Mc Laughlin, James on Wed, 02 Jun 2004 21:32:24 +0200

 RE: Elliptic curves over finite fields

Thanks to everyone who responded.
I guess I was not totally clear.
I use to have access to magma and there was a command there that would output what I wanted directly- something like the following, for example:
p:=1123;
r:=3;
E := EllipticCurve([GF(p,r) | 0, 0, 0, 0, 19 ]);
ord:=Order(E);

ord would be what I wanted.

I have Koblitz's book "Algebraic Aspects of Cryptography" and know the formula some have referred to (Corollary 1.1, page 126):
set N = #E(F_p), a=p+1-N and let \alpha, \hat{\alpha} be the roots of T^2-aT+p. Then
#E(F_p^r) = p^r+1-\alpha^r-\hat{\alpha}^r.

I know a few extra lines of code would do what I wanted. What I wondered was if there a command like "Order" in Magma which would do the same thing. Seems like there should be.

Jimmy.

> Suppose p is a prime.
>
> f=[a1,a2,a3,a4,a6];
> E=ellinit(f);
>
> Then
>
> ellap(E,p)
>
> measures  p+1- #E(F_p) and thus gives a count of #E(F_p).
>
> Is there a way of getting #E(F_p^r), for r>1?
>
> Thanks,
>
> Jimmy Mc Laughlin.
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