Basic Concepts from Probability, Complexity, Algebra and Number Theory - Introduction to Cryptography with Maple - page 111

F 2 8 . This particular polynomial is especially convenient because it is
the one used in the specification of the Advanced Encryption Standard (cf. Chap. 4 ) .
Thus we initialize this field as:

define the field

> F256 := GF(2, 8, xˆ8+xˆ4+xˆ3+x+1):

If the field is defined by the command F := GF(p, n) , i.e., if the irreducible
polynomial is not specified, then Maple chooses an irreducible polynomial which
need not be the “first” one. The irreducible polynomial used by Maple in this case is
given by the command F[extension] .

Exercise 2.36 Define G := GF ( 2 , 8 ) and find the irreducible polynomial used by

Maple to construct this field.

The package GF has several commands to convert field elements between the
different formats they can take. The elements of F 2 8 are binary polynomials of

degree less than 8 and, as we have mentioned before, there is a natural bijective
correspondence between them and the integers in the 0 255 range, and also with

the set of bit strings or lists of length 8 which are the binary representations of these
integers. These bit strings are commonly referred to as bytes so we freely use this
term—as we have already been doing—to denote the integers in the 0

..

255 range
when regarded as elements of this field. The Maple functions that convert between
these integers and polynomials modulo 2 are F256:-input and F256:-output
(assuming that we defined the field F256 with the command above) so that, for
example, in Maple v13 or later we have 9 :
> F256:-input ∼ ([$127 .. 132)]);
[(xˆ6 + xˆ5 + xˆ4 + xˆ3 + xˆ2 +x+1)mod2,xˆ7mod2,(xˆ7 + 1) mod 2,
(xˆ7 + x) mod 2, (xˆ7 +x+1)mod2,(xˆ7 + xˆ2) mod 2]
Now we can go back from binary polynomials to integers as follows:
> F256:-output ∼ (%)

..

[127, 128, 129, 130, 131, 132]
Note that the outputs of the function F256:-input are polynomials modulo 2
(or, more generally, modulo p ), and for doing arithmetic with these polynomials
one should use the modp1 function of Maple. For example, let

> g := F256:-input(7);

(xˆ2 +x+1)mod2

Then, if we want to check whether g is irreducible Irreduc(g) mod 2 would
fail, as g is already a binary polynomial, so we would use modp1 instead:

> modp1(Irreduc(g), 2);

true

Maple also has commands F256:-ConvertIn and F256:-ConvertOut to
convert from modulo 2 polynomials to ordinary polynomials and vice versa. For
example:

9 In Maple versions prior to v13 the elementwise operator ~ is not available and so one would use
map instead to make these conversion functions act on the elements of a list.