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I am searching for the exact definition of High density SIS and Low density SIS, but there is something unclear about it.

SIS problem is to find $x\in \mathbb{Z}^m$ such that for random $A\in\mathbb{Z}_q^{n\times m}$, $Ax=0$ and $\lVert x\rVert < \beta$.

As far as I know, high density SIS is when $n\log q < m$, so that the number of inputs are larger than the output and hence there are many solutions.

My question is :

  1. Is low density SIS parameterized by $n\log q > m $?

  2. If so, are there any specific situations that for some situations like, high density SIS must be used rather than low density one.. and for the others, low density SIS must be used.. ?

Thank you in advance.

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    $\begingroup$ I am not an expert (at all) in SIS, but a slightly educated guess would be: if you want to build a collision-resistant hash function, you will usually be in a setting where there are multiple solutions (since there are many preimages to a given hash). The injective setting is probably more common in encryption-style applications. $\endgroup$ Commented Jan 2, 2023 at 9:02
  • $\begingroup$ @GeoffroyCouteau That would be an answer to my question 2. Thank you for the answer. Have a nice day! $\endgroup$ Commented Jan 4, 2023 at 1:17

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