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
Search WWH ::




Custom Search