2. Start Up and Declareations

In the Mathematica session, When you write

<<GaloisField.m

GaloisField.m was loaded.

then the Galois field package is ready to use. Next, you have to declare a Galois field which you want to use.

There are four types of declarations for finite fields in this package depending on different constructions of finite fields.

I . Prime fields
II . Extension fields over a prime base field. (needs a Galois field object)
III . Extension fields over a declared base field. (needs a Galois field object)
IV . Extension fields over a declared base field. (no objects needed )

When we declare a finite field of type II or III, the object of the finite field must exist as a file; it is called a Galois field object. A Galois field object can be created in precomputation and saved as a file.

Type I

After loading the Galois Field Package by inputting <<GaloisField.m, we declare a finite field with the function DeclareGaloisField.

DeclareGaloisField[K5,5]

K5 was declared.
   Order is 5
   Characteristic is 5

This example declares a prime field of order 5, GF(5). Then we can use the algebra name K5 and constants K5[i] for i=0,1,...,4.

K5[3] + K5[4]
K5[2]

Type II and III

When you declare a type II or III finite field, just use the function DeclareGaloisField

DeclareGaloisField[GF16]

GF16 was declared.
   Order is 16
   Characteristic is 2
   Irreducible Polynomial is {1, 1, 0, 0, 1}
   Base Field is GF2
   Extension Degree is 4
GF2 was declared.
   Order is 2
   Characteristic is 2

If the object GF16 does not exist, you get the following message:

Galois field object, GF16, does not exist.

Then you can make a type II object with the following function:

MakeGaloisField[GF16,2,4,{1,1,0,0,1}]

The first argument is the name of the object (also used as algebra name), the second is the order of the base field (must be a prime number), the third is the extension degree and the fourth argument is the coefficient list of a primitive irreducible polynomial over the base field. The function saves the object as a file with the name of the first argument.

When you make a type III object, the second argument of MakeGaloisField must be an algebra name of the base field.

MakeGaloisField[F16,GF4,2,{2,2,1}]

Of course, the base field must be declared before executing this function. The base field for type III dose not have to be a prime field.

Type IV

Type IV Galois fields do not require objects for them. If a base field has been declared, then you can declare a type IV Galois field directly by the function DeclareGaloisField.

DeclareGaloisField[K16,GF4,2,{2,2,1}]

GF4 has already been declared.
K16 was declared.
   Order is 16
   Characteristic is 2
   Irreducible Polynomial is {2, 2, 1}
   Base Field is GF4
   Extension Degree is 2

Since the fields of this type do not have any table for computations, in each computation on type IV finite field, elements are reformed to polynomials over the base field and compute under the modulo of the irreducible polynomial. Therefore the irreducible polynomial dose not have to be a primitive. So, if you use a reducible polynomial instead of an irreducible polynomial, you can declare a polynomial ring.

Type IV Galois fields are not recommended for fields of an order less than 1000, because of the computations are very slow.