# Defining AES MixColumns in Sage

AES is a famous cipher. It has an operation called MixColumns (See Wikipedia entry Rijndael MixColumns) where operations take place over finite fields.

Actually, there's a specific polynomial $a(x) = a_3x^{3}+a_2x^{2}+a_1x+a_0$ whose coefficients belong to $GF(2^8)$. MixColumns is defined between $a(x)$ and any polynomial $b(x) = b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}$ (whose coefficients belong to $GF(2^8)$ as well): It first multiplies $a(x)$ by $b(x)$ (where the algebraic rules between coefficients are governed by $GF(2^8)$), and then computes the remainder modulo $x^4 + 1$.

I tried to mimic the operations in Sage as follows:

```
R.<x> = PolynomialRing(GF(2), 'x')
S.<y> = QuotientRing(R, R.ideal(x^8+x^4+x^3+x+1))
T.<z> = QuotientRing(S, S.ideal(y^4+1))
```

But Sage displays an error on the last line. The error seems to be originating from the introduction of `z`

. If I replace the last line with, say, `T = QuotientRing(S, S.ideal(y^4+1))`

, the error goes away. Yet this is not what I intended.

Furthermore, the command `S.ideal(y^4+1)`

outputs:

Principal ideal (1) of Univariate Quotient Polynomial Ring in y over Finite Field of size 2 with modulus x^8 + x^4 + x^3 + x + 1

I don't understand why its is the principal ideal (1) rather than the principal ideal (y^4+1).