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10
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414
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th
a
t a
ttack
s
t
a
k
es p
lace
.
I
n
h
ar
m
o
n
y
w
i
th
th
e
s
e
n
s
e
o
f
[
9
]
th
e
Hei
s
e
n
b
er
g
u
n
ce
r
tain
t
y
,
it
is
i
m
p
o
s
s
ib
le
f
o
r
m
ea
s
u
r
in
g
th
e
Q
u
a
n
tu
m
s
tate
s
w
it
h
o
u
t tr
o
u
b
lin
g
m
o
s
t o
f
t
h
e
s
y
s
te
m
.
I
n
th
i
s
w
a
y
,
t
h
e
m
o
v
e
m
e
n
t o
f
th
e
o
p
p
o
s
ite
p
o
s
itio
n
s
o
f
a
p
h
o
to
n
ad
d
itio
n
to
s
tar
t
o
r
en
d
o
f
w
o
r
d
ex
p
er
ien
ce
d
at
th
e
p
o
in
t.
T
h
is
m
a
k
es
ce
r
tain
attac
k
er
s
in
tr
u
s
io
n
s
h
o
u
ld
g
i
v
e
th
at
ca
n
n
o
t c
h
a
n
g
ed
b
ac
k
in
th
e
s
tates o
f
t
h
e
q
u
an
tu
m
p
r
io
r
th
e
y
ar
e
s
e
n
t to
th
e
o
n
e
w
h
o
g
et
s
.
T
h
e
p
r
o
p
er
ty
t
h
at
q
u
an
t
u
m
cr
y
p
to
g
r
ap
h
y
h
a
s
is
th
e
n
o
-
clo
n
i
n
g
p
r
o
p
er
ty
.
I
t
e
x
p
lain
s
th
at
i
s
i
m
p
o
s
s
ib
le
to
g
et
th
e
co
p
y
o
f
th
e
s
i
n
g
le
p
h
o
to
n
th
at
ar
e
n
o
t
g
iv
e
n
ac
ce
s
s
th
e
w
o
r
d
th
at
o
n
e
is
g
o
in
g
to
o
th
er
u
s
er
s
.
T
h
e
m
o
v
e
m
e
n
t to
o
p
p
o
s
ite
p
o
s
itio
n
s
p
h
o
to
n
s
en
s
e
o
f
r
i
g
h
t
g
i
v
es a
d
etailed
ac
co
u
n
t o
f
lig
h
t p
h
o
to
n
s
t
h
at
h
o
w
it c
an
b
e
th
e
ad
j
u
s
t
m
e
n
t t
o
ev
en
ts
o
r
g
a
v
e
o
p
p
o
s
ites
in
s
p
ec
ial
w
a
y
s
.
T
o
s
en
d
th
e
u
n
k
n
o
w
n
q
u
a
n
t
u
m
n
e
w
s
g
iv
e
n
f
ar
a
w
a
y
f
r
o
m
lig
h
t
q
u
an
t
u
m
,
b
y
w
h
o
m
it
h
a
s
ex
is
te
n
ce
th
e
m
ix
ed
p
o
s
itio
n
d
o
in
g
b
y
th
e
u
s
e
o
f
q
u
a
n
t
u
m
m
ix
ed
p
o
s
itio
n
s
Mo
r
eo
v
er
,
th
e
s
tr
u
ctu
r
e
th
at
is
ta
k
en
th
e
q
u
a
n
tu
m
n
e
w
s
g
i
v
en
i
n
t
h
e
p
lace
w
it
h
o
u
t se
n
t to
t
h
e
u
s
er
.
2.
CRYP
T
O
G
RAP
G
Y
T
h
e
ar
t
o
f
w
r
iti
n
g
a
le
v
el
s
tr
e
tch
o
f
teac
h
i
n
g
b
o
o
k
in
to
t
h
e
cip
h
er
teac
h
i
n
g
b
o
o
k
.
T
h
e
u
n
co
m
m
o
n
,
n
o
te
is
m
ad
e
to
a
r
u
le
p
u
t
in
t
o
s
ig
n
s
ex
p
er
ien
ce
d
as
cip
h
er
teac
h
in
g
b
o
o
k
.
T
h
e
p
r
o
ce
s
s
o
f
g
etti
n
g
ch
a
n
g
ed
f
r
o
m
lev
el
s
tr
etc
h
o
f
teac
h
i
n
g
b
o
o
k
to
cip
h
er
teac
h
i
n
g
b
o
o
k
ar
e
e
x
p
er
ien
ce
d
as
e
n
cip
h
er
in
g
o
r
p
r
o
ce
s
s
o
f
ch
an
g
i
n
g
k
n
o
w
led
g
e
in
to
a
s
ec
r
et
f
o
r
m
an
d
p
u
tti
n
g
b
ac
k
to
ea
r
lier
p
o
s
itio
n
th
e
lev
el
s
tr
etc
h
o
f
teac
h
in
g
b
o
o
k
f
r
o
m
cip
h
er
teac
h
in
g
f
o
r
m
a
r
e
ex
p
er
ien
ce
d
as
d
ec
ip
h
er
in
g
.
T
h
e
t
w
o
t
y
p
es
o
f
cr
y
p
to
g
r
ap
h
y
,
s
y
m
m
etr
ic
cr
y
p
to
g
r
ap
h
y
a
n
d
th
e
o
th
er
i
s
as
y
m
m
etr
ic
cr
y
p
to
g
r
ap
h
y
.
2
.
1
.
Sy
mm
et
ric
T
h
e
p
er
s
o
n
w
h
o
s
en
d
s
as
w
el
l
as
t
h
e
p
er
s
o
n
w
h
o
r
ec
eiv
e
s
th
e
d
ata,
u
tili
ze
s
t
h
e
k
e
y
an
d
th
e
s
a
m
e
alg
o
r
ith
m
to
d
o
th
e
cr
y
p
to
g
r
a
p
h
y
.
A
s
s
u
m
e
t
h
at,
A
e
n
cr
y
p
ts
a
lev
el
s
tr
etc
h
o
f
teac
h
i
n
g
b
o
o
k
n
o
te
w
it
h
t
h
e
k
e
y
as
w
e
ll
as
B
d
ec
r
y
p
t
s
th
e
n
o
te
w
h
er
e
A
lice
u
s
es
b
o
t
h
t
h
e
k
e
y
an
d
th
e
al
g
o
r
ith
m
.
T
h
e
k
e
y
s
a
r
e
k
ep
t
s
ec
r
etl
y
,
i
n
w
h
ic
h
th
at
o
n
l
y
A
an
d
B
h
av
e
k
n
o
w
led
g
e
o
f
i
t
[
8
]
.
A
s
an
o
u
tco
m
e
o
f
t
h
at,
th
e
b
est
d
ir
ec
tio
n
f
o
r
h
a
v
in
g
t
h
e
alg
o
r
ith
m
a
n
d
t
h
e
s
ec
r
et
k
e
y
s
i
n
t
h
e
c
h
an
n
el
ar
e
r
eq
u
ested
.
I
n
th
e
a
s
y
m
m
etr
ic,
it i
s
s
tar
ted
t
o
g
et
a
n
s
w
er
to
t
h
e
d
is
tr
ib
u
tio
n
o
f
th
e
k
e
y
s
w
it
h
th
e
h
ar
d
q
u
esti
o
n
lik
e
s
y
m
m
et
r
ic
[
1
0
]
.
Gr
ea
tly
r
esp
ec
ted
lik
e
in
s
ize
al
g
o
r
ith
m
s
h
as t
h
e
f
ac
t
s
f
o
r
m
q
u
alit
y
e
x
a
m
p
le
T
r
ip
le
DE
S (
3
d
es)
an
d
th
e
A
d
v
a
n
ce
d
E
n
cr
y
p
tio
n
Stan
d
ar
d
(
A
E
S)
.
2
.
2
.
Asy
mm
et
ric
T
h
e
p
r
o
ce
s
s
o
f
c
h
a
n
g
i
n
g
t
h
e
k
n
o
w
led
g
e
in
to
a
s
ec
r
et
f
o
r
m
,
w
it
h
w
h
ich
th
e
p
air
o
f
t
h
e
k
e
y
s
i
n
v
o
l
v
ed
.
A
ll
t
h
e
u
s
er
s
w
i
ll
h
a
v
e
th
eir
ex
ac
t
p
u
b
lic
an
d
th
e
p
r
iv
ate
k
e
y
s
.
L
et
u
s
as
s
u
m
e
t
h
at,
if
B
tr
ies
t
o
en
cr
y
p
t
th
e
f
ac
t
s
,
A
w
ill
s
h
ar
e
k
e
y
o
f
p
u
b
lic
w
i
th
B
,
later
B
ca
n
e
n
cr
y
p
t
t
h
e
f
ac
t
s
b
y
u
s
i
n
g
A’
s
k
e
y
[
8
]
.
No
w
,
B
lik
e
s
to
s
h
ar
e
th
e
e
n
cr
y
p
ted
f
ac
ts
w
it
h
A
,
t
h
e
n
A
is
a
llo
w
ed
to
d
ec
r
y
p
t
th
e
f
ac
t
s
b
y
u
s
i
n
g
th
e
k
e
y
o
f
p
r
iv
ate.
T
h
is
is
h
o
w
,
e
n
cr
y
p
tio
n
o
f
t
h
e
f
ac
ts
b
y
th
e
p
r
iv
ate
a
n
d
th
e
u
s
er
w
it
h
t
h
e
p
u
b
lic
k
e
y
h
as
t
h
e
ac
ce
s
s
to
d
ec
r
y
p
tt
h
e
f
ac
t
s
.
3.
F
ACTOR
I
Z
A
T
I
O
N
P
RO
B
L
E
M
Ma
k
i
n
g
t
h
e
d
is
co
v
er
y
o
f
t
h
e
r
o
o
t
am
o
u
n
t
s
h
ar
d
q
u
es
tio
n
o
f
R
S
A
cr
y
p
to
s
y
s
te
m
.
T
h
e
p
u
b
lic
k
e
y
w
h
ic
h
i
s
u
s
ed
a
s
R
S
A
al
g
o
r
it
h
m
[
11
]
.
T
h
e
R
S
A
o
v
er
co
m
e
s
th
e
tr
o
u
b
le
b
y
f
ir
s
t
i
n
r
ati
n
g
o
f
t
h
e
n
u
m
b
er
s
t
h
at
ar
e
p
r
im
e.
P
elzl
an
d
P
aa
r
,
ac
c
o
r
d
in
g
to
th
e
m
th
e
a
s
y
m
m
etr
i
c
alg
o
r
ith
m
s
a
n
d
th
e
R
S
A
t
h
a
t
w
er
e
i
n
p
lace
o
f
s
y
m
m
etr
ic
alg
o
r
it
h
m
s
ar
e
n
o
t
u
s
ed
.
T
h
e
m
ain
u
s
a
g
e
o
f
t
h
e
R
S
A
al
g
o
r
ith
m
is
f
o
r
th
e
s
a
f
est
m
ec
h
a
n
is
m
f
o
r
th
e
k
e
y
e
x
ch
a
n
g
e
an
d
m
o
s
tl
y
u
s
ed
w
ith
t
h
e
li
k
e
al
g
o
r
ith
m
s
l
i
k
e
A
d
v
an
ce
d
E
n
cr
y
p
tio
n
Sta
n
d
ar
d
(
A
E
S).
T
h
e
lik
e
in
s
ize
alg
o
r
it
h
m
d
o
es
t
h
e
tr
u
e,
i
n
f
ac
t
th
e
p
r
o
ce
s
s
o
f
ch
a
n
g
in
g
k
n
o
w
led
g
e
in
to
a
s
ec
r
et
f
o
r
m
a
n
d
p
r
o
ce
s
s
o
f
ch
a
n
g
in
g
k
n
o
w
led
g
e
b
ac
k
i
n
to
s
tar
tin
g
f
o
r
m
ca
r
ef
u
l
w
a
y
.
K
ir
s
ch
,
th
e
a
u
t
h
o
r
illu
s
tr
ated
,
R
S
A
i
s
ba
s
ed
o
n
r
ea
s
o
n
in
g
o
p
en
to
at
tack
,
if
a
tig
h
tl
y
p
r
o
ce
s
s
o
f
d
o
in
g
th
e
f
ac
to
r
izatio
n
in
tr
o
d
u
ce
s
t
h
e
ab
ilit
y
i
n
th
e
cu
r
r
en
tl
y
e
x
i
s
ten
ce
.
T
h
e
k
n
o
w
led
g
e
p
r
o
ce
s
s
i
n
g
m
ac
h
in
e
s
[
11
]
,
n
am
ed
Q
u
an
t
u
m
-
C
o
m
p
u
ter
.
4.
DIS
CR
E
T
E
L
O
G
A
RI
T
H
M
P
RO
B
L
E
M
I
n
th
e
cr
y
p
to
g
r
ap
h
y
p
r
o
b
le
m
s
,
E
llip
tic
C
u
r
v
e
C
r
y
p
to
g
r
ap
h
y
an
d
th
e
Di
f
f
ie
Hell
m
an
[
7
]
w
o
r
k
s
u
n
d
er
th
e
p
r
in
cip
le
o
f
th
e
D
L
P
.
T
h
e
p
r
o
b
lem
b
y
cr
ac
k
in
g
t
h
e
cr
y
p
to
g
r
ap
h
y
en
d
s
u
p
o
n
th
e
d
ete
r
m
in
in
g
r
-
i
n
teg
er
,
w
h
er
e
g
r
=
x
m
o
d
p
.
r
th
e
in
teg
er
is
tab
b
ed
as
t
h
e
D
L
P
f
o
r
x
at
t
h
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w
ir
i
n
g
g
.
I
t
co
u
ld
b
e
r
e
w
r
itte
n
as
r
=
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x
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le
p
r
o
b
lem
to
f
i
n
d
o
u
t,
w
h
e
n
lar
g
e
n
u
m
b
er
o
f
p
ar
a
m
eter
s
ar
e
f
o
u
n
d
.
T
h
e
k
e
y
e
x
c
h
an
g
e
m
ec
h
an
is
m
o
f
t
h
e
Di
f
f
ie
Hel
l
m
a
n
ar
e
th
e
a
s
y
m
m
etr
ical
cr
y
p
to
g
r
ap
h
y
.
I
n
t
h
e
p
u
b
lic
ch
an
n
el
s
,
th
e
k
e
y
s
ar
e
s
h
ar
ed
s
ec
u
r
el
y
.
T
h
e
lar
g
er
k
e
y
s
ize
o
r
th
e
b
its
eq
u
al
to
2
0
4
8
b
its
ap
p
lied
f
o
r
th
e
s
af
er
ex
ch
a
n
g
e
o
f
k
e
y
s
.
T
h
e
E
llip
tic
C
u
r
v
e
C
r
y
p
to
g
r
ap
h
y
is
als
o
th
e
m
e
m
b
er
o
f
p
u
b
lic
k
e
y
f
a
m
i
l
y
.
T
h
e
E
C
C
p
r
o
v
id
es
th
e
s
i
m
ilar
s
ec
u
r
it
y
w
it
h
t
h
e
R
S
A
a
n
d
th
e
D
L
P
[
4
]
.
T
h
e
p
air
(
x:
y
)
is
u
s
ed
b
y
th
e
E
C
C
f
o
r
th
e
eq
u
atio
n
z2
=
y
3
+
b
x
+a
m
o
d
p
w
i
th
t
h
eW
h
er
e
b
;a
∈
Z
p
a
n
d
4
a3
+
2
7
b
3
+
6
=
0
m
o
d
p
.
T
h
e
n
ee
d
s
of
t
h
e
E
C
C
g
r
o
u
p
G
of
c
y
clic
a
n
d
th
e
ele
m
en
t
s
o
f
t
h
e
p
r
i
m
iti
v
es.
T
h
e
E
llip
tic
C
u
r
v
e
C
r
y
p
to
g
r
ap
h
y
i
s
ef
f
icie
n
t
u
n
s
y
m
m
etr
ical
a
n
d
th
e
s
ec
u
r
e
d
s
y
s
te
m
.
5.
E
L
L
I
P
T
I
C
C
URVE C
RYP
T
O
G
RAP
H
Y
T
h
e
n
e
w
l
y
d
esi
g
n
ed
cr
y
p
to
g
r
ap
h
y
al
g
o
r
ith
m
d
ep
e
n
d
s
u
p
o
n
t
h
e
d
o
ts
th
at
ar
e
o
n
t
h
e
lin
e
r
o
u
n
d
a
cir
cle
o
f
n
o
t
b
ased
o
n
r
u
les
d
is
tan
ce
to
th
e
ce
n
ter
o
f
th
e
a
r
ith
m
etic
o
v
er
f
ir
s
t
in
r
ati
n
g
n
u
m
b
er
s
.
T
h
e
E
C
C
s
cien
ce
o
f
k
ee
p
i
n
g
k
n
o
w
led
g
e
s
af
e
a
n
d
s
ec
r
et
m
ak
e
s
r
eq
u
es
t
to
co
m
e
b
et
w
ee
n
t
h
e
s
e
tu
p
a
n
d
th
e
o
p
er
atio
n
an
d
th
e
o
r
g
an
iza
tio
n
s
w
i
th
n
e
w
al
g
o
r
ith
m
p
r
esen
ted
.
I
n
th
e
E
C
C
it
h
as
o
n
l
y
t
h
e
o
n
e
ze
r
o
p
o
in
t
an
d
th
e
i
n
te
g
er
s
p
o
in
ts
w
h
ich
ar
e
n
o
t
a
m
e
m
b
er
o
f
th
e
g
r
o
u
p
co
u
ld
b
e
u
s
e
d
.
I
t
is
a
s
tr
aig
h
t
f
o
r
w
ar
d
en
c
r
y
p
tio
n
t
h
at
t
h
at
o
f
m
ak
in
g
o
r
d
er
s
f
o
r
co
m
p
u
ter
t
h
an
e
llip
tic
c
u
r
v
e
s
c
ien
ce
o
f
k
ee
p
i
n
g
k
n
o
w
led
g
e
s
a
f
e
a
n
d
s
e
cr
et
[
1
2
]
.
T
h
is
E
C
C
alg
o
r
ith
m
h
a
v
e
t
h
e
ch
ie
f
co
n
s
s
id
e
m
ad
e
a
co
m
p
ar
is
o
n
o
f
E
C
C
th
o
s
e
a
r
e
lo
w
in
s
a
f
e
o
f
eq
u
al
p
r
i
m
e
s
th
at
g
iv
e
s
k
n
o
w
led
g
e
b
e
f
o
r
e
ev
en
t
.
W
h
en
co
m
p
ar
ed
to
R
S
A
it
h
as
s
a
m
e
s
ec
u
r
it
y
.
T
h
e
E
C
C
ac
ts
s
ta
g
e
m
o
d
u
latio
n
th
at
p
er
f
o
r
m
th
e
r
o
u
n
d
o
n
el
lip
s
e
o
f
m
an
y
o
p
er
atio
n
s
,
t
h
at
p
u
b
lis
h
ed
in
t
h
e
liter
at
u
r
e
o
f
p
h
y
s
ics
ar
e
in
co
n
f
u
s
ed
s
ta
te
o
f
th
e
P
h
a
s
e
m
o
d
u
latio
n
.
W
ith
t
h
e
w
a
y
o
f
an
alo
g
o
u
s
o
f
Di
f
f
ie
Hell
m
a
n
,
ex
ch
a
n
g
in
g
o
f
k
e
y
s
w
er
e
p
o
s
s
ib
le.
T
h
e
n
e
w
l
y
in
tr
o
d
u
ce
d
alg
o
r
ith
m
ap
p
lied
f
o
r
t
h
e
elec
tr
o
n
ic
s
ig
n
-
m
ar
k
s
,
d
i
s
tr
ib
u
tio
n
o
f
t
h
e
k
e
y
s
,
au
th
e
n
tica
tio
n
s
[
1
3
].
L
et
u
s
a
s
s
u
m
e
th
at
Kan
is
h
an
d
G
an
es
h
w
an
ted
to
s
h
ar
e
m
e
s
s
a
g
es.
I
n
i
tiall
y
Ka
n
is
h
s
elec
t
s
th
e
p
o
f
p
r
im
e
n
u
m
b
er
.
T
h
en
,
t
h
e
i
n
te
g
er
o
f
r
ad
ii
b
a
n
d
a,
w
i
th
th
e
m
a
x
i
m
u
m
o
r
d
er
n
Kan
is
h
s
e
lects
h
er
b
ase
p
air
.
As
alr
ea
d
y
s
a
id
Kan
i
s
h
s
tar
t
s
h
er
m
es
s
ag
e
w
it
h
t
h
e
Di
f
f
ie
Hell
m
an
k
e
y
ex
c
h
a
n
g
e
m
ec
h
a
n
i
s
m
to
h
a
v
e
th
e
s
a
m
e
k
e
y
b
et
w
ee
n
t
h
e
m
.
T
h
e
m
es
s
ag
e
s
th
e
n
b
r
o
k
en
in
to
th
e
b
lo
ck
s
w
ith
p
air
s
b
y
Kan
i
s
h
an
d
th
e
n
en
cr
y
p
ts
t
h
e
b
lo
ck
s
w
i
th
t
h
e
s
a
m
e
k
e
y
s
,
u
p
to
a
ll
t
h
e
m
e
s
s
a
g
es
ar
e
en
co
d
ed
.
L
ater
Ka
n
is
h
s
e
n
d
s
t
h
e
e
n
co
d
ed
m
es
s
ag
e
to
Gan
es
h
.
W
it
h
t
h
e
co
m
m
o
n
l
y
s
h
ar
ed
k
e
y
Gan
e
s
h
d
ec
r
y
p
ts
th
e
m
es
s
a
g
es.
T
h
e
p
air
s
w
it
h
th
e
b
lo
c
k
s
ar
e
d
ec
r
y
p
ted
to
v
ie
w
it a
s
t
h
e
o
r
ig
in
al
m
es
s
ag
e
s
.
6.
Q
UAN
T
UM
CRYP
T
O
G
RA
P
H
Y
T
h
e
b
it
s
tr
in
g
s
m
ad
e
in
a
w
a
y
w
ith
in
t
h
e
ex
c
h
an
g
e
a
m
o
n
g
t
h
e
u
s
er
s
th
at
t
h
e
y
ar
e
n
o
t
d
ir
ec
tl
y
in
ter
ac
ti
n
g
.
Sti
l
l,
t
h
e
u
s
er
s
h
a
v
in
g
a
s
tr
o
n
g
b
elie
f
t
h
at
th
e
m
es
s
ag
e
s
(
b
it
s
tr
i
n
g
s
)
s
h
ar
ed
w
it
h
i
n
th
e
m
ar
e
s
a
f
e
an
d
s
ec
u
r
ed
.
T
h
e
t
w
o
u
s
er
s
g
i
v
en
ac
ce
s
s
,
t
h
at
u
s
u
all
y
b
o
t
h
t
h
e
u
s
er
s
,
to
m
ak
e
ce
r
tain
s
ec
u
r
ed
k
e
y
s
o
r
d
er
b
y
th
e
m
o
v
e
m
e
n
t
to
o
p
p
o
s
ite
p
o
s
itio
n
s
p
r
i
n
cip
le
[
1
4
]
.
T
h
e
p
h
o
to
n
s
t
h
at
p
as
s
ed
ar
e
in
th
ese
w
a
y
s
:
Ver
tical
,
Ho
r
izo
n
tal
an
d
th
e
d
iag
o
n
al
s
as
s
h
o
w
n
i
n
T
ab
le
1
.
T
h
e
in
itial
s
tates
r
ep
r
esen
ted
w
i
th
t
h
e
o
r
ien
tatio
n
ca
lled
th
e
R
ec
ti
lin
ea
r
an
d
t
h
e
n
e
x
t
r
ep
r
esen
ted
as
th
e
o
r
ien
tatio
n
ca
lled
th
e
Diag
o
n
a
l
o
r
ien
tatio
n
.
T
h
e
o
r
ien
tatio
n
s
ar
e
r
ep
r
esen
ted
as+,
/,
\
.
T
ab
le
1
.
States
f
o
r
th
e
p
o
lar
izatio
n
o
f
t
h
e
p
o
to
n
B
a
si
s
0
1
+
×
7.
Q
UAN
T
UM
K
E
Y
DIS
T
RIB
UT
I
O
N
T
h
e
ac
ts
th
at
o
f
f
er
s
t
h
e
s
af
e
l
y
ex
c
h
an
g
i
n
g
th
e
s
ec
r
et
k
e
y
b
et
w
ee
n
t
h
e
u
s
er
s
o
v
er
a
n
u
n
s
af
e
n
ar
r
o
w
w
a
y
[
1
5
].
T
h
e
q
u
an
tu
m
k
e
y
cr
y
p
to
g
r
ap
h
y
lie
s
o
n
th
e
q
u
a
n
tu
m
p
h
y
s
ics
ar
e
b
ased
o
n
th
e
q
u
an
tu
m
la
w
s
t
h
at
ar
e
h
ar
m
f
u
l
f
o
r
in
cr
ea
s
in
g
t
h
e
p
o
w
er
co
n
s
u
m
p
tio
n
s
b
y
th
e
p
o
l
ar
izatio
n
o
f
th
e
q
u
a
n
t
u
m
,
t
h
e
f
ib
er
o
p
tic
ch
an
n
e
ls
u
s
ed
f
o
r
tr
an
s
m
itti
n
g
m
e
s
s
a
g
e
s
.
I
n
th
e
y
ea
r
1
9
8
4
,
th
e
Qu
a
n
t
u
m
Ke
y
D
is
tr
ib
u
tio
n
i
n
itia
ll
y
in
tr
o
d
u
ce
d
,
w
h
er
e
Gills
B
r
ass
ar
d
an
d
C
h
ar
le
s
B
en
n
et
t
f
o
u
n
d
ed
a
p
r
o
to
c
o
l
o
f
B
B
8
4
.
A
f
ter
th
e
d
ev
elo
p
m
e
n
t
o
f
th
is
p
r
o
to
co
l,
m
an
y
o
f
t
h
e
p
r
o
to
co
ls
co
n
s
eq
u
en
t
l
y
in
v
e
n
ted
[
1
6
]
.
I
t is v
er
y
h
ar
d
f
o
r
th
e
attac
k
er
s
to
d
o
ea
v
esd
r
o
p
in
t
h
e
f
ib
er
o
p
tic
ch
an
n
e
l.
W
h
en
t
h
e
m
e
s
s
a
g
e
ar
e
co
r
r
u
p
ted
o
r
attac
k
ed
b
y
th
e
ea
v
esd
r
o
p
p
er
,
it
ca
n
b
e
f
o
u
n
d
th
at
th
e
a
m
o
u
n
t
o
f
m
es
s
ag
e
s
b
ei
n
g
co
r
r
u
p
ted
.
T
h
is
is
u
s
ed
i
n
B
B
8
4
[
17
,
18
]
.
T
h
e
en
tan
g
led
o
b
j
ec
ts
w
i
th
th
e
p
air
s
u
s
ed
i
n
w
h
ic
h
t
h
e
o
b
j
ec
ts
s
h
ar
ed
b
et
w
ee
n
th
e
u
s
er
s
,
th
e
p
r
o
to
co
l
ca
lled
th
e
E
n
ta
n
g
l
e
m
en
t
B
ased
(
E
B
)
.
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.
10
,
No
.
1
,
Feb
r
u
ar
y
2
0
2
0
:
4
0
7
-
414
410
On
e
o
f
t
h
e
o
b
j
ec
ts
is
co
n
s
id
er
ed
f
r
o
m
t
w
o
o
r
m
o
r
e
o
b
j
ec
ts
,
w
h
ic
h
ar
e
th
e
e
n
ta
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t q
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o
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th
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q
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o
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ata
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s
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[
19
]
.
T
h
e
E
9
1
p
r
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to
co
l
is
ex
p
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s
in
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t
h
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r
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A
g
a
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e
d
iv
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in
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ee
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o
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Var
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s
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ase
R
ef
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ce
C
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Dis
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Var
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C
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F
ig
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r
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s
h
o
w
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e
Q
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b
et
w
ee
n
Ka
n
is
h
(
s
en
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er
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d
Gan
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(
r
ec
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er
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Fig
u
r
e
1
.
QKD
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et
w
ee
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k
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h
(
s
en
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er
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ec
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r
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T
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t
h
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th
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ce
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p
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as f
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7
.
1
.
M
a
t
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m
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t
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l ba
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ed
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n
T
h
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m
a
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Dif
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m
atica
l b
ased
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e
n
tat
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t
h
at
m
o
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tl
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r
esear
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a.
L
attice
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ased
cr
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p
to
g
r
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h
y
[4
]
T
h
e
f
ee
b
len
ess
o
f
R
S
A
al
g
o
r
ith
m
k
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t
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e
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a
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n
d
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n
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n
n
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m
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e
r
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ased
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o
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m
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li
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n
ad
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lat
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ased
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p
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y
m
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k
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g
t
h
at
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ased
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n
as
tr
u
e
p
r
o
b
lem
o
f
t
h
e
Sh
o
r
test
Vec
to
r
P
r
o
b
lem
(
SVP
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.
T
h
e
g
iv
en
d
ata
h
er
e
in
d
icate
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ar
e
w
it
h
t
h
e
b
asis
o
f
t
h
e
ar
b
itra
r
y
lattice.
T
h
e
Sh
o
r
tes
t N
o
n
Z
er
o
v
ec
to
r
is
t
h
e
ai
m
o
f
th
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ased
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y
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b.
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lti
v
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iate
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ased
cr
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p
to
g
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y
[
2
0]
T
h
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o
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ith
m
is
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ev
elo
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ed
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t
h
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cr
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ic
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lt
i
n
th
e
m
u
lti
v
ar
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ate
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ased
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y
p
to
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h
y
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T
h
e
b
o
th
d
ig
ita
l
s
ig
n
at
u
r
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an
d
t
h
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en
cr
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tio
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ar
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u
s
ed
i
n
th
e
m
u
lti
v
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iat
e
cr
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to
s
y
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te
m
.
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it
h
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m
u
lti
v
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iate
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ased
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o
l
y
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o
m
ial
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m
an
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t
h
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as
y
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etr
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c
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e
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ied
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h
e
n
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w
w
a
y
o
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t
h
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e
f
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ic
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Ma
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atr
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ltip
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r
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er
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cr
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to
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y
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te
m
a
r
e
u
s
ed
f
o
r
th
e
Di
g
ital
Si
g
n
a
tu
r
e.
T
h
e
s
ig
n
at
u
r
e
w
a
y
s
t
h
at
u
s
ed
ar
e
th
e
r
ai
n
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o
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an
d
v
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ar
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g
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at
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s
ch
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m
es.
T
h
e
r
atio
s
th
at
ar
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b
et
w
e
en
th
e
n
u
m
b
er
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f
t
h
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eq
u
atio
n
s
an
d
th
e
v
ar
iab
les
ar
e
co
n
tain
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i
n
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e
UOV.
T
h
at
m
a
k
es
t
h
e
h
as
h
v
al
u
e
s
th
a
t
a
r
e
th
r
ee
ti
m
e
s
w
ea
k
er
th
a
n
t
h
e
s
ig
n
at
u
r
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W
ith
th
e
R
ai
n
b
o
w
,
i
t h
as
s
m
aller
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atio
th
at
g
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h
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u
tp
u
t a
s
t
h
e
m
i
n
i
m
u
m
k
e
y
s
izes a
n
d
th
e
Di
g
ital Si
g
n
at
u
r
es.
c.
Hash
b
ased
s
i
g
n
atu
r
e
[
2
]
I
n
th
e
y
ea
r
1
9
7
9
,
L
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e
L
a
m
p
o
r
t
in
v
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s
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h
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L
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t
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T
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e
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t
h
e
l
ev
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o
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ec
u
r
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y
w
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th
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e
s
y
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m
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T
h
e
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its
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1
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its
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t
h
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e
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ec
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r
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l.
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h
e
len
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o
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th
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ar
b
itra
r
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a
n
d
th
at
g
i
v
es
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h
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tp
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t
a
s
2
5
6
b
its
.
T
h
e
Op
ti
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al
So
lu
tio
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t
h
at
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itted
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t
h
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e
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SH
A
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.
d.
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e
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ased
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[
2
1
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T
h
e
co
d
e
b
ased
cr
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p
to
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y
m
ak
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f
co
r
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tin
g
t
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co
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r
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cr
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y
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f
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B
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n
g
to
co
p
y
o
r
alter
th
e
m
es
s
ag
e
s
s
e
n
t
b
y
th
e
m
,
t
h
en
t
h
e
u
s
er
s
w
i
ll
g
et
to
k
n
o
w
t
h
at
s
o
m
eo
n
e
i
s
tr
y
i
n
g
to
co
p
y
th
e
s
h
ar
e
s
ec
r
et
m
e
s
s
a
g
es
[
2
3
,
24
]
.
T
h
is
t
y
p
e
o
f
attac
k
is
p
r
ev
en
t
ed
u
s
in
g
t
h
e
B
B
8
4
p
r
o
to
c
ol
.
I
t is illu
s
tr
ated
i
n
Fi
g
u
r
e
2
w
it
h
A
lice
a
n
d
B
o
b
ex
am
p
le
.
Fig
u
r
e
2
.
B
B
8
4
p
r
o
to
co
l
T
h
e
o
th
er
f
e
w
q
u
an
t
u
m
p
r
o
to
co
ls
ar
e
as f
o
llo
w
s
,
a.
B
B
M
[
25
]
–
th
e
en
tan
g
le
m
e
n
t
t
y
p
e
o
f
B
B
8
4
p
r
o
to
co
l
.
b.
E
9
1
[
2
6
]
–
d
ep
en
d
u
p
o
n
Ged
a
n
k
e
n
ex
p
er
i
m
e
n
t a
n
d
i
n
th
e
e
x
ten
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io
n
o
f
B
en
n
ett
a
n
d
B
r
ass
ar
d
.
c.
S
A
R
G0
4
[
22
,
2
3
]
-
s
i
m
ilar
to
B
B
8
4
.
SA
R
G0
4
m
o
r
e
in
u
s
e
th
a
t
o
f
B
B
8
4
o
v
er
th
e
P
h
o
to
n
Nu
m
b
e
r
Sp
litt
in
g
attac
k
(
P
NS)
.
d.
Six
-
Sta
te
P
r
o
to
co
l
[
24
,
27
]
–
also
a
t
y
p
e
o
f
B
B
8
4
p
r
o
t
o
co
l
w
h
ic
h
u
s
es
t
h
e
Si
x
S
tate
p
h
o
t
o
n
p
o
lar
izatio
n
tech
n
iq
u
e.
e.
Six
S
tate
is
also
a
t
y
p
e
o
f
th
e
S
A
R
G0
4
co
d
in
g
tec
h
n
iq
u
e
[
28
].
f.
Sin
g
ap
o
r
e
P
r
o
to
co
l
[
29
]
–
it
is
a
to
m
o
g
r
ap
h
y
r
ep
r
ese
n
tatio
n
w
h
ic
h
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s
h
i
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h
l
y
ef
f
icie
n
t
th
en
t
h
e
Six
Sta
te
P
r
o
to
c
o
l.
g.
B
9
2
P
r
o
to
c
o
l
[
30
]
-
u
s
es
th
e
t
w
o
s
tates
o
f
t
h
e
q
u
a
n
t
u
m
b
y
t
h
e
v
er
y
lo
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n
s
it
y
w
h
ich
i
s
co
h
er
en
t
w
it
h
th
e
p
u
l
s
es o
f
t
h
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li
g
h
t.
8.
Q
UAN
T
UM
CRYP
T
O
G
RA
P
H
Y
AL
G
O
R
I
T
H
M
S
T
h
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u
r
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t c
r
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p
to
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r
ap
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i
v
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b
asis
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Gr
o
v
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d
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e
S
h
o
r
’
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al
g
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ith
m
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if
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lt
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o
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Di
s
cr
ete
L
o
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ar
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h
m
v
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l
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er
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le.
8
.
1
.
Sh
o
r’
s
a
lg
o
rit
h
m
P
eter
Sh
o
r
,
th
e
m
at
h
e
m
aticia
n
“A
l
g
o
r
ith
m
s
f
o
r
t
h
e
Q
u
an
t
u
m
C
o
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p
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ta
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is
cr
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ar
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m
an
d
Facto
r
izatio
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,
q
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a
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llap
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ac
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t
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f
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in
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w
s
t
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f
ac
to
r
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n
p
r
o
ce
s
s
[
31
]
.
L
et
u
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
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p
E
n
g
,
Vo
l.
10
,
No
.
1
,
Feb
r
u
ar
y
2
0
2
0
:
4
0
7
-
414
412
ass
u
m
e,
th
e
n
u
m
b
er
1
5
.
I
t
is
n
ee
d
to
th
e
4
q
u
b
it
r
e
g
is
ter
f
o
r
th
is
n
u
m
b
er
.
T
h
e
r
eg
u
lar
4
b
it
r
eg
is
ter
is
n
ee
d
ed
f
o
r
th
e
co
m
p
u
tat
io
n
o
f
th
e
co
m
p
u
ter
s
.
T
h
e
b
in
ar
y
r
ep
r
esen
t
atio
n
o
f
th
e
n
u
m
b
er
1
5
is
1
1
1
1
.
T
h
e
co
m
p
u
tat
io
n
o
f
th
e
ca
lc
u
latio
n
s
t
h
at
ar
e
p
er
f
o
r
m
ed
o
n
t
h
e
r
eg
is
ter
s
ar
e
d
o
n
e
in
p
ar
allel
b
y
C
astro
an
d
B
o
n
e.
T
h
e
m
aj
o
r
s
tep
th
at
is
d
o
n
e
i
n
Q
u
an
t
u
m
C
r
y
p
t
o
g
r
ap
h
y
to
b
e
p
er
f
o
r
m
ed
.
A
s
ec
o
n
d
4
-
q
u
b
it
r
e
g
is
ter
is
u
s
ed
to
s
to
r
e
th
e
r
e
m
ai
n
d
er
f
r
o
m
t
h
i
s
o
p
er
atio
n
.
T
h
e
s
u
p
er
p
o
s
itio
n
r
esu
lt
s
ar
e
n
o
w
i
n
t
h
e
s
ec
o
n
d
r
e
g
is
ter
.
L
et
u
s
ta
k
e,
X=
2
,
is
s
m
aller
th
a
n
t
h
e
n
u
m
b
er
1
4
b
u
t
lar
g
er
t
h
at
th
e
n
u
m
b
er
1
.
T
h
e
x
is
r
aised
to
th
e
p
o
w
er
o
f
th
e
4
q
u
b
it,
th
a
t
ar
e
th
e
n
u
m
b
er
s
m
a
x
i
m
u
m
o
f
1
5
an
d
ar
e
d
iv
id
ed
b
y
1
5
an
d
th
e
r
e
m
ain
d
er
s
ar
e
s
to
r
ed
in
th
e
r
eg
i
s
ter
s
a
s
s
h
o
w
n
i
n
t
h
e
T
ab
le
2
.
I
n
th
e
o
b
s
er
v
at
io
n
,
th
e
r
esu
lts
ar
e
th
e
n
u
m
b
er
4
i
n
t
h
e
s
eq
u
e
n
ce
.
C
o
n
f
id
en
tl
y
s
a
y
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g
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h
at
s
eq
u
e
n
ce
n
u
m
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er
f
=4
,
w
h
e
n
t
h
e
X
=
2
an
d
t
h
e
n
u
m
b
er
,
n
=
1
5
.
T
o
ca
lcu
la
te
th
e
p
o
s
s
ib
le
f
ac
to
r
o
f
th
e
f
,
th
e
p
o
s
s
ib
le
f
ac
to
r
: P
=
X
f
/2
-
1.
T
ab
le
2
.
4
-
Qu
b
it r
eg
is
ter
s
w
i
t
h
r
e
m
ai
n
d
er
s
R
e
g
i
st
e
r
0
1
2
3
4
5
6
7
8
9
1
1
1
1
1
1
1
0
1
2
3
4
5
R
e
g
i
st
e
r
1
2
4
8
1
2
4
8
1
2
4
8
1
2
4
8
2
T
h
e
f
o
llo
w
i
n
g
ar
e
d
o
n
e
i
n
th
e
alg
o
r
ith
m
ar
e
as f
o
llo
w
s
,
a.
L
et
n
=1
5
,
n
u
m
b
er
to
b
e
f
ac
to
r
ized
.
b.
X
=
n
u
m
b
er
in
r
an
d
o
m
,
c
h
o
o
s
en
f
r
o
m
1
<
x
<n
-
1.
c.
X
th
e
p
o
w
er
w
h
ich
i
s
r
aised
.
T
h
at
ar
e
s
to
r
ed
in
th
e
r
eg
is
ter
s
.
An
d
th
e
n
d
iv
id
ed
b
y
th
e
p
r
i
m
e
n
u
m
b
er
n
.
T
h
e
r
esu
lts
s
o
m
eti
m
es
o
b
tain
ed
w
h
ic
h
ar
e
n
o
t
t
h
e
p
r
i
m
e
n
u
m
b
er
s
.
So
,
r
ep
ea
t
th
e
f
v
a
lu
es
w
i
th
th
e
d
i
f
f
er
en
t
ca
lcu
latio
n
s
.
T
h
e
Dis
cr
ete
L
o
g
ar
ith
m
P
r
o
b
lem
s
ar
e
co
m
p
u
ted
b
y
t
h
e
S
h
o
r
’
s
a
lg
o
r
it
h
m
.
I
t
i
s
s
h
o
w
ed
t
h
at
t
h
e
p
er
f
o
r
m
a
n
ce
o
f
th
e
ca
lcu
latio
n
ar
e
d
o
n
e
f
r
o
m
t
h
e
s
tar
tin
g
r
a
n
d
o
m
s
u
p
er
p
o
s
itio
n
o
f
t
h
e
i
n
te
g
er
o
f
t
w
o
s
tate
s
,
an
d
also
p
er
f
o
r
m
s
th
e
Q
u
an
t
u
m
Fo
u
r
ier
T
r
an
s
f
o
r
m
.
T
h
e
g
iv
e
n
eq
u
ati
o
n
s
ar
e
s
atis
f
ied
b
y
th
e
n
e
w
s
u
p
er
p
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s
itio
n
s
tates
t
o
g
iv
e
t
h
e
h
i
g
h
p
r
o
b
ab
ilit
y
o
f
th
e
t
w
o
i
n
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er
s
.
T
h
e
u
n
k
n
o
w
n
“
e
x
p
o
n
e
n
t”
o
f
th
e
Dis
cr
ete
L
o
g
ar
it
h
m
P
r
o
b
le
m
ca
lcu
lated
u
s
in
g
t
h
is
eq
u
at
i
o
n
to
f
in
d
t
h
e
v
al
u
e
r
.
8
.
2
.
G
ro
v
er
’
s
a
lg
o
rit
h
m
T
h
e
Gr
o
v
er
’
s
a
lg
o
r
it
h
m
t
h
a
t
cr
ea
ted
b
y
t
h
e
m
at
h
e
m
ati
cian
,
L
o
v
Gr
o
v
er
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s
u
s
ed
m
ai
n
l
y
b
y
th
e
Q
u
a
n
tu
m
C
o
m
p
u
ter
f
o
r
s
e
a
r
ch
in
g
t
h
e
u
n
s
o
r
ted
d
atab
ases
[
3
2
]
.
T
h
e
u
n
s
o
r
ted
d
atab
ase
o
f
th
e
N
e
n
tr
ies
a
n
d
in
N
s
ea
r
c
h
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w
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th
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e
Sh
o
r
’
s
al
g
o
r
ith
m
co
u
ld
ab
le
to
f
i
n
d
p
.
T
h
e
s
a
m
e
en
tr
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ld
b
e
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ea
r
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th
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n
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co
m
p
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ter
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ld
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d
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=
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s
ea
r
ch
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s
.
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e
Data
E
n
cr
y
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tio
n
Sta
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d
ar
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th
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o
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r
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s
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ed
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5
6
b
it
k
e
y
.
I
t
is
s
tated
t
h
at
it
n
ee
d
s
th
e
1
8
5
s
ea
r
ch
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f
o
r
f
i
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h
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b
its
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.
T
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n
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m
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o
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k
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y
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its
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in
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p
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e
a
ttack
er
s
.
I
t
r
es
u
lted
t
h
at,
t
h
e
i
n
cr
ea
s
e
in
t
h
e
e
x
p
o
n
e
n
tial
ar
e
h
ig
h
f
o
r
th
e
n
u
m
b
er
o
f
t
h
e
s
ea
r
ch
es
to
cr
ac
k
i
n
g
a
p
ass
w
o
r
d
.
T
h
e
Gr
o
v
er
’
s
alg
o
r
it
h
m
i
s
less
f
ast
t
h
an
th
e
S
h
o
r
’
s
al
g
o
r
ith
m
.
9.
CO
NCLU
SI
O
N
C
o
m
p
ar
ed
w
it
h
cla
s
s
ical
cr
y
p
to
g
r
ap
h
y
,
Q
u
a
n
tu
m
cr
y
p
to
g
r
ap
h
y
h
as
u
l
ti
m
ate
ad
v
a
n
ta
g
es
t
h
at
ar
e
u
n
co
n
d
itio
n
al
s
n
i
f
f
i
n
g
d
etec
t
io
n
an
d
th
e
s
ec
u
r
it
y
.
T
h
ese
ch
ar
ac
ter
is
tics
ca
n
s
o
l
v
e
s
e
cu
r
it
y
p
r
o
b
le
m
f
o
r
th
e
f
u
t
u
r
e
I
n
ter
n
et.
I
n
p
ar
ticu
l
ar
,
it
p
r
o
v
id
es
s
ec
u
r
it
y
f
o
r
v
ar
io
u
s
ap
p
licatio
n
s
li
k
e
I
n
ter
n
et
o
f
th
in
g
s
(
s
en
d
i
n
g
m
es
s
ag
e
s
,
d
ata
ac
ce
s
s
,
s
a
v
i
n
g
th
e
f
ile
s
w
it
h
cr
y
p
to
g
r
ap
h
y
)
Secu
r
it
y
a
n
d
C
o
m
m
u
n
ica
tio
n
Net
w
o
r
k
s
f
o
r
th
e
f
u
tu
r
e
I
n
ter
n
e
t.
Q
u
an
t
u
m
cr
y
p
to
g
r
ap
h
y
r
e
s
u
l
ts
s
h
o
w
t
h
e
u
n
co
n
d
itio
n
al
s
ec
u
r
it
y
an
d
s
n
if
f
i
n
g
(
ea
v
esd
r
o
p
p
in
g
)
d
etec
tio
n
o
f
q
u
an
t
u
m
cr
y
p
to
g
r
ap
h
y
,
w
h
ich
m
ak
e
s
it s
u
itab
le
f
o
r
f
u
tu
r
e
I
n
t
er
n
et.
RE
F
E
R
E
NC
E
S
[1
]
C.
H.
Be
n
n
e
tt
a
n
d
G
.
Bra
ss
a
rd
,
“
Qu
a
n
tu
m
Cr
y
p
to
g
ra
p
h
y
:
p
u
b
li
c
k
e
y
d
istri
b
u
ti
o
n
a
n
d
c
o
i
n
to
ss
i
n
g
,
”
IEE
E
Co
n
fer
e
n
c
e
o
n
C
o
mp
u
ter
,
S
y
ste
ms
,
a
n
d
S
i
g
n
a
l
Pro
c
e
ss
in
g
,
p
p
.
1
7
5
-
90
,
1
9
8
4
.
[2
]
C.
Do
d
s,
e
t
a
l
.
,
“
Ha
sh
Ba
se
d
Dig
i
tal
S
ig
n
a
tu
re
S
c
h
e
m
e
s,”
Cry
p
to
g
r
a
p
h
y
a
n
d
C
o
d
i
n
g
,
v
o
l.
3
7
9
6
,
p
p
.
9
6
-
1
1
5
,
2
0
0
5
.
[3
]
W
.
Diff
ie
a
n
d
M
.
E.
He
ll
m
a
n
,
“
Ne
w
Dire
c
ti
o
n
sin
Cry
p
to
g
ra
p
h
y
,
”
IEE
E
T
ra
n
sa
c
ti
o
n
s i
n
I
n
fo
rm
a
ti
o
n
.
[4
]
D.
M
icc
ian
c
io
,
“
L
a
tt
ice
-
Ba
se
d
Cry
p
to
g
ra
p
h
y
,
”
Po
st
-
Qu
a
n
tu
m Cry
p
to
g
ra
p
h
y
,
v
o
l
.
0
1
5
8
4
8
,
p
p
.
1
4
7
-
1
9
2
,
2
0
0
9
.
[5
]
S
.
V.
M
a
n
ik
a
n
t
h
a
n
a
n
d
T
.
P
a
d
m
a
p
riy
a
,
“
A
S
e
c
u
re
d
M
u
lt
i
-
L
e
v
e
l
Ke
y
M
a
n
a
g
e
m
e
n
t
T
e
c
h
n
iq
u
e
f
o
r
In
ten
sif
ied
W
irele
ss
S
e
n
so
r
Ne
t
w
o
rk
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
Rec
e
n
t
T
e
c
h
n
o
lo
g
y
a
n
d
E
n
g
i
n
e
e
rin
g
,
v
o
l.
7
,
2
0
1
9
.
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
Qu
a
n
tu
m
cryp
to
g
r
a
p
h
y
fo
r
s
e
cu
r
ed
co
mmu
n
ica
tio
n
n
etw
o
r
ks (
B
.
Mu
r
u
g
a
n
a
n
th
a
m)
413
[6
]
D.
Bru
ss
,
e
t
a
l
.,
“
Q
u
a
n
tu
m
c
r
y
p
to
g
ra
p
h
y
:
A
su
rv
e
y
,
”
ACM
Co
mp
u
ti
n
g
S
u
rv
e
y
s
,
v
o
l.
3
9
,
p
p
.
1
-
27
,
2
0
0
7
.
[7
]
C.
H.
Be
n
n
e
t
t,
e
t
a
l
.,
“
Ex
p
e
rim
e
n
t
a
l
q
u
a
n
t
u
m
c
r
y
p
to
g
ra
p
h
y
,
”
J
o
u
rn
a
l
o
fCry
p
t
o
lo
g
y
,
v
o
l.
5
,
p
p
.
3
-
28
,
1
9
9
2
.
[8
]
G
.
J.
S
imm
o
n
,
“
S
y
m
m
e
tri
c
a
n
d
a
sy
m
m
e
tri
c
e
n
c
r
y
p
ti
o
n
,
”
ACM
Co
m
p
u
ti
n
g
S
u
r
veys
,
v
o
l.
1
1
,
p
p
.
3
0
5
-
3
3
0
,
1
9
7
9
.
[9
]
D.
Bru
ss
,
e
t
a
l
.
,
“
Q
u
a
n
tu
m
c
r
y
p
to
g
ra
p
h
y
:
A
su
rv
e
y
,
”
ACM
Co
mp
u
ti
n
g
S
u
rv
e
y
s
,
v
o
l.
3
9
,
p
p
.
1
-
27
,
2
0
0
7
.
[1
0
]
A
.
S
e
n
,
e
t
a
l
.
,
“
Bit
lev
e
ls
y
m
m
e
tri
c
k
e
y
c
r
y
p
to
g
ra
p
h
y
u
sin
g
G
e
n
e
ti
c
A
l
g
o
rit
h
m
,
”
CS
NT
,
p
p
.
37
,
2
0
1
7
.
[1
1
]
N.
P
a
p
a
n
ik
o
la
o
u
,
“
A
n
in
tro
d
u
c
ti
o
n
to
q
u
a
n
tu
m
c
r
y
p
to
g
ra
p
h
y
,
”
ACM
Cro
ss
ro
a
d
s M
a
g
a
zi
n
e
,
v
o
l.
1
1
,
p
p
.
1
-
16
,
2
0
0
5
.
[1
2
]
E.
F
.
De
tt
re
y
a
n
d
E.
A
.
Y
f
a
n
ti
s,
“
A
Ne
w
El
li
p
ti
c
Cu
rv
e
Cr
y
p
to
g
ra
p
h
icA
lg
o
rit
h
m
,
”
Co
mp
u
ti
n
g
a
n
d
C
o
mm
u
n
ic
a
ti
o
n
W
o
rk
sh
o
p
a
n
d
Co
n
fer
e
n
c
e
(
CCW
C),
IEE
E
.
2
0
1
8
.
[1
3
]
R.
Ba
la
m
u
ru
g
a
n
,
e
t
a
l
.
,
“
En
h
a
n
c
i
n
g
se
c
u
rit
y
in
tex
t
m
e
ss
a
g
e
u
sin
g
m
a
tri
x
b
a
se
d
m
a
p
p
in
g
a
n
d
El
g
a
m
a
l
m
e
th
o
d
in
El
li
p
ti
c
Cu
rv
e
Cry
p
to
g
ra
p
h
y
,
”
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
C
o
n
t
e
mp
o
ra
ry
Co
m
p
u
ti
n
g
a
n
d
In
f
o
r
ma
ti
c
s
(
IC3
1
),
IEE
E
,
2
0
1
4
.
[1
4
]
L
.
I.
Jia
n
,
e
t
a
l
.
,
“
A
S
u
rv
e
y
o
n
Qu
a
n
tu
m
Cr
y
p
to
g
ra
p
h
y
,
”
v
ol
.
27,
2
0
1
8
.
[1
5
]
A
.
S
in
g
h
,
“
Ce
n
tralize
d
Ke
y
Distri
b
u
ti
o
n
o
n
Qu
a
n
tu
m
Cr
y
p
to
g
ra
p
h
y
,
”
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
fCo
mp
u
ter
S
c
ien
c
e
a
n
d
M
o
b
i
le Co
mp
u
ti
n
g
(
IJ
CS
M
C)
,
v
o
l.
6
,
2
0
1
7
.
[1
6
]
O
.
Ca
n
g
e
a
,
e
t
a
l
.
,
“
I
m
p
le
m
e
n
ti
n
g
Qu
a
n
tu
m
Cr
y
p
to
g
ra
p
h
y
fo
r
Da
ta
S
e
c
u
rit
y
,
”
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
El
e
c
tro
n
ics
,
Co
m
p
u
ter
s a
n
d
Ar
ti
fi
c
ia
l
I
n
telli
g
e
n
c
e
(
ECA
I)
,
2
0
1
7
.
[1
7
]
Y
.
W
a
n
g
a
n
d
Ku
n
S
h
e
,
“
A
P
ra
c
ti
c
a
l
Qu
a
n
t
u
m
P
u
b
li
c
Ke
y
E
n
c
ry
p
ti
o
n
M
o
d
e
l
,
”
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
In
fo
rm
a
t
io
n
M
a
n
a
g
e
me
n
t
,
2
0
1
7
.
[1
8
]
T
.
Zh
o
u
,
e
t
a
l
.
,
“
Qu
a
n
tu
m
Cry
p
to
g
ra
p
h
y
f
o
r
th
e
F
u
t
u
re
In
te
r
n
e
t
a
n
d
th
e
S
e
c
u
rit
y
A
n
a
l
y
s
is
,
”
2018
.
[1
9
]
R
.
G
o
e
l,
e
t
a
l
.
,
“
Re
se
a
rc
h
Dire
c
ti
o
n
s o
n
Qu
a
n
tu
m
Crp
to
g
ra
p
h
y
,
”
In
ter
n
a
ti
o
n
a
lC
o
n
fer
e
n
c
e
o
n
I
n
f
o
rm
a
ti
o
n
T
e
c
h
n
o
l
o
g
y
(
IT
NG’0
7
)
,
2
0
0
7
.
[2
0
]
J.
Din
g
a
n
d
B.
Y.
Ya
n
g
,
“
M
u
lt
i
v
a
riate
P
u
b
li
c
Ke
y
Cr
y
p
to
g
ra
p
h
y
,
”
Po
st
-
Qu
a
n
t
u
m
Cry
p
to
g
ra
p
h
y
,
p
p
.
1
9
3
-
2
4
1
,
2
0
0
9
.
[2
1
]
R.
Ov
e
rb
e
c
k
a
n
d
N.
S
e
n
d
rier,
“
Co
d
e
-
b
a
se
d
Cry
p
to
g
ra
p
h
y
,
”
Po
s
t
-
Qu
a
n
t
u
m
Cry
p
to
g
ra
p
h
y
,
Be
rli
n
,
He
id
e
lb
e
rg
:
S
p
rin
g
e
r
Be
rli
n
He
id
e
lb
e
rg
,
p
p
.
9
5
-
1
4
5
,
2
0
0
9
.
[2
2
]
V
.
S
c
a
ra
n
i,
e
t
a
l
.
,
“
Qu
a
n
t
u
m
c
r
y
p
to
g
ra
p
h
y
p
ro
to
c
o
ls
ro
b
u
st
a
g
a
in
st
p
h
o
t
o
n
n
u
m
b
e
r
sp
li
tt
i
n
g
a
tt
a
c
k
s
f
o
r
w
e
a
k
l
a
se
r
p
u
lse
im
p
lem
e
n
tatio
n
s,”
P
h
y
sic
a
l
re
v
iew l
e
tt
e
rs
,
v
o
l.
9
2
,
p
p
.
0
5
7
9
0
1
,
2
0
0
4
.
[2
3
]
A
.
Ac
in
,
e
t
a
l
.
,
“
Co
h
e
re
n
t
-
p
u
lse
im
p
le
m
e
n
tatio
n
s
o
f
q
u
a
n
tu
m
c
r
y
p
to
g
ra
p
h
y
p
ro
to
c
o
ls
re
sis
tan
t
to
p
h
o
to
n
-
n
u
m
b
e
r
-
sp
li
tt
i
n
g
a
tt
a
c
k
s,”
Ph
y
sic
a
l
Rev
iew A
,
v
o
l.
6
9
,
p
p
.
0
1
2
3
0
9
,
2
0
0
4
.
[2
4
]
C.
Be
n
n
e
tt
a
n
d
G
.
Bra
ss
a
rd
,
“
Q
u
a
n
tu
m
c
r
y
p
to
g
ra
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414
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