In lattice based cryptography, we say the average-case SIS (short integer solution) problem because it is such kind problem " $A \stackrel{\\\$}{\leftarrow} \mathbb{Z}^{n\times m}_{q}$, finds a vector $s\in \mathbb{Z}^{m}_{q} \backslash \{0\}$ such that $As=0$ and $||s||\leq \beta$"

but not an instance like this " $A \stackrel{\\\$}{\leftarrow} \mathbb{Z}^{n\times m}_{q}$, finds a matirx $B\in \mathbb{Z}^{m\times k}_{q} \backslash \{0\}$ such that $AB=0$ and $||B||\leq \beta$". The latter instance is more general and the former one just includes some cases of the latter one.

And the same case happens in BDD (bounded-distance decoding) problem. We say that LWE problem is an average-case problem because it just includes some cases of BDD problem.

Did I understand that correctly?

In lattice based cryptography, we say the average-case SIS (short integer solution) problem because it is such kind problem " $A \stackrel{\\\$}{\leftarrow} \mathbb{Z}^{n\times m}_{q}$, finds a vector $s\in \mathbb{Z}^{m}_{q} \backslash \{0\}$ such that $As=0$ and $||s||\leq \beta$"

but not instances like this " $A \stackrel{\\\$}{\leftarrow} \mathbb{Z}^{n\times m}_{q}$, finds a matirx $B\in \mathbb{Z}^{m\times k}_{q} \backslash \{0\}$ such that $AB=C$ and $||B||\leq \beta$". The latter instances are more general and the former instances just include some cases of the latter.

And the same case happens in BDD (bounded-distance decoding) problem. We say that LWE problem is an average-case problem because it just includes some cases of BDD problem.

Did I understand that correctly?