  In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are
given two strings $X$ and $Y$ of total length $n$, and we are asked to find a
pair of maximal-length factors, one of $X$ and the other of $Y$, such that
their Hamming distance is at most $k$. Thankachan et al. show that this problem
can be solved in $\mathcal{O}(n \log^k n)$ time and $\mathcal{O}(n)$ space for
constant $k$. We consider the LCF$_k$($\ell$) problem in which we assume that
the sought factors have length at least $\ell$, and the LCF$_k$($\ell$) problem
for $\ell=\Omega(\log^{2k+2} n)$, which we call the Long LCF$_k$ problem. We
use difference covers to reduce the Long LCF$_k$ problem to a task involving
$m=\mathcal{O}(n/\log^{k+1}n)$ synchronized factors. The latter can be solved
in $\mathcal{O}(m \log^{k+1}m)$ time, which results in a linear-time algorithm
for Long LCF$_k$. In general, our solution to LCF$_k$($\ell$) for arbitrary
$\ell$ takes $\mathcal{O}(n + n \log^{k+1} n/\sqrt{\ell})$ time.
