P. S. Chakravarthi on Sun, 28 Nov 2004 17:31:09 +0100

 Re: on generation of primitive polynomials

```These functions still expect the base field to be binary.
which is not what I want.
The base field can be of characteristic 2. i.e. of the form
GF(2^k). But I need to pick up primitive polynomials
in  GF(2^k)[x]..   In fact, it will be great if I can pick up
all primitive polynomials UPTO  a certain degree atleast
taking into consideration, the complexity of the field as
k becomes larger.
Any method or function in pari/gp that can help me?
OR any known efficient algorithms (randomized or
deterministic)?

thank you,
chax.

John Kerl wrote:

```
```I believe you want:

factorff(polcyclo(2^n-1,2,y^4+y+1))

Wrap that with lift(lift()) for legibility.

Here is some sample computation:

\\ Inner modulus, defining GF(16)
im=y^4+y+1;
\\ Inner residue
ir=Mod(1,2)*y;
\\ Primitive element for GF(16).  It has order 15.
ia=Mod(ir,im);

\\ Factor the cyclotomic polynomial.  There are 64 of them.
n=8;
pols=factorff(polcyclo(2^n-1),2,im);
length(pols[,1])

\\ Select an outer modulus from the 64 choices.
om=pols[1,1];
lift(lift(om))
\\ Outer residue.
or=Mod(1,2)*x;
\\ Outer element.  Its order is 255 as we are about to see.
oa=Mod(or,om);
lift(lift(lift(oa^255)))
lift(lift(lift(oa^85)))
lift(lift(lift(oa^51)))
lift(lift(lift(oa^15)))

"P. S. Chakravarthi" <chax@cs.iitm.ernet.in> wrote:

```
```hi all.

I'm a newbie to Pari/gp. Is there a way to generate
all primitive polynomials over fields of the type GF(2^x)
For example (01) + (10) x + (11) x^2  being primitive
over GF(4)  in GF(4^2)  (i.e GF(16))
Please observe that this is  _DIFFERENT_ from using
GF(16)  as GF(2^4)  and asking for primitiveness of
something like  1  +  1.x  +  0.x^2  +  0.x^3  +   1. x^4
I do not seem to find a way. Actually I was suggested
in sci.math.research group to use fxt software ..
the author of which, in turn referred me to pari/gp
Any help would be appreciated.

I have pari/gp installed on my linux comp.

cheers,
chax.

--
```
I might not be the brightest bulb in the chandelier, but I'm pretty good at getting most of the other bulbs to light up.
```  -Jack Welch

```
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--
```
I might not be the brightest bulb in the chandelier, but I'm pretty good at getting most of the other bulbs to light up.
```  -Jack Welch

```