Add polynomials over Galois field
c = gfadd(a,b)
c = gfadd(a,b,p)
c = gfadd(a,b,p,len)
c = gfadd(a,b,field)
Note: This function performs computations in GF(p^{m}) where p is prime. To work in GF(2^{m}), apply the + operator to Galois arrays of equal size. For details, see Example: Addition and Subtraction. |
c = gfadd(a,b)
adds
two GF(2) polynomials, a
and b
,
which can be either polynomial strings or
numeric vectors. If a
and b
are
vectors of the same orientation but different lengths, then the shorter
vector is zero-padded. If a
and b
are
matrices they must be of the same size.
c = gfadd(a,b,p)
adds
two GF(p
) polynomials, where p
is
a prime number. a
, b
, and c
are
row vectors that give the coefficients of the corresponding polynomials
in order of ascending powers. Each coefficient is between 0 and p
-1.
If a
and b
are matrices of the
same size, the function treats each row independently.
c = gfadd(a,b,p,len)
adds
row vectors a
and b
as in the
previous syntax, except that it returns a row vector of length len
.
The output c
is a truncated or extended representation
of the sum. If the row vector corresponding to the sum has fewer than len
entries
(including zeros), extra zeros are added at the end; if it has more
than len
entries, entries from the end are removed.
c = gfadd(a,b,field)
adds
two GF(p^{m}) elements, where m is a positive
integer. a
and b
are the exponential
format of the two elements, relative to some primitive element of
GF(p^{m}). field
is the
matrix listing all elements of GF(p^{m}),
arranged relative to the same primitive element. c
is
the exponential format of the sum, relative to the same primitive
element. See Representing Elements of Galois Fields for an explanation
of these formats. If a
and b
are
matrices of the same size, the function treats each element independently.
In the code below, sum5
is the sum of 2 + 3x + x^{2} and
4 + 2x + 3x^{2} over
GF(5), and linpart
is the degree-one part of sum5
.
sum5 = gfadd([2 3 1],[4 2 3],5) linpart = gfadd([2 3 1],[4 2 3],5,2)
The output is
sum5 = 1 0 4 linpart = 1 0
The code below shows that A^{2} + A^{4} = A^{1}, where A is a root of the primitive polynomial 2 + 2x + x^{2} for GF(9).
p = 3; m = 2; prim_poly = [2 2 1]; field = gftuple([-1:p^m-2]',prim_poly,p); g = gfadd(2,4,field)
The output is
g = 1
Other examples are in Arithmetic in Galois Fields.