Re: Elliptic Curve over binary field

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Fri, Jun 10, 2005 at 10:05:25AM -0700, Hauw Suwito wrote:
> Greetings to PARI experts,
> I'm new to PARI, and would like to know how
> to use ellinit to create elliptic curve over a binary field (F_2^m).
> Any pointer to a document would be also appreciated.

Try the following:

1) Compute a polynomial (in w) defining your field:
F=ffinit(2,m,w);
2) Set a to w mod F
a=Mod(w,F);
3.a) Define your curve in term of a:
E=ellinit([0,0,a,0,a+1]);
3.b) If your curve is defined of F_2, simply multiply it by a^0 instead:
E=ellinit([0,0,1,0,1]*a^0);

Now you can do operations on points of E as usual:
? elladd(E,[1,a],[0,1])
%4 = [Mod(w^2 - 2*w, Mod(1, 2)*w^3 + Mod(1, 2)*w^2 + Mod(1, 2)), Mod(Mod(1, 2),
Mod(1, 2)*w^3 + Mod(1, 2)*w^2 + Mod(1, 2))]

Cheers,
Bill.