A friend asked me the following, pointing out that the method is not very useful (my problem is I do not see why it is not good):
- Consider a person A which chooses $n$ as the public key for the Rabin crypto-system.
- We want to be sure that we are communicating with person A so we send her a random item $r\equiv m^2 \mod n$.
- Person A receives $r$ and decodes it using the factorization of $n$ and finds a square rot $m_1$ of $r$.
- A then sends us $m_1$ and we check $r\equiv m_1^2 \mod n$.
Why is this not useful?