Re: EC arithmetic

[Karim BELABAS <Karim.Belabas@math.u-psud.fr>]

On Tue, 9 Oct 2001, Erik Schlegel wrote:

> Hello!
>
> I'm new in Pari. I need some instructions about EC arithmetic.
>
> 1) GP>E=ellinit(Mod([0,0,0,71,602],1009))
> Is this the only way to define any elliptic curve over finite field
> F_p? What about over F_(p^n) or F_(2^n)?

The elliptic curve module is rather primitive: over F_p^n, you won't be able to
do much besides adding points [and it will be slow]. You can input such a curve
more or less as above, but using POLMODs in INTMODs [ i.e working over F_p^n
understood as Fp[X]/(T) ]

 E = ellinit([0,0,0,71,602] * Mod(Mod(1,1009), Mod(1,1009)*(x^2+1)))

It is advised to write trivial scripts to facilitate such inputs, e.g

global(ff1);

INIT(p,T)= ff1 = Mod(1, Mod(1,p) * T);

toff(x) = x * ff1

and results can be made more readable via repeated lifts, e.g

Lift(x) = lift(lift(lift(x)))  \\ one too many, but just in case...

Hope this helps,

    Karim.
-- 
Karim Belabas                    email: Karim.Belabas@math.u-psud.fr
Dep. de Mathematiques, Bat. 425
Universite Paris-Sud             Tel: (00 33) 1 69 15 57 48
F-91405 Orsay (France)           Fax: (00 33) 1 69 15 60 19
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