Score:0

Finding a basis for q-ary lattices

tr flag

For $A\in \mathbb{Z_q}^{n\times m}$, where $m \geq n$, consider the given two q-ary lattices \begin{align} \Lambda_q^{\bot}{(A)} & = \{\mathbf{x} \in \mathbb{Z}^m: A\mathbf{x} = \mathbf{0}\text{ mod }q\} \\ \Lambda_q{(A)} & = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\}. \end{align}

Compute a basis for the above two lattices

Ievgeni avatar
cn flag
Is it homework?
Hilder Vitor Lima Pereira avatar
us flag
The standard way of computing a basis of $\Lambda_q^\perp(A)$ is by supposing that $A$ contains an invertible block, as it is done [here](https://crypto.stackexchange.com/questions/72064/do-q-ary-lattices-have-parallelogram-kind-of-structure/72068#72068). Then, the basis of $\Lambda_q(A)$ can be obtained by computing computing the [dual basis](https://cims.nyu.edu/~regev/teaching/lattices_fall_2004/ln/DualLattice.pdf), since these two lattices are (the scaled) dual of each other.
mangohost

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.