Exploring
Here is one way we could explore the possibilities in Sage based on your question.
First, it is always a good idea to give names to the objects you are manipulating:
sage: R.<x> = PolynomialRing(GF(2))
sage: F.<t> = R.quotient(x^128 + x^7 + x^2 + x + 1)
sage: a = t^7 + t^2 + t + 1
Then you can investigate the methods you can apply to them. Type
sage: a.
then the TAB key. Among the methods is a method called list.
sage: l = a.list()
sage: len(l)
128
sage: l[:20]
[1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Now you can get a polynomial from a list of coefficients:
sage: p = R(l)
sage: p
x^7 + x^2 + x + 1
You can evaluate this polynomial at 2
sage: p(2)
1
or substitute:
sage: p.subs(x=2)
1
We get 1 because in R, computations are modulo 2.
One way to get around that is to define the polynomial ring over ZZ:
sage: S = PolynomialRing(ZZ,'x')
and create a polynomial in this ring from the list l:
sage: q = S(l)
x^7 + x^2 + x + 1
and we can then evaluate
sage: q(2)
135
or substitute
sage: q.subs(x=2)
135
Another way is to directly change ring by letting
sage: q = p.change_ring(ZZ); q
x^7 + x^2 + x + 1
sage: q(2)
135
In summary, a contracted answer to your question is:
sage: a = t^7 + t^2 + t + 1
sage: R(a.list()).change_ring(ZZ)(2)
135
Working with finite fields
It seems you want to work with the finite field with 2^128 elements.
The easiest way to do that is
sage: K.<t> = GF(2^128); K
Finite Field in t of size 2^128
As it turns out, the minimal polynomial for the generator t in K
is exactly the polynomial you used in your quotient ring.
sage: p = t.minpoly(); p
x^128 + x^7 + x^2 + x + 1
Indeed:
sage: a = t^128; a
t^7 + t^2 + t + 1
Now it is easier to transform a into a polynomial:
sage: q = a.polynomial(); q
t^7 + t^2 + t + 1
We can evaluate this polynomial at 2:
sage: q(2)
1
This is again because q is defined over GF(2). Change ring:
sage: r = q.change_ring(ZZ); r
t^7 + t^2 + t + 1
sage: r(2)
135
