Re: Using libpari

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

* Mike Rosing [2015-02-23 19:39]:
> Hello,
> 
> I am trying to use libpari as a stand alone C program rather than creating
> useful functions for gp.  I can do things easily with gp (well, not so easy
> the first time!) but when I try to do similar things in C I get a lot of
> errors.
> 
> What I would like to do is create GF(2^n) elements in "increasing" order:
> 1
> t
> t + 1 t^2
> t^2 + 1
> t^2 + t
> t^2 + t + 1
> etc..

This is not easy ! The "simple" way is to call ffgen + FF_primroot then
compute its powers via 2^n-2 multiplications.

> I start my program, and things are fine:
> 
> #include <pari/pari.h>
> 
> int main()
> {
>   GEN y, E, f, z, a1, a2;
> 
>   pari_init(2*1024*1024, 5*1024*512);
>   z = ffinit(gen_2, 10, -1);
>   y = ffgen(z, -1);

This creates a polynomial z in F_2[x], and create y as (x mod z(x))

>   pari_printf("%Ps %ld\n", type0(y), lg(y));
> 
> ----------------------
> 
> I can give a2 a constant value like gen_1, but I have no idea how to give it
> a value of y, let alone different t_FFELT values.

For specific values:

a2 = y;  // x mod z
a2 = gadd(y, gen_1); // x+1

More complicated:

A = mkpoln(5, gen_1,gen_0,gen_1,gen_0,gen_0); // x^2+x^4
a2 = gsubst(A, gvar(A), y);  // A(x), assuming A is a t_POL with coefs in Z, say

To go through all finite fields elements in the order you requested, you
need some backtracking algorithm emulating forvec(). It's a nice
exercise, but I'd advise against it if you're beginning with libpari :-)

Another (not too wasteful) posibility is to use something like

  // assume 2^n is representable as an ulong
  ulong i, N = upowuu(2,n);
  GEN vy = concat( gpow(y,gen_0), vecpow(y,n-1) ); // vy[i] = y^(i-1)
  for(i=0; i < N; i++)
  {
    GEN h = RgV_sum( shallowextract(vy, utoi(i)) );
    ...
  }

(Easier in characteristic 2, but digits() allows to generalize ...)

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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