Cryptography Reference
In-Depth Information
where the sum of the points in div(
g
) equals
¥
. Hence,
åå
¢
(3.57)
div(
g
)
=
[
L
]
[
M
]
¢=
=¥
nL
L
nM
Let
f
denote the function such that it starts with a point,
n
is multiplied with the
point, and then the function is applied. Let
P
=
L
+
M
with
ME
Î
those points
P
[]
with
nP
=
L
. Hence, it follows from Eq. (3.55) that
æ
ö æ
ö
÷
÷
ç
åå
ç
÷
÷
ç
ç
¢
n
div(
f
n n
)
=
[
L M
+ -
]
n
[
M
]
=
div(
g
)
(3.58)
÷
÷
ç
ç
÷
÷
ç
ç
ç
÷
ç
÷
è
ø è
ø
M
M
Therefore,
n
(3.59)
=
f
n g
Î
QE
Î
. Then
[]
Let
PEK
()
and
n
n
+= += =
gP
(
Q
)
f nP
( (
Q
))
f nP
(
)
gP
(
)
(3.60)
Hence, we define Weil pairing as
gP
(
+
Q
)
=
Î
μ
eQL
(,)
(3.61)
n
n
gP
()
3.9.4 Bilinear Property
Let
G
1
and
G
2
be two groups of order
q
, where
q
is a large prime. Then the bilinear map
e
:
G
1
´
G
2
G
2
satisfies the following properties.
1. Bilinear: The map
e
:
G
1
´
G
2
G
2
is bilinear if
ab
=
" Î
Î
eaPbQ
(,
) (,
ePQ
) ,
ab
Z
d
PQ G
,
(3.62)
1
2. Nondegenerate: All pairs in
G
1
´
G
1
do not map to the identity of
G
2
. Let
P
be
the generator of
G
1
. Then
e
(
P,P
) is a generator of
G
2
, because
G
1
,
G
2
are prime
order groups.
Î
3. Efficient algorithms exist to compute
e
(
P,Q
), where
PQ
,
.
1
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