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Zoey avatar Crypto

q-ary lattices - proof of dual upto scale

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Zoey

Two lattices are defined as following: \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} T.S.T.

  1. $\Lambda_q{(A)} = q \cdot \Lambda_q^{\bot}{(A)}^*$, where $\Lambda_q^{\bot}{(A)}^*$ is the dual of $\Lambda_q^{\bot}{(A)}$, and
  2. $\Lambda_q^{\bot}{(A)} = q \cdot \Lambda_q{(A)}^*$, where $\Lambda_q{(A)}^*$ is the dual of $\Lambda_q{(A)}$.

In most references I see that it is left as an exercise and I am not able to work out a proof.

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LeoDucas avatar
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LeoDucas
Have you tried expending out the definition of the dual lattice ? A trick you might need at some point is that for $x \in \mathbb R$, the property $x \in \mathbb Z$ can be equivalently re-written as $x = 0 \bmod 1$.
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