Find roots of polynomial over prime Galois field
rt = gfroots(f,m,p)
rt = gfroots(f,prim_poly,p)
[rt,rt_tuple] = gfroots(...)
[rt,rt_tuple,field] = gfroots(...)
Note:
This function performs computations in GF(p^{m}),
where p is prime. To work in GF(2^{m}), use
the |
For all syntaxes, f
is a row vector that
gives the coefficients, in order of ascending powers, of a degree-d
polynomial.
rt = gfroots(f,m,p)
finds
roots in GF(p^m
) of the polynomial that f
represents. rt
is
a column vector each of whose entries is the exponential format of
a root. The exponential format is relative to a root of the default
primitive polynomial for GF(p^m
).
rt = gfroots(f,prim_poly,p)
finds
roots in GF(p
^{m}) of the
polynomial that f
represents. rt
is
a column vector each of whose entries is the exponential format of
a root. The exponential format is relative to a root of the degree-m
primitive polynomial for GF(p
^{m})
that prim_poly
represents.
[rt,rt_tuple] = gfroots(...)
returns
an additional matrix rt_tuple
, whose kth row is
the polynomial format of the root rt(k)
. The polynomial
and exponential formats are both relative to the same primitive element.
[rt,rt_tuple,field] = gfroots(...)
returns
additional matrices rt_tuple
and field
. rt_tuple
is
described in the preceding paragraph. field
gives
the list of elements of the extension field. The list of elements,
the polynomial format, and the exponential format are all relative
to the same primitive element.
Note:
For a description of the various formats that |
Roots of Polynomials contains
a description and example of the use of gfroots
.
The code below finds the polynomial format of the roots of the
primitive polynomial 2 + x^{3} + x^{4} for GF(81).
It then displays the roots in traditional form as polynomials in alph
.
(The output is omitted here.) Because prim_poly
is
both the primitive polynomial and the polynomial whose roots are sought, alph
itself
is a root.
p = 3; m = 4; prim_poly = [2 0 0 1 1]; % A primitive polynomial for GF(81) f = prim_poly; % Find roots of the primitive polynomial. [rt,rt_tuple] = gfroots(f,prim_poly,p); % Display roots as polynomials in alpha. for ii = 1:length(rt_tuple) gfpretty(rt_tuple(ii,:),'alpha') end