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 \(\alpha\) be the largest absolute value of an entry in \(A\), \(B\), and \(C\). The integers involved in the computation are of \(O(n^3\alpha^2)\).