What are field representations?
A field representation defines what bit-patterns are used to represent the various field elements. The representation (and the field) is chosen to make the field arithmetic operations efficient. An analogy is using different representations for integers – for example base 10 or base 2. Depending on the field, various tricks are possible, especially for the case GF(2^m). There are two main possibilities: polynomial and normal basis. Polynomial basis is probably easier to understand. With a polynomial basis, field elements are represented as polynomials. Usually a vector with m+1 components is used to represent a polynomial of degree m. When multiplying, the remainder is taken after dividing the result polynomial by an “irreducible” polynomial [see (14)] If m has factors, the basis can be over a sub-field other than GF(2). This is analagous to performing multi-precision arithmetic on words instead of bits. For small fields such as GF(2^9) field multiplication and division can be performe
