Communications Toolbox

List of All Elements of a Galois Field

Some Galois field functions in this toolbox require an argument that lists all elements of an extension field GF(p^m). This is again relative to a particular primitive element A of GF(p^m). The proper format for the list of elements is that of a matrix having p^m rows, one for each element of the field. The matrix has m columns, one for each coefficient of a power of A in the polynomial format shown in Polynomial Format above. The first row contains only zeros because it corresponds to the zero element in GF(p^m). If k is between 2 and p^m, then the kth row specifies the polynomial format of the element A^k-2.

The minimal polynomial of A aids in the computation of this matrix, because it tells how to express A^m in terms of lower powers of A. For example, the table below lists the elements of GF(3²), where A is a root of the primitive polynomial 2 + 2x + x². This polynomial allows repeated use of the substitution

A² = -2 - 2A = 1 + A

when performing the computations in the middle column of the table.

Elements of GF(9) 

Exponential Format	Polynomial Format	Row of MATLAB Matrix of Elements
A-Inf	0	0 0
A0	1	1 0
A1	A	0 1
A2	1+A	1 1
A3	A + A2 = A + 1 + A = 1 + 2A	1 2
A4	A + 2A2 = A + 2 + 2A = 2	2 0
A5	2A	0 2
A6	2A2 = 2 + 2A	2 2
A7	2A + 2A2 = 2A + 2 + 2A = 2 + A	2 1

Example

An automatic way to generate the matrix whose rows are in the third column of the table above is to use the code below.

• p = 3; m = 2;
  % Use the primitive polynomial 2 + 2x + x^2 for GF(9).
  prim_poly = [2 2 1];
  field = gftuple([-1:p^m-2]',prim_poly,p);
  

The gftuple function is discussed in more detail in Converting and Simplifying Element Formats.

Polynomial Format

Nonuniqueness of Representations