  We consider the problem of recovering the entries of diagonal matrices
$\{U_a\}_a$ for $a = 1,\ldots,t$ from multiple "incomplete" samples $\{W_a\}_a$
of the form $W_a=PU_aQ$, where $P$ and $Q$ are unknown matrices of low rank. We
devise practical algorithms for this problem depending on the ranks of $P$ and
$Q$. This problem finds its motivation in cryptanalysis: we show how to
significantly improve previous algorithms for solving the approximate common
divisor problem and breaking CLT13 cryptographic multilinear maps.
