Re: Question on elliptic curves...

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

* wraithx@morpheus.net [2023-02-11 04:59]:
> Hello,
> 
> I was wondering how to convert the following sage code to Pari/GP?
> I already have a conversion of the first function FindGroupOrderA, but in
> the function FindGroupOrderParam2, I'm not sure how to do the following two
> lines in Pari/GP:
>    P = s*E(-3,3)
>    x,y = P.xy()

  [x,y] = ellmul(E, [-3,3], s);

CAVEAT: If s is a multiple of the point order, so that s*E(-3,3) is the
point at infinity, you will get a 'non existent component' error instead
of a ZeroDivision error in Sage. If you intend to trap it somewhere,
this can be done using iferr(), producing an e_COMPONENT exception.

Cheers,
  
    K.B.


> Thanks for any help you can provide!
> 
> -David C.
> 
> # Example SAGE code:
> def FindGroupOrderA(p,A):
>    K = GF(p)
>    d = K((A+2)/4)
>    a = K(4*d-2)
>    b = K(16*d+2)
>    E = EllipticCurve(K,[0,a/b,0,1/b^2,0])
>    return E.cardinality()
> 
> # for parameter sigma = 2:s
> def FindGroupOrderParam2(p,s):
>    K = GF(p)
>    E = EllipticCurve(K,[0,36])
>    P = s*E(-3,3)
>    x,y = P.xy()
>    x3 = (3*x+y+6)/(2*(y-3))
>    A = -(3*x3^4+6*x3^2-1)/(4*x3^3)
>    return FindGroupOrderA(p, A)
> 
> #=================================
> # Conversion to Pari/GP:
> FindGroupOrderA(p,A)={
>   my(K, d, a, b, E);
>   K = Mod(1,p);
>   d = K*((A+2)/4);
>   a = K*(4*d-2);
>   b = K*(16*d+2);
>   E = ellinit([0,a/b,0,1/b^2,0],K);
>   return(ellcard(E));
> }
> 
> # for parameter sigma = 2:s
> FindGroupOrderParam2(p,s)={
>   my(K,E,P,x,y,x3,A);
>   K = Mod(1,p);
>   E = ellinit([0,36],K);
>   P = s*E(-3,3); //?????
>   x,y = P.xy(); //?????
>   x3 = (3*x+y+6)/(2*(y-3));
>   A = -(3*x3^4+6*x3^2-1)/(4*x3^3);
>   return(FindGroupOrderA(p, A));
> }
> 

    K.B.
--
Pr Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/
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