[GPV] and [MP] (references below) give constructions of the trapdoor function defined by
$$
f_{\mathbf A} (\mathbf x) = \mathbf A \mathbf x,
$$
where $\mathbf A \in \mathbb Z_q^{n \times m}$ is uniformly random, and the domain is $\{ \mathbf x \in \mathbb Z^m \mid \lVert x \rVert \le \beta\}$. Given any $\mathbf y \in \mathbb Z_q^n$, the secret trapdoor allows for computing a preimage $\mathbf x$ of $\mathbf y$, i.e. $\mathbf A \mathbf x = \mathbf y$ and $\lVert \mathbf x \rVert \le \beta$.
The trapdoor in [GPV] is a full-rank set $\mathbf S \in \mathbb Z^{m \times m}$ such that $\mathbf A \mathbf S = 0$. The trapdoor in [MP] is $\mathbf R$ such that $\mathbf A \begin{bmatrix} \mathbf R \\ \mathbf I \end{bmatrix} = \mathbf G$, where $\mathbf G$ is some special public gadget matrix. In any case, each trapdoor has some associated quantity $s$ that describes its quality. For $\mathbf S$, $s =$ max norm of columns of Gram-Schmidt $\tilde{\mathbf S}$ of $\mathbf S$. For $\mathbf R$, $s = $ largest singular value of $\mathbf R$.
Given any $\mathbf y$, a trapdoor of quality $s$ allows for sampling from the discrete Gaussian distribution over $\{\mathbf{x} \mid \mathbf A\mathbf x = \mathbf y\}$ of width $\sigma \ge s\cdot\omega(\sqrt{\log m})$, which gives $\lVert x \rVert \le \sigma\sqrt m$ (with overwhelming probability).
My question is about the reverse. Assume we have an oracle for Gaussian preimage sampling that outputs solutions for $\mathbf A \mathbf x = \mathbf y$ with $\lVert x \rVert \le \beta$ for arbitrarily chosen $\mathbf y$. Can we recover some kind of trapdoor for $\mathbf A$ that lets us compute a preimage $\mathbf x'$ of a random $\mathbf y'$ that's not queried to the oracle, with non-negligible probability?
One obvious attempt is to query for $\mathbf A \mathbf x = \mathbf 0$ to try recover a "short" full-rank set $\mathbf S$ such that $\mathbf A \mathbf S = \mathbf 0$. But this $\mathbf S$ doesn't seem short enough to compute solutions of length $\le \beta$. We can try lattice-reduction. But I don't know what the state-of-the-art lattice reduction is for this problem trying to get shorter basis from an already short basis.
- [GPV] https://eprint.iacr.org/2007/432
- [MP] https://eprint.iacr.org/2011/501