#StackBounty: #lattice-crypto Finding the basis of the transpose of a q-ary lattice

Bounty: 50

Given $q$ and a matrix $A in mathbb{Z}_q^{n times m}$, the $q$-ary lattice is defined as
$$Lambda(A)={x in mathbb{Z}^m:Ax=0 bmod q} $$
An instance of a q-ary lattice and its short basis is computed in Generating short basis for hard random lattices. Once the short basis $T_A$ for $Lambda(A)$ is given, computing the short vector $s$ in $Lambda
is given in SamplePre algorithm.

Is it possible to find a short basis for $Lambda(A^T)$, if we are given a short basis of $ Lambda(A)$?

Basically I want to find the short vector $s^prime$ such that $A^Ts^prime=0 bmod q$.

Get this bounty!!!

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