Correcting matrix products over the ring of integers

Let $A$, $B$, and $C$ be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product $AB$. Given that $C$ is different from $AB$ by at most $k$ entries, we propose an algorithm that uses $O(\sqrt{k}n^2+k^2n)$ operations. Let $α$ be the largest absolute value of an entry in $A$, $B$, and $C$. The integers involved in the computation are of $O(n^3α^2)$.