Cryptography Reference
In-Depth Information
Chapter 14
Zeta Functions
14.1 Elliptic Curves over Finite Fields
Let
E
be an elliptic curve over a finite field
F
q
.Let
N
n
=#
E
(
F
q
n
)
be the number of points on
E
over the field
F
q
n
.The
Z
-function of
E
is
defined to be
Z
E
(
T
)=exp
∞
T
n
.
N
n
n
n
=1
Here exp(
t
)=
t
n
/n
! is the usual exponential function. The
Z
-function
encodes certain arithmetic information about
E
as the coe
cients of a gen-
erating function. The presence of the exponential function is justified by the
simple form for
Z
E
(
T
) in the following result.
PROPOSITION 14.1
Let
E
be an elliptic curve defined over
F
q
,and let
#
E
(
F
q
)=
q
+1
−a
.Then
qT
2
− aT
+1
(1
− T
)(1
− qT
)
.
Z
E
(
T
)=
Factor
X
2
PROOF
− aX
+
q
=(
X − α
)(
X − β
). Theorem 4.12 says that
N
n
=
q
n
+1
− α
n
− β
n
.
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