# Learning with rounding (LWR)

This may be a naive question:

LWR assumption states that for $${A} \stackrel{}{\leftarrow} \mathbb{Z}^{m \times n}_q, s \stackrel{}{\leftarrow} \mathbb{Z}^n_q$$, given $$(A, \lfloor A\cdot s \rfloor_p$$), it is indistinguishable from $$(A, u)$$ with $$u \stackrel{}{\leftarrow} \mathbb{Z}^m_p$$. There seems to be no specific condition on $$m$$ and $$n$$ except $$m,n \geq 1$$.

My question is does this hold true for $$m = n = 1$$? It does not seem right to me, kindly explain.

There seems to be no specific condition on $$m$$ and $$n$$ except $$m,n \geq 1$$.
The LWR assumption isn't that the problem is hard for any arbitrary $$m, n$$, it's there it's true for the specific $$m, n$$ pairs we use in practice. Obviously, for $$m=n=1$$, it's an easy problem; that's not particularly relevant, as we don't use $$m=n=1$$.