Cryptography Reference
In-Depth Information
Pages 24-26 of Charlap and Coley [
118
] present a generalised Weil reciprocity that does
not require the divisors to have disjoint support.
26.2 The Weil pairing
The Weil pairing plays an important role in the study of elliptic curves over number fields,
but tends to be less important in cryptography. For completeness, we briefly sketch its
definition.
Let
E
be an elliptic curve o
ve
r
k
and let
n
∈ N
be coprime to char(
k
). Let
P,Q
∈
E
[
n
].
O
E
). Let
Q
∈
Then there is a function
f
∈ k
(
E
) such that div(
f
)
=
n
(
Q
)
−
n
(
E
(
k
)be
any point such that [
n
]
Q
=
Q
, and so [
n
2
]
Q
=
O
E
. Note that [
n
] is unramified and the
[
n
]
∗
((
Q
)
divisor
D
=
−
(
O
E
)) is equal to
(
Q
+
R
)
−
(
R
)
.
R
∈
E
[
n
]
Since
R
∈
E
[
n
]
R
=
O
E
and [
n
2
]
Q
=
O
E
it
fo
llows from Theorem
7.9.9
that
D
is a
principal divisor. So there is a function
g
[
n
]
∗
((
Q
)
∈ k
(
E
) such that div(
g
)
=
D
=
−
O
E
)). Now, consider the function [
n
]
∗
f
[
n
]. One has div([
n
]
∗
f
)
[
n
]
∗
(div(
f
))
(
=
f
◦
=
=
[
n
]
∗
(
n
(
Q
)
[
n
] and
g
n
have the same divis
or
.
−
n
(
O
E
))
=
nD
. Hence, the functions
f
◦
g
n
. Now, for any point
U
Multiplying
f
by a suitable constant gives
f
◦
[
n
]
=
∈
E
(
k
)
E
[
n
2
]wehave
such that [
n
]
U
∈
P
)
n
g
(
U
)
n
.
g
(
U
+
=
f
([
n
]
U
+
[
n
]
P
)
=
f
([
n
]
U
)
=
In other words,
g
(
U
+
P
)
/g
(
U
)isan
n
th root of unity in
k
.
Lemma 26.2.1
Let the not
a
tion be as above. Then g
(
U
+
P
)
/g
(
U
)
is independent of the
choice of the point U
∈
E
(
k
)
.
Proof
See Section 11.2 of Washington [
560
]. The proof is described as “slightly technical"
and uses the Zariski topology.
k
∈ N
Definition 26.2.2
Let
E
be an elliptic curve over a field
and let
n
be such that
gcd(
n,
char(
k
))
=
1. Define
∗
:
z
n
µ
n
={
z
∈ k
=
1
}
.
The
Weil pairing
is the function
e
n
:
E
[
n
]
×
E
[
n
]
→
µ
n
defined (using the notation above) as
e
n
(
P,Q
)
=
g
(
U
+
P
)
/g
(
U
) for any point
U
∈
E
(
k
),
E
[
n
2
] and where div(
g
)
[
n
]
∗
((
Q
)
U
∈
=
−
(
O
E
)).
