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