Polynomial multiplication over Galois field

How to do the multiplication of two polynomials over Galois field when
the most power of the result exceeds the most big power of the filed
it self?
Example:
let's be A(x) and B(x) defined over GF (2^4) with the primitive
polynomial
p(x) = 1+x+x^4. All coefficients of A(x) and B(x) belong in GF (2^4).

A(x) = 1 + x + x^2
B(x) = 1 + x + x^2 + x^14
result C(X) = 1 + x^2 + x^4 + x^14 + x^15 + x^16 so what to do with
the coffecients of x^15 and x^16 ?

Should the polynomial multiplication over Galois filed be defined like
this :
C(x) = (A(x) * B(x)) mod p(x) in order to get all powers included in
GF(2^4)?

Thanks in advance.

On Sep 22, 2:56&#4294967295;am, MN <mazouz.nezh...@gmail.com> wrote:
> How to do the multiplication of two polynomials over Galois field when
> the most power of the result exceeds the most big power of the filed
> it self?
> Example:
> let's be A(x) and B(x) defined over GF (2^4) with the primitive
> polynomial
> p(x) = 1+x+x^4. All coefficients of A(x) and B(x) belong in GF (2^4).
>
> A(x) = 1 + x + x^2
> B(x) = 1 + x + x^2 + x^14
> result C(X) = 1 + x^2 + x^4 + x^14 + x^15 + x^16 so what to do with
> the coffecients of x^15 and x^16 ?

There is nothing to be done with the coefficients
of x^15 and x^16,  A polynomial defined over
GF(2^4) can have any degree whatsoever, and
the C(x) you have written is indeed the product
A(x)B(x) over GF(2^4).  In fact, C(x) is the product
of A(x) and B(x) over *every* GF(2^m) because the
coefficients of A(x) and B(x) are actually in GF(2),
and so A(x) and B(x) also happen to be polynomials
over every GF(2^m), not just over GF(2^4).

> Should the polynomial multiplication over Galois filed be defined like
> this :
> C(x) = (A(x) * B(x)) mod p(x) in order to get all powers included in
> GF(2^4)?

No.  I think you are mixing up two different notions.
The *elements* of GF(2^4) can be represented as
polynomials of degree 3 or less with coefficients in
GF(2).  If two elements of GF(2^4) are represented
by polynomials a(y) and b(y), then the product of
these two elements is represented by the polynomial
a(y)b(y) mod p(y).  A "polynomial defined over
GF(2^4)" means exactly what you said:  "All coefficients
of A(x) and B(x) belong in GF (2^4)" and so, if you
wanted to, you could represent each coefficient of
A(x) and B(x) as a polynomial of degree 3 or less in y.

Thank you for clear response.