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Galois Field Terminology

The discussion of Galois fields in this document uses a few terms that are not used consistently in the literature. The definitions adopted here appear in van Lint [4].

• A primitive element of GF(2^m) is a cyclic generator of the group of nonzero elements of GF(2^m). This means that every nonzero element of the field can be expressed as the primitive element raised to some integer power.
• A primitive polynomial for GF(2^m) is the minimal polynomial of some primitive element of GF(2^m). That is, it is the binary-coefficient polynomial of smallest nonzero degree having a certain primitive element as a root in GF(2^m). As a consequence, a primitive polynomial has degree m and is irreducible.

The definitions imply that a primitive element is a root of a corresponding primitive polynomial.

Galois Field Computations

Representing Elements of Galois Fields