Normal basis of a large field

We create a normal basis for and observe how exponentiation alters the vector components of an element.

> k<w> := GF(2, 1000);
> time IsNormal(w);
false
Time: 17.590
> time e := NormalElement(k);
Time: 78.279
> time B := [ IsZero(i) select e else Self(i)^2 : i in [0..999] ];
Time: 17.289
> time VS, iso := VectorSpace(k, GF(2), B); 
Time: 1.569
> VS, iso;
Full Vector space of degree 1000 over GF(2)
Mapping from: FldFin: k to ModTupFld: VS

We choose an element of k and map it to VS.

> r := w^947 + w^215 + w^32 + w^2 + 1;    
> rVS := r @ iso;

Now we rotate the vector by three places, and map back to k.

> rot3 := (VS ! Rotate(Eltseq(rVS), 3)) @@ iso;
> rot3;
w^765 + w^764 + w^763 + w^760 + w^201 + w^200 + w^199 + w^197 + w^195 + 
    w^193 + w^191 + w^188 + w^181 + w^180 + w^179 + w^176 + w^72 + 1
> rot3 eq r^(2^3);
true

Previous Group: Lattice of finite fields