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2017-12-19 23:58:48 +0200 | commented answer | Defining AES MixColumns in Sage Yes, in $GF(256)$ we have |

2017-12-19 21:36:46 +0200 | commented answer | Defining AES MixColumns in Sage It must work, but it does not (to my surprise). In your definition, $R$ is the ring of polynomials with coefficients over $GF(256)$, but sage treats it as the ring of polynomials with coefficients over $GF(2)$! Just enter something like |

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2017-12-17 11:55:40 +0200 | commented answer | Defining AES MixColumns in Sage Thanks again. Let me just confirm my understanding: Sage does not provide a way for defining $GF(256)[x] / \langle x^4 + 1 \rangle$, right? |

2017-12-17 04:22:15 +0200 | commented answer | Defining AES MixColumns in Sage Thanks a lot. Could you please answer the first part, too: ln short, how can I define $GF(256)[x] / \langle x^4 + 1 \rangle$ in Sage? I mean, as stated in the question, addition and multiplication of polynomials with coefficients in $GF(256)$ modulo $x^4+1$. |

2017-12-16 21:59:52 +0200 | asked a question | 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: But Sage displays an error on the last line. The error seems to be originating from the introduction of Furthermore, the command
I don't understand why its is the principal ideal (1) rather than the principal ideal (y^4+1). |

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