Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Simplify or convert Galois field element formatting

`tp = gftuple(a,m)`

tp = gftuple(a,prim_poly)

tp = gftuple(a,m,p)

tp = gftuple(a,prim_poly,p)

tp = gftuple(a,prim_poly,p,prim_ck)

[tp,expform] = gftuple(...)

This function performs computations in GF(p^{m}),
where p is prime. To perform equivalent computations in GF(2^{m}),
apply the `.^`

operator and the `log`

function
to Galois arrays. For more information, see Example: Exponentiation and Example: Elementwise Logarithm.

`gftuple`

serves to simplify the polynomial
or exponential format of Galois field elements, or to convert from
one format to another. For an explanation of the formats that `gftuple`

uses,
see Representing Elements of Galois Fields.

In this discussion, the format of an element of GF(p^{m})
is called “simplest” if all exponents of the primitive
element are

Between 0 and m-1 for the polynomial format

Either

`-Inf`

, or between 0 and p^{m-2}, for the exponential format

For all syntaxes, `a`

is a matrix, each row
of which represents an element of a Galois field. The format of `a`

determines
how MATLAB interprets it:

If

`a`

is a column of integers, MATLAB interprets each row as an*exponential*format of an element. Negative integers are equivalent to`-Inf`

in that they all represent the zero element of the field.If

`a`

has more than one column, MATLAB interprets each row as a*polynomial*format of an element. (Each entry of`a`

must be an integer between 0 and`p`

-1.)

The exponential or polynomial formats mentioned above are all
relative to a primitive element specified by the *second* input
argument. The second argument is described below.

`tp = gftuple(a,m)`

returns
the simplest polynomial format of the elements that `a`

represents,
where the kth row of `tp`

corresponds to the kth
row of `a`

. The formats are relative to a root of
the default primitive polynomial for GF(`2^m`

), where `m`

is
a positive integer.

`tp = gftuple(a,prim_poly)`

is
the same as the syntax above, except that `prim_poly`

is
a polynomial character vector or
a row vector that lists the coefficients of a degree `m`

primitive
polynomial for GF(`2^m`

) in order of ascending exponents.

`tp = gftuple(a,m,p)`

is the same as `tp = gftuple(a,m)`

except that 2
is replaced by a prime number `p`

.

`tp = gftuple(a,prim_poly,p)`

is
the same as `tp = gftuple(a,prim_poly)`

except that
2 is replaced by a prime number `p`

.

`tp = gftuple(a,prim_poly,p,prim_ck)`

is
the same as `tp = gftuple(a,prim_poly,p)`

except
that `gftuple`

checks whether `prim_poly`

represents
a polynomial that is indeed primitive. If not, then `gftuple`

generates
an error and `tp`

is not returned. The input argument `prim_ck`

can
be any number or character vector; only its existence matters.

`[tp,expform] = gftuple(...)`

returns
the additional matrix `expform`

. The kth row of `expform`

is
the simplest exponential format of the element that the kth row of `a`

represents.
All other features are as described in earlier parts of this “Description”
section, depending on the input arguments.

As another example, the `gftuple`

command below
generates a list of elements of GF(`p^m`

), arranged
relative to a root of the default primitive polynomial. Some functions
in this toolbox use such a list as an input argument.

p = 5; % Or any prime number m = 4; % Or any positive integer field = gftuple([-1:p^m-2]',m,p);

Finally, the two commands below illustrate the influence of
the *shape* of the input matrix. In the first command,
a column vector is treated as a sequence of elements expressed in
exponential format. In the second command, a row vector is treated
as a single element expressed in polynomial format.

tp1 = gftuple([0; 1],3,3) tp2 = gftuple([0, 0, 0, 1],3,3)

The output is below.

tp1 = 1 0 0 0 1 0 tp2 = 2 1 0

The outputs reflect that, according to the default primitive
polynomial for GF(3^{3}), the relations below
are true.

$$\begin{array}{l}{\alpha}^{0}=1+0\alpha +0{\alpha}^{2}\\ {\alpha}^{1}=0+1\alpha +0{\alpha}^{2}\\ 0+0\alpha +0{\alpha}^{2}+{\alpha}^{3}=2+\alpha +0{\alpha}^{2}\end{array}$$

`gftuple`

uses recursive callbacks to determine
the exponential format.

Was this topic helpful?