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Vo
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IJ
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2252
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8814
A
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
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3
,
No
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2
,
J
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2
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ctio
n
b
et
w
ee
n
s
u
p
er
io
r
Ma
n
d
e
lb
r
o
t
s
et
an
d
s
u
p
er
io
r
J
u
lia
s
et
is
u
t
ilized
to
g
en
er
ate
s
ec
r
et
s
h
ar
ed
k
e
y
b
et
w
ee
n
s
e
n
d
er
an
d
r
ec
eiv
er
.
T
h
is
p
h
en
o
m
en
o
n
is
k
n
o
w
n
as
k
e
y
ag
r
ee
m
e
n
t
b
et
w
ee
n
in
v
o
lv
ed
p
ar
ties
.
2.
P
RO
P
O
SE
D
M
E
T
H
O
D
AND
RE
SU
L
T
DIS
C
USS
I
O
N
A
f
r
ac
tal
i
s
co
n
s
tr
u
cted
b
y
r
e
p
ea
ted
iter
atio
n
o
f
a
f
u
n
ctio
n
an
d
g
e
n
er
ates
a
co
m
p
le
x
s
tr
u
ctu
r
e
a
s
a
r
esu
lta
n
t.
T
h
e
Ma
n
d
elb
r
o
t
an
d
J
u
lia
s
et
ar
e
c
o
n
s
tr
u
cted
u
s
in
g
s
a
m
e
f
u
n
ctio
n
i.e
.
z
2
+c
.
T
h
e
o
n
ly
d
i
f
f
er
e
n
ce
b
et
w
ee
n
t
w
o
i
s
t
h
at
th
e
Ma
n
d
elb
r
o
t
s
et
is
a
s
et
o
f
p
o
in
ts
i
n
co
m
p
le
x
c
-
p
lan
e
s
tar
ti
n
g
at
z
=0
w
h
er
ea
s
J
u
lia
s
et
is
an
i
m
a
g
e
f
o
r
a
f
i
x
ed
c
v
al
u
e
s
tar
tin
g
n
o
n
ze
r
o
z.
I
n
o
u
r
m
et
h
o
d
,
w
e
u
s
ed
s
u
p
e
r
io
r
Ma
n
d
elb
r
o
t
s
et
an
d
s
u
p
er
io
r
J
u
lia
s
et
to
g
en
er
ate
p
u
b
li
c
k
e
y
an
d
p
r
iv
ate
k
e
y
r
esp
ec
ti
v
el
y
at
b
o
th
s
id
es.
T
h
e
eq
u
atio
n
u
s
ed
in
p
r
o
p
o
s
ed
m
et
h
o
d
is
s
u
p
er
io
r
Ma
n
d
elb
r
o
t
f
u
n
ctio
n
“
s
u
p
MS”
(
s
ee
eq
u
a
tio
n
2
&
3
)
:
**
nn
f
z
z
c
e
;
c,
z
Є
Z a
n
d
0
zc
(
2)
1
*
(
)
(
1
)
*
n
n
n
z
s
f
z
s
z
(
3
)
A
n
d
s
u
p
er
io
r
J
u
lia
s
et
“
s
u
p
J
S”
(
s
ee
eq
u
atio
n
4
,
5
&
6
)
:
**
nn
f
z
z
c
e
;
c,
z
Є
Z a
n
d
0
n
z
z
e
(
at
r
ec
eiv
er
s
id
e)
(
4
)
**
nn
f
z
z
c
e
;
c,
z
Є
Z a
n
d
0
k
z
z
d
(
at
s
en
d
er
s
id
e)
(
5)
Eq
u
atio
n
n
o
.
(
6
)
is
co
m
m
o
n
to
b
o
th
s
id
es:
1
*
(
)
(
1
)
*
n
n
n
z
s
f
z
s
z
(
6)
T
h
e
m
et
h
o
d
is
d
ef
i
n
ed
in
f
o
u
r
s
tep
s
:
Step
1
: A
t se
n
d
er
s
id
e
a)
Sen
d
er
ass
u
m
es
e
a
n
d
n
as
p
r
iv
ate
k
e
y
s
a
n
d
c
is
a
g
lo
b
al
v
al
u
e
w
h
ich
e
x
i
s
ts
i
n
s
u
p
er
io
r
Ma
n
d
elb
r
o
t set.
b)
A
p
u
b
lic
k
e
y
n
ze
is
ca
lc
u
lated
b
y
u
s
i
n
g
s
u
p
MS
f
u
n
ctio
n
w
h
ic
h
is
th
e
Ma
n
n
iter
ated
f
o
r
m
o
f
Ma
n
d
elb
r
o
t set.
c)
Sen
d
th
i
s
p
u
b
lic
k
e
y
to
r
ec
eiv
er
.
Step
2
: A
t r
ec
eiv
er
s
id
e
a)
R
ec
eiv
er
a
s
s
u
m
es
k
a
n
d
d
as
p
r
iv
ate
k
e
y
s
an
d
c
is
a
g
lo
b
al
v
alu
e
w
h
ich
ex
i
s
ts
in
s
u
p
er
io
r
Ma
n
d
elb
r
o
t set.
b)
A
p
u
b
lic
k
e
y
k
zd
is
ca
lc
u
lated
b
y
u
s
i
n
g
s
u
p
MS
f
u
n
c
tio
n
w
h
ic
h
is
th
e
Ma
n
n
iter
ated
f
o
r
m
o
f
Ma
n
d
elb
r
o
t set.
c)
Sen
d
th
i
s
p
u
b
lic
k
e
y
to
s
e
n
d
er
.
Step
3
: A
t se
n
d
er
s
id
e
a)
Sen
d
er
f
u
r
t
h
er
ex
ec
u
te
s
s
u
p
JS
f
u
n
c
tio
n
b
y
u
s
i
n
g
e
,
n
a
n
d
r
ec
eiv
er
’
s
p
u
b
lic
k
e
y
a
n
d
o
b
tai
n
ed
a
s
ec
r
et
k
e
y
(
n
ze
)
k
d.
Step
4
: A
t r
ec
eiv
er
s
i
de
a)
No
w
r
ec
eiv
er
ex
ec
u
tes
s
u
p
JS
f
u
n
ctio
n
b
y
u
s
i
n
g
k
,
d
an
d
s
en
d
er
’
s
p
u
b
lic
k
e
y
an
d
o
b
tain
ed
a
s
ec
r
et
k
e
y
(
k
zd
)
n
e
.
I
n
o
u
r
d
is
cu
s
s
io
n
n
a
n
d
k
r
ep
r
esen
ts
t
h
e
n
u
m
b
er
o
f
i
ter
atio
n
s
w
h
ile
e
an
d
d
ar
e
th
e
v
ar
iatio
n
co
n
s
ta
n
ts
a
n
d
ar
e
u
n
k
n
o
w
n
to
p
u
b
lic.
I
n
s
tep
1
an
d
s
tep
2
,
s
en
d
er
an
d
r
ec
eiv
er
e
x
c
h
an
g
e
d
th
eir
p
u
b
lic
k
e
y
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
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A
K
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t P
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p
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th
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r
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te
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ed
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.
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m
p
o
s
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ib
le
to
id
en
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f
y
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e
p
r
iv
ate
v
alu
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ith
t
h
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o
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p
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is
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u
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h
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h
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n
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it
v
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th
at
2
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p
o
s
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le
v
al
u
es c
an
b
e
u
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ed
f
o
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iter
atio
n
.
E
x
a
m
p
le:
An
e
x
a
m
p
le
is
s
h
o
w
n
to
h
o
w
th
e
p
u
b
lic
k
e
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s
ar
e
g
e
n
er
ate
d
b
y
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s
in
g
p
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a
te
v
a
lu
e
s
an
d
s
i
m
ilar
l
y
h
o
w
s
u
p
JS
is
u
s
ed
to
cr
ea
te
s
h
ar
ed
s
ec
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k
e
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n
b
o
th
s
id
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d
er
ass
u
m
es
e
a
s
a
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m
p
lex
v
al
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e,
n
n
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m
b
er
o
f
iter
atio
n
s
a
n
d
a
v
al
u
e
c
,
k
n
o
w
n
to
b
o
th
s
id
es
an
d
in
itial
ized
to
a
co
m
p
le
x
v
al
u
e
ex
is
ted
i
n
s
u
p
er
io
r
Ma
n
d
elb
r
o
t
s
et.
I
n
itial
l
y
s
en
d
er
ca
lcu
late
s
i
ts
p
u
b
lic
k
e
y
b
y
ex
ec
u
t
in
g
s
u
p
MS
f
u
n
ctio
n
a
n
d
o
b
tain
ed
n
ze
in
s
tep
1
.
e
=
0
.
0
1
5
9
2
3
–
0
.
0
3
1
7
9
5
2
3
n
=
4
c
=
0
.
3
2
5
+
1
.
5
1
2
5
s
=
0
.
6
n
ze
=
.
0
0
7
2
6
8
7
8
3
8
2
0
0
1
2
8
0
5
4
7
6
8
4
3
2
7
3
6
7
2
4
2
0
1
4
7
2
7
7
5
6
2
8
2
2
2
6
5
6
2
5
+
0
.
0
5
3
4
0
7
0
8
0
4
6
8
7
6
6
5
7
1
7
5
5
8
7
1
9
6
0
6
7
2
1
5
5
0
0
9
6
1
6
9
3
6
7
2
3
6
3
2
8
1
2
5
Si
m
i
lar
l
y
i
n
s
tep
2
,
r
ec
eiv
er
also
ex
ec
u
tes
s
u
p
MS
f
u
n
ctio
n
an
d
g
en
er
ate
s
its
p
u
b
lic
k
e
y
k
zd
b
y
u
s
i
n
g
p
r
iv
ate
v
al
u
e
s
k
,
d
an
d
c
o
m
m
o
n
v
al
u
e
c
co
m
p
le
x
v
al
u
e
.
k
=
3
s
=
0
.
6
d
=
-
0.
0
5
1
2
4
7
6
1
+
0
.
1
2
9
3
7
6
2
2
k
zd
=
.
0
1
3
6
2
3
2
7
8
1
5
1
9
9
4
2
3
9
5
7
6
9
0
0
7
1
5
5
8
5
1
0
4
8
8
2
8
1
2
5
.
0
2
8
5
3
9
5
2
0
4
9
4
6
3
2
2
8
0
3
3
5
5
9
6
0
1
9
2
2
2
1
4
1
7
9
6
8
7
5
No
w
b
o
th
p
ar
ties
ex
c
h
an
g
e
t
h
eir
p
u
b
lic
k
e
y
s
.
Fo
llo
w
in
g
t
h
is
p
r
o
ce
s
s
is
t
h
e
ca
lcu
lat
io
n
o
f
s
h
ar
ed
k
e
y
s
b
y
u
s
in
g
s
u
p
JS
f
u
n
ctio
n
w
i
th
p
u
b
lic
k
e
y
o
f
eit
h
er
s
id
e
as
in
itial
v
al
u
e
o
f
z
i
n
s
tep
3
an
d
s
tep
4
.
As
a
r
esu
lt
s
ec
r
et
k
e
y
s
(
n
ze
)
k
d
an
d
(
k
zd
)
n
e
ar
e
o
b
tain
ed
.
W
e
ca
n
s
h
o
w
th
at
b
o
th
k
e
y
s
ar
e
in
d
ee
d
th
e
s
a
m
e
at
b
o
th
s
id
es.
Send
er
Si
de:
z=
k
zd
=
.
0
1
3
6
2
3
2
7
8
1
5
1
9
9
4
2
3
9
5
7
6
9
0
0
7
1
5
5
8
5
1
0
4
8
8
2
8
1
2
5
.
0
2
8
5
3
9
5
2
0
4
9
4
6
3
2
2
8
0
3
3
5
5
9
6
0
1
9
2
2
2
1
4
1
7
9
6
8
7
5
e=
0
.
0
1
5
9
2
3
–
0
.
0
3
1
7
9
5
2
3
n
=4
s
=0
.
6
(
k
zd
)
n
e
=.
0
0
0
3
9
7
3
8
0
1
5
6
9
3
1
4
2
1
4
4
4
5
1
8
2
3
7
9
7
0
3
4
0
7
6
5
4
6
7
7
6
7
1
4
4
2
2
3
9
6
6
7
3
3
8
8
7
9
2
7
5
3
4
5
6
3
6
6
9
4
2
4
5
8
0
1
5
4
4
1
8
9
4
5
3
1
2
5
.
0
0
1
0
2
7
6
6
0
3
9
9
8
6
3
0
0
9
0
8
1
5
3
1
6
1
4
9
7
7
0
6
5
8
0
8
7
6
5
5
7
8
3
3
5
1
8
4
6
1
1
2
8
9
6
2
7
2
5
9
1
2
3
5
1
6
4
7
7
6
9
6
8
7
8
5
0
9
5
2
1
4
8
4
3
7
5
Rec
eiv
er
Sid
e:
z=
n
ze
=.
0
0
7
2
6
8
7
8
3
8
2
0
0
1
2
8
0
5
4
7
6
8
4
3
2
7
3
6
7
2
4
2
0
1
4
7
2
7
7
5
6
2
8
2
2
2
6
5
6
2
5
+
0
.
0
5
3
4
0
7
0
8
0
4
6
8
7
6
6
5
7
1
7
5
5
8
7
1
9
6
0
6
7
2
1
5
5
0
0
9
6
1
6
9
3
6
7
2
3
6
3
2
8
1
2
5
s
=0
.
6
k
=3
d=
-
0
.
0
5
1
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I
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RE
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NC
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[1
]
A
.
M
S
,
“
P
u
b
li
c
k
e
y
c
r
y
p
to
g
ra
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y
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.
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.
[3
]
B.
B.
M
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d
e
lb
r
o
t,
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ra
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a
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n
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o
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.
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-
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0
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pp.
30
-
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.
[4
]
Ba
rn
sle
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M
.
,
“
Fra
c
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it
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A
c
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In
c
.
,
S
a
n
Die
g
o
,
C
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.
USA
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,
p
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.
550
.
[5
]
Diff
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W.
,
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t
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,
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sa
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6
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.
22
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No
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6
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pp.
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4
4
-
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.
[6
]
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M
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t
a
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.
,
“
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.
[7
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M
.
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,
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.
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[8
]
M
.
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t
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.
,
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4
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.
279
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.
[9
]
M
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.
261
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7
.
[1
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a
l
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li
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it
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n
,
sp
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g
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r
,
9
9
4
,
pp
.
2
3
5
.
[1
1
]
W.
R.
M
a
n
n
,
“
M
e
a
n
v
a
l
u
e
me
th
o
d
s in
it
e
ra
ti
o
n
s
,
”
P
r
o
c
.
A
m
e
r.
M
a
th
.
S
o
c
.
,
1
9
5
3
,
V
o
l
.
4
,
p
p
.
5
0
6
-
5
1
0
.
[1
2
]
W
.
S
talli
n
g
,
“
Cry
p
to
g
ra
p
h
y
a
n
d
Ne
tw
o
rk
S
e
c
u
rit
y
,
”
P
HI
,
2
0
0
4
.
Evaluation Warning : The document was created with Spire.PDF for Python.