Cryptography Reference
In-Depth Information
varieties over finite fields. The functional equation was proved in the 1960s
by M. Artin, Grothendieck, and Verdier, and the analogue of the Riemann
Hypothesis was proved by Deligne in 1973. Much of Grothendieck's algebraic
geometry was developed for the purpose of proving these conjectures.
Finally, we show how ζ E ( s ) can be defined in a way similar to the Riemann
zeta function. Recall that the Riemann zeta function has the Euler product
expansion
1
p s 1
ζ ( s )=
p
1
when
( s ) > 1. The product is over the prime numbers. We obt ai n ζ E ( s )if
we replace the primes p by points on E . Consider a point P
E ( F q ). Define
deg( P ) to be the smallest n such that P
E ( F q n ). The Frobenius map φ q
acts on P , and it is not dicult to show that the set
P, φ q ( P ) q ( P ) ,...,φ n− 1
S P =
{
( P )
}
q
has exactly n =deg( P )elementsandthat φ q ( P )= P . Eachofthepointsin
S P also has degree n .
PROPOSITION 14.3
Let E be an elliptic curve over F q .Then
1
q s deg( P ) 1
ζ E ( s )=
S P
1
,
where the product is over the points P ∈ E ( F q ) ,butwetake onlyonepoint
fro m
ea ch set S P .
PROOF If deg( P )= m ,then P and all the other points in S P have
coordinates in F q m .Since F q m
F q n if and only if m|n , we see that S P
contributes m points to N n =# E ( F q n ) if and only if m|n , and otherwise it
contributes no points to N n . Therefore,
N n =
m
m.
|
n
S P
deg( P )= m
 
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