$p'q'=N'>N=pq$, $\,\, p',q',p,q$ all primes $\equiv 3 \mod{4}$, $\, \,\,\, c \equiv m^2 \mod{N}$, $\,\,\,\, c' \equiv m^2 \mod{N'}$, to find $0 \leq m \leq N-1$ 'efficiently', i.e. in polynomial time in $N$, without the knowledge of the factorisations of $N$ and $N'$.
(Context: I openly use the Rabin cryptosystem modulo $N$, and a message is sent encoded to me. I have forgotten $p$ and $q$ however so change my modulus and the same message is again encoded and sent to me. Eve intercepts both codes and wishes to decode to find $m$.)