Square roots in finite fields, i.e. mod $p^m$

Square roots in finite fields, i.e. mod $p^m$

Solving the congruence $b\equiv x^2$ mod $p$ is pretty simple using either
closed formulas (e.g. if $p\equiv 3$ mod $4$, then $x=b^{(p+1)/4}$ is a
solution) or the Tonelli-Shanks algorithm.
But what about the congruence $b\equiv x^2$ mod $p^m$ for some
$m\in\mathbb Z_{\ge1}$ ? My idea is the following: Assuming I already know
a solution $x_0$ to $b\equiv x^2$ mod $p$, then in order to solve, say,
$b\equiv x^2$ mod $p^2$ all I need to do is check if one of $x_0,\ x_0+p,\
x_0+2p,\ \ldots,\ x_0+ (p-1)p$ is a solution. So I have to check at most
$p$ values. And in general I can solve $b\equiv x^2$ mod $p^m$ by taking a
solution from $b\equiv x^2$ mod $p^{m-1}$ and, again, checking at most $p$
values.
Is this a viable technique or is my idea rubbish?