The maximum size of $m$ is $n/2$, and the maximum value of $n$ is $10^k-1$. It follows that the size of $m$ is exponential in $k$. Since (when $n>0$) the multiplication algorithm must read the whole of $m$, and that takes time at least proportional to the size of $m$, it follows that the complexity is at least exponential in $k$.
From a theoretical standpoint, the multiplication is thus not efficient for security parameter $k$, since efficient is defined as time bounded by a polynomial of the security parameter.
If the security parameter was $n$, the algorithm could be efficient (and we could prove that by exhibiting a multiplication algorithm with cost bounded by a polynomial in $n$, which is easy: the schoolbook algorithm turns out to be enough; this is left as an exercise to the reader).
In a crypto context, we want the security parameter to be such that it makes the normal user's cost at most polynomial (i.e. efficient), but the attacker's (conjectured) cost at least super-polynomial (hopefully exponential). Thus if our algorithm is to be considered interesting, then
- if that multiplication is to be performed by the user, it must be efficient and thus our parameter can't be $k$.
- if that multiplication is to be performed by an attacker in an attack, but not by the legitimate user, our parameter can be $k$, and probably should be. The multiplication then can be efficient (and is, unless the multiplication algorithm is abysmally inefficient).