Mikael Olofssons Polynomial Service

Primitive Polynomials over GF(2)

Here you find lists of primitive polynomials over the binary field. The files are in text format and they are in some cases also packed.

Contents

• Finding Polynomials
• One Primitive Polynomial per Degree
• All Primitive Polynomials for some Degrees
• Ten Primitive Polynomials per Degree

Finding Polynomials

All polynomials in the files that you can reach from this page have been found using Magma, a programming language designed for calculations in finite structures, for instance finite fields. For information about Magma, we refer to

• The Magma System for Algebra, Number Theory and Geometry.

One Primitive Polynomial per Degree

All you really need to generate all primitive polynomials of a given degree is one primitive polynomial of that degree, and of course the know-how. The polynomials that are available here are those returned by the Magma function PrimitivePolynomial, and nothing else. No attempts have been made to find polynomials with some special property, such as e.g. polynomials with low weight.

For degrees up to 640, one polynomial is listed for every degree. For degrees 642-1232, one polynomial is listed for every even degree. For degrees 1240-1256, one polynomial is listed for degrees that are divisible by 8, i.e. degrees 1240, 1248 and 1256.

All Primitive Polynomials for some Degrees

The polynomials given in the files you find here are listed in no particular order.

Ten Primitive Polynomials per Degree

These polynomials are generated with the only demand that they must be primitive. Nothing has for instance been done to get polynomials of low weight.