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2015-12-25 11:36:31 +0200 | commented answer | Cannot mulyiply polynomial by matrix when ordering is explicitly specified Thank you! |

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2015-12-04 16:43:59 +0200 | asked a question | Cannot mulyiply polynomial by matrix when ordering is explicitly specified Consider the following code: Does anybody know if this is intentional? In my opinion it shouldn't happen because lex ordering is already the implicit default. The bug does not seem to occur with degrevlex ordering, which is also weird. The bug also does not occur with square matrices, which is even weirder. |

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2015-11-28 19:36:54 +0200 | answered a question | Enumerate all solutions to linear system over finite field Okay, I found the solution. Finite vector spaces support iteration, so one can simply do: |

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2015-11-28 19:16:35 +0200 | asked a question | Enumerate all solutions to linear system over finite field Assume I have a matrix $A$ and a vector $b$, both over some finite field $\mathrm{GF}(q)$. I would like to enumerate _all_ solutions to $A x = b$ (there are only finitely many). I can generate a particular solution using |

2015-11-26 23:00:55 +0200 | asked a question | Solving system of MQ equations I would like to use SageMath to solve a system of multivariate quadratic (MQ) equations. The goal is to provide an implementation of the Kipnis--Shamir attack on the MinRank problem (see here, pp. 29--31). Basically, I have matrices $M_j$ over some finite field and I need for form and solve a system of equations that looks like this: $$ \left(\sum\limits_{j = 1}^{k} y_j M_j\right) \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & \vdots \ 0 & 0 & \cdots & 1 \ x_1^{(1)} & x_1^{(2)} & \cdots & x_1^{(n - r)} \ \vdots & \vdots & \vdots & \vdots \ x_r^{(1)} & x_r^{(2)} & \cdots & x_r^{(n - r)} \end{bmatrix} = \mathbf{0} $$ Where the $y$'s and the $x$'s are unknowns. How should I go about achieving this? Thanks in advance. |

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