Kaz Kobara on Sat, 01 Feb 2014 03:31:44 +0100

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RE: Your help needed

Hi Firas,

Thank you for your explanation and examples.
That became clear to me.

A short Weierstrass equation is always singular (and thus never defines an
elliptic curve) in characteristic 2, so you need to use the long form. Then
pass a generator of the desired finite field (as returned by ffgen()) as the
second argument of ellinit(), like so:

(00:07) gp > t = ffgen((t^163+t^7+t^6+t^3+1)*Mod(1,2));
(00:08) gp > E = ellinit([1,0,1,0,1],t);
(00:08) gp > ellgroup(E)
(00:08) gp > E = ellinit([1,0,1,0,1],2);
(00:08) gp > ellgroup(E)

Of course if you want to use an element of F_{2^k}\F_2 in the equation of
your curve you can use it (for example as a polynomial in t) in the array
you pass to ellinit() (then you don't need a second argument):

(00:17) gp > E = ellinit([t,0,t^54+t^3+1,0,1]);
(00:18) gp > ellgroup(E)