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).
