.
1,
p
p
.
5
3
-
6
9
,
1
9
9
7
.
[1
6
]
T
.
L
.
Bo
u
ll
io
n
a
n
d
P
.
L
.
Od
e
ll
,
“
Ge
n
e
ra
li
z
e
d
In
v
e
rse
M
a
tri
c
e
s,” W
il
e
y
,
Ne
wy
o
rk
,
p
p
.
41
-
6
2
,
1
9
7
1
.
[1
7
]
J.
P
i
n
tz
a
n
d
I.
Z
.
P
u
z
sa
,
“
On
L
in
n
ik
’s
a
p
p
ro
x
ima
ti
o
n
to
G
o
ld
b
a
c
h
’
s
p
ro
b
lem
,
”
I.
Acta
Arit
h
ma
ti
ca
,
v
o
l.
1
0
9
,
n
o
.
2
,
p
p
.
1
6
9
-
1
9
4
,
2
0
0
3
.
[18]
M
.
K.
Visw
a
n
a
th
a
n
d
M
.
Ra
n
ji
t
h
k
u
m
a
r,
“
A
P
u
b
l
ic
Ke
y
Cr
y
p
to
s
y
ste
m
Us
in
g
Hill
’s
Cip
h
e
r,
”
J
o
u
rn
a
l
o
f
Disc
re
te
M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
s
&
Cry
p
to
g
ra
p
h
y
,
v
o
l.
1
8
,
n
o
.
1
&
2
,
p
p
.
1
2
9
-
1
3
8
,
2
0
1
5
.
B
I
O
G
RAP
H
I
E
S
O
F
AUTH
O
RS
M.
K
.
V
is
w
a
n
a
t
h
w
a
s
b
o
rn
o
n
8
th
A
p
ril
1
9
5
0
a
t
T
e
ll
ich
e
rry
,
Ke
r
a
la,
In
d
ia.
He
to
o
k
h
is
m
a
st
e
rs
d
e
g
re
e
M
.
S
c
.
in
M
a
th
e
m
a
ti
c
s
f
ro
m
th
e
Un
iv
e
rsit
y
o
f
M
a
d
ra
s
in
1
9
7
1
.
He
jo
in
e
d
a
s
a
T
u
to
r
i
n
M
a
th
e
m
a
ti
c
s
a
t
th
e
M
a
d
ra
s
Ch
ri
stian
Co
ll
e
g
e
imm
e
d
iate
l
y
a
f
ter
c
o
m
p
letin
g
th
e
M
.
S
c
.
d
e
g
re
e
.
He
o
b
tain
e
d
M
.
P
h
il
.
De
g
re
e
(M
a
th
e
m
a
ti
c
s)
in
1
9
7
9
a
n
d
t
h
e
P
h
.
D
.
d
e
g
re
e
(M
a
th
e
m
a
ti
c
s)
f
ro
m
th
e
Un
iv
e
rsit
y
o
f
M
a
d
ra
s
in
th
e
y
e
a
r
1
9
8
7
f
o
r
h
is
th
e
sis
t
it
led
Ha
rm
o
n
ic
A
n
a
l
y
sis
o
n
2,
SP
His
re
se
a
rc
h
in
tere
st
in
c
lu
d
e
Qu
a
n
tu
m
g
ro
u
p
s,
F
u
n
c
ti
o
n
a
l
A
n
a
l
y
sis,
Nu
m
b
e
r
T
h
e
o
r
y
,
Cr
y
p
to
g
ra
p
h
y
a
n
d
A
n
c
ien
t
In
d
ia
n
M
a
th
e
m
a
ti
c
s.
He
re
ti
re
d
a
s
Re
a
d
e
r
in
M
a
t
h
e
m
a
ti
c
s
f
ro
m
th
e
M
a
d
ra
s
Ch
risti
a
n
Co
ll
e
g
e
in
M
a
y
2
0
0
8
a
n
d
th
e
re
a
f
ter
se
r
v
e
d
a
s
P
ro
f
e
ss
o
r
o
f
M
a
th
e
m
a
ti
c
s
a
t
th
e
Ra
jala
k
sh
m
i
En
g
in
e
e
rin
g
Co
ll
e
g
e
,
Ch
e
n
n
a
i
t
il
l
M
a
y
2
0
1
6
.
He
is
a
m
e
m
b
e
r
o
f
th
e
Cr
y
p
to
g
ra
p
h
ic
Re
se
a
rc
h
S
o
c
iet
y
o
f
In
d
ia an
d
th
e
Ke
ra
la M
a
th
e
m
a
ti
c
s a
ss
o
c
iatio
n
.
He
h
a
s p
u
b
li
sh
e
d
2
1
re
se
a
rc
h
a
rti
c
les
in
v
a
rio
u
s
n
a
ti
o
n
a
l
a
n
d
i
n
tern
a
ti
o
n
a
l
jo
u
rn
a
ls.
He
is
a
re
v
iew
e
r
f
o
r
th
e
z
b
M
AT
H
f
o
r
th
e
p
a
st
2
1
y
e
a
r
s.
He
is
m
a
rried
a
n
d
i
s b
les
se
d
w
it
h
tw
o
so
n
s.
M
.
Ra
n
jith
K
u
m
a
r
wa
s
b
o
rn
o
n
1
4
th
Ju
n
e
1
9
8
5
a
t
V
e
l
lo
re
,
T
a
m
il
Na
d
u
,
In
d
ia.
He
is
a
re
s
e
a
rc
h
sc
h
o
lar
in
th
e
De
p
a
rtm
e
n
t
o
f
M
a
t
h
e
m
a
ti
c
s,
Bh
a
ra
th
iar
Un
iv
e
rsity
,
In
d
ia.
He
re
c
e
iv
e
d
th
e
M
.
S
c
.
d
e
g
re
e
in
M
a
th
e
m
a
ti
c
s
f
ro
m
Un
iv
e
rsit
y
o
f
M
a
d
ra
s
(RI
A
S
M
)
in
2
0
0
7
.
He
c
o
m
p
lete
d
M
.
P
h
il
.
M
a
th
e
m
a
ti
c
s
f
ro
m
Un
iv
e
rsit
y
o
f
M
a
d
ra
s
in
th
e
y
e
a
r
2
0
1
0
.
His
re
se
a
rc
h
m
a
in
ly
f
o
c
u
se
s
o
n
Nu
m
b
e
r
T
h
e
o
r
y
a
n
d
Cr
y
p
to
g
ra
p
h
y
.
He
h
a
s
p
u
b
li
sh
e
d
f
iv
e
re
se
a
r
c
h
a
rti
c
les
in
v
a
rio
u
s
n
a
ti
o
n
a
l
a
n
d
i
n
tern
a
ti
o
n
a
l
j
o
u
r
n
a
ls.
Evaluation Warning : The document was created with Spire.PDF for Python.