The GF command returns a module G of procedures and constants for doing arithmetic in the finite field GF(p^k), a Galois Field with $p^k$ elements. The field GF(p^k) is defined by the field extension GF(p)[x]/(a) where a is an irreducible polynomial of degree k over the integers mod p.

If a is not specified, an irreducible polynomial of degree k over the integers mod p is chosen at random.  It can be accessed as the constant G:-extension.  The elements of GF(p^k) are represented using the $\mathrm{modp1}$ representation.

First, you need to define an instance of a Galois field using, for example, G := GF(2, 3). This defines all operations for G, the field of characteristic 2 with 8 elements.

The G:-input and G:-output commands convert from an integer in the range $0..p^k-1$ to the corresponding polynomial and back. Alternatively, G:-ConvertIn and G:-ConvertOut convert an element from GF(p^k) to a Maple sum of products, a univariate polynomial where the variable used is that given in the argument a. Otherwise the name `?` is used. The syntax G(...) can be used instead of G:-ConvertIn(...).

Arithmetic in the field is defined by the following functions.

Field arithmetic can be written very naturally by using a use statement; see the examples below.

The additive and multiplicative identities are given by G:-zero and G:-one.

The G:-trace, G:-norm, and G:-order commands compute the trace, norm and multiplicative order of an element from GF(p^k) respectively.

The G:-random command returns a random element from GF(p^k).

The G:-PrimitiveElement command generates a primitive element at random.

The G:-isPrimitiveElement command tests whether an element from GF(p^k) is a primitive element, being a generator for the multiplicative group GF(p^k) - {0}.

For backwards compatibility, exports of the module returned by the GF command can also be accessed by indexed notation, such as G[':-ConvertIn']. However, using the more modern form G:-ConvertIn, you do not need to quote the name to avoid evaluation.

$\mathrm{G16}≔\mathrm{GF}\left(2\,4\,{\mathrm{\alpha }}^4+\mathrm{\alpha }+1\right)\:$

$a≔\mathrm{G16}:-\mathrm{ConvertIn}\left(\mathrm{\alpha }\right)$

$a≔\mathrm{\alpha }\mathbf{mod}2$

$a≔\mathrm{G16}\left(\mathrm{\alpha }\right)$

$a≔\mathrm{\alpha }\mathbf{mod}2$

$\mathrm{G16}:-\mathrm{`*`}\left(a\,a\right)$

${\mathrm{\alpha }}^2\mathbf{mod}2$

$\mathrm{G16}:-\mathrm{`^`}\left(a\,4\right)$

$\left(\mathrm{\alpha }+1\right)\mathbf{mod}2$

$x≔\mathrm{G16}:-\mathrm{`^`}\left(a\,8\right)$

$x≔\left({\mathrm{\alpha }}^2+1\right)\mathbf{mod}2$

$\mathrm{G16}:-\mathrm{output}\left(x\right)$

$5$

$\mathrm{G16}:-\mathrm{ConvertOut}\left(x\right)$

${\mathrm{\alpha }}^2+1$

$\mathrm{G16}:-\mathrm{isPrimitiveElement}\left(a\right)$

$\mathrm{true}$

$x≔\mathrm{G16}:-\mathrm{`^`}\left(a\,-1\right)$

$x≔\left({\mathrm{\alpha }}^3+1\right)\mathbf{mod}2$

$\mathrm{G16}:-\mathrm{`*`}\left(a\,x\right)$

$1\mathbf{mod}2$

use G16 in
  y := a * ( a + x );
  z := a - y;
  z^3 + y^2
end use;

$\left(\mathrm{\alpha }+1\right)\mathbf{mod}2$

The G(...) syntax for constructing field elements was introduced in Maple 2021.