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I'm new to lattices. According to Lattice Signatures and Bimodal Gaussians in the Rejection Sampling section. In Schnorr, GQ you can simply commit to $y$, use it to hide a secret key $s$. But this doesn't work in lattices. You need to hide the secret key with a small $y$. Turns out, a lot of old lattice-based signatures leaked a part of the secret keys. Instead, we must choose y from a narrow distribution and then perform rejection sampling so that $s$ is not leaked when we add $y$ to it.

Now:

  1. What does there mean to be a narrow distribution?
  2. What does it mean for $y$ to be small?
  3. Why is this a problem in lattices specifically?
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  1. Narrow means that, with high probability, the ouput of the distribution is is small.
  2. Small is meant in the sense of the Euclidean norm of a vector in the Euclidean space $\mathbb R^n$. More concretly, $v$ is informally said small to mean $\|v\| \leq B$ for some value $B > 0$ chosen large enough for a given scheme to work out, but small enough for the lattice problem SIS to remain hard.
  3. It is not a problem outside lattice based crypto because in a finite group $G$, if $x \in G$ is uniform, so is $x+c$. We can't simply do that over an unbounded space such as $\mathbb R^n$.
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