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