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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. So define the polynomial ring over ZZ:

sage: S = PolynomialRing(ZZ,'x')

and compute the polynomial,

sage: q = S(l)
x^7 + x^2 + x + 1

and you can then evaluate

sage: q(2)
135

or substitute

sage: q.subs(x=2)
135

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. So define the polynomial ring over ZZ:

sage: S = PolynomialRing(ZZ,'x')

and compute the polynomial,

sage: q = S(l)
x^7 + x^2 + x + 1

and you can then evaluate

sage: q(2)
135

or substitute

sage: q.subs(x=2)
135

In summary, a contracted answer to your question is:

sage: a = t^7 + t^2 + t + 1
sage: PolynomialRing(ZZ,'x')(a.list())(2)
135

It
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.~~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. So~~ 2.

One way to get around that is to define the polynomial ring over ZZ:

sage: S = PolynomialRing(ZZ,'x')

and ~~compute the polynomial,~~create a polynomial in this ring from the list l:

sage: q = S(l) x^7 + x^2 + x + 1

and ~~you~~ 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: PolynomialRing(ZZ,'x')(a.list())(2) 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